Finite Group Databasebeta

← #4452763

Subgroups — #4452763

Order 512.

3084 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2583C211yesmaximal, characteristic, complemented
3C2×#2504C211yesmaximal, characteristic
4C2×#2502C211yesmaximal, characteristic, complemented
5C2×#2639C211yesmaximal, characteristic
6C22×#398C211yesmaximal, characteristic, complemented
7#17062C214yesmaximal, complemented
8#16964C214yesmaximal, complemented
9#16812C214yesmaximal, complemented
10#16811C214yesmaximal, complemented
11#14946C214yesmaximal, complemented
12#14777C214yesmaximal, complemented
13#13796C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19#2639—28no—
20#2591—28no—
21#2583—28no—
22#2504—28no—
23#2502—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#399C2218yes—
27C2×#398—28no—
28C2×#398C2212yes—
29C2×#398C2214yes—
30C2×#398C2214yes—
31C2×#398C2214yes—
32C2×#397C2212yes—
33C2×#397C2214yes—
34C2×#395C2212yes—
35C2×#394C2212yes—
36C2×#394C2212yes—
37C2×#393C2212yes—
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#389C2214yes—
41C2×#388C2214yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2214yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#384C2212yes—
50C2×#382C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2212yes—
54C22×#95C2211yescharacteristic
55#1445C2218yes—
56#1445C2218yes—
57#1445C2218yes—
58#1444C2218yes—
59#1439C2218yes—
60#1439C2218yes—
61#1439C2218yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C23×Q8C2311yescharacteristic
66C23×D4C2312yes—
67C23×D4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×#121—24no—
77C2×#121—24no—
78C2×#121—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—48no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2318yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2318yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2318yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109#400—28no—
110#400—28no—
111#399—416no—
112#398—28no—
113#398—216no—
114#398—216no—
115#398—216no—
116#397—28no—
117#397—216no—
118#395—28no—
119#394—28no—
120#394—28no—
121#393—28no—
122#391—28no—
123#390—28no—
124#389—216no—
125#388—216no—
126#386—216no—
127#386—28no—
128#386—28no—
129#386—216no—
130#385—28no—
131#385—28no—
132#385—216no—
133#384—28no—
134#382—28no—
135C4×C22⋊C4D418yes—
136C4×C22⋊C4D418yes—
137C2×#95—28no—
138C2×#95D418yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—44no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—216no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—44no—
168C23×C4—22no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D418yes—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C22⋊Q8—416no—
202#121—416no—
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—24no—
210C2×C4⋊C4—216no—
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—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—432no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—216no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4C2×D418yes—
259C2×C22⋊C4C2×D418yes—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42C2×D418yes—
263C2×C42C2×D418yes—
264#95—216no—
265#95—216no—
266#95—28no—
267#95—28no—
268#95—216no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—48no—
274#95—48no—
275#95C2×D418yes—
276#95C2×D418yes—
277#95C2×D418yes—
278#95C2×D418yes—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—28no—
287C24—24no—
288C24—28no—
289C24C22×D412yes—
290C24C22×D412yes—
291C24C22×D412yes—
292C242−1+418yes—
293C24C2511yescharacteristic
294C2×Q8—48no—
295C2×Q8—44no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—416no—
301C2×D4—416no—
302C2×D4—416no—
303C2×D4—44no—
304C2×D4—84no—
305C2×D4—84no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—216no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—432no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4#127116yes—
369C22×C4C22×D418yes—
370C22×C4C22×D418yes—
371C22×C4C22×D418yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4H(5,𝔽2)14yes—
379C4⋊C4—416no—
380C4⋊C4—432no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—88no—
389C4⋊C4—816no—
390C22⋊C4—216no—
391C22⋊C4—216no—
392C22⋊C4—416no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—816no—
399C42—48no—
400C42—48no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—216no—
409C23—216no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—416no—
422C23—416no—
423C23C2×#12714yes—
424C23#53618yes—
425C23#53618yes—
426C23#53618yes—
427C23C23×D411yescharacteristic
428C23C23×D411yescharacteristic
429C23C23×D411yescharacteristic
430C23C2×H(5,𝔽2)12yes—
431C23C2×2−1+414yes—
432C23#58514yes—
433C23#58512yes—
434Q8—88no—
435D4—88no—
436D4—816no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—416no—
456C2×C4—432no—
457C2×C4—416no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—416no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—88no—
490C2×C4—816no—
491C2×C4C2×#12714yes—
492C2×C4C2×#12714yes—
493C22—28no—
494C22—28no—
495C22—24no—
496C22—216no—
497C22—216no—
498C22—24no—
499C22—84no—
500C22—82no—
501C22—84no—
502C22—82no—
503C22—88no—
504C22—88no—
505C22—82no—
506C22#265818yes—
507C22C22×#12711yescharacteristic
508C22C2×#53612yes—
509C22C2×#53612yes—
510C22C2×#53612yes—
511C22#348411yescharacteristic
512C22#348512yes—
513C22#348512yes—
514C22#348512yes—
515C22#348711yescharacteristic
516C22#348812yes—
517C22#348812yes—
518C22#354118yes—
519C22#355318yes—
520C4—24no—
521C4—24no—
522C4—48no—
523C4—48no—
524C4—48no—
525C4—44no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—84no—
531C2—28no—
532C2—162no—
533C2C2×#265811yescharacteristic
534C2#3546311yescharacteristic
535C2#3556511yescharacteristic
536C2#3671612yes—
537C2#3712012yes—
538C2#4140614yes—
539C2#4140814yes—
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×#2583C2semidirect
4. C2×#2502C2semidirect
6. C22×#398C2semidirect
7. #17062C2semidirect
8. #16964C2semidirect
9. #16812C2semidirect
10. #16811C2semidirect
11. #14946C2semidirect
12. #14777C2semidirect
13. #13796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#25832
C2×#25042
C2×#25022
C2×#26392
C22×#3982
#170622
#169642
#168122
#168112
#149462
#147772
#137962