Finite Group Databasebeta

← #620253

Subgroups — #620253

Order 512.

4286 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2623C212yesmaximal, complemented
3C22×#384C211yesmaximal, characteristic, complemented
4#15928C214yesmaximal, complemented
5#15850C214yesmaximal, complemented
6#15430C214yesmaximal, complemented
7#15384C214yesmaximal, complemented
8#14720C212yesmaximal, complemented
9#14719C212yesmaximal
10#14706C212yesmaximal, complemented
11#14342C211yesmaximal, characteristic, complemented
12#14015C214yesmaximal, complemented
13C2×#539—216no—
14C2×#539—28no—
15C22×#126C2212yes—
16C22×#125C2212yes—
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×D4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#2623—216no—
29C2×#400C2214yes—
30C2×#399C2214yes—
31C2×#398C2214yes—
32C2×#397C2214yes—
33C2×#395C2214yes—
34C2×#394C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2214yes—
38C2×#385C2218yes—
39C2×#384—28no—
40C2×#384C2214yes—
41C2×#384C2214yes—
42C2×#381C2212yes—
43#1445C2214yes—
44#1445C2214yes—
45#1444C2218yes—
46#1444C2218yes—
47#1441C2214yes—
48#1441C2214yes—
49#1439C2218yes—
50#1439C2218yes—
51#1437C2214yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2218yes—
55#1436C2214yes—
56C22⋊#95C2212yes—
57C22⋊#95C2212yes—
58C22⋊#95C2214yes—
59C23×Q8C2311yescharacteristic
60C23×D4C2312yes—
61C23×D4C2311yescharacteristic
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68#539—2128no—
69C2×#126—28no—
70C2×#126—28no—
71C2×#126—28no—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—28no—
75C2×#123—28no—
76C2×#123—28no—
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—28no—
80C2×#123D4116yes—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×#121—28no—
87C2×#121—24no—
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—48no—
97C22×C4⋊C4—48no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2318yes—
120C22×C42—24no—
121C22×C42—28no—
122C22×C42C2314yes—
123C22×C42C2312yes—
124#400—216no—
125#399—216no—
126#398—216no—
127#397—216no—
128#395—48no—
129#394—48no—
130#386—216no—
131#386—216no—
132#385—216no—
133#385—416no—
134#384—216no—
135#384—216no—
136#381—28no—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2318yes—
147C25C2412yes—
148C25C2411yescharacteristic
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8—44no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—44no—
159C22×D4C2×D418yes—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—48no—
178C23×C4C2×D418yes—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4C2×D414yes—
182C23×C4#4118yes—
183C23×C4C2418yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2414yes—
193#126—264no—
194#125—264no—
195#123—264no—
196#123—264no—
197C22⋊Q8—264no—
198C22⋊Q8—232no—
199#121—232no—
200C4×D4—264no—
201C2×C4⋊C4—232no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—432no—
234C2×C4⋊C4—432no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×D418yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4C2×D418yes—
264C2×C22⋊C4C2×D418yes—
265C2×C22⋊C4C2×D418yes—
266C2×C42—232no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—216no—
272#95—28no—
273#95—28no—
274#95—216no—
275#95—216no—
276#95—216no—
277#95—216no—
278#95—216no—
279#95—48no—
280#95—44no—
281#95—44no—
282C24—24no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—28no—
288C24—28no—
289C24—24no—
290C24C22×D414yes—
291C24C22×D412yes—
292C24C22×D414yes—
293C24C2×#4114yes—
294C242−1+414yes—
295C24C2511yescharacteristic
296C2×Q8—28no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×D4—216no—
300C2×D4—232no—
301C2×D4—28no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—416no—
305C2×D4—84no—
306C2×D4—88no—
307C22×C4—216no—
308C22×C4—216no—
309C22×C4—232no—
310C22×C4—28no—
311C22×C4—216no—
312C22×C4—28no—
313C22×C4—232no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—232no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—216no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—816no—
364C22×C4#120116yes—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4C22×D414yes—
370C22×C4C22×D412yes—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C2×#4118yes—
375C22×C4H(5,𝔽2)18yes—
376C22×C42−1+418yes—
377C4⋊C4—216no—
378C4⋊C4—28no—
379C4⋊C4—216no—
380C4⋊C4—28no—
381C4⋊C4—216no—
382C4⋊C4—216no—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—432no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—416no—
389C4⋊C4—816no—
390C22⋊C4—216no—
391C22⋊C4—216no—
392C22⋊C4—216no—
393C22⋊C4—216no—
394C22⋊C4—216no—
395C22⋊C4—416no—
396C22⋊C4—432no—
397C22⋊C4—816no—
398C42—232no—
399C42—216no—
400C23—28no—
401C23—24no—
402C23—24no—
403C23—28no—
404C23—28no—
405C23—28no—
406C23—216no—
407C23—24no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—48no—
418C23—44no—
419C23—44no—
420C23—416no—
421C23C2×#12014yes—
422C23#52514yes—
423C23#53614yes—
424C23#547116yes—
425C23C23×D411yescharacteristic
426C23C23×D412yes—
427C23C22×#4112yes—
428C23C2×H(5,𝔽2)14yes—
429C23C2×2−1+412yes—
430C23C2×2−1+414yes—
431Q8—88no—
432D4—432no—
433D4—88no—
434C2×C4—264no—
435C2×C4—232no—
436C2×C4—24no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—88no—
482C2×C4—88no—
483C2×C4—816no—
484C2×C4—816no—
485C2×C4C2×#12018yes—
486C2×C4#539116yescomplemented
487C22—24no—
488C22—28no—
489C22—28no—
490C22—232no—
491C22—24no—
492C22—44no—
493C22—44no—
494C22—416no—
495C22—44no—
496C22—84no—
497C22—84no—
498C22—88no—
499C22#240618yes—
500C22C22×#12011yescharacteristic
501C22C2×#52511yescharacteristic
502C22C2×#53611yescharacteristic
503C22C2×#53914yes—
504C22C2×#54714yes—
505C22#348412yes—
506C22#348612yes—
507C22#349214yes—
508C22#349814yes—
509C22#349914yes—
510C22#355218yes—
511C4—28no—
512C4—216no—
513C4—44no—
514C4—44no—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—88no—
520C4—84no—
521C2—28no—
522C2—84no—
523C2C2×#240611yescharacteristic
524C2#3498912yes—
525C2#3552711yescharacteristic
526C2#3566611yescharacteristic
527C2#3723312yes—
528C2#3934818yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2623C2semidirect
3. C22×#384C2semidirect
4. #15928C2semidirect
5. #15850C2semidirect
6. #15430C2semidirect
7. #15384C2semidirect
8. #14720C2semidirect
10. #14706C2semidirect
11. #14342C2semidirect
12. #14015C2semidirect
486. C2×C4#539semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#26232
C22×#3842
#159282
#158502
#154302
#153842
#147202
#147192
#147062
#143422
#140152