Finite Group Databasebeta

← #388567

Subgroups — #388567

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 779 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23451C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic, complemented
3#21304C211yesmaximal, characteristic, complemented
4C2×#1929C211yesmaximal, characteristic
5#21051C211yesmaximal, characteristic
6#11326C211yesmaximal, characteristic, complemented
7#11317C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11125C211yesmaximal, characteristic
10#11068C211yesmaximal, characteristic
11#10924C211yesmaximal, characteristic
12#10910C211yesmaximal, characteristic, complemented
13#10910C211yesmaximal, characteristic, complemented
14#10887C211yesmaximal, characteristic, complemented
15#10796C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C22×#108C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2297C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#2026C411yescharacteristic, complemented
32#2026C411yescharacteristic, complemented
33#2025C411yescharacteristic, complemented
34#2025C411yescharacteristic, complemented
35#2024C411yescharacteristic, complemented
36#2024C411yescharacteristic, complemented
37#2023C411yescharacteristic, complemented
38#2023C411yescharacteristic, complemented
39#1995—24no—
40#1985—21no—
41#1985—21no—
42#1984—21no—
43#1984—21no—
44#1929—24no—
45#1924—21no—
46#1924—21no—
47#1923—21no—
48#1923—21no—
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1728C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78#533—22no—
79#533—22no—
80#533—41no—
81#533—41no—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#125C2×C411yescharacteristic
85C2×#125C2×C411yescharacteristic
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#108—22no—
97C2×#108—22no—
98C2×#108—22no—
99C2×#108—41no—
100C2×#108—41no—
101C2×#108—41no—
102#424—22no—
103#424—41no—
104#423—22no—
105#423—41no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104—41no—
114C2×#104—41no—
115C2×#104—41no—
116#416—22no—
117#416—22no—
118#416—41no—
119#416—41no—
120C2×#103C2×C411yescharacteristic
121C2×#103C2×C411yescharacteristic
122C2×#103C2×C411yescharacteristic
123C2×#103C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×#102C2×C411yescharacteristic
126C2×#102C2×C411yescharacteristic
127C2×#102C2×C411yescharacteristic
128#408—22no—
129#408—41no—
130#408—41no—
131#408—41no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136#405—22no—
137#405—41no—
138#405—41no—
139#405—41no—
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#390C2×C411yescharacteristic
148#386—21no—
149#386—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152#376—22no—
153#376—41no—
154C2×#95—21no—
155C2×#95—21no—
156#341—41no—
157#341—41no—
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#339—22no—
167#339—22no—
168#339—22no—
169#339—41no—
170#339—41no—
171#339D414yes—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—22no—
179#336—21no—
180#336—21no—
181#336—22no—
182#336—22no—
183#336—41no—
184#336C2×C411yescharacteristic
185#336C2×C411yescharacteristic
186#336C2×C411yescharacteristic
187#336C2×C411yescharacteristic
188#336D411yescharacteristic
189#336D411yescharacteristic
190#336D411yescharacteristic
191#336D411yescharacteristic
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8C22×C411yescharacteristic
207C22×D4—21no—
208C22×D4C22×C411yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4#4111yescharacteristic
213C23×C4#4111yescharacteristic
214C23×C4C2411yescharacteristic
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16C2×D412yes—
238C2×M16#4112yes—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8C22×C411yescharacteristic
268C22×C8C22×C411yescharacteristic
269C22×C8C22×C411yescharacteristic
270C22×C8C22×C411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D411yescharacteristic
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—41no—
278C4⋊Q8—41no—
279#125—22no—
280#125—22no—
281#125—41no—
282#125—41no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286C22⋊Q8—42no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—42no—
291#117—21no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—21no—
301#117C2×D412yes—
302#117#4112yes—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C22×C411yescharacteristic
310C2×C4⋊C4C22×C411yescharacteristic
311C2×C4⋊C4C22×C411yescharacteristic
312C2×C4⋊C4C22×C411yescharacteristic
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C4⋊C4C22×C411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42C22×C411yescharacteristic
340C2×C42C22×C411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4112yes—
345#108—44no—
346#108—44no—
347C4⋊C8—42no—
348C4⋊C8—42no—
349C4⋊C8—42no—
350C4⋊C8—42no—
351#104—44no—
352#104—44no—
353#103—22no—
354#103—22no—
355#103—41no—
356#103—41no—
357#102—22no—
358#102—22no—
359#102—41no—
360#102—41no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—42no—
365#97—42no—
366#97—42no—
367C4×C8—42no—
368C4×C8—42no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—42no—
374#95—42no—
375C24—21no—
376C24C2×#4111yescharacteristic
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4C4×D411yescharacteristic
453C22×C4C4×D411yescharacteristic
454C22×C4C4×D411yescharacteristic
455C22×C4C4×D411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4H(5,𝔽2)11yescharacteristic
463C22×C4H(5,𝔽2)11yescharacteristic
464C22×C4H(5,𝔽2)11yescharacteristic
465C22×C4H(5,𝔽2)11yescharacteristic
466C22×C42−1+411yescharacteristic
467C22×C42−1+411yescharacteristic
468C22×C42−1+411yescharacteristic
469C22×C42−1+411yescharacteristic
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478M16—41no—
479M16—41no—
480M16—41no—
481M16—41no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—41no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—41no—
534C2×C8—41no—
535C2×C8—41no—
536C2×C8—41no—
537C2×C8—41no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4C4×D412yes—
555C4⋊C4C4×D412yes—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—81no—
565C22⋊C4—81no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—21no—
575C42—21no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23C2×C4×D411yescharacteristic
600C23#54211yescharacteristic
601C23#54211yescharacteristic
602C23#54611yescharacteristic
603C23#55111yescharacteristic
604C23#55211yescharacteristic
605C23#55611yescharacteristic
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—41no—
610Q8—84no—
611Q8—84no—
612D4—41no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—81no—
617D4—81no—
618D4—81no—
619D4—81no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—81no—
674C2×C4—81no—
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C4×#4111yescharacteristic
677C2×C4#51811yescharacteristic
678C2×C4#51911yescharacteristic
679C2×C4#52011yescharacteristic
680C2×C4#52011yescharacteristic
681C2×C4#53511yescharacteristic
682C2×C4#53611yescharacteristic
683C2×C4#53611yescharacteristic
684C2×C4#53711yescharacteristic
685C2×C4#53811yescharacteristic
686C2×C4#53811yescharacteristic
687C2×C4#54111yescharacteristic
688C2×C4#54211yescharacteristic
689C2×C4#54211yescharacteristic
690C2×C4#54211yescharacteristic
691C2×C4D4211yescharacteristic
692C2×C4#54611yescharacteristic
693C2×C4#54611yescharacteristic
694C2×C4#54711yescharacteristic
695C2×C4#54811yescharacteristic
696C2×C4#54811yescharacteristic
697C2×C4#55611yescharacteristic
698C2×C4#55911yescharacteristic
699C2×C4#56211yescharacteristic
700C2×C4#56211yescharacteristic
701C2×C4#56311yescharacteristic
702C2×C4#56311yescharacteristic
703C8—41no—
704C8—41no—
705C8—41no—
706C8—41no—
707C8—41no—
708C8—41no—
709C8—42no—
710C8—42no—
711C8—41no—
712C8—41no—
713C8—81no—
714C8—81no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—21no—
719C22—21no—
720C22—21no—
721C22—21no—
722C22—21no—
723C22—21no—
724C22—21no—
725C22—21no—
726C22—21no—
727C22—81no—
728C22—81no—
729C22—81no—
730C22—81no—
731C22—81no—
732C22—81no—
733C22—81no—
734C22#232611yescharacteristic
735C22#235811yescharacteristic
736C22#236211yescharacteristic
737C22#270011yescharacteristic
738C22#270211yescharacteristic
739C22#275411yescharacteristic
740C22#275911yescharacteristic
741C22#295511yescharacteristic
742C22#295511yescharacteristic
743C22#295711yescharacteristic
744C22#295711yescharacteristic
745C4—41no—
746C4—41no—
747C4—42no—
748C4—41no—
749C4—41no—
750C4—42no—
751C4—82no—
752C4—82no—
753C4—81no—
754C4—81no—
755C4—81no—
756C4—81no—
757C4—81no—
758C4#232311yescharacteristic
759C4#233711yescharacteristic
760C4#236711yescharacteristic
761C4#237811yescharacteristic
762C4#267411yescharacteristic
763C4#267611yescharacteristic
764C4#271011yescharacteristic
765C4#271611yescharacteristic
766C2—21no—
767C2—21no—
768C2—21no—
769C2—21no—
770C2—161no—
771C2#1452211yescharacteristic
772C2#2223511yescharacteristic
773C2#2223611yescharacteristic
774C2#2288711yescharacteristic
775C2#2288911yescharacteristic
776C2#2308211yescharacteristic
777C2#2308211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1995C2semidirect
3. #21304C2semidirect
6. #11326C2semidirect
7. #11317C2semidirect
12. #10910C2semidirect
13. #10910C2semidirect
14. #10887C2semidirect
15. #10796C2semidirect
31. #2026C4semidirect
32. #2026C4semidirect
33. #2025C4semidirect
34. #2025C4semidirect
35. #2024C4semidirect
36. #2024C4semidirect
37. #2023C4semidirect
38. #2023C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#234512
C2×#19952
#213042
C2×#19292
#210512
#113262
#113172
#111912
#111252
#110682
#109242
#109102
#108872
#107962