Finite Group Databasebeta

← #7646892

Subgroups — #7646892

Order 512.

8060 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3497C211yesmaximal, characteristic, complemented
2C22×#556C211yesmaximal, characteristic, complemented
3#42889C218yesmaximal, complemented
4#42812C218yesmaximal, complemented
5#39954C2116yesmaximal, complemented
6#37547C214yesmaximal, complemented
7#36961C214yesmaximal, complemented
8#36707C214yesmaximal, complemented
9#35676C212yesmaximal, complemented
10#35666C212yesmaximal, complemented
11#35329C212yesmaximal, complemented
12#35301C214yesmaximal, complemented
13#35168C212yesmaximal, complemented
14#35153C212yesmaximal, complemented
15#34955C212yesmaximal, complemented
16#34059C211yesmaximal, characteristic, complemented
17C22×2−1+4C2212yes—
18#3497—216no—
19C2×#556—216no—
20C2×#556C2218yes—
21C2×#556C2218yes—
22C2×#550C2214yescomplemented
23C2×D4×Q8C2214yescomplemented
24C2×#548C2218yescomplemented
25C2×#537C2212yes—
26C2×#537C2214yes—
27C2×#536C2214yes—
28C2×#535C2214yes—
29C2×#532C2212yes—
30C2×#532C2214yes—
31C22×C4⋊Q8C2211yescharacteristic
32C22×#126C2212yes—
33C22×#126C2212yes—
34C22×#124C2211yescharacteristic
35C22×#124C2211yescharacteristic
36C22×C22⋊Q8C2212yes—
37C22×C4×Q8C2211yescharacteristic
38C22×C4×Q8C2211yescharacteristic
39C22×C4×Q8C2211yescharacteristic
40C22×C4×D4C2211yescharacteristic
41C22×#117C2212yes—
42#2858C2218yes—
43#2854C2218yes—
44#2847C2218yes—
45#2821C2218yescomplemented
46#2818C2218yescomplemented
47#2723C22132yescomplemented
48#2714C22132yescomplemented
49#2692C22132yescomplemented
50#2676C22132yescomplemented
51#2662C22116yescomplemented
52#2658C2214yes—
53#2657C2214yes—
54#2636C22116yescomplemented
55#2628C2218yescomplemented
56#2619C2218yescomplemented
57#2618C2218yes—
58#2617C2218yes—
59#2610C22116yescomplemented
60#2603C22116yescomplemented
61#2598C22116yescomplemented
62#2592C2218yescomplemented
63#2591C2214yes—
64#2591C2214yes—
65#2568C2214yescomplemented
66#2548C2214yescomplemented
67#2540C2218yes—
68#2540C2218yes—
69#2539C2218yes—
70#2539C2218yes—
71#2504C2214yes—
72#2436C22116yes—
73#2436C2214yes—
74#2434C22116yescomplemented
75#2432C22116yescomplemented
76#2431C22116yes—
77#2429C22116yescomplemented
78#2428C22116yes—
79#2427C22116yes—
80#2426C22116yescomplemented
81#2406C2218yes—
82#2406C2218yes—
83#2404C2218yes—
84#2404C2218yes—
85#2398C2218yescomplemented
86#2324C22116yescomplemented
87#2320C2218yescomplemented
88#2316C2212yes—
89#2313C2214yes—
90#2310C2212yes—
91C4×#124C2218yescomplemented
92#2297C2214yes—
93#2297C2214yes—
94#2295C2218yescomplemented
95#2283C2214yes—
96C4×#117C2212yes—
97C2×2−1+4—216no—
98C2×2−1+4—416no—
99C22×#41C2314yes—
100C22×#41C2318yes—
101C22×#41C2318yes—
102C23×Q8C2311yescharacteristic
103C23×Q8C2311yescharacteristic
104C23×Q8C2318yes—
105C23×Q8C2311yescharacteristic
106C23×D4C2311yescharacteristic
107C24×C4C2312yes—
108#556—264no—
109#556—264no—
110#550—232no—
111D4×Q8—232no—
112#548—264no—
113#537—216no—
114#537—232no—
115#536—416no—
116#535—416no—
117#532—232no—
118#532—216no—
119C2×C4⋊Q8—28no—
120C2×C4⋊Q8C2314yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2318yes—
123C2×C4⋊Q8C2318yes—
124C2×#127C2314yes—
125C2×#127C2312yes—
126C2×#127C2318yes—
127C2×#126—216no—
128C2×#126—216no—
129C2×#126C2318yes—
130C2×#126C2318yes—
131C2×#126C23116yes—
132C2×#124—28no—
133C2×#124—28no—
134C2×#124C2314yes—
135C2×#124C2314yes—
136C2×#124C2318yes—
137C2×#124C2314yes—
138C2×#124C2318yes—
139C2×#124C2318yes—
140C2×#124C2318yes—
141C2×#124C2318yes—
142C2×#123C23116yes—
143C2×C22⋊Q8—216no—
144C2×C22⋊Q8C23116yes—
145C2×C22⋊Q8C2318yes—
146C2×C22⋊Q8C23116yes—
147C2×C22⋊Q8C23116yes—
148C2×#121C2314yes—
149C2×#121C2318yes—
150C2×#121C2314yes—
151C2×#121C23116yes—
152C2×#121C2318yes—
153C2×#121C23116yes—
154C2×#121C23116yes—
155C2×#120C23116yes—
156C2×C4×Q8—28no—
157C2×C4×Q8—28no—
158C2×C4×Q8D4116yescomplemented
159C2×C4×Q8C2314yes—
160C2×C4×Q8C2314yes—
161C2×C4×Q8C2318yes—
162C2×C4×Q8C2314yes—
163C2×C4×Q8C2314yes—
164C2×C4×D4—28no—
165C2×C4×D4C2318yes—
166C2×C4×D4C2318yes—
167C2×C4×D4C2314yes—
168C2×C4×D4C2318yes—
169C2×C4×D4C2314yes—
170C2×#117—216no—
171C2×#117C2314yes—
172C2×#117C2318yes—
173C2×#117C2318yes—
174C22×C4⋊C4C2311yescharacteristic
175C22×C4⋊C4C2311yescharacteristic
176C22×C4⋊C4C2312yes—
177C22×C4⋊C4C2312yes—
178C22×C4⋊C4C2311yescharacteristic
179C22×C4⋊C4C2312yes—
180C22×C4⋊C4C2312yes—
181C22×C4⋊C4C2311yescharacteristic
182C22×C22⋊C4C2312yes—
183C22×C22⋊C4C2312yes—
184C22×C22⋊C4C2312yes—
185C22×C22⋊C4C2312yes—
186C22×C22⋊C4C2312yes—
187C22×C42C2312yes—
188C22×C42C2311yescharacteristic
189C22×C42C2311yescharacteristic
190C22×C42C2312yes—
191C22×C42C2311yescharacteristic
192#400C2318yes—
193#400C2318yes—
194#399C23116yes—
195#399C23116yes—
196#398C23116yes—
197#397C23116yes—
198#397C23116yes—
199#396C23132yes—
200#395C23116yes—
201#395C23116yes—
202#394C2318yes—
203#394C23116yes—
204#394C23116yes—
205#394C2318yes—
206#394C23116yes—
207#394C23132yes—
208#393C23116yes—
209#393C23116yes—
210#393C23116yes—
211#392C23116yes—
212#392C23116yes—
213#392C23116yes—
214#392C23116yes—
215#392C23116yes—
216#391C23116yes—
217#391C2318yes—
218#391C2318yes—
219#391C2318yes—
220#390C23116yes—
221#390C23132yes—
222#390C2318yes—
223#390C2318yes—
224#390C23132yes—
225#390C2318yes—
226#389C23116yes—
227#389C23116yes—
228#388C23116yes—
229#388C23116yes—
230#388C23132yes—
231#387C23132yes—
232#386C23132yes—
233#386C2318yes—
234#386C2318yes—
235#386C2318yes—
236#386C2318yes—
237#386C23132yes—
238#386C23132yes—
239#385C23132yes—
240#385C23132yes—
241#383C2318yes—
242#382C2318yes—
243#382C2318yes—
244#381C2318yes—
245#380C2318yes—
246#380C23116yes—
247C4×C4⋊C4C2314yes—
248C4×C4⋊C4C2318yes—
249C4×C4⋊C4C2314yes—
250C4×C22⋊C4C2314yes—
251C4×C22⋊C4C2314yes—
252C4×C22⋊C4C23116yes—
253C4×C22⋊C4C23116yes—
254#376C2314yes—
255#376C2318yes—
256C43C2314yes—
257C25C2412yes—
2582−1+4—4128no—
259C2×#41—232no—
260C2×#41—232no—
261C2×#41—216no—
262C2×#41—216no—
263C2×#41—216no—
264C2×#41—432no—
265C2×#41—416no—
266C2×#41—416no—
267C22×Q8—232no—
268C22×Q8—24no—
269C22×Q8—24no—
270C22×Q8—28no—
271C22×Q8—28no—
272C22×Q8—44no—
273C22×Q8—416no—
274C22×Q8—416no—
275C22×Q8C2×D418yes—
276C22×Q8C2414yes—
277C22×Q8C2414yes—
278C22×Q8C2414yes—
279C22×Q8C2414yes—
280C22×Q8C2418yes—
281C22×D4—24no—
282C22×D4—28no—
283C22×D4C2414yes—
284C22×D4C2418yes—
285C22×D4C2414yes—
286C22×D4C2414yes—
287C22×D4C2418yes—
288C22×D4C2414yes—
289C22×D4C2418yes—
290C22×D4C24132yes—
291C23×C4—28no—
292C23×C4—48no—
293C23×C4C2412yes—
294C23×C4C2411yescharacteristic
295C23×C4C2418yes—
296C23×C4C2418yes—
297C23×C4C2412yes—
298C23×C4C2411yescharacteristic
299C23×C4C2411yescharacteristic
300C23×C4C2411yescharacteristic
301C23×C4C24116yes—
302C23×C4C2411yescharacteristic
303C23×C4C2411yescharacteristic
304C23×C4C2412yes—
305C23×C4C2411yescharacteristic
306C4⋊Q8—232no—
307C4⋊Q8—216no—
308C4⋊Q8—28no—
309C4⋊Q8—416no—
310#127—216no—
311#127—28no—
312#127—416no—
313#126—264no—
314#126—232no—
315#126—232no—
316#124—232no—
317#124—216no—
318#124—216no—
319#124—232no—
320#124—216no—
321#124—416no—
322#124—416no—
323#124—416no—
324#123—432no—
325C22⋊Q8—264no—
326C22⋊Q8—232no—
327C22⋊Q8—264no—
328C22⋊Q8—432no—
329#121—216no—
330#121—232no—
331#121—216no—
332#121—232no—
333#121—264no—
334#121—432no—
335#121—432no—
336#120—432no—
337C4×Q8—232no—
338C4×Q8—216no—
339C4×Q8—216no—
340C4×Q8C2×D4132yes—
341C4×Q8C2×D4132yes—
342C4×D4—232no—
343C4×D4—232no—
344C4×D4—216no—
345C4×D4—232no—
346C4×D4—216no—
347#117—216no—
348#117—232no—
349#117—232no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4C2×D418yes—
357C2×C4⋊C4C2×D4116yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C2418yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C2418yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C24116yes—
371C2×C4⋊C4C24132yes—
372C2×C4⋊C4C2414yes—
373C2×C4⋊C4C2418yes—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—28no—
378C2×C22⋊C4—28no—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2414yes—
381C2×C22⋊C4C2414yes—
382C2×C22⋊C4C2418yes—
383C2×C22⋊C4C2418yes—
384C2×C22⋊C4C2418yes—
385C2×C22⋊C4C2418yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C24116yes—
388C2×C22⋊C4C2418yes—
389C2×C22⋊C4C24116yes—
390C2×C22⋊C4C24132yes—
391C2×C22⋊C4C24132yes—
392C2×C22⋊C4C24116yes—
393C2×C22⋊C4C24116yes—
394C2×C22⋊C4C24116yes—
395C2×C42—28no—
396C2×C42—24no—
397C2×C42—24no—
398C2×C42C2×D4116yes—
399C2×C42C2×D418yes—
400C2×C42C2418yes—
401C2×C42C2418yes—
402C2×C42C2414yes—
403C2×C42C2414yes—
404C2×C42C2418yes—
405C2×C42C2418yes—
406C2×C42C2414yes—
407C2×C42C2414yes—
408C2×C42C2412yes—
409C2×C42C2412yes—
410#95C2418yes—
411#95C2418yes—
412#95C24116yes—
413#95C24116yes—
414#95C24116yes—
415#95C24116yes—
416#95C24116yes—
417#95C24116yes—
418#95C24116yes—
419C24—24no—
420C24—28no—
421C24—44no—
422C24—44no—
423C24C2514yes—
424C24C2518yes—
425C24C2511yescharacteristic
426#41—4128no—
427#41—4128no—
428#41—464no—
429C2×Q8—232no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—216no—
433C2×Q8—28no—
434C2×Q8—216no—
435C2×Q8—216no—
436C2×Q8—28no—
437C2×Q8—28no—
438C2×Q8—28no—
439C2×Q8—416no—
440C2×Q8—44no—
441C2×Q8—416no—
442C2×Q8—4128no—
443C2×Q8—416no—
444C2×Q8C22×D4116yes—
445C2×D4—216no—
446C2×D4—216no—
447C2×D4—216no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—216no—
451C2×D4—28no—
452C2×D4—216no—
453C2×D4—28no—
454C2×D4—216no—
455C2×D4—28no—
456C2×D4—28no—
457C2×D4—48no—
458C2×D4—416no—
459C2×D4—416no—
460C2×D4—48no—
461C2×D4—432no—
462C2×D4—432no—
463C2×D4—432no—
464C2×D4—416no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—216no—
474C22×C4—232no—
475C22×C4—216no—
476C22×C4—216no—
477C22×C4—216no—
478C22×C4—216no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—416no—
487C22×C4—416no—
488C22×C4—432no—
489C22×C4C22×D418yes—
490C22×C4C22×D418yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4H(5,𝔽2)18yes—
495C22×C4C25116yes—
496C22×C4C2512yes—
497C22×C4C2512yes—
498C22×C4C2512yes—
499C22×C4C2518yes—
500C22×C4C2514yes—
501C22×C4C2511yescharacteristic
502C22×C4C2514yes—
503C22×C4C2512yes—
504C22×C4C2514yes—
505C22×C4C2511yescharacteristic
506C22×C4C2512yes—
507C22×C4C2512yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—28no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—216no—
516C4⋊C4—28no—
517C4⋊C4—216no—
518C4⋊C4—28no—
519C4⋊C4—216no—
520C4⋊C4—416no—
521C4⋊C4—432no—
522C4⋊C4C22×D4116yes—
523C4⋊C4C22×D4132yes—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—232no—
533C22⋊C4—216no—
534C22⋊C4—232no—
535C22⋊C4—416no—
536C22⋊C4—416no—
537C22⋊C4—432no—
538C22⋊C4—432no—
539C22⋊C4—416no—
540C42—216no—
541C42—28no—
542C42—28no—
543C42—216no—
544C42—48no—
545C42—48no—
546C42C22×D4116yes—
547C42C22×D4116yes—
548C42C22×D418yes—
549C42C22×D418yes—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—216no—
558C23—28no—
559C23—216no—
560C23—28no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—816no—
565C23C23×D412yes—
566C23C2×H(5,𝔽2)14yes—
567C23C2611yescharacteristic
568Q8—232no—
569Q8—416no—
570Q8—432no—
571Q8—416no—
572Q8—416no—
573D4—416no—
574D4—432no—
575D4—432no—
576D4—416no—
577D4—416no—
578D4—816no—
579D4—88no—
580D4—1632no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—216no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—216no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—216no—
600C2×C4—24no—
601C2×C4—216no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—432no—
608C2×C4—464no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—432no—
614C2×C4#535116yes—
615C2×C4C23×D418yes—
616C2×C4C23×D412yes—
617C2×C4C23×D412yes—
618C2×C4C23×D414yes—
619C2×C4C23×D418yes—
620C2×C4C23×D414yes—
621C2×C4C2×H(5,𝔽2)18yes—
622C2×C4#58514yes—
623C22—22no—
624C22—22no—
625C22—28no—
626C22—44no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—416no—
633C22—416no—
634C22—416no—
635C22—816no—
636C22C2×#53514yes—
637C22C24×D411yescharacteristic
638C22C22×H(5,𝔽2)12yes—
639C22C2×#58512yes—
640C222−1+612yes—
641C4—24no—
642C4—28no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—28no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—48no—
651C4—44no—
652C4—416no—
653C4C2×#53518yes—
654C2—24no—
655C2—88no—
656C2—84no—
657C2C22×#53511yescharacteristic
658C2#6281111yescharacteristic
659C2#6284811yescharacteristic
660C2#6293114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3497C2semidirect
2. C22×#556C2semidirect
3. #42889C2semidirect
4. #42812C2semidirect
5. #39954C2semidirect
6. #37547C2semidirect
7. #36961C2semidirect
8. #36707C2semidirect
9. #35676C2semidirect
10. #35666C2semidirect
11. #35329C2semidirect
12. #35301C2semidirect
13. #35168C2semidirect
14. #35153C2semidirect
15. #34955C2semidirect
16. #34059C2semidirect
22. C2×#550C22semidirect
23. C2×D4×Q8C22semidirect
24. C2×#548C22semidirect
45. #2821C22semidirect
46. #2818C22semidirect
47. #2723C22semidirect
48. #2714C22semidirect
49. #2692C22semidirect
50. #2676C22semidirect
51. #2662C22semidirect
54. #2636C22semidirect
55. #2628C22semidirect
56. #2619C22semidirect
59. #2610C22semidirect
60. #2603C22semidirect
61. #2598C22semidirect
62. #2592C22semidirect
65. #2568C22semidirect
66. #2548C22semidirect
74. #2434C22semidirect
75. #2432C22semidirect
77. #2429C22semidirect
80. #2426C22semidirect
85. #2398C22semidirect
86. #2324C22semidirect
87. #2320C22semidirect
91. C4×#124C22semidirect
94. #2295C22semidirect
158. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34972
C22×#5562
#428892
#428122
#399542
#375472
#369612
#367072
#356762
#356662
#353292
#353012
#351682
#351532
#349552
#340592