Finite Group Databasebeta

← #6463938

Subgroups — #6463938

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35265C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#29138C214yesmaximal, complemented
4#29135C214yesmaximal, complemented
5#29132C212yesmaximal, complemented
6#29129C212yesmaximal, complemented
7#29126C212yesmaximal, complemented
8#29123C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic
10#23744C211yesmaximal, characteristic
11#23507C211yesmaximal, characteristic
12C2×#2043C211yesmaximal, characteristic
13#21090C212yesmaximal
14#21088C212yesmaximal
15#20651C212yesmaximal
16#20651C212yesmaximal
17#20430C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519C2212yes—
20C2×#424C2212yes—
21C2×#419C2212yes—
22C2×#419C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C2×#405C2212yes—
26#2864C2212yes—
27#2859C2212yes—
28#2647C2212yes—
29#2647C2212yes—
30#2509—21no—
31#2508—21no—
32#2507—21no—
33#2506—21no—
34#2505—22no—
35#2473—22no—
36#2365—21no—
37#2365—21no—
38#2365—21no—
39#2365—21no—
40#2344—22no—
41#2344—22no—
42#2327C2214yes—
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2316C2211yescharacteristic
47#2310C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2212yes—
50#2296C2211yescharacteristic
51C2×#389C2211yescharacteristic
52#2285C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2101C2212yes—
55#2101C2212yes—
56#2101C2214yes—
57#2100C2211yescharacteristic
58#2100C2211yescharacteristic
59#2100C2212yes—
60#2099C2211yescharacteristic
61#2099C2211yescharacteristic
62#2099C2212yes—
63#2043C2212yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2214yes—
68#2043C2212yes—
69#2043C2212yes—
70#1935C2214yes—
71#1935C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1805C2214yes—
79#1805C2214yes—
80#1805C2214yes—
81#1805C2214yes—
82#1803C2214yes—
83#1803C2214yes—
84#1803C2214yes—
85#1803C2214yes—
86#1768C2214yes—
87#1767C2212yes—
88C4×C4⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92#1735C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8C2312yes—
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#125—22no—
104#519—28no—
105#519—28no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2314yes—
110C2×#117—24no—
111C2×#117C2312yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#424—28no—
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2314yes—
125#419—28no—
126#419—44no—
127C2×#103—28no—
128C2×#103—44no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137#405—44no—
138C2×#97C2314yes—
139C2×C4×C8C2314yes—
140#400—22no—
141#400—22no—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#391—22no—
149#391—22no—
150#391C2314yes—
151#391C2314yes—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—24no—
184#385—21no—
185#385—21no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#382—22no—
196#382C2312yes—
197#382C2312yes—
198#382C2312yes—
199#381—24no—
200#380—21no—
201#380—21no—
202#380—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217#376C2312yes—
218C2×#95—21no—
219C43C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8#4118yes—
244C22×Q8C2414yes—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264#125—84no—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×Q8—48no—
270C4×Q8—48no—
271#117—22no—
272#117—22no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—22no—
277#117—22no—
278#117—416no—
279#117#4118yes—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C4⋊C8—28no—
346C4⋊C8—28no—
347#103—28no—
348#103—28no—
349#103—28no—
350#103—28no—
351#103—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355#97—44no—
356C4×C8—44no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24C22×D411yescharacteristic
382C2×Q8—28no—
383C2×Q8—24no—
384C2×Q8—216no—
385C2×Q8—216no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8C2×#4118yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C22×C4C2511yescharacteristic
433M16—48no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—48no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4#12414yes—
486C4⋊C4#12414yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4#12418yes—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—44no—
504C42—48no—
505C42—48no—
506C42—42no—
507C42C2×#4118yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#52512yes—
513C23C2×#12412yes—
514C23#53712yes—
515C23C23×D411yescharacteristic
516Q8—28no—
517Q8—28no—
518Q8—416no—
519Q8—44no—
520Q8—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—416no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—162no—
573C2×C4#52512yes—
574C2×C4C2×#12414yes—
575C2×C4C2×#12414yes—
576C2×C4C2×#12412yes—
577C2×C4#53712yes—
578C2×C4#55218yes—
579C2×C4#55318yes—
580C2×C4#57712yes—
581C2×C4#57812yes—
582C2×C4C22×#4112yes—
583C2×C4C22×#4112yes—
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C8—42no—
587C8—42no—
588C8—44no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22#256214yes—
596C22C2×#52511yescharacteristic
597C22C22×#12411yescharacteristic
598C22C2×#53711yescharacteristic
599C22C2×#57711yescharacteristic
600C22C2×#57811yescharacteristic
601C22#358611yescharacteristic
602C22#358611yescharacteristic
603C4—22no—
604C4—22no—
605C4—24no—
606C4—24no—
607C4—48no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4—81no—
618C4—82no—
619C4—81no—
620C4#256214yes—
621C4C2×#55212yes—
622C4C2×#55312yes—
623C2—24no—
624C2C2×#256211yescharacteristic
625C2#6058211yescharacteristic
626C2#6058311yescharacteristic
627C2#6062311yescharacteristic
628C2#6062711yescharacteristic
629C2#6092311yescharacteristic
630C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29138C2semidirect
4. #29135C2semidirect
5. #29132C2semidirect
6. #29129C2semidirect
7. #29126C2semidirect
8. #29123C2semidirect

Maximal subgroups

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

subgroupindex
#352652
#342092
#291382
#291352
#291322
#291292
#291262
#291232
#237472
#237442
#235072
C2×#20432
#210902
#210882
#206512
#204302