Finite Group Databasebeta

← #4417970

Subgroups — #4417970

Order 512.

3330 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2665C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3#17080C212yesmaximal, complemented
4#16898C212yesmaximal, complemented
5#16301C214yesmaximal, complemented
6#15429C214yesmaximal, complemented
7#15323C214yesmaximal, complemented
8#15177C212yesmaximal, complemented
9#14812C212yesmaximal, complemented
10#14758C214yesmaximal, complemented
11#14675C211yesmaximal, characteristic, complemented
12#14353C211yesmaximal, characteristic, complemented
13#14337C211yesmaximal, characteristic
14#13965C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2212yes—
19#2665—28no—
20C2×#400C2212yes—
21C2×#396C2212yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25#2339—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#387C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#381C2211yescharacteristic
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2212yes—
45#1446C2212yes—
46#1446C2211yescharacteristic
47#1446C2211yescharacteristic
48#1446C2214yes—
49#1445C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2212yes—
70#1438C2212yes—
71#1438C2211yescharacteristic
72#1438C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—48no—
80C22×#41Q818yescomplemented
81C23×Q8C2312yes—
82C23×Q8C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2312yes—
85C24×C4C2314yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—48no—
92C2×C4×D4—28no—
93C2×C4×D4—48no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2312yes—
117#400—28no—
118#396—28no—
119#396—48no—
120#395—28no—
121#393—28no—
122#391—28no—
123#391—28no—
124#387—28no—
125#386—216no—
126#386—48no—
127#385—48no—
128#385—48no—
129#381—24no—
130#381—28no—
131#380—24no—
132#380—24no—
133#380—24no—
134#380—44no—
135#380—48no—
136#380—48no—
137#380—44no—
138C4×C22⋊C4D418yes—
139C2×#95—24no—
140C2×#95—28no—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C25C2411yescharacteristic
170C25C2412yes—
171C2×#41—232no—
172C2×#41—432no—
173C2×#41—432no—
174C2×#41—432no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8C2×Q814yes—
184C22×D4—28no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—44no—
191C22×D4C2×Q814yes—
192C22×D4C2×Q818yes—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×Q818yes—
212C23×C4C2×Q814yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C22⋊Q8—464no—
228C4×D4—432no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C2×D418yes—
286C2×C42—44no—
287C2×C42C2×D418yes—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24C22×D412yes—
329C24C22×Q812yes—
330C24C22×Q814yes—
331C24C2×#4112yes—
332C24H(5,𝔽2)18yes—
333C24C2511yescharacteristic
334#41—4128no—
335C2×Q8—28no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—416no—
340C2×Q8—48no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—88no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4—216no—
349C2×D4—28no—
350C2×D4—48no—
351C2×D4—416no—
352C2×D4—416no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—216no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4C22⋊Q8116yescomplemented
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×Q814yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q812yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C42−1+414yes—
440C22×C42−1+414yes—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—416no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C22⋊C4—216no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—84no—
458C22⋊C4—82no—
459C22⋊C4—84no—
460C22⋊C4—82no—
461C22⋊C4—84no—
462C22⋊C4—88no—
463C42—48no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—28no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C2×C22⋊Q814yes—
500C23#54318yes—
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)14yes—
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+412yes—
508C23C2×2−1+412yes—
509Q8—416no—
510Q8—816no—
511D4—232no—
512D4—432no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—416no—
534C2×C4—432no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—832no—
580C2×C4—816no—
581C2×C4C2×C22⋊Q814yes—
582C2×C4C2×C22⋊Q814yes—
583C2×C4#53718yes—
584C22—216no—
585C22—28no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—44no—
598C22—48no—
599C22—44no—
600C22—44no—
601C22C22×C22⋊Q811yescharacteristic
602C22C2×#53712yes—
603C22C2×#54312yes—
604C22#348412yes—
605C22#349012yes—
606C22#349212yes—
607C22#349811yescharacteristic
608C22#349911yescharacteristic
609C22#350312yes—
610C22#350312yes—
611C22#350811yescharacteristic
612C22#350911yescharacteristic
613C22#353214yes—
614C22#353614yes—
615C22#353614yes—
616C22#354414yes—
617C4—28no—
618C4—24no—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—84no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C2—48no—
631C2—44no—
632C2#3498211yescharacteristic
633C2#3528011yescharacteristic
634C2#3553211yescharacteristic
635C2#4048812yes—
636C2#4199714yes—
637C2#4242812yes—
638C2#4788112yes—
639C2#4959112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2665C2semidirect
3. #17080C2semidirect
4. #16898C2semidirect
5. #16301C2semidirect
6. #15429C2semidirect
7. #15323C2semidirect
8. #15177C2semidirect
9. #14812C2semidirect
10. #14758C2semidirect
11. #14675C2semidirect
12. #14353C2semidirect
14. #13965C2semidirect
80. C22×#41Q8semidirect
427. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26652
C2×#23392
#170802
#168982
#163012
#154292
#153232
#151772
#148122
#147582
#146752
#143532
#143372
#139652
#139562