Finite Group Databasebeta

← #7626807

Subgroups — #7626807

Order 512.

6446 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2#34435C218yesmaximal, complemented
3#34435C218yesmaximal, complemented
4#34433C218yesmaximal
5#34403C214yesmaximal, complemented
6#34399C214yesmaximal, complemented
7#34394C214yesmaximal, complemented
8#34272C214yesmaximal, complemented
9#34271C212yesmaximal, complemented
10#34231C214yesmaximal, complemented
11#34225C214yesmaximal, complemented
12#34221C214yesmaximal, complemented
13#34180C212yesmaximal
14#34179C211yesmaximal, characteristic
15#34109C212yesmaximal
16C2×#2346C211yesmaximal, characteristic
17C4×#544C212yesmaximal, complemented
18#3458C4132yescomplemented
19C2×#544C2211yescharacteristic
20C2×#544C2211yescharacteristic
21C2×#544C2212yes—
22C2×#544C2214yes—
23C2×#544C2214yes—
24C2×#543C2214yes—
25C2×#533C2214yes—
26C2×#533C2214yes—
27C2×#533C2214yes—
28C22×#117C2211yescharacteristic
29C22×#117C2212yes—
30#2382C22116yescomplemented
31#2378C22116yes—
32#2378C22116yes—
33#2371C22116yescomplemented
34#2370C2218yescomplemented
35#2370C2218yescomplemented
36#2370C2218yescomplemented
37#2370C2218yescomplemented
38#2369C2218yescomplemented
39#2369C2218yescomplemented
40#2368C22116yes—
41#2365C2218yescomplemented
42#2365C2218yescomplemented
43#2362C22116yes—
44#2362C22116yes—
45#2360C22116yes—
46#2356C2218yescomplemented
47#2356C2218yescomplemented
48#2356C2218yescomplemented
49#2351C2218yescomplemented
50#2347C2212yes—
51#2347C2212yes—
52#2347C2214yes—
53#2347C2214yes—
54#2347C2214yes—
55#2346C2214yes—
56#2346C2218yes—
57#2346C2214yes—
58#2346C2212yes—
59#2346C2212yes—
60#2346C2214yes—
61#2346C2218yes—
62#2345C2214yes—
63#2345C2214yes—
64#2344C2214yes—
65#2344C2214yes—
66#2343C2214yes—
67#2343C2214yes—
68#2343C2218yes—
69#2342C2218yes—
70#2342C2218yes—
71#2332C22116yes—
72#2332C22116yes—
73#2331C2218yescomplemented
74#2331C2218yescomplemented
75#2331C2218yescomplemented
76#2331C2218yescomplemented
77#2330C2218yescomplemented
78#2330C2218yescomplemented
79#2329C2218yes—
80#2329C2218yes—
81#2327C2218yescomplemented
82#2327C2218yescomplemented
83#2323C22116yes—
84#2323C22116yes—
85#2320C2214yescomplemented
86#2320C2218yescomplemented
87#2320C2218yescomplemented
88#2320C2214yescomplemented
89#2317C2214yes—
90#2316C2212yes—
91#2314C2218yes—
92#2312C2214yes—
93C4×C4⋊Q8C2218yescomplemented
94C4×C4⋊Q8C2218yescomplemented
95C4×#125C2218yescomplemented
96C4×#125C2218yescomplemented
97C4×C22⋊Q8C2218yes—
98C4×C22⋊Q8C2218yes—
99#2298C2218yes—
100#2296C2214yes—
101C2×#389C2212yes—
102C2×#388C2214yes—
103#2285C2214yes—
104#2283C2214yes—
105#2283C2214yes—
106#2283C2214yes—
107C2×#382C2212yes—
108#2280C2212yes—
109C2×#380C2211yescharacteristic
110#2276C2214yescomplemented
111#2276C2214yescomplemented
112#2274C2212yes—
113#2274C2214yes—
114C4×#117C2214yes—
115C4×#117C2214yes—
116C2×C4×C4⋊C4C2212yes—
117C2×C4×C22⋊C4C2211yescharacteristic
118C2×#376C2212yes—
119C24×C4C2311yescharacteristic
120#544C2×C4116yes—
121#544C2×C4116yes—
122#544C2×C4132yes—
123#544C2×C4164yescomplemented
124#544C2×C4164yescomplemented
125#543C2×C4164yes—
126#533C2×C4164yes—
127#533C2×C4164yescomplemented
128#533C2×C4164yescomplemented
129C2×C4⋊Q8C2314yes—
130C2×C4⋊Q8C2314yes—
131C2×C4⋊Q8C2314yes—
132C2×C4⋊Q8C2314yes—
133C2×C4⋊Q8C2314yes—
134C2×C4⋊Q8C2314yes—
135C2×#125C2314yes—
136C2×#125C2314yes—
137C2×#125C2314yes—
138C2×#125C2314yes—
139C2×#125C2318yes—
140C2×C22⋊Q8C2318yes—
141C2×C22⋊Q8C2318yes—
142C2×C22⋊Q8C2314yes—
143C2×C22⋊Q8C2314yes—
144C2×C22⋊Q8C2318yes—
145C2×C22⋊Q8C2318yes—
146C2×C22⋊Q8C2318yes—
147C2×C4×Q8C2318yes—
148C2×C4×Q8C2318yes—
149C2×C4×Q8C23116yes—
150C2×C4×Q8C2314yes—
151C2×C4×Q8C2314yes—
152C2×C4×Q8C2318yes—
153C2×#117C2×C4116yes—
154C2×#117C2×C4132yes—
155C2×#117C2314yes—
156C2×#117C2318yes—
157C2×#117C2318yes—
158C2×#117C2314yes—
159C2×#117C2312yes—
160C2×#117C2314yes—
161C2×#117C2314yes—
162C2×#117C2314yes—
163C2×#117C2312yes—
164C2×#117C2312yes—
165C2×#117C2312yes—
166C22×C4⋊C4C2312yes—
167C22×C4⋊C4C2314yes—
168C22×C4⋊C4C2311yescharacteristic
169C22×C4⋊C4C2312yes—
170C22×C4⋊C4C2312yes—
171C22×C4⋊C4C2311yescharacteristic
172C22×C22⋊C4C2312yes—
173C22×C22⋊C4C2312yes—
174C22×C22⋊C4C2312yes—
175C22×C22⋊C4C2314yes—
176C22×C42C2312yes—
177C22×C42C2314yes—
178C22×C42C2312yes—
179C22×C42C2314yes—
180#391C23116yes—
181#391C23116yes—
182#391C2318yes—
183#391C2318yes—
184#391C23116yes—
185#391C23116yes—
186#391C23116yes—
187#389C2314yes—
188#389C2318yes—
189#389C2318yes—
190#389C2318yes—
191#389C2314yes—
192#389C2318yes—
193#389C2318yes—
194#389C23116yes—
195#389C23116yes—
196#389C2318yes—
197#389C23116yes—
198#389C2318yes—
199#389C23116yes—
200#389C23116yes—
201#389C2314yes—
202#389C2318yes—
203#389C2314yes—
204#388C23116yes—
205#388C23116yes—
206#388C23116yes—
207#388C23116yes—
208#388C2318yes—
209#388C2318yes—
210#388C23116yes—
211#388C23116yes—
212#388C2318yes—
213#388C2318yes—
214#387C23116yes—
215#387C23116yes—
216#387C23116yes—
217#387C23116yes—
218#387C23116yes—
219#387C23116yes—
220#387C23116yes—
221#387C23116yes—
222#387C23116yes—
223#387C23116yes—
224#387C23116yes—
225#387C2318yes—
226#387C2318yes—
227#387C23116yes—
228#387C23116yes—
229#387C23116yes—
230#387C2318yes—
231#387C2318yes—
232#385C23116yes—
233#385C2318yes—
234#385C2318yes—
235#384C2318yes—
236#383C2314yes—
237#383C2318yes—
238#383C2314yes—
239#383C2318yes—
240#382C2318yes—
241#382C2314yes—
242#382C2318yes—
243#382C2314yes—
244#382C2318yes—
245#382C2318yes—
246#382C2318yes—
247#381C2318yes—
248#381C2318yes—
249#381C2314yes—
250#380C2318yes—
251#380C2314yes—
252#380C2314yes—
253#380C2318yes—
254#380C2318yes—
255#380C2314yes—
256#380C2312yes—
257#380C2312yes—
258#379C2314yes—
259C4×C4⋊C4C2318yes—
260C4×C4⋊C4C2314yes—
261C4×C4⋊C4C2318yes—
262C4×C4⋊C4C23116yes—
263C4×C4⋊C4C2318yes—
264C4×C4⋊C4C23116yes—
265C4×C4⋊C4C2318yes—
266C4×C4⋊C4C23116yes—
267C4×C4⋊C4C2314yes—
268C4×C4⋊C4C23116yes—
269C4×C4⋊C4C2318yes—
270C4×C4⋊C4C2318yes—
271C4×C4⋊C4C2318yes—
272C4×C4⋊C4C2318yes—
273C4×C4⋊C4C2314yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2314yes—
276C4×C4⋊C4C2314yes—
277C4×C4⋊C4C23116yes—
278C4×C4⋊C4C2314yes—
279C4×C4⋊C4C2318yes—
280C4×C4⋊C4C2314yes—
281C4×C4⋊C4C2318yes—
282C4×C4⋊C4C2314yes—
283C4×C4⋊C4C2318yes—
284C4×C4⋊C4C2318yes—
285C4×C22⋊C4C2318yes—
286C4×C22⋊C4C2318yes—
287C4×C22⋊C4C23116yes—
288C4×C22⋊C4C2318yes—
289C4×C22⋊C4C2318yes—
290C4×C22⋊C4C2318yes—
291C4×C22⋊C4C2314yes—
292C4×C22⋊C4C2314yes—
293C4×C22⋊C4C2312yes—
294C4×C22⋊C4C2312yes—
295C4×C22⋊C4C2314yes—
296#376C2314yes—
297#376C2314yes—
298#376C2318yes—
299#376C2318yes—
300#376C2318yes—
301#376C2318yes—
302#376C2314yes—
303#376C2314yes—
304C2×#95C2314yes—
305C2×#95C2311yescharacteristic
306C2×#95C2312yes—
307C2×#95C2311yescharacteristic
308C2×#95C2314yes—
309C43C2318yes—
310C25C2411yescharacteristic
311C22×Q8C2414yes—
312C22×Q8C2414yes—
313C22×Q8C2418yes—
314C23×C4—28no—
315C23×C4C22×C418yes—
316C23×C4C2412yes—
317C23×C4C2412yes—
318C23×C4C2411yescharacteristic
319C23×C4C2412yes—
320C23×C4C2412yes—
321C23×C4C2414yes—
322C23×C4C2412yes—
323C23×C4C2414yes—
324C23×C4C2414yes—
325C23×C4C2412yes—
326C23×C4C2411yescharacteristic
327C4⋊Q8C22×C4132yes—
328C4⋊Q8C22×C4132yes—
329C4⋊Q8C22×C4132yes—
330C4⋊Q8C22×C4132yes—
331C4⋊Q8C22×C4132yes—
332C4⋊Q8C22×C4132yes—
333#125C22×C4164yes—
334#125C22×C4132yes—
335#125C22×C4132yes—
336#125C22×C4132yes—
337#125C22×C4132yes—
338C22⋊Q8C22×C4132yes—
339C22⋊Q8C22×C4132yes—
340C22⋊Q8C22×C4164yes—
341C22⋊Q8C22×C4164yes—
342C22⋊Q8C22×C4164yes—
343C22⋊Q8C22×C4164yes—
344C22⋊Q8C22×C4164yes—
345C4×Q8—416no—
346C4×Q8—416no—
347C4×Q8—816no—
348C4×Q8C22×C4132yes—
349C4×Q8C22×C4132yes—
350C4×Q8C22×C4164yes—
351#117—216no—
352#117—416no—
353#117—88no—
354#117C22×C4132yes—
355#117C22×C4132yes—
356#117C22×C4116yes—
357#117C22×C4116yes—
358#117C22×C4116yes—
359#117C22×C4116yes—
360#117C22×C4132yes—
361#117C22×C4132yes—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—84no—
364C2×C4⋊C4C22×C418yes—
365C2×C4⋊C4C22×C418yes—
366C2×C4⋊C4C22×C4116yes—
367C2×C4⋊C4C22×C4116yes—
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C24116yes—
373C2×C4⋊C4C24116yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C2418yes—
377C2×C4⋊C4C2418yes—
378C2×C4⋊C4C24116yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2414yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2414yes—
386C2×C4⋊C4C2414yes—
387C2×C4⋊C4C2414yes—
388C2×C4⋊C4C2414yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2414yes—
392C2×C4⋊C4C2418yes—
393C2×C4⋊C4C2418yes—
394C2×C4⋊C4C2418yes—
395C2×C22⋊C4—44no—
396C2×C22⋊C4—82no—
397C2×C22⋊C4C22×C4116yes—
398C2×C22⋊C4C22×C4132yes—
399C2×C22⋊C4C2414yes—
400C2×C22⋊C4C2418yes—
401C2×C22⋊C4C2418yes—
402C2×C22⋊C4C2414yes—
403C2×C22⋊C4C2414yes—
404C2×C22⋊C4C2418yes—
405C2×C22⋊C4C2414yes—
406C2×C22⋊C4C2414yes—
407C2×C22⋊C4C2414yes—
408C2×C22⋊C4C2418yes—
409C2×C22⋊C4C2414yes—
410C2×C22⋊C4C2414yes—
411C2×C42—28no—
412C2×C42—48no—
413C2×C42C22×C4116yes—
414C2×C42C22×C4132yes—
415C2×C42C2418yes—
416C2×C42C2418yes—
417C2×C42C2418yes—
418C2×C42C2414yes—
419C2×C42C2414yes—
420C2×C42C24116yes—
421C2×C42C2418yes—
422C2×C42C2418yes—
423C2×C42C2418yes—
424C2×C42C24116yes—
425C2×C42C2418yes—
426C2×C42C2418yes—
427C2×C42C2418yes—
428C2×C42C2414yes—
429C2×C42C2414yes—
430C2×C42C2414yes—
431C2×C42C2414yes—
432C2×C42C2414yes—
433C2×C42C2414yes—
434#95C2411yescharacteristic
435#95C2411yescharacteristic
436#95C2412yes—
437#95C2414yes—
438#95C2412yes—
439#95C2418yes—
440#95C2414yes—
441#95C2418yes—
442#95C2412yes—
443#95C24116yes—
444#95C2414yes—
445#95C2418yes—
446#95C2414yes—
447#95C2418yes—
448#95C2418yes—
449#95C2418yes—
450#95C2418yes—
451#95C2418yes—
452#95C2418yes—
453#95C24116yes—
454#95C24116yes—
455#95C24116yes—
456#95C24116yes—
457#95C24116yes—
458#95C24116yes—
459#95C2418yes—
460#95C2418yes—
461C24—24no—
462C24—42no—
463C24—41no—
464C24—41no—
465C24C23×C414yes—
466C24C2512yes—
467C24C2511yescharacteristic
468C2×Q8—48no—
469C2×Q8—48no—
470C2×Q8—48no—
471C2×Q8—48no—
472C2×Q8C23×C4116yes—
473C2×Q8C23×C4116yes—
474C2×Q8C23×C4132yes—
475C22×C4—24no—
476C22×C4—232no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—28no—
482C22×C4—24no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4C23×C414yes—
495C22×C4C23×C418yes—
496C22×C4C23×C418yes—
497C22×C4C23×C4116yes—
498C22×C4C23×C4116yes—
499C22×C4C23×C418yes—
500C22×C4C23×C418yes—
501C22×C4C23×C414yes—
502C22×C42−1+418yes—
503C22×C4C2512yes—
504C22×C4C2518yes—
505C22×C4C2514yes—
506C22×C4C2518yes—
507C22×C4C2518yes—
508C22×C4C2511yescharacteristic
509C22×C4C2511yescharacteristic
510C22×C4C2514yes—
511C22×C4C2514yes—
512C22×C4C2514yes—
513C22×C4C2514yes—
514C22×C4C2514yes—
515C22×C4C2514yes—
516C22×C4C2512yes—
517C22×C4C2512yes—
518C4⋊C4—216no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—416no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—416no—
526C4⋊C4—416no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4C23×C4116yes—
531C4⋊C4C23×C4116yes—
532C4⋊C4C23×C4116yes—
533C4⋊C4C23×C4116yes—
534C4⋊C4C23×C418yes—
535C4⋊C4C23×C418yes—
536C4⋊C4C23×C418yes—
537C4⋊C4C23×C418yes—
538C4⋊C4C23×C4132yes—
539C4⋊C4C23×C4132yes—
540C4⋊C4C23×C4132yes—
541C4⋊C4C23×C4132yes—
542C4⋊C4C23×C4132yes—
543C4⋊C4C23×C4116yes—
544C4⋊C4C23×C4116yes—
545C22⋊C4—216no—
546C22⋊C4—28no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4C23×C4132yes—
553C22⋊C4C23×C4116yes—
554C22⋊C4C23×C4116yes—
555C22⋊C4C23×C4116yes—
556C22⋊C4C23×C4116yes—
557C42—28no—
558C42—28no—
559C42—216no—
560C42—216no—
561C42—216no—
562C42—48no—
563C42—48no—
564C42—48no—
565C42—48no—
566C42—416no—
567C42—416no—
568C42—48no—
569C42—48no—
570C42C23×C4116yes—
571C42C23×C4116yes—
572C42C23×C4116yes—
573C42C23×C4116yes—
574C42C23×C4116yes—
575C42C23×C4116yes—
576C23—22no—
577C23—21no—
578C23—21no—
579C23—24no—
580C23—24no—
581C23—22no—
582C23—22no—
583C23—28no—
584C23—44no—
585C23—44no—
586C23—84no—
587C23—84no—
588C23C24×C414yes—
589C23C24×C412yes—
590C23C2×2−1+414yes—
591C23C2611yescharacteristic
592Q8—416no—
593Q8—416no—
594Q8—432no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—216no—
610C2×C4—216no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—22no—
626C2×C4—48no—
627C2×C4—416no—
628C2×C4—48no—
629C2×C4—416no—
630C2×C4—48no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4#519116yes—
634C2×C4C24×C412yes—
635C2×C4C24×C412yes—
636C2×C4C24×C418yes—
637C2×C4C24×C418yes—
638C2×C4C24×C418yes—
639C2×C4C24×C418yes—
640C2×C4C24×C418yes—
641C2×C4C24×C418yes—
642C2×C4C24×C414yes—
643C2×C4C2×2−1+418yes—
644C2×C4#58514yes—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—44no—
653C22—44no—
654C22—88no—
655C22C2×#51914yes—
656C22C25×C411yescharacteristic
657C22C22×2−1+412yes—
658C22C2×#58512yes—
659C22H(7,𝔽2)11yescharacteristic
660C222−1+611yescharacteristic
661C4—22no—
662C4—22no—
663C4—28no—
664C4—28no—
665C4—28no—
666C4—28no—
667C4—28no—
668C4—28no—
669C4—24no—
670C4—48no—
671C4—42no—
672C4—48no—
673C4—44no—
674C4—48no—
675C4C2×#51918yes—
676C2—24no—
677C2—44no—
678C2C22×#51911yescharacteristic
679C2C4×#58511yescharacteristic
680C2#6265411yescharacteristic
681C2#6292812yes—
682C2#6292912yes—
subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC25C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34435C2semidirect
3. #34435C2semidirect
5. #34403C2semidirect
6. #34399C2semidirect
7. #34394C2semidirect
8. #34272C2semidirect
9. #34271C2semidirect
10. #34231C2semidirect
11. #34225C2semidirect
12. #34221C2semidirect
17. C4×#544C2semidirect
18. #3458C4semidirect
30. #2382C22semidirect
33. #2371C22semidirect
34. #2370C22semidirect
35. #2370C22semidirect
36. #2370C22semidirect
37. #2370C22semidirect
38. #2369C22semidirect
39. #2369C22semidirect
41. #2365C22semidirect
42. #2365C22semidirect
46. #2356C22semidirect
47. #2356C22semidirect
48. #2356C22semidirect
49. #2351C22semidirect
73. #2331C22semidirect
74. #2331C22semidirect
75. #2331C22semidirect
76. #2331C22semidirect
77. #2330C22semidirect
78. #2330C22semidirect
81. #2327C22semidirect
82. #2327C22semidirect
85. #2320C22semidirect
86. #2320C22semidirect
87. #2320C22semidirect
88. #2320C22semidirect
93. C4×C4⋊Q8C22semidirect
94. C4×C4⋊Q8C22semidirect
95. C4×#125C22semidirect
96. C4×#125C22semidirect
110. #2276C22semidirect
111. #2276C22semidirect
123. #544C2×C4semidirect
124. #544C2×C4semidirect
127. #533C2×C4semidirect
128. #533C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
#344352
#344332
#344032
#343992
#343942
#342722
#342712
#342312
#342252
#342212
#341802
#341792
#341092
C2×#23462
C4×#5442