Finite Group Databasebeta

← #4452715

Subgroups — #4452715

Order 512.

2886 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic
3#16827C214yesmaximal, complemented
4#16826C214yesmaximal, complemented
5#16524C214yesmaximal, complemented
6#16522C214yesmaximal, complemented
7#15092C212yesmaximal
8#14955C212yesmaximal
9#14874C212yesmaximal
10#14786C212yesmaximal
11#13966C211yesmaximal, characteristic
12#13965C211yesmaximal, characteristic
13#13881C211yesmaximal, characteristic
14#13880C211yesmaximal, characteristic
15#13796C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2646—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#395—28no—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2214yes—
47#1446C2214yes—
48#1445C2214yes—
49#1444C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8C2314yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117#400—28no—
118#400—28no—
119#398—28no—
120#398—28no—
121#398—48no—
122#395—216no—
123#395—216no—
124#395—48no—
125#395—48no—
126#385—28no—
127#385—28no—
128#385—44no—
129#385—44no—
130#385—44no—
131#385—44no—
132#384—44no—
133#382—44no—
134#380—24no—
135#380—24no—
136C4×C22⋊C4Q818yescomplemented
137C4×C22⋊C4Q818yescomplemented
138C2×#95—24no—
139C2×#95—24no—
140C2×#95D418yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2411yescharacteristic
171C22×Q8—28no—
172C22×Q8—28no—
173C22×Q8—28no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—416no—
179C22×Q8—416no—
180C22×Q8—416no—
181C22×Q8—464no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q814yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4C2×Q818yes—
278C2×C22⋊C4C2×Q818yes—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×Q818yescomplemented
284C2×C42C2×Q818yescomplemented
285#95—28no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C2×D418yes—
311#95C2×D418yes—
312#95C2×Q818yescomplemented
313#95C2×Q818yescomplemented
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D412yes—
319C24C22×Q812yes—
320C24C22×Q812yes—
321C24H(5,𝔽2)14yes—
322C242−1+414yes—
323C24C2511yescharacteristic
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—416no—
327C2×Q8—416no—
328C2×Q8—416no—
329C2×Q8—416no—
330C2×Q8—4128no—
331C2×Q8—416no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—44no—
407C22×C4—416no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—416no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—88no—
418C22×C4C4⋊Q8116yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×Q818yes—
424C22×C4C22×Q818yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q812yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C42−1+414yes—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C42—48no—
453C42—48no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23C2×C4⋊Q814yes—
467C23#53514yes—
468C23#53614yes—
469C23#54418yes—
470C23#54418yes—
471C23C23×D411yescharacteristic
472C23C23×Q811yescharacteristic
473C23C23×Q811yescharacteristic
474C23C2×H(5,𝔽2)12yes—
475C23C2×H(5,𝔽2)12yes—
476C23C2×H(5,𝔽2)12yes—
477C23C2×2−1+412yes—
478C23C2×2−1+412yes—
479C23#58512yes—
480Q8—464no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—416no—
508C2×C4—416no—
509C2×C4—416no—
510C2×C4—416no—
511C2×C4—416no—
512C2×C4—416no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—88no—
557C2×C4C2×C4⋊Q814yes—
558C2×C4C2×C4⋊Q814yes—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—216no—
564C22—216no—
565C22—24no—
566C22—24no—
567C22#266018yes—
568C22C22×C4⋊Q811yescharacteristic
569C22C2×#53511yescharacteristic
570C22C2×#53611yescharacteristic
571C22C2×#54412yes—
572C22C2×#54412yes—
573C22#348411yescharacteristic
574C22#348411yescharacteristic
575C22#348511yescharacteristic
576C22#348611yescharacteristic
577C22#350112yes—
578C22#350112yes—
579C22#350312yes—
580C22#350312yes—
581C22#353214yes—
582C22#353514yes—
583C22#354114yes—
584C22#355214yes—
585C4—24no—
586C4—24no—
587C4—44no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—48no—
592C4—48no—
593C4—44no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C2—28no—
600C2C2×#266011yescharacteristic
601C2#3539111yescharacteristic
602C2#3557611yescharacteristic
603C2#3670311yescharacteristic
604C2#3671611yescharacteristic
605C2#3711811yescharacteristic
606C2#3723311yescharacteristic
607C2#4061414yes—
608C2#4061614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16827C2semidirect
4. #16826C2semidirect
5. #16524C2semidirect
6. #16522C2semidirect
136. C4×C22⋊C4Q8semidirect
137. C4×C22⋊C4Q8semidirect
283. C2×C42C2×Q8semidirect
284. C2×C42C2×Q8semidirect
312. #95C2×Q8semidirect
313. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3952
#168272
#168262
#165242
#165222
#150922
#149552
#148742
#147862
#139662
#139652
#138812
#138802
#137962