Finite Group Databasebeta

← #6621134

Subgroups — #6621134

Order 512.

1481 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic
3#33202C212yesmaximal
4#33196C211yesmaximal, characteristic
5#28522C212yesmaximal
6#28512C211yesmaximal, characteristic
7#28284C211yesmaximal, characteristic
8#28283C212yesmaximal
9#26539C214yesmaximal
10#26527C212yesmaximal
11#26100C214yesmaximal
12#26089C212yesmaximal
13#22729C212yesmaximal
14#22722C211yesmaximal, characteristic
15#22547C212yesmaximal
16#21615C212yesmaximal
17C4×#477C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#475C2212yes—
23C2×#475C2212yes—
24C2×#446C2212yes—
25#2868C2211yescharacteristic
26#2862C2212yes—
27#2582—21no—
28#2571C2212yes—
29#2570C2214yes—
30#2568—21no—
31#2511—22no—
32#2446—22no—
33#2370—21no—
34#2370—22no—
35#2365—21no—
36#2365C2211yescharacteristic
37#2365C2212yes—
38#2355—21no—
39#2355C2211yescharacteristic
40#2355C2212yes—
41#2355C2212yes—
42#2351—21no—
43Q8×C4⋊C4C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#125—21no—
49C4×#125—22no—
50C42×Q8—21no—
51C42×Q8C2211yescharacteristic
52C42×Q8C2212yes—
53C42×Q8C2211yescharacteristic
54#2097C2214yes—
55#2097C2214yes—
56#2096C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#1988C2212yes—
61#1987C2211yescharacteristic
62#1981C2214yes—
63#1981C2212yes—
64#1960C2214yes—
65#1956C2212yes—
66#1949C2214yes—
67#1949C2214yes—
68#1948C2214yes—
69#1948C2212yes—
70#1948C2212yes—
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1940C2214yes—
74#1939C2212yes—
75#1921C2212yes—
76#1920C2214yes—
77#1863C2212yes—
78#1863C2214yes—
79#1847C2214yes—
80#1844C2212yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1805C2214yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95#557—24no—
96#557Q818yescomplemented
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×#125—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118#477—24no—
119#477—44no—
120#475—28no—
121#475—44no—
122#475—44no—
123#446—28no—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#400—24no—
148#398—24no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389C2312yes—
171#389C2314yes—
172#387—24no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2312yes—
188#382—21no—
189#382—22no—
190#382—22no—
191#382C2314yes—
192#382C2314yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C43—22no—
224C43—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251C4⋊Q8—82no—
252C4⋊Q8C2×Q814yes—
253C4⋊Q8C2×Q818yes—
254#125—216no—
255#125—28no—
256C4×Q8—28no—
257C4×Q8—24no—
258C4×Q8—22no—
259C4×Q8—24no—
260C4×Q8—22no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×Q8—41no—
268C4×Q8—42no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—84no—
272C4×Q8C2×Q814yes—
273C4×Q8C2×Q818yes—
274C4×Q8C2×Q814yes—
275C4×Q8#4114yes—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—24no—
340#107—24no—
341#107—44no—
342#106—28no—
343#106—44no—
344#106—44no—
345C4⋊C8—24no—
346C4⋊C8—28no—
347C4⋊C8—22no—
348C4⋊C8—24no—
349C4⋊C8—42no—
350C4⋊C8—44no—
351C4⋊C8#4114yes—
352#103—28no—
353#103—24no—
354#103—42no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#97—28no—
360C4×C8—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8C22⋊Q814yes—
402C2×Q8C22×Q812yes—
403C2×Q8C22×Q814yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2511yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8C2×#4114yes—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—28no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—82no—
469C4⋊C4—81no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4C22⋊Q818yes—
474C4⋊C4C22⋊Q814yes—
475C4⋊C4C22×Q812yes—
476C4⋊C42−1+412yes—
477C42—28no—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—24no—
482C42—22no—
483C42—21no—
484C42—22no—
485C42—21no—
486C42—22no—
487C42—41no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—44no—
492C42—44no—
493C42—44no—
494C42—42no—
495C42—42no—
496C42—44no—
497C42—42no—
498C42—81no—
499C42—82no—
500C42—84no—
501C42#13714yes—
502C42C22×Q814yes—
503C42C22×Q812yes—
504C42C2×#4112yes—
505C42C2×#4114yes—
506C42C2×#4112yes—
507C42H(5,𝔽2)12yes—
508C23C23×D411yescharacteristic
509Q8—28no—
510Q8—24no—
511Q8—22no—
512Q8—22no—
513Q8—24no—
514Q8—42no—
515Q8—41no—
516Q8—41no—
517Q8—42no—
518Q8—84no—
519Q8—88no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4C2×C22⋊Q814yes—
567C2×C4C2×C22⋊Q812yes—
568C2×C4C2×C22⋊Q814yes—
569C2×C4C2×C22⋊Q812yes—
570C2×C4#55012yes—
571C2×C4#55012yes—
572C2×C4C2×#13712yes—
573C2×C4C2×#13712yes—
574C2×C4C2×#13712yes—
575C2×C4#57512yes—
576C2×C4#57712yes—
577C2×C4C23×Q811yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C2×H(5,𝔽2)11yescharacteristic
581C2×C4C2×2−1+411yescharacteristic
582C2×C4#58511yescharacteristic
583C8—24no—
584C8—42no—
585C8—44no—
586C22C22×C22⋊Q811yescharacteristic
587C22C2×#55011yescharacteristic
588C22#348911yescharacteristic
589C22C22×#13711yescharacteristic
590C22C2×#57511yescharacteristic
591C22C2×#57711yescharacteristic
592C22#358611yescharacteristic
593C4—24no—
594C4—24no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—82no—
606C4—84no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4#307814yes—
611C4#340014yes—
612C4Q8×#4111yescharacteristic
613C4#348211yescharacteristic
614C4#350311yescharacteristic
615C4#351711yescharacteristic
616C2#3644511yescharacteristic
617C2C2×#307811yescharacteristic
618C2#6049711yescharacteristic
619C2C2×#340011yescharacteristic
620C2#6140311yescharacteristic
621C2#6211311yescharacteristic
622C2#6213411yescharacteristic
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

Complements

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

NH ≅ G/Ntype
96. #557Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×#5572
#332022
#331962
#285222
#285122
#282842
#282832
#265392
#265272
#261002
#260892
#227292
#227222
#225472
#216152
C4×#4772