Finite Group Databasebeta

← #8429486

Subgroups — #8429486

Order 512.

5027 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#59220C212yesmaximal, complemented
3#59127C214yesmaximal, complemented
4#55190C218yesmaximal, complemented
5#53032C214yesmaximal, complemented
6#53009C218yesmaximal, complemented
7#42812C214yesmaximal, complemented
8#42548C218yesmaximal, complemented
9#39608C214yesmaximal, complemented
10#39124C214yesmaximal, complemented
11#38943C214yesmaximal, complemented
12#37642C212yesmaximal, complemented
13#37617C214yesmaximal, complemented
14#36188C211yesmaximal, characteristic, complemented
15D4×C4⋊Q8C212yesmaximal, complemented
16#34943C211yesmaximal, characteristic, complemented
17#34932C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×D4×Q8C2214yes—
20C2×D4×Q8C2214yes—
21C22×C4⋊Q8C2214yes—
22C2×#529C2214yes—
23#2868C2211yescharacteristic
24#2857C2214yes—
25#2857C2212yes—
26#2844C2218yes—
27#2844C2218yes—
28#2835C22116yes—
29#2821C2218yes—
30#2821C22116yes—
31#2809C2214yes—
32#2782C2218yes—
33#2782C2218yes—
34#2769C22116yes—
35#2763C22116yes—
36#2744C22116yes—
37#2723C22116yes—
38#2723C22116yes—
39#2696C22116yes—
40#2692C22116yes—
41#2677C22116yes—
42#2658C2214yes—
43#2642C2218yes—
44#2636C2218yes—
45#2628C2218yes—
46#2628C2218yes—
47#2628C2218yes—
48#2628C2218yes—
49#2623C22116yes—
50#2598C2218yes—
51#2572C2214yes—
52#2571C2211yescharacteristic
53#2571C2218yes—
54#2571C2212yes—
55#2568C2212yes—
56#2568C2214yes—
57#2563C2218yes—
58#2563C2212yes—
59#2553C2218yes—
60#2552C2212yes—
61#2551C2218yes—
62#2551C2214yes—
63#2551C2214yes—
64#2547C2214yes—
65#2547C2212yes—
66#2546C2214yes—
67#2546C2214yes—
68#2542C2212yes—
69#2540C2214yes—
70#2537C2211yescharacteristic
71#2537C2212yes—
72#2535C2218yes—
73#2508C2218yes—
74#2508C2218yes—
75#2499C2214yes—
76#2497C2218yes—
77#2496C2214yes—
78#2496C2218yes—
79#2495C2214yes—
80#2495C2218yes—
81#2489C2218yes—
82#2489C2214yes—
83#2487C2218yes—
84#2487C2218yes—
85#2452C2218yes—
86#2449C22116yes—
87#2440C2218yes—
88#2440C2218yes—
89#2440C2218yes—
90#2436C2218yes—
91#2434C2218yes—
92#2434C2218yes—
93#2429C2218yes—
94#2427C2214yes—
95#2422C2214yes—
96#2420C2214yes—
97#2409C2218yes—
98#2398C2214yes—
99#2395C2218yes—
100#2369C2212yes—
101#2365C2214yes—
102#2361C2218yes—
103#2355C2214yes—
104D4×C4⋊C4C2214yes—
105D4×C4⋊C4C2214yes—
106#2324C2218yes—
107#2320C2214yes—
108C4×C4⋊Q8C2212yes—
109C4×C4⋊Q8C2211yescharacteristic
110C4×C22⋊Q8C2214yes—
111C4×#121C2214yes—
112C4×#121C2214yes—
113#2285C2212yes—
114#2285C2214yes—
115#2283C2214yes—
116C42×D4C2212yes—
117C42×D4C2211yescharacteristic
118C23×Q8C2318yes—
119#556—48no—
120D4×Q8—416no—
121D4×Q8—416no—
122C2×C4⋊Q8—416no—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2314yes—
126C2×C4⋊Q8C2318yes—
127C2×C4⋊Q8C2318yes—
128C2×C4⋊Q8C2318yes—
129C2×C4⋊Q8C2312yes—
130C2×C4⋊Q8C2314yes—
131C2×C4⋊Q8C2318yes—
132C2×C4⋊Q8C2318yes—
133#529—416no—
134C2×#126C2318yes—
135C2×#126C2318yes—
136C2×#125C2314yes—
137C2×#124C2314yes—
138C2×#124C2314yes—
139C2×#123C2318yes—
140C2×C22⋊Q8C2314yes—
141C2×C22⋊Q8C2318yes—
142C2×C22⋊Q8C2314yes—
143C2×C22⋊Q8C2314yes—
144C2×C22⋊Q8C23116yes—
145C2×C22⋊Q8C23116yes—
146C2×#121C2314yes—
147C2×#121C2314yes—
148C2×#121C2314yes—
149C2×C4×Q8C2314yes—
150C2×C4×Q8C2314yes—
151C2×C4×Q8C2312yes—
152C2×C4×D4C2312yes—
153C2×C4×D4C2312yes—
154C2×C4×D4C2312yes—
155C2×C4×D4C2311yescharacteristic
156C2×C4×D4C2312yes—
157C2×C4×D4C2318yes—
158C2×C4×D4C2318yes—
159C2×C4×D4C2318yes—
160C2×C4×D4C2314yes—
161C2×#117C2318yes—
162C2×#117C2314yes—
163C22×C4⋊C4C2318yes—
164C22×C4⋊C4C2318yes—
165C22×C42C2314yes—
166C22×C42C2312yes—
167#400C2314yes—
168#400C2318yes—
169#400C23116yes—
170#400C23116yes—
171#400C2318yes—
172#400C2318yes—
173#400C2314yes—
174#400C23116yes—
175#400C23116yes—
176#400C2318yes—
177#400C23116yes—
178#399C2318yes—
179#399C23116yes—
180#399C2318yes—
181#399C2318yes—
182#399C2318yes—
183#398C23116yes—
184#398C23116yes—
185#398C23116yes—
186#398C2318yes—
187#397C2318yes—
188#397C2318yes—
189#397C23116yes—
190#397C2318yes—
191#397C2318yes—
192#397C23116yes—
193#396C23116yes—
194#395C23116yes—
195#395C23116yes—
196#395C23116yes—
197#395C23116yes—
198#395C23116yes—
199#394C2314yes—
200#394C2318yes—
201#394C2314yes—
202#394C2318yes—
203#393C23116yes—
204#393C2318yes—
205#393C23116yes—
206#393C23116yes—
207#392C2318yes—
208#391C2314yes—
209#391C2312yes—
210#391C2314yes—
211#391C2318yes—
212#391C2318yes—
213#391C2318yes—
214#391C2318yes—
215#391C2314yes—
216#390C2314yes—
217#390C23116yes—
218#390C2314yes—
219#389C2314yes—
220#389C2314yes—
221#389C2314yes—
222#389C2318yes—
223#389C2318yes—
224#389C2318yes—
225#389C2314yes—
226#389C23116yes—
227#389C2314yes—
228#389C2318yes—
229#389C2318yes—
230#389C2318yes—
231#389C2318yes—
232#389C2314yes—
233#388C2318yes—
234#388C23116yes—
235#388C23116yes—
236#388C2318yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C2318yes—
241#386C2318yes—
242#386C23116yes—
243#386C2318yes—
244#386C23116yes—
245#386C23116yes—
246#385C23116yes—
247#385C23116yes—
248#385C23116yes—
249#385C23116yes—
250#385C23116yes—
251#385C2318yes—
252#384C2311yescharacteristic
253#384C2312yes—
254#384C2314yes—
255#384C2312yes—
256#384C2314yes—
257#384C2312yes—
258#384C2314yes—
259#383C2318yes—
260#383C2318yes—
261#382C23116yes—
262#382C2314yes—
263#382C2314yes—
264#382C2318yes—
265#382C2314yes—
266#382C2318yes—
267#382C2318yes—
268#382C2318yes—
269#381C2318yes—
270#380C2314yes—
271#380C2314yes—
272#380C2318yes—
273#380C2318yes—
274#380C2314yes—
275#380C2318yes—
276#380C2314yes—
277#380C2318yes—
278#380C2314yes—
279#380C2318yes—
280#380C2314yes—
281#380C2314yes—
282C4×C4⋊C4C2314yes—
283C4×C4⋊C4C2314yes—
284C4×C4⋊C4C2314yes—
285C4×C4⋊C4C2312yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2311yescharacteristic
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2318yes—
290C4×C4⋊C4C2318yes—
291C4×C4⋊C4C2312yes—
292C4×C4⋊C4C2312yes—
293C4×C4⋊C4C2312yes—
294C4×C22⋊C4C2314yes—
295C4×C22⋊C4C2318yes—
296C4×C22⋊C4C2314yes—
297C4×C22⋊C4C2314yes—
298C4×C22⋊C4C2314yes—
299C4×C22⋊C4C2318yes—
300C4×C22⋊C4C2312yes—
301#376C2314yes—
302C43C2311yescharacteristic
303C22×Q8—416no—
304C22×Q8—416no—
305C22×Q8—416no—
306C22×Q8C2414yes—
307C22×Q8C2414yes—
308C22×Q8C2412yes—
309C22×Q8C24116yes—
310C22×D4C2411yescharacteristic
311C22×D4C2414yes—
312C22×D4C2412yes—
313C22×D4C2418yes—
314C23×C4C2414yes—
315C23×C4C2414yes—
316C23×C4C2414yes—
317C23×C4C2414yes—
318C23×C4C2412yes—
319C23×C4C24116yes—
320C4⋊Q8—28no—
321C4⋊Q8—216no—
322C4⋊Q8—24no—
323C4⋊Q8—416no—
324C4⋊Q8—416no—
325C4⋊Q8—416no—
326C4⋊Q8—416no—
327C4⋊Q8—416no—
328C4⋊Q8—48no—
329C4⋊Q8—44no—
330#126—416no—
331#126—416no—
332#125—48no—
333#124—48no—
334#124—48no—
335#123—416no—
336C22⋊Q8—216no—
337C22⋊Q8—432no—
338C22⋊Q8—432no—
339C22⋊Q8—48no—
340C22⋊Q8—48no—
341C22⋊Q8—416no—
342#121—216no—
343#121—216no—
344#121—48no—
345C4×Q8—216no—
346C4×Q8—216no—
347C4×Q8—44no—
348C4×D4—28no—
349C4×D4—24no—
350C4×D4—28no—
351C4×D4—28no—
352C4×D4—28no—
353C4×D4—416no—
354C4×D4—416no—
355C4×D4—416no—
356C4×D4—48no—
357#117—416no—
358#117—48no—
359C2×C4⋊C4—416no—
360C2×C4⋊C4—416no—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C2418yes—
372C2×C4⋊C4C2414yes—
373C2×C4⋊C4C2414yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C24116yes—
377C2×C4⋊C4C24116yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2414yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C2418yes—
383C2×C4⋊C4C2418yes—
384C2×C4⋊C4C2414yes—
385C2×C4⋊C4C2414yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C24116yes—
388C2×C4⋊C4C2414yes—
389C2×C4⋊C4C2414yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2411yescharacteristic
392C2×C22⋊C4C2418yes—
393C2×C22⋊C4C2414yes—
394C2×C22⋊C4C2414yes—
395C2×C22⋊C4C2412yes—
396C2×C22⋊C4C2414yes—
397C2×C22⋊C4C2414yes—
398C2×C22⋊C4C2418yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2418yes—
401C2×C22⋊C4C2418yes—
402C2×C22⋊C4C2418yes—
403C2×C22⋊C4C24116yes—
404C2×C22⋊C4C2418yes—
405C2×C22⋊C4C2418yes—
406C2×C22⋊C4C2418yes—
407C2×C22⋊C4C2418yes—
408C2×C22⋊C4C2418yes—
409C2×C42—216no—
410C2×C42—28no—
411C2×C42C2414yes—
412C2×C42C2418yes—
413C2×C42C2418yes—
414C2×C42C2414yes—
415C2×C42C2411yescharacteristic
416C2×C42C2412yes—
417C2×C42C2414yes—
418C2×C42C2414yes—
419C2×C42C2414yes—
420C2×C42C2412yes—
421C2×C42C2411yescharacteristic
422C2×C42C2418yes—
423C2×C42C2412yes—
424C2×C42C2411yescharacteristic
425#95C2414yes—
426#95C2418yes—
427#95C2414yes—
428#95C2414yes—
429#95C2414yes—
430#95C2414yes—
431#95C2414yes—
432#95C2412yes—
433#95C2418yes—
434#95C2418yes—
435#95C2418yes—
436#95C2418yes—
437#95C2418yes—
438#95C2418yes—
439#95C2418yes—
440#95C24116yes—
441#95C2418yes—
442#95C2418yes—
443#95C2418yes—
444#95C24116yes—
445#95C24116yes—
446#95C24116yes—
447#95C24116yes—
448#95C24116yes—
449C24C2512yes—
450C24C2514yes—
451C2×Q8—28no—
452C2×Q8—28no—
453C2×Q8—28no—
454C2×Q8—216no—
455C2×Q8—28no—
456C2×Q8—24no—
457C2×Q8—42no—
458C2×Q8—416no—
459C2×Q8—416no—
460C2×Q8—416no—
461C2×Q8—42no—
462C2×Q8—864no—
463C2×D4—216no—
464C2×D4—216no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—24no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—48no—
472C2×D42−1+418yes—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—416no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4C2514yes—
492C22×C4C2514yes—
493C22×C4C2514yes—
494C22×C4C2511yescharacteristic
495C22×C4C2514yes—
496C22×C4C2518yes—
497C22×C4C2514yes—
498C22×C4C2514yes—
499C22×C4C2518yes—
500C22×C4C2518yes—
501C22×C4C2512yes—
502C22×C4C2512yes—
503C22×C4C2511yescharacteristic
504C22×C4C2511yescharacteristic
505C22×C4C2512yes—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—416no—
522C4⋊C4—416no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—416no—
535C4⋊C4—44no—
536C4⋊C4H(5,𝔽2)14yes—
537C22⋊C4—28no—
538C22⋊C4—216no—
539C22⋊C4—216no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—216no—
544C22⋊C4—28no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—416no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C42—28no—
555C42—28no—
556C42—28no—
557C42—216no—
558C42—216no—
559C42—22no—
560C42—28no—
561C42—28no—
562C42—24no—
563C42—24no—
564C42—41no—
565C42—48no—
566C42H(5,𝔽2)18yes—
567C422−1+414yes—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—22no—
575C23—24no—
576C23—22no—
577C23—24no—
578C23—22no—
579C23C2×2−1+418yes—
580C23C2611yescharacteristic
581Q8—416no—
582Q8—416no—
583Q8—84no—
584Q8—832no—
585D4—216no—
586D4—416no—
587D4—44no—
588D4—816no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—216no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—48no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—832no—
628C2×C4C2×H(5,𝔽2)14yes—
629C2×C4C2×H(5,𝔽2)14yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×H(5,𝔽2)14yes—
632C2×C4C2×H(5,𝔽2)14yes—
633C2×C4C2×2−1+412yes—
634C2×C4C2×2−1+414yes—
635C2×C4C2×2−1+414yes—
636C22—28no—
637C22—28no—
638C22—28no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22C22×H(5,𝔽2)12yes—
649C22C22×H(5,𝔽2)11yescharacteristic
650C22C22×2−1+411yescharacteristic
651C22C22×2−1+412yes—
652C222−1+611yescharacteristic
653C4—22no—
654C4—24no—
655C4—24no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—48no—
660C4—48no—
661C4—41no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—48no—
666C4#354914yes—
667C4#355218yes—
668C2—48no—
669C2—82no—
670C2C2×#355212yes—
671C2C2×#354911yescharacteristic
672C2#6291511yescharacteristic
673C2#6294212yes—
674C2#6301611yescharacteristic
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. #59869C2semidirect
2. #59220C2semidirect
3. #59127C2semidirect
4. #55190C2semidirect
5. #53032C2semidirect
6. #53009C2semidirect
7. #42812C2semidirect
8. #42548C2semidirect
9. #39608C2semidirect
10. #39124C2semidirect
11. #38943C2semidirect
12. #37642C2semidirect
13. #37617C2semidirect
14. #36188C2semidirect
15. D4×C4⋊Q8C2semidirect
16. #34943C2semidirect
17. #34932C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#592202
#591272
#551902
#530322
#530092
#428122
#425482
#396082
#391242
#389432
#376422
#376172
#361882
D4×C4⋊Q82
#349432
#349322