Finite Group Databasebeta

← #7630284

Subgroups — #7630284

Order 512.

5194 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#36640C212yesmaximal
3#36574C214yesmaximal, complemented
4#36571C218yesmaximal
5#36567C214yesmaximal, complemented
6#36563C212yesmaximal, complemented
7#36560C212yesmaximal, complemented
8#36556C214yesmaximal, complemented
9#36552C214yesmaximal
10#36545C214yesmaximal
11#34953C212yesmaximal
12#34951C214yesmaximal, complemented
13#34920C218yesmaximal, complemented
14#34815C214yesmaximal
15C4×#557C214yesmaximal, complemented
16#34145C212yesmaximal
17C4×#533C211yesmaximal, characteristic
18C4×#533C212yesmaximal
19C2×C42×Q8C211yesmaximal, characteristic
20C2×#557C2214yes—
21C2×#533C2212yes—
22C2×#533C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2864C2218yes—
28#2862C2212yes—
29#2859C2214yes—
30#2859C2214yes—
31#2859C2212yes—
32#2859C2212yes—
33#2857C2214yes—
34#2857C2214yes—
35#2851C2218yes—
36#2851C2218yes—
37#2590C22116yes—
38#2588C22116yes—
39#2585C22116yes—
40#2584C2218yes—
41#2584C2218yes—
42#2583C2218yes—
43#2582C2214yes—
44#2581C2218yes—
45#2568C2214yes—
46#2567C2214yes—
47#2565C2218yes—
48#2552C2214yes—
49#2550C2218yes—
50#2546C2214yes—
51#2546C2214yes—
52#2520C22116yes—
53#2511C2218yes—
54#2509C2218yes—
55#2509C2218yes—
56#2508C2218yes—
57#2508C2218yes—
58#2507C2218yes—
59#2506C2218yes—
60#2506C2218yes—
61#2505C22116yes—
62#2502C2214yes—
63#2473C22116yes—
64#2462C22116yes—
65#2453C2218yes—
66#2453C2218yes—
67#2451C2218yes—
68#2446C2218yes—
69#2434C2214yes—
70#2402C22116yes—
71#2399C2218yes—
72#2370C2218yes—
73#2370C22116yes—
74#2370C2218yes—
75#2369C2214yes—
76#2368C2218yes—
77#2365C2214yes—
78#2365C2218yes—
79#2365C2214yes—
80#2360C2218yes—
81#2355C2214yes—
82#2355C2218yes—
83#2351C2218yes—
84Q8×C4⋊C4C2214yes—
85C4×C4⋊Q8C2214yes—
86C4×C4⋊Q8C2214yes—
87C4×C4⋊Q8C2218yes—
88C4×C4⋊Q8C2212yes—
89C4×C4⋊Q8C2214yes—
90C4×C4⋊Q8C2214yes—
91C4×C4⋊Q8C2212yes—
92C4×#125C2218yes—
93C4×#125C2218yes—
94C4×#125C22116yes—
95C4×#125C2212yes—
96C4×#125C2218yes—
97C4×#125C2212yes—
98C4×C22⋊Q8C2214yes—
99C4×C22⋊Q8C2218yes—
100C4×C22⋊Q8C2218yes—
101C4×C22⋊Q8C2214yes—
102#2296C2214yes—
103#2296C2214yes—
104#2296C2214yes—
105#2296C2212yes—
106#2295C2218yes—
107#2295C2214yes—
108#2295C2214yes—
109#2295C2218yes—
110#2285C2211yescharacteristic
111#2285C2212yes—
112#2285C2212yes—
113C42×Q8C2214yes—
114C42×Q8C2214yes—
115C42×Q8C2218yes—
116C42×Q8C2212yes—
117C42×Q8C2214yes—
118C42×Q8C2212yes—
119C42×Q8C2212yes—
120C42×Q8C2212yes—
121C42×Q8C2212yes—
122C42×Q8C2212yes—
123C4×#117C2212yes—
124C4×#117C2214yes—
125C2×C4×C4⋊C4C2211yescharacteristic
126C2×C4×C4⋊C4C2212yes—
127C2×C4×C4⋊C4C2214yes—
128C2×C4×C4⋊C4C2212yes—
129C2×C43C2212yes—
130C2×C43C2211yescharacteristic
131C23×Q8C2311yescharacteristic
132#557—232no—
133#533—216no—
134#533Q8116yescomplemented
135C2×C4⋊Q8C2314yes—
136C2×C4⋊Q8C2318yes—
137C2×C4⋊Q8C2312yes—
138C2×C4⋊Q8C2312yes—
139C2×C4⋊Q8C2314yes—
140C2×C4⋊Q8C2312yes—
141C2×#125C2318yes—
142C2×#125C23116yes—
143C2×#125C2314yes—
144C2×#125C2318yes—
145C2×#125C2312yes—
146C2×C22⋊Q8C2314yes—
147C2×C22⋊Q8C2314yes—
148C2×C22⋊Q8C2314yes—
149C2×C4×Q8—216no—
150C2×C4×Q8—28no—
151C2×C4×Q8C2312yes—
152C2×C4×Q8C2314yes—
153C2×C4×Q8C2314yes—
154C2×C4×Q8C2318yes—
155C2×C4×Q8C2314yes—
156C2×C4×Q8C2312yes—
157C2×C4×Q8C2314yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2312yes—
160C2×C4×Q8C2318yes—
161C2×C4×Q8C2312yes—
162C2×C4×Q8C2312yes—
163C2×C4×Q8C2312yes—
164C2×C4×Q8C2314yes—
165C2×C4×Q8C2318yes—
166C2×C4×Q8C2314yes—
167C2×C4×Q8C2314yes—
168C2×C4×Q8C2312yes—
169C2×#117C2314yes—
170C2×#117C2312yes—
171C2×#117C2312yes—
172C22×C4⋊C4C2312yes—
173C22×C4⋊C4C2312yes—
174C22×C4⋊C4C2314yes—
175C22×C4⋊C4C2311yescharacteristic
176C22×C42C2314yes—
177C22×C42C2312yes—
178C22×C42C2312yes—
179C22×C42C2312yes—
180C22×C42C2311yescharacteristic
181C22×C42C2314yes—
182C22×C42C2312yes—
183C22×C42C2311yescharacteristic
184C22×C42C2311yescharacteristic
185#400C23116yes—
186#400C23116yes—
187#400C23116yes—
188#400C23116yes—
189#398C23116yes—
190#398C23116yes—
191#398C23116yes—
192#398C23116yes—
193#398C23116yes—
194#398C23116yes—
195#395C23116yes—
196#395C23116yes—
197#391C2314yes—
198#391C2318yes—
199#391C2318yes—
200#391C2318yes—
201#391C2314yes—
202#391C2318yes—
203#391C2318yes—
204#391C2318yes—
205#391C2318yes—
206#391C23116yes—
207#389C2318yes—
208#389C2318yes—
209#389C2314yes—
210#389C2314yes—
211#389C2318yes—
212#389C2318yes—
213#389C23116yes—
214#389C23116yes—
215#389C2318yes—
216#389C2318yes—
217#389C2318yes—
218#389C23116yes—
219#389C23116yes—
220#389C2314yes—
221#389C2318yes—
222#389C2318yes—
223#389C2318yes—
224#389C2318yes—
225#389C2314yes—
226#389C2318yes—
227#389C2318yes—
228#389C2318yes—
229#389C23116yes—
230#389C2318yes—
231#389C2318yes—
232#389C2318yes—
233#387C23116yes—
234#387C23116yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C2318yes—
240#387C23116yes—
241#387C2318yes—
242#387C23116yes—
243#387C23116yes—
244#387C23116yes—
245#387C23116yes—
246#387C23116yes—
247#387C23116yes—
248#385C2318yes—
249#385C2318yes—
250#385C2318yes—
251#385C2318yes—
252#385C23116yes—
253#385C2318yes—
254#384C2312yes—
255#384C2312yes—
256#384C2312yes—
257#384C2314yes—
258#384C2318yes—
259#384C2314yes—
260#384C2312yes—
261#384C2312yes—
262#382C2314yes—
263#382C2312yes—
264#382C2318yes—
265#382C2318yes—
266#382C23116yes—
267#382C2314yes—
268#380C2314yes—
269#380C2318yes—
270#380C2318yes—
271C4×C4⋊C4C2314yes—
272C4×C4⋊C4C2314yes—
273C4×C4⋊C4C2318yes—
274C4×C4⋊C4C23116yes—
275C4×C4⋊C4C2318yes—
276C4×C4⋊C4C2318yes—
277C4×C4⋊C4C2318yes—
278C4×C4⋊C4C2318yes—
279C4×C4⋊C4C2314yes—
280C4×C4⋊C4C2314yes—
281C4×C4⋊C4C2318yes—
282C4×C4⋊C4C2318yes—
283C4×C4⋊C4C23116yes—
284C4×C4⋊C4C2318yes—
285C4×C4⋊C4C23116yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C23116yes—
288C4×C4⋊C4C2318yes—
289C4×C4⋊C4C2314yes—
290C4×C4⋊C4C2318yes—
291C4×C4⋊C4C2314yes—
292C4×C4⋊C4C2314yes—
293C4×C4⋊C4C2312yes—
294C4×C4⋊C4C2314yes—
295C4×C4⋊C4C2314yes—
296C4×C4⋊C4C2314yes—
297C4×C4⋊C4C2312yes—
298C4×C4⋊C4C2318yes—
299C4×C4⋊C4C2314yes—
300C4×C4⋊C4C23116yes—
301C4×C4⋊C4C2312yes—
302C4×C4⋊C4C2312yes—
303C4×C4⋊C4C2312yes—
304C4×C4⋊C4C2314yes—
305C4×C4⋊C4C2314yes—
306C4×C4⋊C4C2314yes—
307C4×C4⋊C4C2314yes—
308C4×C4⋊C4C2314yes—
309C4×C4⋊C4C2314yes—
310C4×C4⋊C4C2314yes—
311C4×C4⋊C4C2312yes—
312C4×C4⋊C4C2314yes—
313C4×C4⋊C4C2311yescharacteristic
314C4×C4⋊C4C2311yescharacteristic
315C4×C4⋊C4C2312yes—
316C4×C22⋊C4C2314yes—
317C4×C22⋊C4C2314yes—
318C4×C22⋊C4C23116yes—
319#376C2314yes—
320#376C2318yes—
321C2×#95C2312yes—
322C2×#95C2314yes—
323C43C2311yescharacteristic
324C43C2311yescharacteristic
325C43C2314yes—
326C43C2314yes—
327C43C2314yes—
328C43C2318yes—
329C43C2311yescharacteristic
330C43C2314yes—
331C43C2311yescharacteristic
332C43C2312yes—
333C43C2312yes—
334C43C2314yes—
335C22×Q8—28no—
336C22×Q8—24no—
337C22×Q8C2412yes—
338C22×Q8C2414yes—
339C22×Q8C2414yes—
340C22×Q8C2412yes—
341C22×Q8C2414yes—
342C23×C4C2412yes—
343C23×C4C2414yes—
344C23×C4C2412yes—
345C23×C4C2412yes—
346C23×C4C2411yescharacteristic
347C23×C4C2412yes—
348C23×C4C2411yescharacteristic
349C23×C4C2411yescharacteristic
350C4⋊Q8—232no—
351C4⋊Q8—28no—
352C4⋊Q8—28no—
353C4⋊Q8—216no—
354C4⋊Q8—48no—
355C4⋊Q8C2×Q8116yescomplemented
356#125—264no—
357#125—216no—
358#125—232no—
359#125—416no—
360#125C2×Q8116yescomplemented
361C22⋊Q8—216no—
362C22⋊Q8—216no—
363C22⋊Q8C2×Q8132yes—
364C4×Q8—28no—
365C4×Q8—28no—
366C4×Q8—216no—
367C4×Q8—232no—
368C4×Q8—216no—
369C4×Q8—28no—
370C4×Q8—28no—
371C4×Q8—28no—
372C4×Q8—28no—
373C4×Q8—28no—
374C4×Q8—216no—
375C4×Q8—216no—
376C4×Q8—232no—
377C4×Q8—216no—
378C4×Q8—48no—
379C4×Q8—416no—
380C4×Q8C2×Q8132yescomplemented
381C4×Q8#41116yes—
382#117—216no—
383#117—28no—
384#117C2×Q8116yes—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—216no—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2414yes—
391C2×C4⋊C4C2414yes—
392C2×C4⋊C4C2414yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2414yes—
395C2×C4⋊C4C2414yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2414yes—
398C2×C4⋊C4C2414yes—
399C2×C4⋊C4C2414yes—
400C2×C4⋊C4C2414yes—
401C2×C4⋊C4C2418yes—
402C2×C4⋊C4C2418yes—
403C2×C4⋊C4C2414yes—
404C2×C4⋊C4C2414yes—
405C2×C4⋊C4C2414yes—
406C2×C4⋊C4C2414yes—
407C2×C4⋊C4C24116yes—
408C2×C4⋊C4C2418yes—
409C2×C4⋊C4C2418yes—
410C2×C4⋊C4C2418yes—
411C2×C4⋊C4C24116yes—
412C2×C4⋊C4C2418yes—
413C2×C4⋊C4C24116yes—
414C2×C4⋊C4C2418yes—
415C2×C4⋊C4C24116yes—
416C2×C4⋊C4C24116yes—
417C2×C4⋊C4C24116yes—
418C2×C4⋊C4C2418yes—
419C2×C4⋊C4C2414yes—
420C2×C4⋊C4C2418yes—
421C2×C4⋊C4C2414yes—
422C2×C4⋊C4C2412yes—
423C2×C4⋊C4C2414yes—
424C2×C22⋊C4C2418yes—
425C2×C22⋊C4C2414yes—
426C2×C22⋊C4C2414yes—
427C2×C42—28no—
428C2×C42—28no—
429C2×C42—24no—
430C2×C42—216no—
431C2×C42—28no—
432C2×C42—48no—
433C2×C42—44no—
434C2×C42C2×Q818yes—
435C2×C42#4118yes—
436C2×C42C24116yes—
437C2×C42C2418yes—
438C2×C42C2414yes—
439C2×C42C2412yes—
440C2×C42C2414yes—
441C2×C42C2412yes—
442C2×C42C2414yes—
443C2×C42C2412yes—
444C2×C42C2414yes—
445C2×C42C2412yes—
446C2×C42C2414yes—
447C2×C42C2414yes—
448C2×C42C2412yes—
449C2×C42C2414yes—
450C2×C42C2411yescharacteristic
451C2×C42C2411yescharacteristic
452C2×C42C2418yes—
453C2×C42C2414yes—
454C2×C42C2414yes—
455C2×C42C2412yes—
456C2×C42C2412yes—
457C2×C42C2412yes—
458C2×C42C2418yes—
459C2×C42C24116yes—
460C2×C42C2414yes—
461C2×C42C2412yes—
462C2×C42C2414yes—
463C2×C42C2412yes—
464C2×C42C2411yescharacteristic
465C2×C42C2411yescharacteristic
466C2×C42C2414yes—
467C2×C42C2412yes—
468C2×C42C2414yes—
469C2×C42C2412yes—
470C2×C42C2411yescharacteristic
471C2×C42C2411yescharacteristic
472C2×C42C2411yescharacteristic
473C2×C42C2411yescharacteristic
474#95C2412yes—
475#95C2414yes—
476#95C2412yes—
477#95C2414yes—
478#95C2414yes—
479#95C2414yes—
480#95C2414yes—
481#95C2418yes—
482#95C24116yes—
483#95C24116yes—
484#95C2418yes—
485#95C2418yes—
486#95C24116yes—
487#95C24116yes—
488#95C2418yes—
489#95C2418yes—
490#95C24116yes—
491#95C24116yes—
492#95C24116yes—
493#95C24116yes—
494#95C24116yes—
495#95C24116yes—
496C24C2511yescharacteristic
497C2×Q8—232no—
498C2×Q8—28no—
499C2×Q8—28no—
500C2×Q8—24no—
501C2×Q8—24no—
502C2×Q8—24no—
503C2×Q8—28no—
504C2×Q8—216no—
505C2×Q8—28no—
506C2×Q8—28no—
507C2×Q8—48no—
508C2×Q8C22×Q8116yes—
509C2×Q8C2×#4118yes—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—28no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—24no—
516C22×C4—28no—
517C22×C4—24no—
518C22×C4—28no—
519C22×C4—24no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—44no—
524C22×C4C22×Q818yes—
525C22×C4C22×Q814yes—
526C22×C4C2×#4114yes—
527C22×C4C2×#4114yes—
528C22×C4C2×#4114yes—
529C22×C42−1+414yes—
530C22×C4C2512yes—
531C22×C4C2514yes—
532C22×C4C2518yes—
533C22×C4C2514yes—
534C22×C4C2518yes—
535C22×C4C2518yes—
536C22×C4C2512yes—
537C22×C4C2518yes—
538C22×C4C2511yescharacteristic
539C22×C4C2514yes—
540C22×C4C2512yes—
541C22×C4C2514yes—
542C22×C4C2511yescharacteristic
543C22×C4C2511yescharacteristic
544C22×C4C2511yescharacteristic
545C22×C4C2512yes—
546C22×C4C2511yescharacteristic
547C22×C4C2511yescharacteristic
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—28no—
551C4⋊C4—28no—
552C4⋊C4—28no—
553C4⋊C4—28no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—28no—
558C4⋊C4—28no—
559C4⋊C4—216no—
560C4⋊C4—216no—
561C4⋊C4—28no—
562C4⋊C4—28no—
563C4⋊C4—216no—
564C4⋊C4—28no—
565C4⋊C4—232no—
566C4⋊C4—232no—
567C4⋊C4—216no—
568C4⋊C4—232no—
569C4⋊C4—216no—
570C4⋊C4—28no—
571C4⋊C4—28no—
572C4⋊C4—216no—
573C4⋊C4—48no—
574C4⋊C4—416no—
575C4⋊C4—416no—
576C4⋊C4—416no—
577C4⋊C4—48no—
578C4⋊C4C22×Q8116yes—
579C4⋊C4C22×Q8132yes—
580C4⋊C4C22×Q8116yes—
581C4⋊C4C2×#4118yes—
582C4⋊C4C2×#41116yes—
583C22⋊C4—216no—
584C22⋊C4—28no—
585C22⋊C4C22×Q8116yes—
586C42—28no—
587C42—24no—
588C42—28no—
589C42—24no—
590C42—28no—
591C42—28no—
592C42—28no—
593C42—24no—
594C42—216no—
595C42—232no—
596C42—216no—
597C42—24no—
598C42—22no—
599C42—24no—
600C42—28no—
601C42—24no—
602C42—22no—
603C42—28no—
604C42—28no—
605C42—24no—
606C42—28no—
607C42—24no—
608C42—42no—
609C42—44no—
610C42—42no—
611C42—44no—
612C42—48no—
613C42—416no—
614C42C22×Q818yes—
615C42C22×Q814yes—
616C42C22×Q814yes—
617C42C22×Q8116yes—
618C42C2×#41116yes—
619C42C2×#4118yes—
620C42C2×#4114yes—
621C42C2×#4114yes—
622C42C2×#4114yes—
623C42C2×#4114yes—
624C23—22no—
625C23—22no—
626C23C23×Q812yes—
627C23C22×#4112yes—
628C23C2×2−1+412yes—
629C23C2611yescharacteristic
630Q8—216no—
631Q8—216no—
632Q8—232no—
633Q8—416no—
634Q8—88no—
635C2×C4—216no—
636C2×C4—22no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—216no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—216no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—28no—
661C2×C4—22no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—28no—
666C2×C4—216no—
667C2×C4—44no—
668C2×C4—416no—
669C2×C4—48no—
670C2×C4—48no—
671C2×C4#557116yescomplemented
672C2×C4C23×Q814yes—
673C2×C4C23×Q8116yes—
674C2×C4C23×Q812yes—
675C2×C4C23×Q812yes—
676C2×C4C23×Q814yes—
677C2×C4C22×#4118yes—
678C2×C4C22×#4114yes—
679C2×C4C22×#4118yes—
680C2×C4C22×#4112yes—
681C2×C4C22×#4112yes—
682C2×C4C22×#4112yes—
683C2×C4C22×#4114yes—
684C2×C4C22×#4112yes—
685C2×C4C22×#4112yes—
686C2×C4C22×#4112yes—
687C2×C4C22×#4114yes—
688C2×C4C2×2−1+412yes—
689C2×C4C2×2−1+412yes—
690C2×C4#58514yes—
691C22—24no—
692C22—22no—
693C22—22no—
694C22—24no—
695C22C2×#55714yes—
696C22C24×Q811yescharacteristic
697C22C23×#4112yes—
698C22C23×#4111yescharacteristic
699C22C22×2−1+411yescharacteristic
700C22C2×#58512yes—
701C4—28no—
702C4—24no—
703C4—22no—
704C4—22no—
705C4—28no—
706C4—22no—
707C4—24no—
708C4—216no—
709C4—48no—
710C4—42no—
711C4—44no—
712C4—44no—
713C4—48no—
714C4—48no—
715C4#344414yes—
716C4C2×#55714yes—
717C4C2×#55714yes—
718C4Q8×#4118yes—
719C2—24no—
720C2C2×#344411yescharacteristic
721C2C22×#55711yescharacteristic
722C2C2×Q8×#4112yes—
723C2#6280512yes—
724C2#6281511yescharacteristic
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
3. #36574C2semidirect
5. #36567C2semidirect
6. #36563C2semidirect
7. #36560C2semidirect
8. #36556C2semidirect
12. #34951C2semidirect
13. #34920C2semidirect
15. C4×#557C2semidirect
134. #533Q8semidirect
355. C4⋊Q8C2×Q8semidirect
360. #125C2×Q8semidirect
380. C4×Q8C2×Q8semidirect
671. C2×C4#557semidirect

Maximal subgroups

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

subgroupindex
#366452
#366402
#365742
#365712
#365672
#365632
#365602
#365562
#365522
#365452
#349532
#349512
#349202
#348152
C4×#5572
#341452
C4×#5332
C2×C42×Q82