Finite Group Databasebeta

← #390204

Subgroups — #390204

Order 512.

927 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2#21147C211yesmaximal, characteristic, complemented
3C2×#1929C211yesmaximal, characteristic
4#21052C211yesmaximal, characteristic
5C2×#1922C211yesmaximal, characteristic, complemented
6#11291C211yesmaximal, characteristic, complemented
7#11233C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#11068C211yesmaximal, characteristic
11#10923C211yesmaximal, characteristic
12#10910C211yesmaximal, characteristic, complemented
13#10909C211yesmaximal, characteristic, complemented
14#10878C211yesmaximal, characteristic, complemented
15#10832C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#408C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30#2297C2211yescharacteristic
31#2296C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#2068C411yescharacteristic, complemented
34#2068C411yescharacteristic, complemented
35#2067C411yescharacteristic, complemented
36#2067C411yescharacteristic, complemented
37#2066C411yescharacteristic, complemented
38#2066C411yescharacteristic, complemented
39#2065C411yescharacteristic, complemented
40#2065C411yescharacteristic, complemented
41#1940—21no—
42#1940—21no—
43#1939—21no—
44#1939—21no—
45#1929—24no—
46#1924—21no—
47#1924—21no—
48#1923—21no—
49#1923—21no—
50#1922—24no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54#1735C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64#1728C2211yescharacteristic
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
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—41no—
89#427—22no—
90#427—22no—
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95#424—22no—
96#424—41no—
97#424—41no—
98#424—41no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8—41no—
107C2×C4⋊C8C2×C411yescharacteristic
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C411yescharacteristic
110C2×C4⋊C8C2×C411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113C2×C4⋊C8D411yescharacteristic
114C2×C4⋊C8D411yescharacteristic
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—22no—
118C2×#104—41no—
119C2×#104—41no—
120C2×#104—41no—
121#416—22no—
122#416—22no—
123#416—41no—
124#416—41no—
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133#408—22no—
134#408—41no—
135#408—41no—
136#408—41no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141#405—22no—
142#405—22no—
143#405—41no—
144#405—41no—
145C2×#97—21no—
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149#391C2×C411yescharacteristic
150#391C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#390C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#389C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#386—21no—
158#386—21no—
159#385—21no—
160#385—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C43—41no—
166C43—41no—
167#341—22no—
168#341—22no—
169#341—41no—
170#340C2×C411yescharacteristic
171#340C2×C411yescharacteristic
172#340C2×C411yescharacteristic
173#340C2×C411yescharacteristic
174#339—22no—
175#339—22no—
176#339—41no—
177#339—41no—
178#339—41no—
179#339D414yes—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—22no—
185#336—21no—
186#336—21no—
187#336—22no—
188#336—21no—
189#336—21no—
190#336—22no—
191#336—22no—
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×Q8C22×C411yescharacteristic
205C22×D4—21no—
206C22×D4C22×C411yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4#4111yescharacteristic
211C23×C4#4111yescharacteristic
212C23×C4C2411yescharacteristic
213C2×M16—21no—
214C2×M16—21no—
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×M16C2×D412yes—
236C2×M16#4112yes—
237C22×C8—21no—
238C22×C8—21no—
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×C8C22×C411yescharacteristic
266C22×C8C22×C411yescharacteristic
267C22×C8C22×C411yescharacteristic
268C22×C8C22×C411yescharacteristic
269C22×C8C2×D411yescharacteristic
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8C2×D411yescharacteristic
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117C2×D412yes—
284#117#4112yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—24no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42#4112yes—
331#108—44no—
332#107—22no—
333#107—22no—
334#106—22no—
335#106—22no—
336C4⋊C8—41no—
337C4⋊C8—41no—
338C4⋊C8—41no—
339C4⋊C8—41no—
340C4⋊C8—41no—
341C4⋊C8—41no—
342C4⋊C8—41no—
343C4⋊C8—41no—
344C4⋊C8—42no—
345C4⋊C8—42no—
346C4⋊C8—42no—
347C4⋊C8—42no—
348#104—44no—
349#104—44no—
350#103—22no—
351#103—22no—
352#103—41no—
353#103—41no—
354#102—22no—
355#102—22no—
356#102—41no—
357#102—41no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362#97—42no—
363#97—42no—
364C4×C8—42no—
365C4×C8—42no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374C24—21no—
375C24C2×#4111yescharacteristic
376#41—41no—
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—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—41no—
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×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C4×D411yescharacteristic
448C22×C4C4×D411yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C42−1+411yescharacteristic
460C22×C42−1+411yescharacteristic
461C22×C42−1+411yescharacteristic
462C22×C42−1+411yescharacteristic
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478M16—41no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
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—22no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
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—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C4⋊C4C4×D412yes—
546C4⋊C4C4×D412yes—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—48no—
572C42—48no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23C2×C4×D411yescharacteristic
593C23#53811yescharacteristic
594C23#53911yescharacteristic
595C23#54711yescharacteristic
596C23#55211yescharacteristic
597C23#55311yescharacteristic
598C23#55611yescharacteristic
599Q8—41no—
600Q8—41no—
601Q8—41no—
602Q8—41no—
603D4—41no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—81no—
608D4—81no—
609D4—81no—
610D4—81no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
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—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4#44312yes—
665C2×C4#44412yes—
666C2×C4C2×C4×D411yescharacteristic
667C2×C4C4×#4111yescharacteristic
668C2×C4#51811yescharacteristic
669C2×C4#51911yescharacteristic
670C2×C4#52011yescharacteristic
671C2×C4#52011yescharacteristic
672C2×C4#53411yescharacteristic
673C2×C4#53611yescharacteristic
674C2×C4#53711yescharacteristic
675C2×C4#53711yescharacteristic
676C2×C4#53811yescharacteristic
677C2×C4#53911yescharacteristic
678C2×C4#54111yescharacteristic
679C2×C4#54111yescharacteristic
680C2×C4#54211yescharacteristic
681C2×C4#54211yescharacteristic
682C2×C4D4211yescharacteristic
683C2×C4#54611yescharacteristic
684C2×C4#54711yescharacteristic
685C2×C4#54811yescharacteristic
686C2×C4#54811yescharacteristic
687C2×C4#55011yescharacteristic
688C2×C4#55311yescharacteristic
689C2×C4#55511yescharacteristic
690C2×C4#56211yescharacteristic
691C2×C4#56211yescharacteristic
692C2×C4#56311yescharacteristic
693C2×C4#56311yescharacteristic
694C8—41no—
695C8—41no—
696C8—41no—
697C8—41no—
698C8—42no—
699C8—42no—
700C8—41no—
701C8—41no—
702C8—41no—
703C8—41no—
704C8—81no—
705C8—81no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—21no—
717C22—21no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22—81no—
725C22#232511yescharacteristic
726C22#236111yescharacteristic
727C22#236211yescharacteristic
728C22#271511yescharacteristic
729C22#272111yescharacteristic
730C22#277911yescharacteristic
731C22#278311yescharacteristic
732C22C2×#44311yescharacteristic
733C22C2×#44411yescharacteristic
734C22#295611yescharacteristic
735C22#295611yescharacteristic
736C4—22no—
737C4—22no—
738C4—44no—
739C4—44no—
740C4—41no—
741C4—41no—
742C4—42no—
743C4—42no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4—81no—
749C4#232311yescharacteristic
750C4#233411yescharacteristic
751C4#236311yescharacteristic
752C4#237811yescharacteristic
753C4#268111yescharacteristic
754C4#268211yescharacteristic
755C4#272511yescharacteristic
756C4#273111yescharacteristic
757C2—21no—
758C2—21no—
759C2—21no—
760C2—21no—
761C2—161no—
762C2#1456511yescharacteristic
763C2#2220411yescharacteristic
764C2#2220711yescharacteristic
765C2#2301511yescharacteristic
766C2#2301611yescharacteristic
767C2#2308011yescharacteristic
768C2#2308111yescharacteristic
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 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21147C2semidirect
5. C2×#1922C2semidirect
6. #11291C2semidirect
7. #11233C2semidirect
12. #10910C2semidirect
13. #10909C2semidirect
14. #10878C2semidirect
15. #10832C2semidirect
33. #2068C4semidirect
34. #2068C4semidirect
35. #2067C4semidirect
36. #2067C4semidirect
37. #2066C4semidirect
38. #2066C4semidirect
39. #2065C4semidirect
40. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
#211472
C2×#19292
#210522
C2×#19222
#112912
#112332
#111912
#110682
#109232
#109102
#109092
#108782
#108322