Finite Group Databasebeta

← #369901

Subgroups — #369901

Order 512.

869 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#21086C211yesmaximal, characteristic
2#21048C211yesmaximal, characteristic
3#20439C211yesmaximal, characteristic, complemented
4#11291C211yesmaximal, characteristic, complemented
5#11291C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11068C211yesmaximal, characteristic
8#10934C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10933C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic, complemented
12#10886C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10798C211yesmaximal, characteristic, complemented
15#10705C211yesmaximal, characteristic, complemented
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C22×#104C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C2×#416C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#408C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2297C2211yescharacteristic
30#2296C2211yescharacteristic
31#1936C411yescharacteristic, complemented
32#1936C411yescharacteristic, complemented
33#1936C411yescharacteristic, complemented
34#1936C411yescharacteristic, complemented
35#1935C411yescharacteristic, complemented
36#1935C411yescharacteristic, complemented
37#1935C411yescharacteristic, complemented
38#1935C411yescharacteristic, complemented
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1771—21no—
44#1771—21no—
45#1770—21no—
46C8×C4⋊C4—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1735C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61C2×#336C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
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×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#424—22no—
83#424—22no—
84#424—22no—
85#424—41no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×#104—22no—
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—41no—
95C2×#104—41no—
96#416—22no—
97#416—22no—
98#416—41no—
99#416—41no—
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#408—22no—
109#408—22no—
110#408—41no—
111#408—41no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116#405—22no—
117#405—22no—
118#405—22no—
119#405—41no—
120C4×M16—22no—
121C4×M16—22no—
122C4×M16—41no—
123C4×M16—41no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×#97—41no—
129C2×#97—41no—
130C2×#97C2×C411yescharacteristic
131C2×#97C2×C411yescharacteristic
132C2×#97C2×C411yescharacteristic
133C2×#97C2×C411yescharacteristic
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#390C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#386—21no—
147#386—21no—
148#385—21no—
149#385—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C2×#95—21no—
155C2×#95—21no—
156#341—22no—
157#341—22no—
158#341—22no—
159#341—22no—
160#340C2×C411yescharacteristic
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#339—22no—
169#339—22no—
170#339—22no—
171#339—22no—
172#339—41no—
173#339—41no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—41no—
179#336—41no—
180#336—41no—
181#336—41no—
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#4112yes—
226C2×M16#4112yes—
227C2×M16#4112yes—
228C2×M16#4112yes—
229C2×M16#4112yes—
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—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—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×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C4⋊C8—24no—
320C4⋊C8—24no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323#104—44no—
324#104—44no—
325#103—22no—
326#103—22no—
327#103—41no—
328#103—41no—
329#102—22no—
330#102—22no—
331#102—41no—
332#102—41no—
333C22⋊C8—42no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337#97—24no—
338#97—41no—
339#97—41no—
340#97—41no—
341#97—41no—
342#97—42no—
343#97—42no—
344#97—42no—
345C4×C8—24no—
346C4×C8—42no—
347C4×C8—42no—
348C4×C8—42no—
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—
358#95—42no—
359C24—21no—
360C24C2×#4111yescharacteristic
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—
376#41—41no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—41no—
409C22×C4—41no—
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×C4—41no—
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C42−1+411yescharacteristic
448M16—22no—
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—22no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16#12614yes—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
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—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
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—
515C2×C8—41no—
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—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—81no—
529C4⋊C4—81no—
530C4⋊C4#11712yes—
531C4⋊C4#11712yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—22no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
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#52911yescharacteristic
577C23#52911yescharacteristic
578C23#55311yescharacteristic
579C23#55311yescharacteristic
580C23#55611yescharacteristic
581C23#55911yescharacteristic
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—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
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—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
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—81no—
649C2×C4C2×#11711yescharacteristic
650C2×C4C4×#4111yescharacteristic
651C2×C4C4×#4111yescharacteristic
652C2×C4C4×#4111yescharacteristic
653C2×C4#51811yescharacteristic
654C2×C4#52011yescharacteristic
655C2×C4#52511yescharacteristic
656C2×C4#52511yescharacteristic
657C2×C4C2×#12611yescharacteristic
658C2×C4#52911yescharacteristic
659C2×C4#52911yescharacteristic
660C2×C4#52911yescharacteristic
661C2×C4#53811yescharacteristic
662C2×C4#53911yescharacteristic
663C2×C4#54011yescharacteristic
664C2×C4#54211yescharacteristic
665C2×C4#54611yescharacteristic
666C2×C4#54611yescharacteristic
667C2×C4#54611yescharacteristic
668C2×C4#54811yescharacteristic
669C2×C4#55211yescharacteristic
670C2×C4#55211yescharacteristic
671C2×C4#55311yescharacteristic
672C2×C4#55311yescharacteristic
673C2×C4#55511yescharacteristic
674C2×C4#55511yescharacteristic
675C2×C4#55911yescharacteristic
676C2×C4#55911yescharacteristic
677C8—24no—
678C8—41no—
679C8—41no—
680C8—41no—
681C8—41no—
682C8—42no—
683C8—41no—
684C8—41no—
685C8—42no—
686C8—41no—
687C8—41no—
688C8—42no—
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#229011yescharacteristic
709C22#236411yescharacteristic
710C22#236711yescharacteristic
711C22#251311yescharacteristic
712C22#252511yescharacteristic
713C22#259311yescharacteristic
714C22#259611yescharacteristic
715C22#295511yescharacteristic
716C22#295511yescharacteristic
717C22#295611yescharacteristic
718C22#295611yescharacteristic
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—42no—
724C4—42no—
725C4—82no—
726C4—82no—
727C4—81no—
728C4—81no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4C4×#12611yescharacteristic
733C4#233511yescharacteristic
734C4#234611yescharacteristic
735C4#235711yescharacteristic
736C4#245811yescharacteristic
737C4#245911yescharacteristic
738C4#247611yescharacteristic
739C4#248111yescharacteristic
740C2—21no—
741C2—21no—
742C2—21no—
743C2—21no—
744C2—161no—
745C2#1403711yescharacteristic
746C2#2113811yescharacteristic
747C2#2113911yescharacteristic
748C2#2314811yescharacteristic
749C2#2314811yescharacteristic
750C2#2323611yescharacteristic
751C2#2323611yescharacteristic
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. #20439C2semidirect
4. #11291C2semidirect
5. #11291C2semidirect
11. #10887C2semidirect
12. #10886C2semidirect
13. #10809C2semidirect
14. #10798C2semidirect
15. #10705C2semidirect
31. #1936C4semidirect
32. #1936C4semidirect
33. #1936C4semidirect
34. #1936C4semidirect
35. #1935C4semidirect
36. #1935C4semidirect
37. #1935C4semidirect
38. #1935C4semidirect

Maximal subgroups

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

subgroupindex
#210862
#210482
#204392
#112912
#111912
#110682
#109342
#109332
#108872
#108862
#108092
#107982
#107052