Finite Group Databasebeta

← #7628463

Subgroups — #7628463

Order 512.

4898 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C214yesmaximal
2#35221C218yesmaximal, complemented
3#35208C214yesmaximal, complemented
4#35199C214yesmaximal, complemented
5#35179C212yesmaximal
6#35173C211yesmaximal, characteristic
7#35167C211yesmaximal, characteristic
8#35159C212yesmaximal
9#35154C212yesmaximal
10#35015C214yesmaximal
11#35013C214yesmaximal
12#34951C214yesmaximal, complemented
13#34950C214yesmaximal, complemented
14#34920C218yesmaximal, complemented
15#34595C214yesmaximal
16C4×#544C212yesmaximal
17#34059C212yesmaximal
18C4×#533C212yesmaximal
19C2×C42×Q8C211yesmaximal, characteristic
20C2×#544C2212yes—
21C2×#533C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#2864C2218yes—
26#2863C2218yes—
27#2863C2218yes—
28#2862C2214yes—
29#2859C2214yes—
30#2857C2214yes—
31#2857C2214yes—
32#2855C2218yes—
33#2855C2218yes—
34#2851C2218yes—
35#2666C22116yes—
36#2666C22116yes—
37#2659C2214yes—
38#2659C2214yes—
39#2646C2218yes—
40#2627C2218yes—
41#2627C2218yes—
42#2590C22116yes—
43#2588C22116yes—
44#2586C22116yes—
45#2585C22116yes—
46#2584C2218yes—
47#2582C2214yes—
48#2582C2214yes—
49#2581C2218yes—
50#2568C2214yes—
51#2567C2214yes—
52#2567C2214yes—
53#2565C2218yes—
54#2552C2214yes—
55#2551C2214yes—
56#2550C2218yes—
57#2546C2214yes—
58#2520C22116yes—
59#2511C2218yes—
60#2509C2218yes—
61#2508C2218yes—
62#2507C2218yes—
63#2507C2218yes—
64#2506C2218yes—
65#2462C22116yes—
66#2453C2218yes—
67#2453C2218yes—
68#2451C2218yes—
69#2451C2218yes—
70#2451C2218yes—
71#2446C2218yes—
72#2402C22116yes—
73#2399C22116yes—
74#2331C2218yes—
75#2330C2218yes—
76#2329C2218yes—
77#2320C2218yes—
78#2320C2218yes—
79#2317C2214yes—
80#2316C2212yes—
81#2316C2212yes—
82#2316C2212yes—
83#2313C2218yes—
84#2310C2214yes—
85C4×C4⋊Q8C2214yes—
86C4×C4⋊Q8C2214yes—
87C4×C4⋊Q8C2218yes—
88C4×C4⋊Q8C2212yes—
89C4×C4⋊Q8C2212yes—
90C4×C4⋊Q8C2212yes—
91C4×C4⋊Q8C2212yes—
92C4×#125C2218yes—
93C4×#125C2218yes—
94C4×#125C2214yes—
95C4×#125C2214yes—
96C4×C22⋊Q8C2214yes—
97C4×C22⋊Q8C2218yes—
98C4×C22⋊Q8C2214yes—
99C4×C22⋊Q8C2214yes—
100C4×C22⋊Q8C2214yes—
101#2296C2214yes—
102#2295C2218yes—
103#2295C2218yes—
104#2295C2218yes—
105#2295C2218yes—
106#2295C2214yes—
107#2295C2214yes—
108#2285C2212yes—
109#2283C2218yes—
110C42×Q8C2214yes—
111C42×Q8C2214yes—
112C42×Q8C2214yes—
113C42×Q8C2214yes—
114C42×Q8C2214yes—
115C42×Q8C2214yes—
116C4×#117C2214yes—
117C2×C4×C4⋊C4C2212yes—
118C2×C4×C4⋊C4C2211yescharacteristic
119C2×C4×C4⋊C4C2211yescharacteristic
120C2×C4×C4⋊C4C2211yescharacteristic
121C2×C4×C4⋊C4C2212yes—
122C2×C4×C4⋊C4C2212yes—
123C2×C43C2212yes—
124C2×C43C2211yescharacteristic
125C23×Q8C2311yescharacteristic
126#544—216no—
127#533—216no—
128C2×C4⋊Q8C2312yes—
129C2×C4⋊Q8C2312yes—
130C2×C4⋊Q8C2318yes—
131C2×C4⋊Q8C2314yes—
132C2×C4⋊Q8C2314yes—
133C2×#125C2314yes—
134C2×#125C2318yes—
135C2×#125C2318yes—
136C2×C22⋊Q8C2318yes—
137C2×C22⋊Q8C2314yes—
138C2×C22⋊Q8C2314yes—
139C2×C4×Q8—28no—
140C2×C4×Q8—28no—
141C2×C4×Q8Q8116yescomplemented
142C2×C4×Q8C2314yes—
143C2×C4×Q8C2314yes—
144C2×C4×Q8C2314yes—
145C2×C4×Q8C2314yes—
146C2×C4×Q8C2314yes—
147C2×C4×Q8C2314yes—
148C2×C4×Q8C2314yes—
149C2×C4×Q8C2318yes—
150C2×C4×Q8C2314yes—
151C2×C4×Q8C2314yes—
152C2×C4×Q8C2314yes—
153C2×C4×Q8C2314yes—
154C2×C4×Q8C2314yes—
155C2×#117C2314yes—
156C2×#117C2314yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2311yescharacteristic
162C22×C4⋊C4C2311yescharacteristic
163C22×C42C2312yes—
164C22×C42C2312yes—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167C22×C42C2312yes—
168C22×C42C2312yes—
169C22×C42C2311yescharacteristic
170C22×C42C2311yescharacteristic
171C22×C42C2312yes—
172C22×C42C2311yescharacteristic
173C22×C42C2311yescharacteristic
174C22×C42C2312yes—
175C22×C42C2311yescharacteristic
176#400C23116yes—
177#400C23116yes—
178#400C23116yes—
179#400C23116yes—
180#398C23116yes—
181#398C23116yes—
182#398C23116yes—
183#398C23116yes—
184#398C23116yes—
185#398C23116yes—
186#398C23116yes—
187#398C23116yes—
188#395C23116yes—
189#395C23116yes—
190#395C23116yes—
191#395C23116yes—
192#391C2318yes—
193#391C23116yes—
194#391C2314yes—
195#391C2318yes—
196#391C2314yes—
197#391C2314yes—
198#391C2314yes—
199#391C2318yes—
200#391C2318yes—
201#391C2318yes—
202#391C2318yes—
203#391C2318yes—
204#391C2318yes—
205#391C2318yes—
206#391C2318yes—
207#391C2318yes—
208#389C2314yes—
209#389C2314yes—
210#389C2314yes—
211#389C2314yes—
212#389C2318yes—
213#389C2318yes—
214#389C2318yes—
215#389C2318yes—
216#389C2318yes—
217#389C2318yes—
218#389C2318yes—
219#389C23116yes—
220#389C2314yes—
221#389C2318yes—
222#389C2318yes—
223#389C2318yes—
224#389C2314yes—
225#389C2314yes—
226#389C2314yes—
227#389C2318yes—
228#389C2318yes—
229#389C2318yes—
230#389C2318yes—
231#389C2318yes—
232#389C2318yes—
233#389C2318yes—
234#389C2318yes—
235#389C2318yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C2318yes—
240#387C23116yes—
241#387C2318yes—
242#387C2318yes—
243#387C2318yes—
244#387C23116yes—
245#387C23116yes—
246#387C23116yes—
247#387C23116yes—
248#387C23116yes—
249#387C23116yes—
250#387C23116yes—
251#387C23116yes—
252#387C23116yes—
253#385C23116yes—
254#385C2318yes—
255#385C2318yes—
256#384C2312yes—
257#384C2312yes—
258#384C2318yes—
259#384C2314yes—
260#384C2314yes—
261#383C23116yes—
262#382C2314yes—
263#382C2318yes—
264#382C2318yes—
265#382C2318yes—
266#382C2318yes—
267#381C23116yes—
268#380C2318yes—
269#380C2318yes—
270#380C2318yes—
271#380C2314yes—
272#380C2314yes—
273C4×C4⋊C4C2314yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2314yes—
276C4×C4⋊C4C23116yes—
277C4×C4⋊C4C2318yes—
278C4×C4⋊C4C2314yes—
279C4×C4⋊C4C2314yes—
280C4×C4⋊C4C2314yes—
281C4×C4⋊C4C2318yes—
282C4×C4⋊C4C2318yes—
283C4×C4⋊C4C2318yes—
284C4×C4⋊C4C2318yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C2314yes—
287C4×C4⋊C4C2314yes—
288C4×C4⋊C4C2314yes—
289C4×C4⋊C4C2314yes—
290C4×C4⋊C4C2314yes—
291C4×C4⋊C4C2314yes—
292C4×C4⋊C4C2314yes—
293C4×C4⋊C4C2314yes—
294C4×C4⋊C4C2318yes—
295C4×C4⋊C4C2314yes—
296C4×C4⋊C4C2318yes—
297C4×C4⋊C4C2318yes—
298C4×C4⋊C4C2318yes—
299C4×C4⋊C4C2318yes—
300C4×C4⋊C4C2318yes—
301C4×C4⋊C4C2318yes—
302C4×C4⋊C4C2318yes—
303C4×C4⋊C4C2314yes—
304C4×C4⋊C4C2314yes—
305C4×C22⋊C4C2318yes—
306C4×C22⋊C4C2318yes—
307#376C2318yes—
308#376C23116yes—
309C2×#95C2312yes—
310C2×#95C2312yes—
311C2×#95C2312yes—
312C43C2312yes—
313C43C2312yes—
314C43C2314yes—
315C43C2318yes—
316C43C2312yes—
317C43C2312yes—
318C43C2314yes—
319C43C2314yes—
320C43C2314yes—
321C22×Q8—24no—
322C22×Q8—24no—
323C22×Q8C2×Q818yes—
324C22×Q8C2414yes—
325C22×Q8C2414yes—
326C22×Q8C2414yes—
327C22×Q8C2418yes—
328C23×C4C2411yescharacteristic
329C23×C4C2412yes—
330C23×C4C2411yescharacteristic
331C23×C4C2411yescharacteristic
332C23×C4C2412yes—
333C23×C4C2411yescharacteristic
334C23×C4C2411yescharacteristic
335C23×C4C2412yes—
336C23×C4C2412yes—
337C23×C4C2411yescharacteristic
338C23×C4C2411yescharacteristic
339C4⋊Q8—232no—
340C4⋊Q8—28no—
341C4⋊Q8—28no—
342C4⋊Q8—48no—
343C4⋊Q8—48no—
344#125—232no—
345#125—216no—
346#125—416no—
347C22⋊Q8—216no—
348C22⋊Q8—216no—
349C22⋊Q8—232no—
350C4×Q8—216no—
351C4×Q8—216no—
352C4×Q8—216no—
353C4×Q8—216no—
354C4×Q8—216no—
355C4×Q8—216no—
356C4×Q8—216no—
357C4×Q8—48no—
358C4×Q8—416no—
359C4×Q8—48no—
360C4×Q8—48no—
361C4×Q8C2×Q8132yes—
362C4×Q8C2×Q8132yes—
363#117—216no—
364#117—216no—
365C2×C4⋊C4—28no—
366C2×C4⋊C4—28no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4C2×Q8116yes—
369C2×C4⋊C4C2×Q818yes—
370C2×C4⋊C4#4118yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2414yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2414yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2414yes—
381C2×C4⋊C4C2414yes—
382C2×C4⋊C4C2418yes—
383C2×C4⋊C4C2414yes—
384C2×C4⋊C4C2414yes—
385C2×C4⋊C4C2418yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C24116yes—
388C2×C4⋊C4C2414yes—
389C2×C4⋊C4C2414yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2418yes—
392C2×C4⋊C4C2418yes—
393C2×C4⋊C4C24116yes—
394C2×C4⋊C4C2418yes—
395C2×C4⋊C4C2418yes—
396C2×C4⋊C4C24116yes—
397C2×C4⋊C4C2418yes—
398C2×C4⋊C4C2418yes—
399C2×C4⋊C4C2418yes—
400C2×C4⋊C4C2418yes—
401C2×C4⋊C4C2414yes—
402C2×C4⋊C4C2414yes—
403C2×C22⋊C4C2418yes—
404C2×C22⋊C4C2418yes—
405C2×C42—28no—
406C2×C42—24no—
407C2×C42—28no—
408C2×C42—24no—
409C2×C42—42no—
410C2×C42—42no—
411C2×C42—44no—
412C2×C42—42no—
413C2×C42—44no—
414C2×C42—44no—
415C2×C42C2×Q8116yes—
416C2×C42C2×Q818yes—
417C2×C42#4118yes—
418C2×C42C2418yes—
419C2×C42C2418yes—
420C2×C42C2412yes—
421C2×C42C2412yes—
422C2×C42C2412yes—
423C2×C42C2412yes—
424C2×C42C2414yes—
425C2×C42C2414yes—
426C2×C42C2414yes—
427C2×C42C2414yes—
428C2×C42C2414yes—
429C2×C42C2414yes—
430C2×C42C2411yescharacteristic
431C2×C42C2411yescharacteristic
432C2×C42C2418yes—
433C2×C42C2414yes—
434C2×C42C2412yes—
435C2×C42C2418yes—
436C2×C42C2411yescharacteristic
437C2×C42C2418yes—
438C2×C42C2418yes—
439C2×C42C2412yes—
440C2×C42C2412yes—
441C2×C42C2414yes—
442C2×C42C2412yes—
443C2×C42C2414yes—
444C2×C42C2414yes—
445C2×C42C2412yes—
446C2×C42C2414yes—
447C2×C42C2412yes—
448C2×C42C2414yes—
449C2×C42C2412yes—
450C2×C42C2412yes—
451C2×C42C2411yescharacteristic
452#95C2418yes—
453#95C2418yes—
454#95C2418yes—
455#95C2418yes—
456#95C2418yes—
457#95C2418yes—
458#95C2418yes—
459#95C24116yes—
460#95C24116yes—
461#95C24116yes—
462#95C24116yes—
463#95C2418yes—
464#95C2418yes—
465#95C2418yes—
466#95C2418yes—
467#95C2418yes—
468#95C2418yes—
469#95C24116yes—
470#95C24116yes—
471#95C24116yes—
472#95C24116yes—
473#95C24116yes—
474C24C2511yescharacteristic
475C2×Q8—232no—
476C2×Q8—28no—
477C2×Q8—28no—
478C2×Q8—28no—
479C2×Q8—28no—
480C2×Q8—216no—
481C2×Q8—28no—
482C2×Q8—28no—
483C2×Q8—44no—
484C2×Q8—44no—
485C2×Q8—48no—
486C2×Q8—48no—
487C2×Q8C22×Q8116yes—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4C22×Q814yes—
512C22×C4C22×Q818yes—
513C22×C4C22×Q814yes—
514C22×C4C22×Q818yes—
515C22×C4C22×Q814yes—
516C22×C4C2×#4114yes—
517C22×C4C2×#4118yes—
518C22×C4C2×#4114yes—
519C22×C4C2×#4114yes—
520C22×C4C2×#4114yes—
521C22×C4C2518yes—
522C22×C4C2518yes—
523C22×C4C2518yes—
524C22×C4C2514yes—
525C22×C4C2512yes—
526C22×C4C2518yes—
527C22×C4C2511yescharacteristic
528C22×C4C2514yes—
529C22×C4C2512yes—
530C22×C4C2514yes—
531C22×C4C2511yescharacteristic
532C22×C4C2514yes—
533C22×C4C2512yes—
534C22×C4C2511yescharacteristic
535C22×C4C2511yescharacteristic
536C22×C4C2512yes—
537C22×C4C2511yescharacteristic
538C22×C4C2511yescharacteristic
539C4⋊C4—216no—
540C4⋊C4—216no—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—28no—
544C4⋊C4—28no—
545C4⋊C4—28no—
546C4⋊C4—216no—
547C4⋊C4—216no—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—216no—
551C4⋊C4—232no—
552C4⋊C4—28no—
553C4⋊C4—216no—
554C4⋊C4—216no—
555C4⋊C4—28no—
556C4⋊C4—48no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—48no—
560C4⋊C4—416no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—48no—
564C4⋊C4—416no—
565C4⋊C4—48no—
566C4⋊C4—48no—
567C4⋊C4—48no—
568C4⋊C4C22×Q8132yes—
569C4⋊C4C22×Q8116yes—
570C4⋊C4C2×#41116yes—
571C22⋊C4—216no—
572C22⋊C4—216no—
573C42—28no—
574C42—28no—
575C42—24no—
576C42—24no—
577C42—28no—
578C42—28no—
579C42—216no—
580C42—216no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—216no—
585C42—24no—
586C42—24no—
587C42—28no—
588C42—24no—
589C42—28no—
590C42—28no—
591C42—216no—
592C42—42no—
593C42—42no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42—42no—
598C42—48no—
599C42—48no—
600C42C22×Q8116yes—
601C42C22×Q818yes—
602C42C22×Q8116yes—
603C42C22×Q818yes—
604C42C2×#4118yes—
605C42C2×#4118yes—
606C422−1+414yes—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23C23×Q812yes—
612C23C22×#4112yes—
613C23C22×#4112yes—
614C23C2611yescharacteristic
615Q8—232no—
616Q8—416no—
617Q8—416no—
618Q8—816no—
619C2×C4—28no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—216no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—28no—
659C2×C4—44no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—48no—
665C2×C4#533116yes—
666C2×C4C23×Q812yes—
667C2×C4C23×Q814yes—
668C2×C4C23×Q818yes—
669C2×C4C23×Q818yes—
670C2×C4C23×Q814yes—
671C2×C4C23×Q812yes—
672C2×C4C22×#4112yes—
673C2×C4C22×#4118yes—
674C2×C4C22×#4114yes—
675C2×C4C22×#4114yes—
676C2×C4C22×#4114yes—
677C2×C4C22×#4112yes—
678C2×C4C2×2−1+412yes—
679C2×C4C2×2−1+412yes—
680C2×C4C2×2−1+412yes—
681C2×C4#58512yes—
682C2×C4#58512yes—
683C2×C4#58512yes—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22C2×#53314yes—
691C22C24×Q811yescharacteristic
692C22C23×#4111yescharacteristic
693C22C23×#4111yescharacteristic
694C22C22×2−1+411yescharacteristic
695C22C2×#58511yescharacteristic
696C22C2×#58511yescharacteristic
697C22C2×#58511yescharacteristic
698C4—28no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—24no—
703C4—28no—
704C4—28no—
705C4—22no—
706C4—24no—
707C4—42no—
708C4—48no—
709C4—48no—
710C4—48no—
711C4—44no—
712C4—48no—
713C4C2×#53318yes—
714C4#346114yes—
715C4#350214yes—
716C2—24no—
717C2C22×#53311yescharacteristic
718C2C2×#346111yescharacteristic
719C2#6268611yescharacteristic
720C2#6278111yescharacteristic
721C2C2×#350211yescharacteristic
722C2#6278811yescharacteristic
723C2#6281311yescharacteristic
subgroupG/H
centerC22×C4C25
commutator, Frattini, socle, cosocleC23C26
prime radicalC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35221C2semidirect
3. #35208C2semidirect
4. #35199C2semidirect
12. #34951C2semidirect
13. #34950C2semidirect
14. #34920C2semidirect
141. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
#352212
#352082
#351992
#351792
#351732
#351672
#351592
#351542
#350152
#350132
#349512
#349502
#349202
#345952
C4×#5442
#340592
C4×#5332
C2×C42×Q82