Finite Group Databasebeta

← #361750

Subgroups — #361750

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#416C211yesmaximal, characteristic
2C2×#1825C211yesmaximal, characteristic, complemented
3C2×#1808C211yesmaximal, characteristic, complemented
4C2×#1788C211yesmaximal, characteristic
5C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6#13939C211yesmaximal, characteristic, complemented
7#13937C211yesmaximal, characteristic
8#11124C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#10926C212yesmaximal, complemented
11#10925C212yesmaximal, complemented
12C23×#41C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C2×#416—24no—
17C2×#416—24no—
18C2×#416C418yescomplemented
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C4×C8C2212yes—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1825—24no—
33#1808—24no—
34#1788—24no—
35C8×C22⋊C4—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1440—22no—
41#1438—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C22⋊#95—22no—
46C22×#41—22no—
47C22×#41—24no—
48C22×#41—24no—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—42no—
54C22×#41—42no—
55C22×#41C2×C414yes—
56C23×Q8C2311yescharacteristic
57C23×D4—21no—
58C23×D4C2311yescharacteristic
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C23×C8—42no—
62C23×C8—42no—
63C23×C8C2×C414yes—
64C23×C8D414yes—
65C23×C8D414yes—
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117C2×C414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2312yes—
86C2×#107—24no—
87C2×#107—42no—
88C2×#107—42no—
89#416—232no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103C2×C414yes—
98C2×#103C2×C414yes—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102C2×C414yes—
107C2×#102C2×C414yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8D414yes—
111C2×C4×C8—24no—
112C2×C4×C8—42no—
113C2×C4×C8—42no—
114#391—24no—
115#391—42no—
116#390—24no—
117#390—42no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—24no—
126C2×#95—21no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95D412yes—
130#340—24no—
131#340—24no—
132#340—24no—
133#336—24no—
134C25—21no—
135C25C2×D411yescharacteristic
136C2×#41—216no—
137C2×#41—216no—
138C2×#41—28no—
139C2×#41—28no—
140C2×#41—28no—
141C2×#41—28no—
142C2×#41—44no—
143C2×#41—44no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—48no—
151C2×#41—48no—
152C2×#41—44no—
153C2×#41—44no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8C22×C412yes—
163C22×Q8#4114yes—
164C22×D4—24no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22×C412yes—
178C22×D4#4114yes—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22×C412yes—
194C23×C4C2×D414yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4#4112yes—
200C23×C4#4112yes—
201C23×C4C2411yescharacteristic
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8—48no—
215C22×C8—48no—
216C22×C8C22⋊C4116yescomplemented
217C22×C8C22×C412yes—
218C22×C8C22×C412yes—
219C22×C8C2×D414yes—
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224#117—28no—
225#117—28no—
226#117—44no—
227#117—44no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—84no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273#107—48no—
274#103—28no—
275#103—28no—
276#103—48no—
277#102—28no—
278#102—28no—
279#102—48no—
280C22⋊C8—28no—
281C4×C8—48no—
282#95—24no—
283#95—48no—
284#95—42no—
285#95—42no—
286#95—48no—
287#95—84no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—41no—
296C24—41no—
297C24—42no—
298C24—44no—
299C24—42no—
300C24—42no—
301C24C4×D412yes—
302C24#12112yes—
303C24#12712yes—
304C24C22×D411yescharacteristic
305#41—232no—
306#41—416no—
307#41—416no—
308#41—432no—
309#41—416no—
310#41—432no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8#11714yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—48no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4#11714yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
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—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4C2×C22⋊C414yes—
404C22×C4C4×D414yes—
405C22×C4C4×D412yes—
406C22×C4#12114yes—
407C22×C4#12112yes—
408C22×C4#12414yes—
409C22×C4#12712yes—
410C22×C4#13518yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)12yes—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—416no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8C2×C22⋊C414yes—
438C2×C8C2×C22⋊C414yes—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C22⋊C4—216no—
456C22⋊C4—48no—
457C22⋊C4—816no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—48no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—48no—
485C23—44no—
486C23—48no—
487C23—44no—
488C23—42no—
489C23—82no—
490C23—81no—
491C23—81no—
492C23—84no—
493C23—84no—
494C23—82no—
495C23—82no—
496C23#39014yes—
497C23C2×C4×D411yescharacteristic
498C23C2×#12111yescharacteristic
499C23C2×#12711yescharacteristic
500C23#54612yes—
501C23#54612yes—
502C23C2×#13514yes—
503C23#57612yes—
504C23#57812yes—
505Q8—24no—
506Q8—24no—
507Q8—44no—
508Q8—42no—
509Q8—42no—
510Q8—44no—
511D4—24no—
512D4—24no—
513D4—432no—
514D4—44no—
515D4—42no—
516D4—42no—
517D4—44no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—432no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4#39014yes—
561C2×C4C22×C22⋊C411yescharacteristic
562C2×C4C2×#11711yescharacteristic
563C2×C4C2×#11711yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4#51811yescharacteristic
566C2×C4#51811yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4C2×#12411yescharacteristic
571C2×C4#53412yes—
572C2×C4#53412yes—
573C2×C4#54612yes—
574C2×C4#54612yes—
575C2×C4#55112yes—
576C2×C4#55112yes—
577C2×C4#55112yes—
578C2×C4#55112yes—
579C8—24no—
580C8—24no—
581C8—44no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—44no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—88no—
596C22—82no—
597C22—82no—
598C22C2×#39011yescharacteristic
599C22#234211yescharacteristic
600C22#234211yescharacteristic
601C22#253412yes—
602C22#253912yes—
603C22C4×#13512yes—
604C22#295112yes—
605C22#305012yes—
606C22#306112yes—
607C22#314712yes—
608C22#315511yescharacteristic
609C22#315611yescharacteristic
610C22#329714yes—
611C22#330014yes—
612C22#330214yes—
613C22#330514yes—
614C4—24no—
615C4—44no—
616C4—48no—
617C4—82no—
618C4—82no—
619C4—162no—
620C4—162no—
621C4#228011yescharacteristic
622C4#230811yescharacteristic
623C4#234211yescharacteristic
624C4#234211yescharacteristic
625C4#253412yes—
626C4#256112yes—
627C2—44no—
628C2—84no—
629C2#1390311yescharacteristic
630C2#2122411yescharacteristic
631C2#2123011yescharacteristic
632C2#2257312yes—
633C2#2257612yes—
634C2#2809614yes—
635C2#2833712yes—
636C2#2833812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1825C2semidirect
3. C2×#1808C2semidirect
5. C2×C8×C22⋊C4C2semidirect
6. #13939C2semidirect
8. #11124C2semidirect
9. #11123C2semidirect
10. #10926C2semidirect
11. #10925C2semidirect
18. C2×#416C4semidirect
216. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4162
C2×#18252
C2×#18082
C2×#17882
C2×C8×C22⋊C42
#139392
#139372
#111242
#111232
#109262
#109252