Finite Group Databasebeta

← #6806339

Subgroups — #6806339

Order 512.

1200 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39753C211yesmaximal, characteristic
2#39635C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic
4#31477C212yesmaximal
5#31476C212yesmaximal
6#31473C212yesmaximal
7#31472C212yesmaximal
8#30232C212yesmaximal
9#29210C211yesmaximal, characteristic
10#29187C212yesmaximal
11#28729C212yesmaximal
12#28717C211yesmaximal, characteristic
13#26113C211yesmaximal, characteristic
14#26112C211yesmaximal, characteristic
15#26109C211yesmaximal, characteristic
16#26108C211yesmaximal, characteristic
17#24997C211yesmaximal, characteristic
18#24996C211yesmaximal, characteristic
19#24995C211yesmaximal, characteristic
20#24994C211yesmaximal, characteristic
21#22229C212yesmaximal
22C4×#491C211yesmaximal, characteristic
23#21343C211yesmaximal, characteristic
24C2×#491C2211yescharacteristic
25C2×#487C2212yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2802—22no—
29#2802—22no—
30#2791—22no—
31#2713—22no—
32#2660C2212yes—
33#2647C2212yes—
34#2582C2211yescharacteristic
35#2582C2211yescharacteristic
36#2571C2211yescharacteristic
37#2570C2212yes—
38#2565—21no—
39#2552C2211yescharacteristic
40#2552C2212yes—
41#2550—21no—
42#2511—21no—
43#2511—21no—
44#2507—21no—
45#2507—21no—
46#2506—21no—
47#2453C2212yes—
48#2453C2212yes—
49#2451—21no—
50#2355C2211yescharacteristic
51Q8×C4⋊C4C2211yescharacteristic
52Q8×C4⋊C4C2211yescharacteristic
53Q8×C4⋊C4C2211yescharacteristic
54#2331C2212yes—
55#2327C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125C2211yescharacteristic
58#2295C2211yescharacteristic
59#2101C2212yes—
60#2101C2212yes—
61#2101C2212yes—
62#2101C2212yes—
63#2100C2212yes—
64#2100C2212yes—
65#2099C2212yes—
66#2099C2212yes—
67#2088C2212yes—
68#2088C2212yes—
69#2087C2212yes—
70#2086C2212yes—
71#2074C2212yes—
72#2074C2212yes—
73#2072C2212yes—
74#2072C2212yes—
75#2072C2212yes—
76#2072C2212yes—
77#2059C2212yes—
78#2059C2212yes—
79#1993C2212yes—
80#1993C2212yes—
81#1992C2212yes—
82#1992C2212yes—
83#1992C2211yescharacteristic
84#1991C2212yes—
85#1990C2212yes—
86#1984C2211yescharacteristic
87#1984C2211yescharacteristic
88#1960C2211yescharacteristic
89#1948C2211yescharacteristic
90#1948C2211yescharacteristic
91#1947C2211yescharacteristic
92#1947C2211yescharacteristic
93#1939C2211yescharacteristic
94#1939C2211yescharacteristic
95#1935C2211yescharacteristic
96#1935C2211yescharacteristic
97#1935C2211yescharacteristic
98#1935C2211yescharacteristic
99#1928C2212yes—
100#1928C2212yes—
101#1927C2212yes—
102#1927C2212yes—
103#1921C2212yes—
104#1920C2212yes—
105#1919C2212yes—
106#1918C2212yes—
107#1917C2212yes—
108#1917C2212yes—
109#1865C2212yes—
110#1865C2212yes—
111#1863C2212yes—
112#1863C2212yes—
113#1846C2212yes—
114#1836C2211yescharacteristic
115#1834C2211yescharacteristic
116#1834C2211yescharacteristic
117#1834C2211yescharacteristic
118#1834C2211yescharacteristic
119#1805C2212yes—
120#1805C2212yes—
121#1803C2211yescharacteristic
122#1803C2212yes—
123#1803C2211yescharacteristic
124#1803C2211yescharacteristic
125#1803C2211yescharacteristic
126#1803C2212yes—
127#1768C2212yes—
128C4×C4⋊C8C2211yescharacteristic
129C4×#103C2211yescharacteristic
130C4×#103C2211yescharacteristic
131C4×#103C2211yescharacteristic
132C4×#103C2211yescharacteristic
133#1746C2212yes—
134C4×#97C2211yescharacteristic
135C2×C4⋊Q8—21no—
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C2311yescharacteristic
138C2×C4⋊Q8C2312yes—
139C2×#125—21no—
140C2×#125—22no—
141C2×#125C2311yescharacteristic
142C2×#125C2312yes—
143C2×C4×Q8—21no—
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2312yes—
147C2×C4×Q8C2312yes—
148C2×C4×Q8C2311yescharacteristic
149#491—24no—
150#487—28no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2311yescharacteristic
160C2×#103C2311yescharacteristic
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2311yescharacteristic
166C2×#103C2311yescharacteristic
167C2×#103C2311yescharacteristic
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2311yescharacteristic
171C2×#97C2311yescharacteristic
172C2×#97C2311yescharacteristic
173C2×#97C2311yescharacteristic
174C2×#97C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2312yes—
178#400—22no—
179#400—22no—
180#400—22no—
181#400—22no—
182#400—22no—
183#400—22no—
184#398—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#395—22no—
197#395—22no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—22no—
202#391—22no—
203#391D411yescharacteristic
204#391D411yescharacteristic
205#391C2311yescharacteristic
206#391C2311yescharacteristic
207#391C2312yes—
208#391C2312yes—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—22no—
213#389—21no—
214#389—21no—
215#389—22no—
216#389—22no—
217#389—22no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389—22no—
225#389—22no—
226#389D411yescharacteristic
227#389D411yescharacteristic
228#389C2311yescharacteristic
229#389C2311yescharacteristic
230#389C2312yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#389C2312yes—
236#387—22no—
237#387—21no—
238#387—21no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387D412yes—
250#387D412yes—
251#384—21no—
252#384—21no—
253#384—21no—
254#384—21no—
255#384C2311yescharacteristic
256#384C2312yes—
257#384C2311yescharacteristic
258#384C2311yescharacteristic
259#384C2312yes—
260#384C2311yescharacteristic
261#384C2311yescharacteristic
262#384C2312yes—
263#382—21no—
264#382C2311yescharacteristic
265#382C2312yes—
266#382C2312yes—
267#382C2312yes—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4C2311yescharacteristic
276C4×C4⋊C4C2311yescharacteristic
277C4×C4⋊C4C2312yes—
278C4×C4⋊C4C2312yes—
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281#376C2312yes—
282C43C2311yescharacteristic
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#340C2312yes—
291#340C2312yes—
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2311yescharacteristic
296#340C2311yescharacteristic
297#340C2311yescharacteristic
298#340C2311yescharacteristic
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#336C2312yes—
308#336C2312yes—
309#336C2312yes—
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314C22×Q8—22no—
315C22×Q8C2×D411yescharacteristic
316C22×Q8C2411yescharacteristic
317C22×Q8C2411yescharacteristic
318C22×C8C2411yescharacteristic
319C22×C8C2411yescharacteristic
320C22×C8C2412yes—
321C22×C8C2412yes—
322C22×C8C2411yescharacteristic
323C22×C8C2411yescharacteristic
324C4⋊Q8—24no—
325C4⋊Q8—22no—
326C4⋊Q8—24no—
327C4⋊Q8—82no—
328C4⋊Q8—82no—
329C4⋊Q8#4114yes—
330#125—22no—
331#125—22no—
332#125—42no—
333#125—42no—
334#125—82no—
335#125—84no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—42no—
343C4×Q8—42no—
344C4×Q8—42no—
345C4×Q8—42no—
346C4×Q8—82no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D411yescharacteristic
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2411yescharacteristic
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2412yes—
411C2×C42C2412yes—
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415#107—42no—
416#107—42no—
417#106—42no—
418#106—42no—
419C4⋊C8—22no—
420C4⋊C8—22no—
421C4⋊C8—42no—
422C4⋊C8—42no—
423#103—24no—
424#103—24no—
425#103—24no—
426#103—24no—
427#103—22no—
428#103—22no—
429#103—22no—
430#103—22no—
431#103—41no—
432#103—41no—
433#103—41no—
434#103—41no—
435#97—22no—
436#97—22no—
437#97—41no—
438#97—41no—
439C4×C8—24no—
440C4×C8—22no—
441C4×C8—22no—
442#95—21no—
443#95—21no—
444#95—22no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—22no—
452#95—22no—
453#95—21no—
454#95—21no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95C2×D412yes—
471#95C2×D412yes—
472#95C2×D412yes—
473#95C2×D412yes—
474#95C2×D412yes—
475#95C2×D412yes—
476#95C2×D412yes—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—44no—
491C2×Q8—82no—
492C2×Q8—82no—
493C2×Q8C2×#4112yes—
494C2×Q8C2×#4112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C2511yescharacteristic
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—22no—
535C2×C8—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—44no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—44no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—81no—
577C4⋊C4—81no—
578C4⋊C4—81no—
579C4⋊C4—81no—
580C4⋊C4—82no—
581C4⋊C4—82no—
582C4⋊C4—82no—
583C4⋊C4—82no—
584C4⋊C4—82no—
585C4⋊C4—82no—
586C4⋊C4—82no—
587C4⋊C4—82no—
588C4⋊C4—82no—
589C4⋊C4—82no—
590C4⋊C4—82no—
591C4⋊C4—82no—
592C4⋊C4—82no—
593C4⋊C4—82no—
594C4⋊C4—81no—
595C4⋊C4—81no—
596C4⋊C4—82no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—21no—
602C42—21no—
603C42—22no—
604C42—22no—
605C42—41no—
606C42—41no—
607C42—42no—
608C42—42no—
609C42—42no—
610C42—42no—
611C42—82no—
612C42—82no—
613C42—82no—
614C42—82no—
615C42C2×#4112yes—
616C422−1+412yes—
617C422−1+412yes—
618C422−1+412yes—
619C23C23×D411yescharacteristic
620Q8—22no—
621Q8—22no—
622Q8—22no—
623Q8—22no—
624Q8—44no—
625Q8—41no—
626Q8—41no—
627Q8—41no—
628Q8—41no—
629Q8—164no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4#53612yes—
691C2×C4#53612yes—
692C2×C4#54812yes—
693C2×C4#54812yes—
694C2×C4#57712yes—
695C2×C4#57812yes—
696C2×C4C22×#4111yescharacteristic
697C2×C4C2×2−1+411yescharacteristic
698C2×C4C2×2−1+411yescharacteristic
699C2×C4C2×2−1+411yescharacteristic
700C2×C4#58511yescharacteristic
701C2×C4#58511yescharacteristic
702C8—42no—
703C8—42no—
704C8—41no—
705C8—41no—
706C8—41no—
707C8—41no—
708C22C2×#53611yescharacteristic
709C22C2×#54811yescharacteristic
710C22#350011yescharacteristic
711C22C2×#57711yescharacteristic
712C22C2×#57811yescharacteristic
713C22#358511yescharacteristic
714C22#358611yescharacteristic
715C4—22no—
716C4—22no—
717C4—22no—
718C4—42no—
719C4—42no—
720C4—41no—
721C4—42no—
722C4—41no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—81no—
727C4—82no—
728C4—81no—
729C4—82no—
730C4—81no—
731C4—82no—
732C4—82no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4#324214yes—
737C4#351511yescharacteristic
738C4#351511yescharacteristic
739C4#355511yescharacteristic
740C4#355511yescharacteristic
741C2#3917311yescharacteristic
742C2C2×#324211yescharacteristic
743C2#6088711yescharacteristic
744C2#6134111yescharacteristic
745C2#6134211yescharacteristic
746C2#6261611yescharacteristic
747C2#6261911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#397532
#396352
#332972
#314772
#314762
#314732
#314722
#302322
#292102
#291872
#287292
#287172
#261132
#261122
#261092
#261082
#249972
#249962
#249952
#249942
#222292
C4×#4912
#213432