Finite Group Databasebeta

← #7626700

Subgroups — #7626700

Order 512.

6272 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C211yesmaximal, characteristic
2#34491C218yesmaximal, complemented
3#34490C218yesmaximal, complemented
4#34490C218yesmaximal, complemented
5#34432C218yesmaximal
6#34389C214yesmaximal
7#34328C218yesmaximal, complemented
8#34271C212yesmaximal
9#34254C214yesmaximal
10#34221C214yesmaximal
11#34215C212yesmaximal
12#34145C212yesmaximal
13C2×#2356C211yesmaximal, characteristic
14C4×#544C212yesmaximal
15#34059C211yesmaximal, characteristic
16#3502C4132yescomplemented
17C2×Q82C2214yes—
18C2×#557C2218yes—
19C2×#557C2214yes—
20C2×#544C2212yes—
21C2×#544C2214yes—
22C2×#533C2214yes—
23C2×#533C2212yes—
24C2×#519C2212yes—
25C22×C4×Q8C2211yescharacteristic
26#2382C22116yes—
27#2381C2218yes—
28#2381C2218yes—
29#2381C22116yes—
30#2371C2218yes—
31#2371C22116yes—
32#2371C2218yes—
33#2370C2214yes—
34#2370C2218yes—
35#2370C2214yes—
36#2370C2218yes—
37#2369C2214yes—
38#2369C2218yes—
39#2368C2214yes—
40#2368C2218yes—
41#2368C2214yes—
42#2368C2218yes—
43#2365C22116yes—
44#2365C2218yes—
45#2365C2214yes—
46#2360C2218yes—
47#2356C2214yes—
48#2356C22116yes—
49#2356C2218yes—
50#2356C2214yes—
51#2356C2218yes—
52#2356C22116yes—
53#2356C2218yes—
54#2355C2218yes—
55#2355C2218yes—
56#2355C2218yes—
57#2355C2218yes—
58#2343C22116yes—
59#2343C22116yes—
60#2336C22116yes—
61#2336C22116yes—
62#2336C22116yes—
63#2331C2218yes—
64#2331C22116yes—
65#2331C2214yes—
66#2331C2214yes—
67#2331C2218yes—
68#2330C2218yes—
69#2329C2218yes—
70#2329C2218yes—
71#2327C22116yes—
72#2327C2218yes—
73#2320C2218yes—
74#2320C22116yes—
75#2320C2218yes—
76#2320C2218yes—
77#2320C2218yes—
78#2317C2214yes—
79#2316C2211yescharacteristic
80#2316C2211yescharacteristic
81#2313C2214yes—
82#2310C2212yes—
83C4×C4⋊Q8C2218yes—
84C4×#125C2218yes—
85C4×#125C2218yes—
86C4×C22⋊Q8C2218yes—
87C4×C22⋊Q8C2218yes—
88#2296C2214yes—
89#2295C2218yes—
90#2295C2218yes—
91#2295C2218yes—
92#2295C2214yes—
93#2295C2218yes—
94C2×#391C2212yes—
95C2×#387C2214yes—
96#2283C2214yes—
97C2×#383C2212yes—
98C2×#382C2211yescharacteristic
99#2276C2218yes—
100#2274C2214yes—
101C42×Q8C2214yes—
102C4×#117C2212yes—
103C4×#117C2214yes—
104C2×C4×C4⋊C4C2212yes—
105C2×C4×C4⋊C4C2211yescharacteristic
106C2×#376C2212yes—
107C23×Q8C2311yescharacteristic
108Q82C2×C4164yescomplemented
109#557C2×C41128yescomplemented
110#557C2×C4164yescomplemented
111#544C2×C4132yes—
112#544C2×C4164yes—
113#533C2×C4164yes—
114#533C2×C4132yes—
115C2×C4⋊Q8C2318yes—
116C2×C4⋊Q8C2314yes—
117C2×C4⋊Q8C23116yes—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2314yes—
121C2×#125C2314yes—
122C2×#125C2318yes—
123C2×#125C2318yes—
124C2×#125C23116yes—
125C2×C22⋊Q8C2318yes—
126C2×C22⋊Q8C2314yes—
127C2×C22⋊Q8C2314yes—
128#519C2×C4132yes—
129C2×C4×Q8C2×C4116yes—
130C2×C4×Q8C2318yes—
131C2×C4×Q8C2318yes—
132C2×C4×Q8C2318yes—
133C2×C4×Q8C2314yes—
134C2×C4×Q8C2318yes—
135C2×C4×Q8C2318yes—
136C2×C4×Q8C2318yes—
137C2×C4×Q8C2318yes—
138C2×C4×Q8C2314yes—
139C2×C4×Q8C2314yes—
140C2×C4×Q8C2314yes—
141C2×#117C2312yes—
142C2×#117C2312yes—
143C2×#117C2314yes—
144C2×#117C2318yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2314yes—
151C22×C42C2312yes—
152C22×C42C2311yescharacteristic
153#391C23116yes—
154#391C2318yes—
155#391C2318yes—
156#391C2318yes—
157#391C2318yes—
158#391C2318yes—
159#391C2318yes—
160#391C23116yes—
161#391C23116yes—
162#391C23116yes—
163#391C2318yes—
164#391C2318yes—
165#391C2318yes—
166#389C2318yes—
167#389C2318yes—
168#389C2318yes—
169#389C23116yes—
170#389C2318yes—
171#389C2314yes—
172#389C2314yes—
173#389C23116yes—
174#389C2318yes—
175#389C2318yes—
176#389C2318yes—
177#389C2318yes—
178#389C2318yes—
179#389C2314yes—
180#389C2314yes—
181#387C23116yes—
182#387C23116yes—
183#387C23116yes—
184#387C23116yes—
185#387C23116yes—
186#387C23116yes—
187#387C2318yes—
188#387C2318yes—
189#387C23116yes—
190#387C23116yes—
191#387C23116yes—
192#387C23116yes—
193#387C23116yes—
194#387C23116yes—
195#387C23116yes—
196#387C2318yes—
197#387C2318yes—
198#387C23116yes—
199#387C23116yes—
200#387C23116yes—
201#387C23116yes—
202#387C23116yes—
203#387C23116yes—
204#387C23116yes—
205#387C23116yes—
206#387C23116yes—
207#387C2318yes—
208#387C2318yes—
209#385C23116yes—
210#385C2318yes—
211#385C2318yes—
212#384C2318yes—
213#383C23116yes—
214#383C23116yes—
215#383C2318yes—
216#383C2318yes—
217#383C2318yes—
218#382C2314yes—
219#382C2318yes—
220#382C2318yes—
221#382C23116yes—
222#382C23116yes—
223#382C2314yes—
224#381C2318yes—
225#380C2314yes—
226#380C2314yes—
227C4×C4⋊C4C2318yes—
228C4×C4⋊C4C2314yes—
229C4×C4⋊C4C2318yes—
230C4×C4⋊C4C23116yes—
231C4×C4⋊C4C23116yes—
232C4×C4⋊C4C2318yes—
233C4×C4⋊C4C2318yes—
234C4×C4⋊C4C23116yes—
235C4×C4⋊C4C2314yes—
236C4×C4⋊C4C2314yes—
237C4×C4⋊C4C2318yes—
238C4×C4⋊C4C23116yes—
239C4×C4⋊C4C2318yes—
240C4×C4⋊C4C2318yes—
241C4×C4⋊C4C2314yes—
242C4×C4⋊C4C2318yes—
243C4×C4⋊C4C2314yes—
244C4×C4⋊C4C23116yes—
245C4×C4⋊C4C2318yes—
246C4×C4⋊C4C2318yes—
247C4×C4⋊C4C2318yes—
248C4×C4⋊C4C23116yes—
249C4×C4⋊C4C2314yes—
250C4×C4⋊C4C2318yes—
251C4×C4⋊C4C2318yes—
252C4×C4⋊C4C2314yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2318yes—
256C4×C22⋊C4C2314yes—
257C4×C22⋊C4C2318yes—
258C4×C22⋊C4C23116yes—
259C4×C22⋊C4C2318yes—
260C4×C22⋊C4C2314yes—
261#376C2318yes—
262#376C2314yes—
263#376C2314yes—
264#376C23116yes—
265#376C2318yes—
266#376C23116yes—
267#376C2314yes—
268#376C2318yes—
269C2×#95C2314yes—
270C2×#95C2314yes—
271C2×#95C2314yes—
272C2×#95C2311yescharacteristic
273C2×#95C2312yes—
274C2×#95C2311yescharacteristic
275C2×#95C2312yes—
276C43C2314yes—
277C43C2318yes—
278C22×Q8—24no—
279C22×Q8—24no—
280C22×Q8C22×C418yes—
281C22×Q8C2414yes—
282C22×Q8C2418yes—
283C22×Q8C2414yes—
284C22×Q8C2414yes—
285C23×C4C2412yes—
286C23×C4C2412yes—
287C23×C4C2412yes—
288C23×C4C2414yes—
289C23×C4C2411yescharacteristic
290C23×C4C2412yes—
291C23×C4C2411yescharacteristic
292C23×C4C2411yescharacteristic
293C4⋊Q8C22×C4116yes—
294C4⋊Q8C22×C4132yes—
295C4⋊Q8C22×C4132yes—
296C4⋊Q8C22×C4116yes—
297C4⋊Q8C22×C4164yes—
298C4⋊Q8C22×C41128yes—
299#125C22×C4132yes—
300#125C22×C4164yes—
301#125C22×C4164yes—
302#125C22×C41128yes—
303C22⋊Q8C22×C4132yes—
304C22⋊Q8C22×C4164yes—
305C22⋊Q8C22×C4132yes—
306C4×Q8—416no—
307C4×Q8—416no—
308C4×Q8—416no—
309C4×Q8C22×C4132yes—
310C4×Q8C22×C4132yes—
311C4×Q8C22×C4164yes—
312C4×Q8C22×C4164yes—
313C4×Q8C22×C4164yes—
314C4×Q8C22×C4132yes—
315C4×Q8C22×C4164yes—
316C4×Q8C22×C4132yes—
317#117—28no—
318#117—28no—
319#117C22×C4164yes—
320#117C22×C4132yes—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C22×C4116yes—
324C2×C4⋊C4C22×C418yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2418yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2418yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2418yes—
332C2×C4⋊C4C2418yes—
333C2×C4⋊C4C2418yes—
334C2×C4⋊C4C24116yes—
335C2×C4⋊C4C2418yes—
336C2×C4⋊C4C24116yes—
337C2×C4⋊C4C2418yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2418yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2418yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2418yes—
344C2×C4⋊C4C2418yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2418yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C24116yes—
349C2×C4⋊C4C2418yes—
350C2×C22⋊C4C2414yes—
351C2×C22⋊C4C2414yes—
352C2×C22⋊C4C2418yes—
353C2×C42—28no—
354C2×C42—48no—
355C2×C42C22×C4116yes—
356C2×C42C22×C418yes—
357C2×C42C2414yes—
358C2×C42C2414yes—
359C2×C42C2418yes—
360C2×C42C2418yes—
361C2×C42C2414yes—
362C2×C42C2414yes—
363C2×C42C2418yes—
364C2×C42C2418yes—
365C2×C42C24116yes—
366C2×C42C2418yes—
367C2×C42C2418yes—
368C2×C42C2418yes—
369C2×C42C2418yes—
370C2×C42C24116yes—
371C2×C42C2418yes—
372C2×C42C24116yes—
373C2×C42C2412yes—
374C2×C42C2414yes—
375C2×C42C2412yes—
376C2×C42C2414yes—
377#95C2411yescharacteristic
378#95C2411yescharacteristic
379#95C2418yes—
380#95C2412yes—
381#95C2414yes—
382#95C2418yes—
383#95C2414yes—
384#95C24116yes—
385#95C2412yes—
386#95C2418yes—
387#95C2412yes—
388#95C24116yes—
389#95C2418yes—
390#95C24116yes—
391#95C2418yes—
392#95C2418yes—
393#95C2418yes—
394#95C2418yes—
395#95C2418yes—
396#95C2418yes—
397#95C2418yes—
398#95C2418yes—
399#95C2418yes—
400#95C24116yes—
401#95C24116yes—
402#95C24116yes—
403#95C24116yes—
404#95C24116yes—
405#95C2418yes—
406#95C2418yes—
407C24C2511yescharacteristic
408C2×Q8—232no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8C23×C4116yes—
416C2×Q8C23×C4116yes—
417C2×Q8C23×C4132yes—
418C2×Q8C23×C4116yes—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4C23×C418yes—
434C22×C4C23×C414yes—
435C22×C4C23×C414yes—
436C22×C4C23×C418yes—
437C22×C4C23×C414yes—
438C22×C42−1+414yes—
439C22×C42−1+414yes—
440C22×C4C2512yes—
441C22×C4C2514yes—
442C22×C4C2518yes—
443C22×C4C2518yes—
444C22×C4C2518yes—
445C22×C4C2514yes—
446C22×C4C2511yescharacteristic
447C22×C4C2511yescharacteristic
448C22×C4C2518yes—
449C22×C4C2518yes—
450C22×C4C2514yes—
451C22×C4C2512yes—
452C22×C4C2512yes—
453C22×C4C2512yes—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4C23×C4132yes—
468C4⋊C4C23×C4116yes—
469C4⋊C4C23×C4132yes—
470C4⋊C4C23×C4116yes—
471C4⋊C4C23×C4132yes—
472C4⋊C4C23×C4116yes—
473C4⋊C4C23×C4132yes—
474C4⋊C4C23×C4116yes—
475C4⋊C4C23×C4132yes—
476C4⋊C4C23×C4116yes—
477C4⋊C4C23×C4164yes—
478C4⋊C4C23×C4132yes—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4C23×C4132yes—
482C42—28no—
483C42—28no—
484C42—28no—
485C42—28no—
486C42—216no—
487C42—216no—
488C42—48no—
489C42—48no—
490C42—48no—
491C42—416no—
492C42—48no—
493C42—416no—
494C42—48no—
495C42—48no—
496C42C23×C4116yes—
497C42C23×C418yes—
498C42C23×C4116yes—
499C42C23×C418yes—
500C42C23×C4164yes—
501C42C23×C4132yes—
502C23—22no—
503C23—22no—
504C23C24×C412yes—
505C23C2×2−1+412yes—
506C23C2×2−1+412yes—
507C23C2611yescharacteristic
508Q8—232no—
509Q8—432no—
510Q8—416no—
511Q8—416no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—216no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—84no—
545C2×C4—88no—
546C2×C4#519116yes—
547C2×C4C24×C412yes—
548C2×C4C24×C418yes—
549C2×C4C24×C412yes—
550C2×C4C24×C4116yes—
551C2×C4C24×C4116yes—
552C2×C4C24×C418yes—
553C2×C4C24×C414yes—
554C2×C4C24×C414yes—
555C2×C4C2×2−1+414yes—
556C2×C4C2×2−1+414yes—
557C2×C4#58514yes—
558C22—22no—
559C22—22no—
560C22—24no—
561C22—24no—
562C22C2×#51914yes—
563C22C25×C411yescharacteristic
564C22C22×2−1+411yescharacteristic
565C22C22×2−1+411yescharacteristic
566C22C2×#58512yes—
567C222−1+612yes—
568C4—28no—
569C4—28no—
570C4—24no—
571C4—216no—
572C4—22no—
573C4—22no—
574C4—216no—
575C4—24no—
576C4—48no—
577C4—42no—
578C4—44no—
579C4—48no—
580C4—48no—
581C4C2×#51918yes—
582C2—24no—
583C2C22×#51911yescharacteristic
584C2C4×#58511yescharacteristic
585C2#6265311yescharacteristic
586C2#6292812yes—
587C2#6293212yes—
subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC24C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34491C2semidirect
3. #34490C2semidirect
4. #34490C2semidirect
7. #34328C2semidirect
16. #3502C4semidirect
108. Q82C2×C4semidirect
109. #557C2×C4semidirect
110. #557C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35022
#344912
#344902
#344322
#343892
#343282
#342712
#342542
#342212
#342152
#341452
C2×#23562
C4×#5442
#340592