Finite Group Databasebeta

← #636929

Subgroups — #636929

Order 512.

4238 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#2487C212yesmaximal, complemented
3C22×#389C211yesmaximal, characteristic, complemented
4#16674C214yesmaximal, complemented
5#16634C212yesmaximal, complemented
6#16510C212yesmaximal, complemented
7#15158C212yesmaximal, complemented
8#15129C212yesmaximal, complemented
9#14912C214yesmaximal, complemented
10#14720C212yesmaximal, complemented
11#14719C212yesmaximal
12#14685C211yesmaximal, characteristic, complemented
13#14264C214yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C2×#529—28no—
16C2×#529—216no—
17C22×#126C2212yes—
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#123C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×#117C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2212yes—
30C23×C42C2211yescharacteristic
31#2487—216no—
32C2×#400C2214yes—
33C2×#399C2214yes—
34C2×#395C2214yes—
35C2×#394C2214yes—
36C2×#389—28no—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#381C2212yes—
46C2×#380C2214yes—
47C2×#380C2214yes—
48C2×#380C2212yes—
49C22×#95C2212yes—
50#1444C2218yes—
51#1442C2218yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2218yes—
55#1441C2214yes—
56#1440C2218yes—
57#1439C2214yes—
58#1439C2218yes—
59#1439C2218yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C23×Q8C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76#529—2128no—
77C2×#126—28no—
78C2×#126—28no—
79C2×#126—28no—
80C2×#125—28no—
81C2×#125—24no—
82C2×#124—28no—
83C2×#124—24no—
84C2×#123—28no—
85C2×#123—28no—
86C2×#123—28no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—28no—
89C2×#121—28no—
90C2×#121—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—28no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—24no—
97C2×C4×D4D4116yes—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—48no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2318yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2318yes—
128C22×C22⋊C4C2318yes—
129C22×C42—28no—
130C22×C42D418yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#400—216no—
137#399—216no—
138#395—48no—
139#394—48no—
140#389—216no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#386—216no—
146#386—48no—
147#385—216no—
148#381—44no—
149#380—216no—
150#380—28no—
151#380—216no—
152C2×#95—28no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×Q8—24no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—42no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4C2×D418yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D418yes—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D418yes—
209C23×C4C2418yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#126—264no—
222#125—232no—
223#124—232no—
224#123—264no—
225C22⋊Q8—232no—
226#121—232no—
227C4×Q8—232no—
228C4×D4—232no—
229C4×D4—264no—
230#117—264no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
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—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—432no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4#4118yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4C2×D418yes—
294C2×C22⋊C4C2×D418yes—
295C2×C42—232no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42C2×D414yes—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307#95—28no—
308#95—28no—
309#95—216no—
310#95—28no—
311#95—216no—
312#95—216no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—48no—
318#95—48no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24C22×D414yes—
334C24C22×D414yes—
335C24C22×D412yes—
336C24H(5,𝔽2)12yes—
337C242−1+414yes—
338C242−1+412yes—
339C24C2511yescharacteristic
340C2×Q8—28no—
341C2×Q8—416no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×D4—28no—
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4—416no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—44no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—232no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—232no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—44no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—816no—
428C22×C4#120116yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C2×#4118yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—816no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—416no—
464C22⋊C4—816no—
465C22⋊C4—816no—
466C42—28no—
467C42—28no—
468C42—28no—
469C42—28no—
470C42—216no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×#12014yes—
507C23#53614yes—
508C23#53618yes—
509C23#53718yes—
510C23#53714yes—
511C23C23×D411yescharacteristic
512C23C23×D412yes—
513C23C22×#4112yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×2−1+412yes—
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+412yes—
519C23#58512yes—
520Q8—416no—
521D4—416no—
522D4—432no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—232no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—416no—
550C2×C4—432no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—432no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—816no—
585C2×C4C2×#12014yes—
586C2×C4C2×#12014yes—
587C2×C4#52514yes—
588C2×C4#53514yes—
589C2×C4#53614yes—
590C2×C4#547116yes—
591C22—216no—
592C22—216no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—44no—
600C22—48no—
601C22—48no—
602C22—44no—
603C22—44no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22#261618yes—
611C22C22×#12011yescharacteristic
612C22C2×#52511yescharacteristic
613C22C2×#53511yescharacteristic
614C22C2×#53612yes—
615C22C2×#53611yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#53712yes—
618C22C2×#53711yescharacteristic
619C22#344911yescharacteristic
620C22C2×#54714yes—
621C22#348814yes—
622C22#351614yes—
623C22#351912yes—
624C22#352012yes—
625C22#355214yes—
626C22#355912yes—
627C22#356312yes—
628C4—24no—
629C4—24no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—48no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—88no—
638C4—84no—
639C4#240618yes—
640C2—28no—
641C2—84no—
642C2C2×#261611yescharacteristic
643C2C2×#240611yescharacteristic
644C2#3523611yescharacteristic
645C2#3524511yescharacteristic
646C2#3525011yescharacteristic
647C2#3754911yescharacteristic
648C2#3761711yescharacteristic
649C2#3804014yes—
650C2#3804214yes—
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×#2487C2semidirect
3. C22×#389C2semidirect
4. #16674C2semidirect
5. #16634C2semidirect
6. #16510C2semidirect
7. #15158C2semidirect
8. #15129C2semidirect
9. #14912C2semidirect
10. #14720C2semidirect
12. #14685C2semidirect
13. #14264C2semidirect
14. #13872C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#24872
C22×#3892
#166742
#166342
#165102
#151582
#151292
#149122
#147202
#147192
#146852
#142642
#138722