Finite Group Databasebeta

← #4453329

Subgroups — #4453329

Order 512.

4546 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2409C212yesmaximal, complemented
3C22×#395C211yesmaximal, characteristic, complemented
4#16656C214yesmaximal, complemented
5#16608C214yesmaximal, complemented
6#16254C214yesmaximal, complemented
7#16251C214yesmaximal, complemented
8#14912C214yesmaximal, complemented
9#14720C212yesmaximal, complemented
10#14719C212yesmaximal
11#14339C211yesmaximal, characteristic, complemented
12#13960C212yesmaximal, complemented
13C24×Q8C2211yescharacteristic
14C2×#539—216no—
15C2×#539—28no—
16C22×#126C2212yes—
17C22×#125C2212yes—
18C22×#123C2212yes—
19C22×#123C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2214yes—
23C22×#121C2211yescharacteristic
24C22×C4×D4C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27#2409—216no—
28C2×#395—28no—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#395C2214yes—
33C2×#394C2214yes—
34C2×#393C2214yes—
35C2×#391C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43#1444C2218yes—
44#1442C2218yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2218yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2218yes—
52#1439C2214yes—
53#1439C2218yes—
54#1439C2218yes—
55#1434C2214yes—
56C22⋊#95C2212yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C23×Q8—24no—
60C23×Q8—28no—
61C23×Q8—48no—
62C23×Q8C2314yes—
63C23×Q8C2311yescharacteristic
64C23×D4C2312yes—
65C23×D4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72#539—2128no—
73C2×#126—28no—
74C2×#126—28no—
75C2×#126—28no—
76C2×#125—28no—
77C2×#125—28no—
78C2×#125—28no—
79C2×#123—28no—
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—28no—
84C2×#123—28no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—416no—
92C2×#121—28no—
93C2×#121—24no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4D4116yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—48no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2318yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2318yes—
121C22×C42C2312yes—
122#395—216no—
123#395—216no—
124#395—216no—
125#395—48no—
126#394—48no—
127#393—216no—
128#391—216no—
129#386—216no—
130#386—48no—
131#385—216no—
132#381—44no—
133#380—28no—
134C2×#95—28no—
135C2×#95—24no—
136C2×#95C2314yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2314yes—
143C2×#95C2318yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C25C2412yes—
148C25C2411yescharacteristic
149C22×Q8—28no—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—28no—
153C22×Q8—28no—
154C22×Q8—216no—
155C22×Q8—48no—
156C22×Q8—44no—
157C22×Q8—432no—
158C22×Q8—464no—
159C22×Q8—416no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—44no—
169C22×D4C2×D418yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—48no—
189C23×C4C2×D414yes—
190C23×C4C2×D414yes—
191C23×C4C2×D418yes—
192C23×C4C2×D418yes—
193C23×C4C2×D414yes—
194C23×C4C2418yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204#126—264no—
205#125—264no—
206#123—264no—
207#123—264no—
208C22⋊Q8—264no—
209C22⋊Q8—232no—
210C22⋊Q8—464no—
211#121—232no—
212C4×D4—264no—
213C2×C4⋊C4—232no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—432no—
245C2×C4⋊C4C2×D418yes—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D418yes—
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—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4C2×D418yes—
271C2×C22⋊C4C2×D418yes—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42C2×D418yes—
275#95—28no—
276#95—216no—
277#95—28no—
278#95—216no—
279#95—216no—
280#95—216no—
281#95—48no—
282#95—48no—
283#95—44no—
284#95—48no—
285#95—44no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—28no—
291C24—22no—
292C24—28no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24C22×D414yes—
297C24C22×D414yes—
298C24C22×D412yes—
299C24H(5,𝔽2)12yes—
300C242−1+414yes—
301C242−1+412yes—
302C24C2511yescharacteristic
303C2×Q8—232no—
304C2×Q8—28no—
305C2×Q8—432no—
306C2×Q8—432no—
307C2×Q8—416no—
308C2×Q8—4128no—
309C2×Q8—416no—
310C2×Q8—88no—
311C2×Q8—84no—
312C2×D4—216no—
313C2×D4—216no—
314C2×D4—216no—
315C2×D4—28no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—84no—
321C2×D4—88no—
322C22×C4—28no—
323C22×C4—216no—
324C22×C4—216no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—416no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—432no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4#120116yes—
386C22×C4C22×D414yes—
387C22×C4C22×D414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D412yes—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4C22×D414yes—
393C22×C4C22×D414yes—
394C22×C4C22×D414yes—
395C22×C4H(5,𝔽2)18yes—
396C22×C42−1+418yes—
397C4⋊C4—28no—
398C4⋊C4—216no—
399C4⋊C4—216no—
400C4⋊C4—28no—
401C4⋊C4—216no—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—416no—
406C4⋊C4—416no—
407C4⋊C4—432no—
408C4⋊C4—416no—
409C22⋊C4—216no—
410C22⋊C4—216no—
411C22⋊C4—216no—
412C22⋊C4—216no—
413C22⋊C4—216no—
414C22⋊C4—432no—
415C22⋊C4—416no—
416C22⋊C4—816no—
417C42—216no—
418C23—24no—
419C23—28no—
420C23—24no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—216no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—28no—
429C23—28no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—48no—
439C23—48no—
440C23—48no—
441C23—48no—
442C23C2×#12014yes—
443C23#53614yes—
444C23#53618yes—
445C23#53718yes—
446C23#53714yes—
447C23C23×D411yescharacteristic
448C23C23×D412yes—
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×H(5,𝔽2)14yes—
451C23C2×2−1+411yescharacteristic
452C23C2×2−1+412yes—
453C23C2×2−1+414yes—
454Q8—464no—
455Q8—88no—
456D4—432no—
457D4—88no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—216no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—416no—
481C2×C4—432no—
482C2×C4—432no—
483C2×C4—416no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—816no—
513C2×C4—88no—
514C2×C4C2×#12018yes—
515C22—24no—
516C22—216no—
517C22—216no—
518C22—24no—
519C22—28no—
520C22—28no—
521C22—44no—
522C22—48no—
523C22—48no—
524C22—44no—
525C22—44no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22#261618yes—
531C22C22×#12011yescharacteristic
532C22C2×#53612yes—
533C22C2×#53611yescharacteristic
534C22C2×#53712yes—
535C22C2×#53711yescharacteristic
536C22#348414yes—
537C22#348412yes—
538C22#348612yes—
539C22#349914yes—
540C22#353614yes—
541C22#353814yes—
542C22#355218yes—
543C4—28no—
544C4—48no—
545C4—42no—
546C4—44no—
547C4—42no—
548C4—44no—
549C4—44no—
550C4—44no—
551C4—48no—
552C4—88no—
553C2—28no—
554C2—84no—
555C2C2×#261611yescharacteristic
556C2#3534611yescharacteristic
557C2#3566611yescharacteristic
558C2#3671912yes—
559C2#3723312yes—
560C2#4049314yes—
561C2#4049514yes—
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×#2409C2semidirect
3. C22×#395C2semidirect
4. #16656C2semidirect
5. #16608C2semidirect
6. #16254C2semidirect
7. #16251C2semidirect
8. #14912C2semidirect
9. #14720C2semidirect
11. #14339C2semidirect
12. #13960C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24092
C22×#3952
#166562
#166082
#162542
#162512
#149122
#147202
#147192
#143392
#139602