Finite Group Databasebeta

← #7674484

Subgroups — #7674484

Order 512.

4645 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45436C211yesmaximal, characteristic, complemented
2#45432C212yesmaximal, complemented
3#45427C211yesmaximal, characteristic, complemented
4#45228C214yesmaximal, complemented
5#45217C214yesmaximal, complemented
6#45170C214yesmaximal, complemented
7#40120C218yesmaximal, complemented
8#40115C218yesmaximal, complemented
9#38825C218yesmaximal, complemented
10#38818C218yesmaximal, complemented
11#35635C214yesmaximal, complemented
12#35209C214yesmaximal, complemented
13#34643C211yesmaximal, characteristic
14#34318C214yesmaximal, complemented
15C4×#541C212yesmaximal, complemented
16C2×#552C2214yes—
17C2×#541C2212yes—
18C2×#541C2214yes—
19C2×#529C2212yes—
20#2860C2218yes—
21#2859C2218yes—
22#2795C2218yes—
23#2792C2218yes—
24#2783C2218yes—
25#2783C2218yes—
26#2781C2218yes—
27#2775C2218yes—
28#2769C2218yes—
29#2769C2218yes—
30#2769C2218yes—
31#2756C2218yes—
32#2727C2218yes—
33#2721C2218yes—
34#2721C2218yes—
35#2721C2218yes—
36#2705C2218yes—
37#2694C2218yes—
38#2664C2214yes—
39#2664C2218yes—
40#2664C2218yes—
41#2663C2214yes—
42#2662C2214yes—
43#2661C2214yes—
44#2612C2218yes—
45#2584C2214yes—
46#2574C2218yes—
47#2568C2214yes—
48#2562C2214yes—
49#2559C2218yes—
50#2558C2218yes—
51#2556C2218yes—
52#2554C2218yes—
53#2553C2218yes—
54#2552C2214yes—
55#2552C2218yes—
56#2546C2214yes—
57#2542C2214yes—
58#2542C2214yes—
59#2538C2214yes—
60#2531C2218yes—
61#2529C2218yes—
62#2521C2218yes—
63#2519C2218yes—
64#2519C2218yes—
65#2519C2218yes—
66#2518C2218yes—
67#2518C2218yes—
68#2512C2218yes—
69#2508C2218yes—
70#2507C2218yes—
71#2506C2218yes—
72#2506C2218yes—
73#2497C2218yes—
74#2496C2218yes—
75#2495C2218yes—
76#2489C2214yes—
77#2489C2214yes—
78#2479C2218yes—
79#2471C2218yes—
80#2441C2218yes—
81#2441C2218yes—
82#2429C2214yes—
83#2429C2214yes—
84#2417C2214yes—
85#2417C2214yes—
86#2415C2214yes—
87#2415C2214yes—
88#2399C2211yescharacteristic
89#2398C2212yes—
90#2397C2211yescharacteristic
91#2392C2214yes—
92#2374C2218yes—
93#2373C2218yes—
94#2370C2214yes—
95#2370C2218yes—
96#2362C2214yes—
97#2360C2214yes—
98#2356C2214yes—
99#2352C2214yes—
100#2347C2214yes—
101#2347C2218yes—
102#2345C2214yes—
103#2336C2218yes—
104#2336C2218yes—
105#2335C2218yes—
106#2334C2218yes—
107#2330C2212yes—
108#2327C2212yes—
109#2323C2214yes—
110#2323C2214yes—
111#2323C2214yes—
112#2320C2214yes—
113#2314C2212yes—
114C4×C4⋊Q8C2212yes—
115C4×#125C2212yes—
116C4×#125C2218yes—
117C4×#126C2218yes—
118C4×#126C2218yes—
119C4×C22⋊Q8C2214yes—
120#2298C2218yes—
121#2293C2214yes—
122#2274C2214yes—
123C42×Q8C2214yes—
124C4×#117C2211yescharacteristic
125#552—232no—
126#541—216no—
127#541—416no—
128C2×C4⋊Q8C2312yes—
129C2×C4⋊Q8C2314yes—
130#529—48no—
131C2×#126C2318yes—
132C2×#126C2318yes—
133C2×#126C2318yes—
134C2×#126C2318yes—
135C2×#126C2314yes—
136C2×#125C2318yes—
137C2×#125C2318yes—
138C2×#125C2314yes—
139C2×#125C2318yes—
140C2×#125C2318yes—
141C2×#125C2312yes—
142C2×#125C2312yes—
143C2×#124C2312yes—
144C2×#123C2318yes—
145C2×#123C2314yes—
146C2×C22⋊Q8C2314yes—
147C2×C22⋊Q8C2314yes—
148C2×C22⋊Q8C2314yes—
149C2×C22⋊Q8C2312yes—
150C2×C22⋊Q8C2314yes—
151C2×C22⋊Q8C2314yes—
152C2×#121C2312yes—
153C2×C4×Q8C2314yes—
154C2×C4×Q8C2314yes—
155C2×C4×Q8C2314yes—
156C2×C4×Q8C2312yes—
157C2×C4×D4C2314yes—
158C2×C4×D4C2314yes—
159C2×C4×D4C2312yes—
160C2×#117C2312yes—
161C2×#117C2314yes—
162C2×#117C2314yes—
163C2×#117C2318yes—
164C22×C4⋊C4C2314yes—
165C22×C42C2311yescharacteristic
166#400C2318yes—
167#400C2318yes—
168#400C2318yes—
169#400C2314yes—
170#400C2318yes—
171#400C2318yes—
172#399C2314yes—
173#399C2318yes—
174#399C2318yes—
175#399C2318yes—
176#399C2318yes—
177#398C2318yes—
178#398C2318yes—
179#398C2318yes—
180#398C2314yes—
181#398C2314yes—
182#398C2318yes—
183#398C2318yes—
184#398C2318yes—
185#398C2318yes—
186#398C2318yes—
187#398C2318yes—
188#397C2314yes—
189#397C2314yes—
190#397C2318yes—
191#397C2318yes—
192#397C2318yes—
193#397C2318yes—
194#396C2318yes—
195#396C2314yes—
196#396C2314yes—
197#396C2318yes—
198#396C2318yes—
199#396C2318yes—
200#396C2318yes—
201#396C2318yes—
202#395C2318yes—
203#395C2314yes—
204#394C2318yes—
205#394C2314yes—
206#394C2314yes—
207#394C2314yes—
208#394C2314yes—
209#394C2318yes—
210#394C2318yes—
211#393C2314yes—
212#392C2314yes—
213#392C2314yes—
214#391C2314yes—
215#391C2314yes—
216#391C2318yes—
217#391C2314yes—
218#391C2314yes—
219#390C2314yes—
220#390C2318yes—
221#390C2318yes—
222#390C2314yes—
223#390C2318yes—
224#389C2314yes—
225#389C2318yes—
226#389C2318yes—
227#389C2318yes—
228#389C2314yes—
229#389C2314yes—
230#389C2314yes—
231#389C2318yes—
232#389C2318yes—
233#389C2318yes—
234#389C2318yes—
235#389C2318yes—
236#389C2314yes—
237#389C2318yes—
238#389C2318yes—
239#389C2318yes—
240#389C2318yes—
241#389C2318yes—
242#389C2318yes—
243#389C2318yes—
244#389C2314yes—
245#389C2318yes—
246#389C2318yes—
247#388C2314yes—
248#388C2318yes—
249#388C2318yes—
250#388C2318yes—
251#388C2318yes—
252#388C2318yes—
253#388C2318yes—
254#388C2318yes—
255#388C2318yes—
256#388C2318yes—
257#388C2318yes—
258#388C2318yes—
259#388C2318yes—
260#388C2318yes—
261#388C2318yes—
262#388C2314yes—
263#388C2318yes—
264#388C2318yes—
265#388C2318yes—
266#388C2318yes—
267#388C2318yes—
268#388C2314yes—
269#388C2314yes—
270#388C2318yes—
271#387C2318yes—
272#387C2318yes—
273#387C2318yes—
274#387C2318yes—
275#387C2318yes—
276#387C2318yes—
277#387C2318yes—
278#387C2318yes—
279#387C2318yes—
280#387C2318yes—
281#387C2318yes—
282#387C2318yes—
283#387C2318yes—
284#387C2318yes—
285#387C2318yes—
286#387C2318yes—
287#387C2318yes—
288#387C2318yes—
289#387C2318yes—
290#387C2318yes—
291#386C2318yes—
292#386C2318yes—
293#386C2318yes—
294#385C2318yes—
295#385C2318yes—
296#384C2312yes—
297#384C2318yes—
298#384C2318yes—
299#384C2318yes—
300#383C2318yes—
301#383C2318yes—
302#383C2318yes—
303#382C2312yes—
304#382C2318yes—
305#382C2318yes—
306#382C2318yes—
307#382C2318yes—
308#380C2314yes—
309#380C2314yes—
310#380C2314yes—
311#380C2318yes—
312#380C2318yes—
313#380C2318yes—
314C4×C4⋊C4C2318yes—
315C4×C4⋊C4C2318yes—
316C4×C4⋊C4C2318yes—
317C4×C4⋊C4C2318yes—
318C4×C4⋊C4C2318yes—
319C4×C4⋊C4C2318yes—
320C4×C4⋊C4C2314yes—
321C4×C4⋊C4C2318yes—
322C4×C4⋊C4C2314yes—
323C4×C4⋊C4C2314yes—
324C4×C4⋊C4C2314yes—
325C4×C4⋊C4C2318yes—
326C4×C4⋊C4C2314yes—
327C4×C4⋊C4C2314yes—
328C4×C4⋊C4C2318yes—
329C4×C4⋊C4C2318yes—
330C4×C4⋊C4C2318yes—
331C4×C4⋊C4C2314yes—
332C4×C22⋊C4C2314yes—
333C4×C22⋊C4C2318yes—
334C4×C22⋊C4C2318yes—
335C4×C22⋊C4C2318yes—
336C4×C22⋊C4C2318yes—
337C4×C22⋊C4C2318yes—
338C4×C22⋊C4C2318yes—
339C4×C22⋊C4C2314yes—
340C4×C22⋊C4C2314yes—
341#376C2311yescharacteristic
342#376C2311yescharacteristic
343#376C2318yes—
344C43C2318yes—
345C43C2312yes—
346C22×Q8C2414yes—
347C22×Q8C2412yes—
348C22×D4C2414yes—
349C22×D4C2414yes—
350C22×D4C2412yes—
351C23×C4C2414yes—
352C23×C4C2411yescharacteristic
353C23×C4C2418yes—
354C23×C4C2412yes—
355C4⋊Q8—28no—
356C4⋊Q8—48no—
357#126—232no—
358#126—232no—
359#126—48no—
360#126—416no—
361#126—416no—
362#125—232no—
363#125—232no—
364#125—44no—
365#125—48no—
366#125—416no—
367#125—416no—
368#125#41116yes—
369#124—44no—
370#123—232no—
371#123—48no—
372C22⋊Q8—216no—
373C22⋊Q8—216no—
374C22⋊Q8—216no—
375C22⋊Q8—44no—
376C22⋊Q8—48no—
377C22⋊Q8—48no—
378#121—44no—
379C4×Q8—216no—
380C4×Q8—216no—
381C4×Q8—216no—
382C4×Q8—44no—
383C4×D4—216no—
384C4×D4—44no—
385C4×D4—48no—
386#117—216no—
387#117—232no—
388#117—28no—
389#117—216no—
390C2×C4⋊C4—216no—
391C2×C4⋊C4C2414yes—
392C2×C4⋊C4C2418yes—
393C2×C4⋊C4C2414yes—
394C2×C4⋊C4C2418yes—
395C2×C4⋊C4C2414yes—
396C2×C4⋊C4C2414yes—
397C2×C4⋊C4C2414yes—
398C2×C4⋊C4C2414yes—
399C2×C4⋊C4C2418yes—
400C2×C4⋊C4C2418yes—
401C2×C4⋊C4C2418yes—
402C2×C4⋊C4C2418yes—
403C2×C4⋊C4C2418yes—
404C2×C4⋊C4C2418yes—
405C2×C4⋊C4C2412yes—
406C2×C4⋊C4C2414yes—
407C2×C4⋊C4C2412yes—
408C2×C4⋊C4C2414yes—
409C2×C4⋊C4C2418yes—
410C2×C4⋊C4C2418yes—
411C2×C4⋊C4C2418yes—
412C2×C4⋊C4C2418yes—
413C2×C4⋊C4C2418yes—
414C2×C4⋊C4C2418yes—
415C2×C4⋊C4C2418yes—
416C2×C4⋊C4C2418yes—
417C2×C4⋊C4C2418yes—
418C2×C4⋊C4C2418yes—
419C2×C4⋊C4C2414yes—
420C2×C4⋊C4C2418yes—
421C2×C4⋊C4C2414yes—
422C2×C4⋊C4C2414yes—
423C2×C4⋊C4C2418yes—
424C2×C22⋊C4C2414yes—
425C2×C22⋊C4C2414yes—
426C2×C22⋊C4C2418yes—
427C2×C22⋊C4C2414yes—
428C2×C22⋊C4C2418yes—
429C2×C22⋊C4C2414yes—
430C2×C22⋊C4C2414yes—
431C2×C22⋊C4C2418yes—
432C2×C22⋊C4C2418yes—
433C2×C22⋊C4C2414yes—
434C2×C22⋊C4C2414yes—
435C2×C22⋊C4C2414yes—
436C2×C22⋊C4C2418yes—
437C2×C22⋊C4C2418yes—
438C2×C22⋊C4C2414yes—
439C2×C22⋊C4C2414yes—
440C2×C22⋊C4C2418yes—
441C2×C22⋊C4C2414yes—
442C2×C22⋊C4C2418yes—
443C2×C22⋊C4C2418yes—
444C2×C22⋊C4C2414yes—
445C2×C42—24no—
446C2×C42C2418yes—
447C2×C42C2414yes—
448C2×C42C2412yes—
449C2×C42C2418yes—
450C2×C42C2411yescharacteristic
451C2×C42C2418yes—
452C2×C42C2418yes—
453C2×C42C2412yes—
454C2×C42C2414yes—
455C2×C42C2418yes—
456C2×C42C2414yes—
457C2×C42C2418yes—
458C2×C42C2418yes—
459C2×C42C2418yes—
460C2×C42C2418yes—
461C2×C42C2412yes—
462C2×C42C2411yescharacteristic
463#95C2414yes—
464#95C2414yes—
465#95C2418yes—
466#95C2414yes—
467#95C2418yes—
468#95C2418yes—
469#95C2418yes—
470#95C2418yes—
471#95C2418yes—
472#95C2418yes—
473#95C2414yes—
474#95C2414yes—
475#95C2418yes—
476#95C2418yes—
477#95C2414yes—
478#95C2414yes—
479#95C2418yes—
480#95C2418yes—
481#95C2418yes—
482#95C2418yes—
483#95C2414yes—
484#95C2418yes—
485#95C2418yes—
486#95C2418yes—
487#95C2418yes—
488#95C2418yes—
489#95C2418yes—
490#95C2418yes—
491#95C2418yes—
492C24C2511yescharacteristic
493C24C2514yes—
494C2×Q8—28no—
495C2×Q8—28no—
496C2×Q8—28no—
497C2×Q8—42no—
498C2×Q8—44no—
499C2×D4—28no—
500C2×D4—44no—
501C2×D4—42no—
502C2×D4—44no—
503C2×D4—44no—
504C2×D4—44no—
505C2×D4—44no—
506C2×D4—44no—
507C22×C4—28no—
508C22×C4—28no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—28no—
512C22×C4—216no—
513C22×C4—24no—
514C22×C4—44no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4C2518yes—
518C22×C4C2512yes—
519C22×C4C2514yes—
520C22×C4C2514yes—
521C22×C4C2514yes—
522C22×C4C2518yes—
523C22×C4C2518yes—
524C22×C4C2514yes—
525C22×C4C2511yescharacteristic
526C22×C4C2514yes—
527C22×C4C2518yes—
528C22×C4C2512yes—
529C22×C4C2511yescharacteristic
530C4⋊C4—28no—
531C4⋊C4—216no—
532C4⋊C4—216no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—216no—
536C4⋊C4—216no—
537C4⋊C4—216no—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—216no—
544C4⋊C4—44no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—44no—
550C4⋊C4—48no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—48no—
556C4⋊C4—48no—
557C4⋊C4—48no—
558C4⋊C4—48no—
559C4⋊C4—48no—
560C4⋊C4—48no—
561C4⋊C4C2×#41116yes—
562C4⋊C4C2×#41132yes—
563C22⋊C4—28no—
564C22⋊C4—216no—
565C22⋊C4—216no—
566C22⋊C4—28no—
567C22⋊C4—28no—
568C22⋊C4—216no—
569C22⋊C4—216no—
570C22⋊C4—28no—
571C22⋊C4—28no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—48no—
575C22⋊C4—48no—
576C22⋊C4—48no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—48no—
583C22⋊C4—48no—
584C42—216no—
585C42—24no—
586C42—216no—
587C42—216no—
588C42—22no—
589C42—24no—
590C42—28no—
591C42—216no—
592C42—44no—
593C42—48no—
594C42—44no—
595C42—48no—
596C42—48no—
597C42—48no—
598C42—48no—
599C42C2×#4118yes—
600C42H(5,𝔽2)14yes—
601C23—22no—
602C23—24no—
603C23—24no—
604C23—24no—
605C23—21no—
606C23—24no—
607C23—24no—
608C23—24no—
609C23—24no—
610C23—24no—
611C23C2611yescharacteristic
612Q8—416no—
613Q8—84no—
614D4—88no—
615D4—84no—
616D4—88no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—416no—
648C2×C4—44no—
649C2×C4—816no—
650C2×C4—84no—
651C2×C4C22×#4118yes—
652C2×C4C22×#41116yes—
653C2×C4C22×#4114yes—
654C2×C4C2×H(5,𝔽2)14yes—
655C2×C4C2×H(5,𝔽2)12yes—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—41no—
665C22—44no—
666C22C23×#4112yes—
667C22C22×H(5,𝔽2)11yescharacteristic
668C222−1+614yes—
669C4—216no—
670C4—28no—
671C4—24no—
672C4—44no—
673C4—48no—
674C4—48no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—41no—
680C4—48no—
681C4#348314yes—
682C2—84no—
683C2—81no—
684C2C2×#348311yescharacteristic
685C2#6286114yes—
686C2#6288812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radical#34643C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #45436C2semidirect
2. #45432C2semidirect
3. #45427C2semidirect
4. #45228C2semidirect
5. #45217C2semidirect
6. #45170C2semidirect
7. #40120C2semidirect
8. #40115C2semidirect
9. #38825C2semidirect
10. #38818C2semidirect
11. #35635C2semidirect
12. #35209C2semidirect
14. #34318C2semidirect
15. C4×#541C2semidirect

Maximal subgroups

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

subgroupindex
#454362
#454322
#454272
#452282
#452172
#451702
#401202
#401152
#388252
#388182
#356352
#352092
#346432
#343182
C4×#5412