Finite Group Databasebeta

← #7630930

Subgroups — #7630930

Order 512.

4794 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36023C218yesmaximal, complemented
2#36013C218yesmaximal
3#35818C218yesmaximal, complemented
4#35789C218yesmaximal, complemented
5#35339C211yesmaximal, characteristic, complemented
6#35324C214yesmaximal, complemented
7#35321C214yesmaximal, complemented
8#35320C211yesmaximal, characteristic, complemented
9#35314C214yesmaximal, complemented
10#35313C212yesmaximal, complemented
11#35311C212yesmaximal
12#34634C211yesmaximal, characteristic
13#34629C214yesmaximal, complemented
14C2×#2621C211yesmaximal, characteristic
15#34254C212yesmaximal, complemented
16#34215C211yesmaximal, characteristic, complemented
17#34209C211yesmaximal, characteristic, complemented
18#34203C212yesmaximal
19#33996C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C2×#519C2212yes—
24C22×#117C2211yescharacteristic
25C22×#117C2211yescharacteristic
26C22×#117C2211yescharacteristic
27#2864C2214yescomplemented
28#2863C2218yescomplemented
29#2862C2214yescomplemented
30#2855C2218yes—
31#2655C22116yescomplemented
32#2652C22116yes—
33#2647C2214yescomplemented
34#2645C2218yescomplemented
35#2642C2214yescomplemented
36#2629C2212yes—
37#2629C2212yes—
38#2628C2212yes—
39#2628C2212yes—
40#2627C2214yes—
41#2626C2214yes—
42#2625C2218yes—
43#2621C2212yes—
44#2621C2214yes—
45#2621C2214yes—
46#2621C2212yes—
47#2621C2212yes—
48#2621C2212yes—
49#2588C22116yescomplemented
50#2585C22116yescomplemented
51#2575C22116yes—
52#2573C22116yes—
53#2570C2218yescomplemented
54#2569C2218yes—
55#2568C2218yescomplemented
56#2528C22116yescomplemented
57#2527C22116yes—
58#2524C22116yescomplemented
59#2524C22116yescomplemented
60#2523C22116yes—
61#2520C22116yescomplemented
62#2517C22116yes—
63#2515C22116yes—
64#2514C2218yescomplemented
65#2511C2218yescomplemented
66#2503C2218yes—
67#2503C2214yes—
68#2502C2214yes—
69#2499C2214yes—
70#2499C2214yes—
71#2493C22116yes—
72#2484C2218yes—
73#2483C2218yes—
74#2402C22116yescomplemented
75#2401C22116yes—
76#2396C2214yes—
77#2396C2214yes—
78C2×#398C2214yes—
79C2×#397C2214yes—
80#2368C2218yes—
81#2365C2218yescomplemented
82#2362C2218yes—
83#2360C2218yes—
84#2356C2218yescomplemented
85#2355C2218yescomplemented
86#2355C2218yescomplemented
87#2344C2218yes—
88#2344C2218yes—
89#2343C22116yes—
90#2342C2218yes—
91#2331C2214yescomplemented
92#2327C2214yescomplemented
93#2323C2218yes—
94#2320C2218yescomplemented
95#2317C2212yes—
96#2317C2212yes—
97#2316C2212yes—
98#2310C2212yes—
99#2310C2212yes—
100#2310C2212yes—
101C4×C4⋊Q8C2214yescomplemented
102C4×#125C2214yescomplemented
103C4×C22⋊Q8C2218yes—
104#2296C2212yes—
105#2295C2218yescomplemented
106C2×#391C2212yes—
107#2285C2211yescharacteristic
108C2×#382C2211yescharacteristic
109#2280C2212yes—
110#2274C2212yes—
111#2274C2214yes—
112#2274C2214yes—
113#2274C2214yes—
114#2273C2214yes—
115C4×#117C2211yescharacteristic
116C4×#117C2212yes—
117C4×#117C2211yescharacteristic
118C4×#117C2211yescharacteristic
119C4×#117C2211yescharacteristic
120C4×#117C2212yes—
121C4×#117C2212yes—
122C4×#117C2212yes—
123C23×Q8C2312yes—
124C24×C4C2311yescharacteristic
125#533—28no—
126C2×C4⋊Q8—28no—
127C2×C4⋊Q8C2312yes—
128C2×C4⋊Q8C2314yes—
129C2×C4⋊Q8C2312yes—
130C2×C4⋊Q8C2312yes—
131C2×#125C2312yes—
132C2×C22⋊Q8—216no—
133C2×C22⋊Q8C2314yes—
134C2×C22⋊Q8C2314yes—
135C2×C22⋊Q8C2314yes—
136C2×C22⋊Q8C2314yes—
137C2×C22⋊Q8C2314yes—
138#519—216no—
139C2×C4×Q8C2318yes—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2318yes—
142C2×#117—28no—
143C2×#117—28no—
144C2×#117—28no—
145C2×#117C2311yescharacteristic
146C2×#117C2311yescharacteristic
147C2×#117C2311yescharacteristic
148C2×#117C2312yes—
149C2×#117C2312yes—
150C2×#117C2312yes—
151C2×#117C2318yes—
152C2×#117C2312yes—
153C2×#117C2312yes—
154C2×#117C2311yescharacteristic
155C2×#117C2311yescharacteristic
156C2×#117C2311yescharacteristic
157C2×#117C2311yescharacteristic
158C2×#117C2312yes—
159C2×#117C2312yes—
160C2×#117C2312yes—
161C2×#117C2312yes—
162C2×#117C2312yes—
163C2×#117C2311yescharacteristic
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2314yes—
166C22×C4⋊C4C2314yes—
167C22×C4⋊C4C2312yes—
168C22×C4⋊C4C2312yes—
169C22×C4⋊C4C2312yes—
170C22×C22⋊C4C2312yes—
171C22×C22⋊C4C2314yes—
172C22×C22⋊C4C2312yes—
173C22×C22⋊C4C2312yes—
174C22×C42C2312yes—
175C22×C42C2311yescharacteristic
176C22×C42C2312yes—
177C22×C42C2311yescharacteristic
178#401C23116yes—
179#400C23116yes—
180#400C23116yes—
181#400C23116yes—
182#400C23116yes—
183#400C23116yes—
184#400C23116yes—
185#400C2314yes—
186#400C2314yes—
187#400C23116yes—
188#399C2314yes—
189#399C2314yes—
190#398C23116yes—
191#398C2318yes—
192#398C2318yes—
193#398C2318yes—
194#398C2318yes—
195#398C2318yes—
196#398C23116yes—
197#398C2318yes—
198#397C2318yes—
199#397C2318yes—
200#397C2318yes—
201#397C2318yes—
202#397C23116yes—
203#397C23116yes—
204#396C2318yes—
205#396C2318yes—
206#395C2314yes—
207#395C2314yes—
208#393C2314yes—
209#393C2314yes—
210#391C2318yes—
211#391C2314yes—
212#391C2314yes—
213#391C2318yes—
214#391C2318yes—
215#391C2318yes—
216#391C2314yes—
217#391C2314yes—
218#391C23116yes—
219#391C2314yes—
220#391C23116yes—
221#391C2318yes—
222#391C2314yes—
223#389C2318yes—
224#389C23116yes—
225#389C2318yes—
226#389C2318yes—
227#389C2318yes—
228#389C2314yes—
229#389C23116yes—
230#389C2314yes—
231#388C23116yes—
232#388C23116yes—
233#387C23116yes—
234#387C23116yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C23116yes—
241#387C23116yes—
242#387C23116yes—
243#387C23116yes—
244#387C23116yes—
245#387C23116yes—
246#387C23116yes—
247#387C23116yes—
248#387C23116yes—
249#387C23116yes—
250#385C2318yes—
251#385C2318yes—
252#385C23116yes—
253#385C23116yes—
254#385C2318yes—
255#384C2312yes—
256#384C2314yes—
257#384C2314yes—
258#383C23116yes—
259#383C23116yes—
260#382C2312yes—
261#382C23116yes—
262#382C2314yes—
263#382C2314yes—
264#382C2314yes—
265#382C23116yes—
266#382C2312yes—
267#382C2312yes—
268#381C2318yes—
269#381C2318yes—
270#381C2318yes—
271#381C23116yes—
272#381C2314yes—
273#380C2318yes—
274#380C2314yes—
275#380C2314yes—
276#380C2314yes—
277#379C2314yes—
278C4×C4⋊C4C23116yes—
279C4×C4⋊C4C2318yes—
280C4×C4⋊C4C23116yes—
281C4×C4⋊C4C2314yes—
282C4×C4⋊C4C2314yes—
283C4×C4⋊C4C23116yes—
284C4×C4⋊C4C2318yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C23116yes—
287C4×C4⋊C4C2318yes—
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2314yes—
290C4×C4⋊C4C2318yes—
291C4×C4⋊C4C2318yes—
292C4×C4⋊C4C2314yes—
293C4×C4⋊C4C2318yes—
294C4×C22⋊C4C2314yes—
295C4×C22⋊C4C2318yes—
296C4×C22⋊C4C2318yes—
297C4×C22⋊C4C2318yes—
298C4×C22⋊C4C23116yes—
299C4×C22⋊C4C2314yes—
300C4×C22⋊C4C2314yes—
301C4×C22⋊C4C2314yes—
302C4×C22⋊C4C2314yes—
303C4×C22⋊C4C2318yes—
304C4×C22⋊C4C2314yes—
305C4×C22⋊C4C2318yes—
306#376C2312yes—
307#376C2312yes—
308#376C23116yes—
309#376C2312yes—
310#376C2312yes—
311C2×#95C2314yes—
312C2×#95C2318yes—
313C43C2312yes—
314C43C2312yes—
315C43C2312yes—
316C43C2312yes—
317C25C2411yescharacteristic
318C22×Q8—28no—
319C22×Q8—28no—
320C22×Q8—28no—
321C22×Q8C2414yes—
322C22×Q8C2414yes—
323C22×Q8C2414yes—
324C23×C4—24no—
325C23×C4—24no—
326C23×C4#4118yes—
327C23×C4C2414yes—
328C23×C4C2411yescharacteristic
329C23×C4C2411yescharacteristic
330C23×C4C2411yescharacteristic
331C23×C4C2411yescharacteristic
332C23×C4C2411yescharacteristic
333C23×C4C2414yes—
334C23×C4C2412yes—
335C23×C4C2414yes—
336C23×C4C2414yes—
337C23×C4C2412yes—
338C23×C4C2411yescharacteristic
339C4⋊Q8—216no—
340C4⋊Q8—28no—
341C4⋊Q8—28no—
342C4⋊Q8—28no—
343#125—28no—
344C22⋊Q8—216no—
345C22⋊Q8—216no—
346C22⋊Q8—216no—
347C22⋊Q8—216no—
348C22⋊Q8—216no—
349C4×Q8—232no—
350C4×Q8—232no—
351C4×Q8—216no—
352#117—28no—
353#117—28no—
354#117—28no—
355#117—24no—
356#117—24no—
357#117—24no—
358#117—28no—
359#117—28no—
360#117—28no—
361#117—28no—
362#117—232no—
363#117—28no—
364#117—28no—
365#117—28no—
366#117—24no—
367#117—24no—
368#117—24no—
369#117—24no—
370#117#4118yes—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—216no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—216no—
376C2×C4⋊C4—28no—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2414yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2414yes—
386C2×C4⋊C4C2414yes—
387C2×C4⋊C4C2414yes—
388C2×C4⋊C4C2418yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2418yes—
392C2×C4⋊C4C2414yes—
393C2×C4⋊C4C24116yes—
394C2×C4⋊C4C2414yes—
395C2×C4⋊C4C2418yes—
396C2×C4⋊C4C2414yes—
397C2×C4⋊C4C2418yes—
398C2×C4⋊C4C2418yes—
399C2×C4⋊C4C24116yes—
400C2×C4⋊C4C2418yes—
401C2×C4⋊C4C24116yes—
402C2×C4⋊C4C24116yes—
403C2×C4⋊C4C2412yes—
404C2×C22⋊C4—28no—
405C2×C22⋊C4—216no—
406C2×C22⋊C4—28no—
407C2×C22⋊C4—28no—
408C2×C22⋊C4C2412yes—
409C2×C22⋊C4C2412yes—
410C2×C22⋊C4C2412yes—
411C2×C22⋊C4C2414yes—
412C2×C22⋊C4C2412yes—
413C2×C22⋊C4C2412yes—
414C2×C22⋊C4C2412yes—
415C2×C22⋊C4C2412yes—
416C2×C22⋊C4C2414yes—
417C2×C22⋊C4C2418yes—
418C2×C22⋊C4C2418yes—
419C2×C22⋊C4C2414yes—
420C2×C22⋊C4C2414yes—
421C2×C22⋊C4C2418yes—
422C2×C22⋊C4C2418yes—
423C2×C22⋊C4C2412yes—
424C2×C42—28no—
425C2×C42—24no—
426C2×C42—28no—
427C2×C42#4118yes—
428C2×C42C2412yes—
429C2×C42C2412yes—
430C2×C42C2412yes—
431C2×C42C2411yescharacteristic
432C2×C42C2411yescharacteristic
433C2×C42C2414yes—
434C2×C42C2412yes—
435C2×C42C2414yes—
436C2×C42C24116yes—
437C2×C42C2414yes—
438C2×C42C2414yes—
439C2×C42C2418yes—
440C2×C42C2418yes—
441C2×C42C2418yes—
442C2×C42C2418yes—
443C2×C42C24116yes—
444C2×C42C2412yes—
445C2×C42C2412yes—
446C2×C42C2411yescharacteristic
447C2×C42C2411yescharacteristic
448#95C24116yes—
449#95C2418yes—
450#95C2418yes—
451#95C24116yes—
452#95C2418yes—
453#95C2418yes—
454#95C2418yes—
455#95C2418yes—
456#95C2418yes—
457#95C24116yes—
458#95C24116yes—
459#95C2418yes—
460#95C24116yes—
461#95C2418yes—
462#95C24116yes—
463#95C24116yes—
464#95C2414yes—
465#95C2414yes—
466#95C24116yes—
467#95C24116yes—
468#95C24116yes—
469#95C24116yes—
470#95C24116yes—
471#95C2418yes—
472C24—22no—
473C24—22no—
474C24—44no—
475C24C2×#4114yes—
476C24C2511yescharacteristic
477C24C2511yescharacteristic
478C24C2511yescharacteristic
479C2×Q8—28no—
480C2×Q8—28no—
481C2×Q8—28no—
482C2×Q8—28no—
483C2×Q8—28no—
484C2×Q8—28no—
485C2×Q8—28no—
486C2×Q8—48no—
487C2×Q8—432no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—232no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—28no—
500C22×C4—28no—
501C22×C4—28no—
502C22×C4—28no—
503C22×C4—24no—
504C22×C4—28no—
505C22×C4—28no—
506C22×C4—44no—
507C22×C4—48no—
508C22×C4—44no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4C2×#4114yes—
512C22×C4C2×#4114yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4114yes—
515C22×C4C2×#4114yes—
516C22×C4C2×#4118yes—
517C22×C4C2×#4114yes—
518C22×C4C2×#4114yes—
519C22×C4C2×#4114yes—
520C22×C4C2512yes—
521C22×C4C2512yes—
522C22×C4C2514yes—
523C22×C4C2514yes—
524C22×C4C2514yes—
525C22×C4C2518yes—
526C22×C4C2518yes—
527C22×C4C2518yes—
528C22×C4C2518yes—
529C22×C4C2512yes—
530C22×C4C2514yes—
531C22×C4C2512yes—
532C22×C4C2511yescharacteristic
533C22×C4C2511yescharacteristic
534C22×C4C2511yescharacteristic
535C22×C4C2511yescharacteristic
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—28no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—28no—
546C4⋊C4—28no—
547C4⋊C4—216no—
548C4⋊C4—216no—
549C4⋊C4—216no—
550C4⋊C4—216no—
551C4⋊C4—28no—
552C4⋊C4—232no—
553C4⋊C4—28no—
554C4⋊C4—216no—
555C4⋊C4—28no—
556C4⋊C4—48no—
557C4⋊C4—48no—
558C4⋊C4—416no—
559C4⋊C4—416no—
560C4⋊C4—416no—
561C4⋊C4—48no—
562C4⋊C4C2×#4118yes—
563C22⋊C4—28no—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—216no—
568C22⋊C4—216no—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—28no—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—48no—
577C22⋊C4—48no—
578C22⋊C4C2×#4118yes—
579C42—28no—
580C42—28no—
581C42—24no—
582C42—24no—
583C42—24no—
584C42—22no—
585C42—22no—
586C42—24no—
587C42—28no—
588C42—232no—
589C42—28no—
590C42—48no—
591C42—416no—
592C42—48no—
593C42—48no—
594C42—48no—
595C42C2×#4118yes—
596C42C2×#4118yes—
597C42C2×#4114yes—
598C42C2×#4114yes—
599C23—24no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—24no—
605C23—24no—
606C23—28no—
607C23—48no—
608C23—84no—
609C23—84no—
610C23C22×#4112yes—
611C23C22×#4112yes—
612C23C22×#4112yes—
613C23C22×#4112yes—
614C23C22×#4112yes—
615C23C2611yescharacteristic
616Q8—416no—
617Q8—416no—
618Q8—88no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—216no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—48no—
653C2×C4—416no—
654C2×C4—88no—
655C2×C4—84no—
656C2×C4—88no—
657C2×C4—88no—
658C2×C4#529116yes—
659C2×C4C22×#4118yes—
660C2×C4C22×#4112yes—
661C2×C4C22×#4114yes—
662C2×C4C22×#4114yes—
663C2×C4C22×#4112yes—
664C2×C4C22×#4112yes—
665C2×C4C22×#4112yes—
666C2×C4C22×#4112yes—
667C2×C4C22×#4112yes—
668C2×C4C22×#4112yes—
669C2×C4C22×#4112yes—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—88no—
682C22C2×#52914yes—
683C22C23×#4111yescharacteristic
684C22C23×#4111yescharacteristic
685C22C23×#4111yescharacteristic
686C222−1+614yes—
687C4—24no—
688C4—24no—
689C4—28no—
690C4—22no—
691C4—22no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—48no—
697C4—48no—
698C4—48no—
699C4—44no—
700C4—48no—
701C4C2×#52914yes—
702C4C2×#52914yes—
703C2—24no—
704C2—42no—
705C2—42no—
706C2C22×#52911yescharacteristic
707C2#6286012yes—
708C2#6286012yes—
709C2#6286112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36023C2semidirect
3. #35818C2semidirect
4. #35789C2semidirect
5. #35339C2semidirect
6. #35324C2semidirect
7. #35321C2semidirect
8. #35320C2semidirect
9. #35314C2semidirect
10. #35313C2semidirect
13. #34629C2semidirect
15. #34254C2semidirect
16. #34215C2semidirect
17. #34209C2semidirect
27. #2864C22semidirect
28. #2863C22semidirect
29. #2862C22semidirect
31. #2655C22semidirect
33. #2647C22semidirect
34. #2645C22semidirect
35. #2642C22semidirect
49. #2588C22semidirect
50. #2585C22semidirect
53. #2570C22semidirect
55. #2568C22semidirect
56. #2528C22semidirect
58. #2524C22semidirect
59. #2524C22semidirect
61. #2520C22semidirect
64. #2514C22semidirect
65. #2511C22semidirect
74. #2402C22semidirect
81. #2365C22semidirect
84. #2356C22semidirect
85. #2355C22semidirect
86. #2355C22semidirect
91. #2331C22semidirect
92. #2327C22semidirect
94. #2320C22semidirect
101. C4×C4⋊Q8C22semidirect
102. C4×#125C22semidirect
105. #2295C22semidirect

Maximal subgroups

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

subgroupindex
#360232
#360132
#358182
#357892
#353392
#353242
#353212
#353202
#353142
#353132
#353112
#346342
#346292
C2×#26212
#342542
#342152
#342092
#342032
#339962