Finite Group Databasebeta

← #6971192

Subgroups — #6971192

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36290C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#32777C212yesmaximal, complemented
4#31146C214yesmaximal, complemented
5#30649C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#29756C212yesmaximal, complemented
8#29695C212yesmaximal, complemented
9#29319C211yesmaximal, characteristic, complemented
10#28755C211yesmaximal, characteristic, complemented
11#26527C212yesmaximal, complemented
12#26465C212yesmaximal, complemented
13#26236C212yesmaximal
14#26234C212yesmaximal
15#24064C212yesmaximal, complemented
16#23838C211yesmaximal, characteristic, complemented
17#22012C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#491C2212yes—
20C2×#484C2214yes—
21C2×#483C2214yes—
22C2×#479C2214yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2571C2211yescharacteristic
28#2552—21no—
29#2552C2211yescharacteristic
30#2542C2212yes—
31#2538—21no—
32#2509—21no—
33#2507—21no—
34#2505—22no—
35#2473—22no—
36#2442C2212yes—
37#2442C2212yes—
38#2441—22no—
39#2429C2214yes—
40#2423—22no—
41#2365C2212yes—
42#2354—21no—
43Q8×C4⋊C4—21no—
44#2349—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47D4×C4⋊C4C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#125C2211yescharacteristic
50C2×#389C2212yes—
51C42×D4C2211yescharacteristic
52#2097C2214yes—
53#2085C2212yes—
54#2084C2212yes—
55#2084C2211yescharacteristic
56#2083C2211yescharacteristic
57#2066C2214yes—
58#2066C2214yes—
59#2059C2212yes—
60#2058C2211yescharacteristic
61#2057C2211yescharacteristic
62#2057C2212yes—
63#2043C2212yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2214yes—
68#2036C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2035C2214yes—
72#2026C2214yes—
73#1981C2212yes—
74#1981C2212yes—
75#1960C2212yes—
76#1956C2212yes—
77#1956C2212yes—
78#1954C2212yes—
79#1948C2212yes—
80#1947C2212yes—
81#1943C2214yes—
82#1939C2212yes—
83#1924C2214yes—
84#1872C2214yes—
85#1872C2214yes—
86#1863C2214yes—
87#1863C2214yes—
88#1817C2212yes—
89#1805C2214yes—
90#1797C2212yes—
91C4×#107C2212yes—
92C4×#106C2212yes—
93#1767C2211yescharacteristic
94#1744C2211yescharacteristic
95#533—48no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—22no—
101C2×#125—21no—
102C2×#125C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C42C2312yes—
120#491—44no—
121#484—48no—
122#483—48no—
123#479—48no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2312yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—22no—
146#400—22no—
147#399—24no—
148#398—24no—
149#398—22no—
150#398—22no—
151#398—22no—
152#396—24no—
153#391—21no—
154#391C2314yes—
155#390—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#387—21no—
178#387—21no—
179#387—22no—
180#387—22no—
181#387—24no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382—21no—
198#382—21no—
199#382C2311yescharacteristic
200#382C2312yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C2×#95—22no—
221C43C2311yescharacteristic
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8C2414yes—
235C22×D4C2×D411yescharacteristic
236C23×C4—22no—
237C23×C4—22no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—44no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250#125—22no—
251#125—41no—
252#125—44no—
253#125—48no—
254#125—82no—
255C22⋊Q8—28no—
256C22⋊Q8—44no—
257C22⋊Q8—48no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—28no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—48no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—42no—
267C4×D4—44no—
268C4×D4—44no—
269#117—48no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—84no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—24no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—44no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—28no—
337#107—24no—
338#107—42no—
339#106—24no—
340#106—24no—
341#106—44no—
342C4⋊C8—24no—
343C4⋊C8—44no—
344#103—28no—
345#103—44no—
346#103—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352C22⋊C8—44no—
353C22⋊C8—44no—
354#97—42no—
355C4×C8—24no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—24no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D412yes—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4#13714yes—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4H(5,𝔽2)12yes—
478C4⋊C4H(5,𝔽2)14yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—84no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—24no—
492C42—22no—
493C42—21no—
494C42—42no—
495C42—42no—
496C42—48no—
497C42—42no—
498C42—42no—
499C42—44no—
500C422−1+414yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23C2×#13714yes—
508C23C23×D411yescharacteristic
509Q8—48no—
510Q8—44no—
511Q8—44no—
512Q8—44no—
513Q8—44no—
514D4—28no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—164no—
569C2×C4#53612yes—
570C2×C4#53612yes—
571C2×C4C2×#13712yes—
572C2×C4#57512yes—
573C2×C4#57712yes—
574C2×C4#57812yes—
575C2×C4C2×H(5,𝔽2)11yescharacteristic
576C2×C4C2×H(5,𝔽2)12yes—
577C2×C4C2×2−1+412yes—
578C2×C4#58511yescharacteristic
579C8—44no—
580C8—42no—
581C8—42no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×#53611yescharacteristic
590C22#348411yescharacteristic
591C22#348811yescharacteristic
592C22C22×#13711yescharacteristic
593C22C2×#57511yescharacteristic
594C22C2×#57711yescharacteristic
595C22C2×#57811yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—42no—
606C4—82no—
607C4—82no—
608C4—81no—
609C4—81no—
610C4—82no—
611C4—81no—
612C4—84no—
613C4#323614yes—
614C4#354812yes—
615C4#355412yes—
616C2—44no—
617C2#4247911yescharacteristic
618C2C2×#323611yescharacteristic
619C2#6087811yescharacteristic
620C2#6160211yescharacteristic
621C2#6160411yescharacteristic
622C2#6198011yescharacteristic
623C2#6199811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32777C2semidirect
4. #31146C2semidirect
5. #30649C2semidirect
6. #30220C2semidirect
7. #29756C2semidirect
8. #29695C2semidirect
9. #29319C2semidirect
10. #28755C2semidirect
11. #26527C2semidirect
12. #26465C2semidirect
15. #24064C2semidirect
16. #23838C2semidirect

Maximal subgroups

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

subgroupindex
#362902
#361342
#327772
#311462
#306492
#302202
#297562
#296952
#293192
#287552
#265272
#264652
#262362
#262342
#240642
#238382
#220122