Finite Group Databasebeta

← #4433054

Subgroups — #4433054

Order 512.

2408 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×#2820C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×#2632C211yesmaximal, characteristic
4C2×#2356C211yesmaximal, characteristic, complemented
5#16679C212yesmaximal, complemented
6#16676C212yesmaximal, complemented
7#16628C214yesmaximal, complemented
8#15360C214yesmaximal, complemented
9#15346C214yesmaximal
10#14910C212yesmaximal, complemented
11#14894C212yesmaximal, complemented
12#14788C214yesmaximal, complemented
13#14296C211yesmaximal, characteristic, complemented
14#14289C211yesmaximal, characteristic, complemented
15#14153C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2820—28no—
20#2632—28no—
21#2584—28no—
22C2×#399C2212yes—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#394C2212yes—
31C2×#394C2212yes—
32C2×#393C2212yes—
33#2356—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2212yes—
37C2×#389C2214yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#383C2212yes—
46C2×#383C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×#376C2212yes—
52#1445C2214yes—
53#1445C2211yescharacteristic
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1445C2211yescharacteristic
57#1444C2214yes—
58#1444C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C2×#125—24no—
83C2×#125—24no—
84C2×#125—24no—
85C2×#124—24no—
86C2×#124—24no—
87C2×#124—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8D418yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#399—28no—
122#399—28no—
123#398—28no—
124#397—28no—
125#396—216no—
126#395—28no—
127#395—28no—
128#395—28no—
129#394—28no—
130#394—28no—
131#393—28no—
132#391—28no—
133#391—44no—
134#390—44no—
135#389—216no—
136#389—28no—
137#389—28no—
138#389—48no—
139#388—28no—
140#388—28no—
141#388—48no—
142#387—216no—
143#383—24no—
144#383—28no—
145#382D418yes—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149#376—28no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2412yes—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8C2×D414yes—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—42no—
191C22×D4—42no—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206#125—416no—
207#124—416no—
208C4×Q8—232no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
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⋊C4C2×D418yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D418yes—
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—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42C2×D418yes—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291C2×C42#4114yes—
292#95—22no—
293#95—28no—
294#95—22no—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95C2×D418yes—
317#95C2×D418yes—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24C2511yescharacteristic
330C2×Q8—28no—
331C2×Q8—48no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—44no—
337C2×Q8—84no—
338C2×Q8—82no—
339C2×Q8—82no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—84no—
343C2×D4—82no—
344C2×D4—82no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
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—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4C22×D418yes—
407C22×C4C22×D414yes—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C42−1+418yes—
420C4⋊C4—216no—
421C4⋊C4—28no—
422C4⋊C4—216no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—88no—
445C42—216no—
446C42—416no—
447C42—48no—
448C42—84no—
449C42#121116yes—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23C23×D411yescharacteristic
474C23C23×D411yescharacteristic
475C23C22×#4111yescharacteristic
476C23C2×H(5,𝔽2)12yes—
477C23C2×2−1+414yes—
478C23#58512yes—
479C23#58512yes—
480C23#58512yes—
481Q8—416no—
482Q8—88no—
483D4—88no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
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—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4C2×#12114yes—
557C2×C4C2×#12114yes—
558C2×C4C2×#12114yes—
559C22—84no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22—84no—
564C22—84no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22C22×#12111yescharacteristic
571C22#344912yes—
572C22#346012yes—
573C22#348511yescharacteristic
574C22#348612yes—
575C22#348711yescharacteristic
576C22#348812yes—
577C22#349312yes—
578C22#349811yescharacteristic
579C22#349912yes—
580C22#350011yescharacteristic
581C22#350912yes—
582C22#353514yes—
583C22#354114yes—
584C22#355714yes—
585C22#356514yes—
586C4—24no—
587C4—24no—
588C4—24no—
589C4—44no—
590C4—44no—
591C4—48no—
592C4—44no—
593C4—48no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C2—162no—
601C2#3515011yescharacteristic
602C2#3525611yescharacteristic
603C2#3566711yescharacteristic
604C2#4174714yes—
605C2#4195712yes—
606C2#4215412yes—
607C2#5106612yes—
608C2#5106912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2584C2semidirect
4. C2×#2356C2semidirect
5. #16679C2semidirect
6. #16676C2semidirect
7. #16628C2semidirect
8. #15360C2semidirect
10. #14910C2semidirect
11. #14894C2semidirect
12. #14788C2semidirect
13. #14296C2semidirect
14. #14289C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25842
C2×#26322
C2×#23562
#166792
#166762
#166282
#153602
#153462
#149102
#148942
#147882
#142962
#142892
#141532