Finite Group Databasebeta

← #612427

Subgroups — #612427

Order 512.

8830 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#525C211yesmaximal, characteristic
2C24×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2849C212yesmaximal, complemented
4C2×#2406C212yesmaximal, complemented
5#15069C214yesmaximal, complemented
6#14720C214yesmaximal, complemented
7#14719C214yesmaximal, complemented
8#14694C212yesmaximal, complemented
9#14681C211yesmaximal, characteristic, complemented
10#14015C218yesmaximal, complemented
11#13870C212yesmaximal, complemented
12C23×#41C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C2×#525—28no—
17C2×#525—216no—
18C22×#123C2212yes—
19C22×#123C2214yes—
20C22×C22⋊Q8C2212yes—
21C22×#121C2212yes—
22C22×#120C2212yes—
23C22×C4×D4C2214yes—
24C22×#117C2211yescharacteristic
25C23×C4⋊C4—28no—
26C23×C4⋊C4—216no—
27C23×C4⋊C4C2214yes—
28C23×C4⋊C4C2214yes—
29C23×C4⋊C4C2212yes—
30C23×C4⋊C4C2212yes—
31#2849—216no—
32#2406—216no—
33C2×#395C2214yescomplemented
34C2×#394C2214yescomplemented
35C2×#386C2218yes—
36C2×#386C2218yes—
37C2×#386C2214yes—
38C2×#385C2218yes—
39C2×#385C2218yescomplemented
40C2×#381C2214yes—
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#379C2212yes—
44#1444C22116yescomplemented
45#1441C2218yescomplemented
46#1439C22116yescomplemented
47#1436C2218yescomplemented
48C22⋊#95C2214yes—
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2218yes—
52C26C2311yescharacteristic
53C22×#41—28no—
54C22×#41—24no—
55C22×#41—28no—
56C22×#41—44no—
57C23×Q8C2311yescharacteristic
58C23×D4C2312yes—
59C23×D4C2314yes—
60C23×D4C2311yescharacteristic
61C24×C4—24no—
62C24×C4—28no—
63C24×C4—216no—
64C24×C4—24no—
65C24×C4—28no—
66C24×C4—28no—
67C24×C4—28no—
68C24×C4—28no—
69C24×C4—44no—
70C24×C4—44no—
71C24×C4D418yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2314yes—
76C24×C4C2314yes—
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81#525—2128no—
82C2×#123—28no—
83C2×#123—28no—
84C2×#123—216no—
85C2×#123—216no—
86C2×#123—28no—
87C2×#123—48no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—216no—
90C2×#121—216no—
91C2×#121—28no—
92C2×#120—28no—
93C2×#120—28no—
94C2×#120D4116yes—
95C2×C4×D4—216no—
96C2×C4×D4—216no—
97C2×C4×D4—216no—
98C2×#117—24no—
99C2×#117—28no—
100C22×C4⋊C4—2128no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—216no—
104C22×C4⋊C4—216no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—232no—
109C22×C4⋊C4—232no—
110C22×C4⋊C4D4116yescomplemented
111C22×C4⋊C4C2318yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2318yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125#395—216no—
126#394—216no—
127#386—232no—
128#386—216no—
129#386—232no—
130#385—232no—
131#385—232no—
132#381—28no—
133#381—216no—
134#381—44no—
135#379—28no—
136C2×#95C2314yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C23116yes—
140C2×#95C23116yes—
141C2×#95C2318yes—
142C25—24no—
143C25—28no—
144C25—24no—
145C25—28no—
146C25C2×D414yes—
147C25C2×D418yes—
148C25C2412yes—
149C25C2411yescharacteristic
150C25C2412yes—
151C2×#41—232no—
152C2×#41—464no—
153C2×#41—416no—
154C2×#41—416no—
155C22×Q8—24no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—42no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—28no—
165C22×D4—216no—
166C22×D4—28no—
167C22×D4—216no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—42no—
171C22×D4C2×D4116yes—
172C22×D4C2×D418yes—
173C23×C4—232no—
174C23×C4—264no—
175C23×C4—2128no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—216no—
183C23×C4—216no—
184C23×C4—216no—
185C23×C4—232no—
186C23×C4—232no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—232no—
199C23×C4—216no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—216no—
205C23×C4—28no—
206C23×C4—416no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—48no—
210C23×C4—44no—
211C23×C4—464no—
212C23×C4—464no—
213C23×C4—416no—
214C23×C4—416no—
215C23×C4C2×D414yes—
216C23×C4C2×D418yes—
217C23×C4C2×D418yes—
218C23×C4C2×D418yes—
219C23×C4C2×D418yes—
220C23×C4C2×D414yes—
221C23×C4C2×D418yes—
222C23×C4#41116yescomplemented
223C23×C4C2418yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231#123—264no—
232#123—464no—
233C22⋊Q8—264no—
234#121—264no—
235#120—264no—
236C4×D4—2128no—
237#117—232no—
238C2×C4⋊C4—264no—
239C2×C4⋊C4—2128no—
240C2×C4⋊C4—2128no—
241C2×C4⋊C4—264no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4C2×D4116yes—
258C2×C4⋊C4C2×D4116yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—232no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4C2×D418yes—
278C2×C22⋊C4C2×D4116yes—
279C2×C42—28no—
280C2×C42—24no—
281#95—28no—
282#95—232no—
283#95—28no—
284#95—216no—
285#95—232no—
286#95—48no—
287C24—232no—
288C24—232no—
289C24—264no—
290C24—28no—
291C24—216no—
292C24—28no—
293C24—216no—
294C24—232no—
295C24—264no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—28no—
306C24—24no—
307C24—28no—
308C24#120116yes—
309C24C22×D414yes—
310C24C22×D412yes—
311C24C22×D414yes—
312C24C22×D414yes—
313C24C22×D414yes—
314C24C22×D418yes—
315C24C2×#4118yes—
316C24H(5,𝔽2)14yes—
317C242−1+418yes—
318C24C2511yescharacteristic
319#41—4128no—
320C2×Q8—28no—
321C2×Q8—416no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×D4—28no—
325C2×D4—232no—
326C2×D4—216no—
327C2×D4—232no—
328C2×D4—264no—
329C2×D4—416no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—416no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—432no—
337C2×D4—44no—
338C22×C4—232no—
339C22×C4—2256no—
340C22×C4—264no—
341C22×C4—232no—
342C22×C4—232no—
343C22×C4—232no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—232no—
347C22×C4—28no—
348C22×C4—232no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—232no—
357C22×C4—264no—
358C22×C4—232no—
359C22×C4—216no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—232no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—416no—
392C22×C4—4128no—
393C22×C4—432no—
394C22×C4—432no—
395C22×C4—464no—
396C22×C4—416no—
397C22×C4—48no—
398C22×C4—4128no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—48no—
406C22×C4#120116yes—
407C22×C4C22×D418yes—
408C22×C4C22×D414yes—
409C22×C4C22×D418yes—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D418yes—
416C22×C4C2×#41116yes—
417C4⋊C4—264no—
418C4⋊C4—216no—
419C4⋊C4—232no—
420C4⋊C4—232no—
421C4⋊C4—232no—
422C4⋊C4—432no—
423C22⋊C4—232no—
424C22⋊C4—216no—
425C22⋊C4—232no—
426C22⋊C4—432no—
427C22⋊C4—416no—
428C22⋊C4—432no—
429C22⋊C4—432no—
430C42—216no—
431C23—232no—
432C23—2256no—
433C23—232no—
434C23—264no—
435C23—232no—
436C23—264no—
437C23—24no—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—216no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—216no—
446C23—216no—
447C23—216no—
448C23—28no—
449C23—216no—
450C23—216no—
451C23—216no—
452C23—216no—
453C23—24no—
454C23—28no—
455C23—216no—
456C23—42no—
457C23—48no—
458C23—44no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—48no—
463C23—48no—
464C23—416no—
465C23C2×#12014yes—
466C23C2×#12014yes—
467C23C2×#12018yes—
468C23#52518yes—
469C23#53414yes—
470C23#53618yes—
471C23D42116yes—
472C23#547132yes—
473C23C23×D411yescharacteristic
474C23C23×D412yes—
475C23C23×D412yes—
476C23C22×#4114yes—
477C23C2×H(5,𝔽2)12yes—
478C23C2×2−1+414yes—
479Q8—416no—
480D4—416no—
481D4—432no—
482D4—464no—
483C2×C4—264no—
484C2×C4—2128no—
485C2×C4—232no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—216no—
491C2×C4—216no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—216no—
503C2×C4—28no—
504C2×C4—216no—
505C2×C4—432no—
506C2×C4—432no—
507C2×C4—432no—
508C2×C4—464no—
509C2×C4—416no—
510C2×C4—432no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—416no—
524C2×C4—48no—
525C2×C4—416no—
526C2×C4C2×#12014yes—
527C2×C4C2×#12014yes—
528C2×C4#525116yescomplemented
529C22—2128no—
530C22—24no—
531C22—28no—
532C22—232no—
533C22—264no—
534C22—28no—
535C22—24no—
536C22—28no—
537C22—28no—
538C22—28no—
539C22—28no—
540C22—28no—
541C22—28no—
542C22—416no—
543C22—44no—
544C22—48no—
545C22—82no—
546C22—88no—
547C22—82no—
548C22—84no—
549C22#240318yes—
550C22#2406116yescomplemented
551C22C22×#12011yescharacteristic
552C22C22×#12011yescharacteristic
553C22C2×#52514yes—
554C22C2×#52512yes—
555C22C2×#53411yescharacteristic
556C22C2×#53612yes—
557C22C2×D4214yes—
558C22C2×#54718yes—
559C22#348614yes—
560C22#349218yes—
561C4—216no—
562C4—24no—
563C4—24no—
564C4—44no—
565C4—48no—
566C4—44no—
567C4—48no—
568C4—88no—
569C2—28no—
570C2—216no—
571C2—84no—
572C2C2×#240311yescharacteristic
573C2C2×#240612yes—
574C2#3505712yes—
575C2#3524211yescharacteristic
576C2#3610318yes—
577C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#525C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C24×C4⋊C4C2semidirect
3. C2×#2849C2semidirect
4. C2×#2406C2semidirect
5. #15069C2semidirect
6. #14720C2semidirect
7. #14719C2semidirect
8. #14694C2semidirect
9. #14681C2semidirect
10. #14015C2semidirect
11. #13870C2semidirect
33. C2×#395C22semidirect
34. C2×#394C22semidirect
39. C2×#385C22semidirect
44. #1444C22semidirect
45. #1441C22semidirect
46. #1439C22semidirect
47. #1436C22semidirect
110. C22×C4⋊C4D4semidirect
222. C23×C4#41semidirect
528. C2×C4#525semidirect
550. C22#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#5252
C24×C4⋊C42
C2×#28492
C2×#24062
#150692
#147202
#147192
#146942
#146812
#140152
#138702