Finite Group Databasebeta

← #6624977

Subgroups — #6624977

Order 512.

1739 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33147C211yesmaximal, characteristic
4#32960C212yesmaximal
5#29112C212yesmaximal
6#28121C212yesmaximal
7#26910C214yesmaximal, complemented
8#26772C214yesmaximal, complemented
9#26461C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic
11#25323C212yesmaximal, complemented
12#24212C211yesmaximal, characteristic
13#23222C212yesmaximal, complemented
14#23004C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16C4×#484C211yesmaximal, characteristic
17#21455C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#484C2211yescharacteristic
22C2×#479C2214yes—
23C2×#462C2212yes—
24C2×#451C2212yes—
25C2×#441C2214yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2590—22no—
29#2568C2211yescharacteristic
30#2552C2211yescharacteristic
31#2511—21no—
32#2508C2212yes—
33#2505—22no—
34#2502—21no—
35#2453C2212yes—
36#2365—22no—
37#2365—21no—
38#2365—21no—
39#2365C2211yescharacteristic
40#2365C2214yes—
41#2365C2212yes—
42#2365C2212yes—
43#2360—21no—
44#2355—21no—
45Q8×C4⋊C4C2211yescharacteristic
46C4×C22⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#387—22no—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2074C2214yes—
55#2068C2214yes—
56#2056C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2026C2214yes—
60#2016C2212yes—
61#1988C2212yes—
62#1981C2212yes—
63#1956C2212yes—
64#1948C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1935C2214yes—
68#1928C2212yes—
69#1928C2214yes—
70#1921C2212yes—
71#1919C2214yes—
72#1917C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1872C2211yescharacteristic
76#1865C2214yes—
77#1865C2214yes—
78#1863C2214yes—
79#1853C2212yes—
80#1852C2211yescharacteristic
81#1824C2211yescharacteristic
82#1824C2212yes—
83#1805C2214yes—
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—48no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—22no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#487—28no—
126#484—24no—
127#479—48no—
128#462—28no—
129#451—28no—
130#441—48no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#400—22no—
153#398—22no—
154#398—22no—
155#398—22no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2314yes—
164#391C2314yes—
165#391C2311yescharacteristic
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2311yescharacteristic
178#387—22no—
179#387—21no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—21no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#382—22no—
197#382C2312yes—
198#380—21no—
199#380C2311yescharacteristic
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8C2411yescharacteristic
243C22×Q8C2414yes—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C2×Q16—28no—
253C2×Q16—28no—
254C2×Q16—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8#4118yes—
264#125—44no—
265C22⋊Q8—22no—
266C22⋊Q8—44no—
267C22⋊Q8#4114yes—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8—48no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42—48no—
333C2×C42—44no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—24no—
343#107—24no—
344#106—28no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347#103—28no—
348#103—24no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—24no—
353#103—44no—
354#103—44no—
355#103—44no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8—22no—
359#97—44no—
360C4×C8—24no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24C22×D411yescharacteristic
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—416no—
397C2×Q8—416no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C2×#4118yes—
404C2×Q8C2×#4112yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4#13714yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449Q16—416no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×#4112yes—
495C4⋊C4H(5,𝔽2)14yes—
496C22⋊C4—24no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—24no—
504C42—42no—
505C42—42no—
506C42—48no—
507C42—48no—
508C42—44no—
509C42—44no—
510C42—42no—
511C42—44no—
512C42—42no—
513C42—82no—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#54812yes—
520C23C2×#13712yes—
521C23#57812yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—28no—
525Q8—28no—
526Q8—416no—
527Q8—44no—
528Q8—42no—
529Q8—44no—
530Q8—84no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—416no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#52512yes—
589C2×C4#52512yes—
590C2×C4#54812yes—
591C2×C4C2×#13712yes—
592C2×C4C2×#13712yes—
593C2×C4#57512yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4112yes—
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×2−1+411yescharacteristic
598C8—42no—
599C8—44no—
600C8—42no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22#337414yes—
608C22C2×#52511yescharacteristic
609C22C2×#54811yescharacteristic
610C22#348411yescharacteristic
611C22C22×#13711yescharacteristic
612C22C2×#57511yescharacteristic
613C22C2×#57811yescharacteristic
614C22#358611yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—44no—
624C4—41no—
625C4—48no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4#311714yes—
633C4#348112yes—
634C4#352012yes—
635C2—24no—
636C2#3652811yescharacteristic
637C2C2×#311711yescharacteristic
638C2#6058311yescharacteristic
639C2C2×#337411yescharacteristic
640C2#6132311yescharacteristic
641C2#6158411yescharacteristic
642C2#6159511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #26910C2semidirect
8. #26772C2semidirect
9. #26461C2semidirect
11. #25323C2semidirect
13. #23222C2semidirect
14. #23004C2semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#23652
#331472
#329602
#291122
#281212
#269102
#267722
#264612
#258562
#253232
#242122
#232222
#230042
#226292
C4×#4842
#214552