Finite Group Databasebeta

← #6627776

Subgroups — #6627776

Order 512.

1501 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic
4#33027C212yesmaximal
5#29357C212yesmaximal
6#29287C211yesmaximal, characteristic
7#29257C212yesmaximal
8#28748C211yesmaximal, characteristic
9#26714C214yesmaximal
10#26317C212yesmaximal
11#26183C214yesmaximal
12#25931C212yesmaximal
13#23222C212yesmaximal
14#23004C212yesmaximal
15#22825C211yesmaximal, characteristic
16C4×#477C211yesmaximal, characteristic
17#21506C212yesmaximal
18C2×#557C2211yescharacteristic
19C2×#487C2212yes—
20C2×#477C2211yescharacteristic
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#441C2214yes—
25#2868C2211yescharacteristic
26#2862C2212yes—
27#2582—21no—
28#2571—21no—
29#2570—22no—
30#2568C2212yes—
31#2511—22no—
32#2446C2214yes—
33#2370—22no—
34#2370C2212yes—
35#2365—21no—
36#2365C2211yescharacteristic
37#2365C2212yes—
38#2355—21no—
39#2355—21no—
40#2355—21no—
41#2355C2211yescharacteristic
42#2351C2212yes—
43Q8×C4⋊C4C2211yescharacteristic
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#125—22no—
49C4×#125C2212yes—
50C42×Q8—21no—
51C42×Q8C2212yes—
52C42×Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2097C2214yes—
55#2088C2214yes—
56#2074C2214yes—
57#2043C2214yes—
58#2026C2214yes—
59#2026C2214yes—
60#1988C2212yes—
61#1987C2212yes—
62#1987C2211yescharacteristic
63#1984C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1956C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1935C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1928C2214yes—
77#1927C2214yes—
78#1921C2212yes—
79#1917C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1865C2214yes—
84#1863C2212yes—
85#1863C2214yes—
86#1842C2211yescharacteristic
87#1831C2212yes—
88#1803C2211yescharacteristic
89C4×#107C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95C22×Q16C2314yes—
96#557—24no—
97#557—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×#125C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2311yescharacteristic
119#487—28no—
120#477—24no—
121#462—28no—
122#462—28no—
123#462—44no—
124#441—48no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#400—24no—
145#400—24no—
146#398—24no—
147#391—21no—
148#391—22no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—22no—
154#391C2314yes—
155#391C2314yes—
156#391C2314yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—24no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#384—21no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—22no—
187#382—22no—
188#382—22no—
189#382—22no—
190#382C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C43—22no—
222C43—21no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2414yes—
240C22×Q8C2411yescharacteristic
241C2×Q16—28no—
242C2×Q16—28no—
243C2×Q16—28no—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C4⋊Q8—28no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8#4114yes—
258#125—216no—
259#125—24no—
260#125—24no—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—42no—
272C4×Q8—42no—
273C4×Q8—41no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8#4114yes—
280C4×Q8#4114yes—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
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—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—24no—
345#107—42no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—42no—
352#103—24no—
353#103—24no—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—28no—
358#103—44no—
359#103—44no—
360#103—44no—
361#97—44no—
362C4×C8—24no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C2×#4112yes—
404C2×Q8C2×#4112yes—
405C2×Q8H(5,𝔽2)12yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426Q16—416no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C2×#4112yes—
478C4⋊C4C2×#4114yes—
479C4⋊C4C2×#4112yes—
480C42—28no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—22no—
492C42—21no—
493C42—41no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—44no—
499C42—44no—
500C42—42no—
501C42—44no—
502C42—42no—
503C42—81no—
504C42—82no—
505C42—84no—
506C42#13714yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C422−1+412yes—
511C23C23×D411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—44no—
516Q8—48no—
517Q8—44no—
518Q8—44no—
519Q8—44no—
520Q8—44no—
521Q8—84no—
522Q8#55914yes—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4#52512yes—
572C2×C4#52512yes—
573C2×C4#54812yes—
574C2×C4#54812yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4#57512yes—
579C2×C4#57812yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×H(5,𝔽2)11yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C2×C4#58511yescharacteristic
586C8—44no—
587C8—42no—
588C8—42no—
589C22C2×#52511yescharacteristic
590C22C2×#54811yescharacteristic
591C22#348811yescharacteristic
592C22C22×#13711yescharacteristic
593C22C2×#57511yescharacteristic
594C22C2×#57811yescharacteristic
595C22#358611yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—24no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—82no—
609C4—81no—
610C4—82no—
611C4—82no—
612C4—84no—
613C4#311614yes—
614C4#337414yes—
615C4C2×#55911yescharacteristic
616C4#348011yescharacteristic
617C4#350711yescharacteristic
618C4#352211yescharacteristic
619C2#3658011yescharacteristic
620C2C2×#311611yescharacteristic
621C2#6059411yescharacteristic
622C2C2×#337411yescharacteristic
623C2#6132311yescharacteristic
624C2#6196411yescharacteristic
625C2#6199811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ 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
#365982
C4×#5572
#331632
#330272
#293572
#292872
#292572
#287482
#267142
#263172
#261832
#259312
#232222
#230042
#228252
C4×#4772
#215062