Finite Group Databasebeta

← #385698

Subgroups — #385698

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#21051C211yesmaximal, characteristic
3#20754C211yesmaximal, characteristic, complemented
4#11376C211yesmaximal, characteristic, complemented
5#11269C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11075C211yesmaximal, characteristic
8#11068C211yesmaximal, characteristic
9#11068C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic, complemented
12#10887C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10874C211yesmaximal, characteristic, complemented
15#10852C211yesmaximal, characteristic, complemented
16C2×#427C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1928C411yescharacteristic, complemented
32#1928C411yescharacteristic, complemented
33#1927C411yescharacteristic, complemented
34#1927C411yescharacteristic, complemented
35#1926C411yescharacteristic, complemented
36#1926C411yescharacteristic, complemented
37#1925C411yescharacteristic, complemented
38#1925C411yescharacteristic, complemented
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1832—21no—
44#1832—21no—
45#1831—21no—
46#1831—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#427—41no—
84#427—41no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89#424—22no—
90#424—22no—
91#424—41no—
92#424—41no—
93#423—22no—
94#423—41no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×#104—22no—
102C2×#104—41no—
103C2×#104—41no—
104C2×#104—41no—
105C2×#104—41no—
106C2×#104—41no—
107#416—22no—
108#416—22no—
109#416—41no—
110#416—41no—
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119#408—22no—
120#408—41no—
121#408—41no—
122#408—41no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127#405—41no—
128#405—41no—
129C2×#97—21no—
130C2×C4×C8—21no—
131C2×C4×C8—41no—
132C2×C4×C8—41no—
133C2×C4×C8—41no—
134#391C2×C411yescharacteristic
135#391C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#386—21no—
139#386—21no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—41no—
142C4×C4⋊C4C2×C411yescharacteristic
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150#376—21no—
151C2×#95—21no—
152C43—21no—
153#341—22no—
154#341—41no—
155#341—41no—
156#341—41no—
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#339—22no—
162#339—22no—
163#339—41no—
164#339—41no—
165#339—41no—
166#339—41no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—22no—
176#336—41no—
177#336—41no—
178#336C2×C411yescharacteristic
179#336C2×C411yescharacteristic
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336D411yescharacteristic
183#336D411yescharacteristic
184#336D411yescharacteristic
185#336D411yescharacteristic
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C22×Q8C22×C411yescharacteristic
199C22×D4—21no—
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4#4111yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
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#4112yes—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
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×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C22×C411yescharacteristic
262C22×C8C22×C411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117#4112yes—
278#117#4112yes—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—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—21no—
316C2×C42—21no—
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42#4112yes—
322#107—41no—
323#107—41no—
324#106—41no—
325#106—41no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332#104—44no—
333#104—44no—
334#103—22no—
335#103—22no—
336#103—41no—
337#103—41no—
338#102—22no—
339#102—22no—
340#102—41no—
341#102—41no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346#97—42no—
347C4×C8—42no—
348C4×C8—44no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—42no—
358C24—21no—
359C24C2×#4111yescharacteristic
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C42−1+411yescharacteristic
445C22×C42−1+411yescharacteristic
446C22×C42−1+411yescharacteristic
447M16—22no—
448M16—22no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—22no—
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—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
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—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—48no—
526C4⋊C4#11712yes—
527C4⋊C4#11712yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—42no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C22⋊C4—81no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—42no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—81no—
563C42—81no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×C4×D411yescharacteristic
576C23#54011yescharacteristic
577C23#54211yescharacteristic
578C23#55111yescharacteristic
579C23#55211yescharacteristic
580C23#55211yescharacteristic
581C23#55311yescharacteristic
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—81no—
591D4—81no—
592D4—81no—
593D4—81no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4#44312yes—
649C2×C4C2×#11711yescharacteristic
650C2×C4C4×#4111yescharacteristic
651C2×C4#51811yescharacteristic
652C2×C4#52011yescharacteristic
653C2×C4#52011yescharacteristic
654C2×C4#52011yescharacteristic
655C2×C4#53511yescharacteristic
656C2×C4#53711yescharacteristic
657C2×C4#53811yescharacteristic
658C2×C4#53911yescharacteristic
659C2×C4#54011yescharacteristic
660C2×C4#54111yescharacteristic
661C2×C4#54111yescharacteristic
662C2×C4#54211yescharacteristic
663C2×C4#54211yescharacteristic
664C2×C4#54211yescharacteristic
665C2×C4#54611yescharacteristic
666C2×C4#54611yescharacteristic
667C2×C4#54711yescharacteristic
668C2×C4#54811yescharacteristic
669C2×C4#55311yescharacteristic
670C2×C4#55311yescharacteristic
671C2×C4#55911yescharacteristic
672C2×C4#55911yescharacteristic
673C2×C4H(3,𝔽4)11yescharacteristic
674C2×C4#56211yescharacteristic
675C2×C4#56211yescharacteristic
676C2×C4#56311yescharacteristic
677C8—41no—
678C8—41no—
679C8—41no—
680C8—41no—
681C8—41no—
682C8—41no—
683C8—42no—
684C8—41no—
685C8—41no—
686C8—81no—
687C8—81no—
688C8—81no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22#232411yescharacteristic
709C22#235211yescharacteristic
710C22#235911yescharacteristic
711C22#275311yescharacteristic
712C22#275411yescharacteristic
713C22#277211yescharacteristic
714C22#277311yescharacteristic
715C22C2×#44311yescharacteristic
716C22#295511yescharacteristic
717C22#295611yescharacteristic
718C22#295711yescharacteristic
719C4—22no—
720C4—44no—
721C4—42no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—42no—
726C4—82no—
727C4—81no—
728C4—81no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4#231411yescharacteristic
733C4#233411yescharacteristic
734C4#235711yescharacteristic
735C4#237911yescharacteristic
736C4#271011yescharacteristic
737C4#271111yescharacteristic
738C4#271211yescharacteristic
739C4#272511yescharacteristic
740C2—21no—
741C2—21no—
742C2—21no—
743C2—21no—
744C2—161no—
745C2#1441211yescharacteristic
746C2#2216511yescharacteristic
747C2#2216711yescharacteristic
748C2#2278311yescharacteristic
749C2#2278411yescharacteristic
750C2#2294811yescharacteristic
751C2#2294911yescharacteristic
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
3. #20754C2semidirect
4. #11376C2semidirect
5. #11269C2semidirect
11. #10909C2semidirect
12. #10887C2semidirect
13. #10882C2semidirect
14. #10874C2semidirect
15. #10852C2semidirect
31. #1928C4semidirect
32. #1928C4semidirect
33. #1927C4semidirect
34. #1927C4semidirect
35. #1926C4semidirect
36. #1926C4semidirect
37. #1925C4semidirect
38. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#210512
#207542
#113762
#112692
#111912
#110752
#110682
#109292
#109092
#108872
#108822
#108742
#108522