Finite Group Databasebeta

← #7674637

Subgroups — #7674637

Order 512.

4872 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#44462C218yesmaximal, complemented
2#43511C218yesmaximal, complemented
3#39822C218yesmaximal, complemented
4#39819C218yesmaximal, complemented
5#39553C218yesmaximal, complemented
6#37947C218yesmaximal, complemented
7#35673C212yesmaximal, complemented
8#35630C212yesmaximal, complemented
9#35339C211yesmaximal, characteristic, complemented
10#35316C211yesmaximal, characteristic, complemented
11#35306C212yesmaximal
12C2×#2621C211yesmaximal, characteristic, complemented
13#34209C211yesmaximal, characteristic, complemented
14#34203C212yesmaximal, complemented
15#34189C211yesmaximal, characteristic, complemented
16#34065C211yesmaximal, characteristic
17#34052C211yesmaximal, characteristic, complemented
18C2×#536C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#525C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C2×#519C2212yes—
23C2×C4×#41C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C22×#117C2212yes—
26#2864C2214yescomplemented
27#2857C2214yes—
28#2857C2214yescomplemented
29#2850C2214yes—
30#2802C22116yescomplemented
31#2796C22116yes—
32#2787C22116yes—
33#2777C22116yescomplemented
34#2776C22116yescomplemented
35#2770C22116yes—
36#2746C22116yes—
37#2739C22116yescomplemented
38#2647C2214yescomplemented
39#2621C2214yes—
40#2621C2214yes—
41#2621C2218yes—
42#2621C2214yes—
43#2621C2214yescomplemented
44#2620C2214yes—
45#2606C22116yescomplemented
46#2597C22116yes—
47#2594C22116yescomplemented
48#2590C22116yescomplemented
49#2578C22116yes—
50#2577C22116yes—
51#2570C2218yescomplemented
52#2569C2218yes—
53#2547C2214yescomplemented
54#2546C2214yescomplemented
55#2530C22116yes—
56#2522C22116yescomplemented
57#2514C2218yescomplemented
58#2514C2218yescomplemented
59#2511C2218yescomplemented
60#2511C2218yescomplemented
61#2498C22116yes—
62#2493C22116yes—
63#2486C2218yes—
64#2448C2218yescomplemented
65#2446C2218yescomplemented
66#2423C2218yescomplemented
67#2421C2218yescomplemented
68#2417C2218yes—
69#2415C2218yes—
70#2413C2218yes—
71#2412C2218yes—
72C2×#398C2214yes—
73C2×#397C2214yes—
74#2365C2218yescomplemented
75#2362C2218yes—
76#2360C2218yes—
77#2357C2214yescomplemented
78#2356C2214yescomplemented
79#2353C2218yes—
80#2344C2218yes—
81#2344C2218yes—
82#2342C2218yes—
83#2327C2214yescomplemented
84#2324C2214yes—
85#2323C2214yes—
86#2323C2218yes—
87#2322C2214yes—
88#2321C2214yes—
89#2317C2214yes—
90#2317C2214yes—
91#2317C2212yes—
92#2316C2214yes—
93#2316C2212yes—
94#2315C2214yes—
95#2314C2214yescomplemented
96#2311C2212yes—
97#2310C2212yes—
98#2310C2212yes—
99C4×C4⋊Q8C2214yescomplemented
100C4×C22⋊Q8C2218yes—
101#2297C2211yescharacteristic
102#2296C2218yes—
103#2296C2211yescharacteristic
104C2×#391C2212yes—
105#2293C2214yes—
106#2290C2214yes—
107#2285C2211yescharacteristic
108#2285C2212yes—
109#2285C2211yescharacteristic
110#2283C2214yescomplemented
111C2×#382C2211yescharacteristic
112#2280C2211yescharacteristic
113#2274C2212yes—
114C4×#117C2211yescharacteristic
115C22×#41C2311yescharacteristic
116C23×Q8C2312yes—
117C24×C4C2311yescharacteristic
118#536—48no—
119C2×C4⋊Q8—28no—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2314yes—
123C2×C4⋊Q8C2314yes—
124C2×#124C2314yes—
125#525—28no—
126C2×#123C2312yes—
127C2×#123C2318yes—
128C2×C22⋊Q8—216no—
129C2×C22⋊Q8C2314yes—
130C2×C22⋊Q8C2312yes—
131C2×C22⋊Q8C2314yes—
132C2×C22⋊Q8C2314yes—
133C2×C22⋊Q8C2314yes—
134C2×C22⋊Q8C2318yes—
135C2×#121C2312yes—
136C2×#120C2312yes—
137#519—48no—
138C4×#41—28no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2318yes—
141C2×C4×Q8C2318yes—
142C2×C4×D4C2314yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2314yes—
145C2×#117—28no—
146C2×#117—48no—
147C2×#117C2311yescharacteristic
148C2×#117C2311yescharacteristic
149C2×#117C2312yes—
150C2×#117C2312yes—
151C2×#117C2312yes—
152C2×#117C2311yescharacteristic
153C2×#117C2312yes—
154C2×#117C2312yes—
155C2×#117C2314yes—
156C2×#117C2314yes—
157C2×#117C2314yes—
158C2×#117C2314yes—
159C2×#117C2318yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2314yes—
163C22×C4⋊C4C2314yes—
164C22×C4⋊C4C2314yes—
165C22×C22⋊C4C2312yes—
166C22×C22⋊C4C2314yes—
167C22×C22⋊C4C2314yes—
168C22×C42C2312yes—
169C22×C42C2311yescharacteristic
170C22×C42C2314yes—
171C22×C42C2311yescharacteristic
172#401C23116yes—
173#400C23116yes—
174#400C23116yes—
175#400C23116yes—
176#400C23116yes—
177#400C23116yes—
178#399C23116yes—
179#398C23116yes—
180#398C2318yes—
181#398C2318yes—
182#398C23116yes—
183#398C23116yes—
184#398C23116yes—
185#398C23116yes—
186#397C23116yes—
187#397C23116yes—
188#397C23116yes—
189#397C23116yes—
190#397C2318yes—
191#397C2318yes—
192#396C23116yes—
193#394C23116yes—
194#391C2318yes—
195#391C2318yes—
196#391C2314yes—
197#391C2314yes—
198#391C2318yes—
199#391C2314yes—
200#391C2312yes—
201#391C2314yes—
202#391C2314yes—
203#391C2318yes—
204#390C2314yes—
205#390C2318yes—
206#390C2312yes—
207#389C23116yes—
208#389C2318yes—
209#389C2318yes—
210#389C2318yes—
211#389C2318yes—
212#389C2318yes—
213#389C2318yes—
214#389C2318yes—
215#389C2318yes—
216#389C2314yes—
217#389C23116yes—
218#389C23116yes—
219#389C23116yes—
220#388C23116yes—
221#388C2318yes—
222#388C2318yes—
223#388C23116yes—
224#388C2318yes—
225#388C23116yes—
226#388C23116yes—
227#388C23116yes—
228#388C23116yes—
229#387C23116yes—
230#387C23116yes—
231#387C23116yes—
232#387C23116yes—
233#387C23116yes—
234#387C2318yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C23116yes—
241#387C23116yes—
242#386C2318yes—
243#386C2314yes—
244#386C2318yes—
245#386C23116yes—
246#385C2318yes—
247#385C2318yes—
248#385C2318yes—
249#385C2318yes—
250#385C2318yes—
251#385C2314yes—
252#385C2318yes—
253#385C23116yes—
254#384C2312yes—
255#384C2314yes—
256#384C2312yes—
257#384C2314yes—
258#383C23116yes—
259#383C2318yes—
260#382C2312yes—
261#382C2314yes—
262#382C2314yes—
263#382C2312yes—
264#382C2312yes—
265#382C2314yes—
266#382C2312yes—
267#381C2318yes—
268#381C2318yes—
269#381C2312yes—
270#381C2318yes—
271#380C2318yes—
272#380C2314yes—
273#380C2318yes—
274#380C2314yes—
275#380C2318yes—
276#380C23116yes—
277#380C23116yes—
278#380C2314yes—
279#379C2312yes—
280C4×C4⋊C4C23116yes—
281C4×C4⋊C4C2318yes—
282C4×C4⋊C4C2314yes—
283C4×C4⋊C4C2314yes—
284C4×C4⋊C4C2318yes—
285C4×C4⋊C4C23116yes—
286C4×C4⋊C4C2314yes—
287C4×C4⋊C4C2318yes—
288C4×C4⋊C4C2314yes—
289C4×C22⋊C4C2314yes—
290C4×C22⋊C4C23116yes—
291C4×C22⋊C4C2318yes—
292C4×C22⋊C4C2318yes—
293C4×C22⋊C4C2314yes—
294C4×C22⋊C4C2314yes—
295#376C2312yes—
296#376C2318yes—
297C2×#95C2314yes—
298C2×#95C2318yes—
299C43C2312yes—
300C25C2411yescharacteristic
301C2×#41—24no—
302C2×#41—24no—
303C2×#41—24no—
304C22×Q8—28no—
305C22×Q8—28no—
306C22×Q8—28no—
307C22×Q8C2414yes—
308C22×Q8C2414yes—
309C22×Q8C2411yescharacteristic
310C22×D4C2412yes—
311C22×D4C2411yescharacteristic
312C22×D4C2414yes—
313C23×C4—24no—
314C23×C4—24no—
315C23×C4#4118yes—
316C23×C4C2414yes—
317C23×C4C2412yes—
318C23×C4C2411yescharacteristic
319C23×C4C2412yes—
320C23×C4C2412yes—
321C23×C4C2414yes—
322C23×C4C2412yes—
323C23×C4C2414yes—
324C23×C4C2418yes—
325C23×C4C2412yes—
326C23×C4C2411yescharacteristic
327C4⋊Q8—28no—
328C4⋊Q8—28no—
329C4⋊Q8—216no—
330C4⋊Q8—48no—
331#124—48no—
332#123—28no—
333#123—416no—
334C22⋊Q8—216no—
335C22⋊Q8—216no—
336C22⋊Q8—216no—
337C22⋊Q8—216no—
338C22⋊Q8—28no—
339C22⋊Q8—416no—
340#121—28no—
341#120—28no—
342C4×Q8—28no—
343C4×Q8—416no—
344C4×Q8—416no—
345C4×D4—216no—
346C4×D4—216no—
347C4×D4—28no—
348#117—24no—
349#117—24no—
350#117—28no—
351#117—28no—
352#117—24no—
353#117—28no—
354#117—28no—
355#117—216no—
356#117—28no—
357#117—48no—
358#117—48no—
359#117—416no—
360#117—48no—
361C2×C4⋊C4—28no—
362C2×C4⋊C4—216no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—216no—
365C2×C4⋊C4—48no—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2412yes—
380C2×C4⋊C4C2414yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C24116yes—
383C2×C4⋊C4C24116yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C2418yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C2418yes—
388C2×C4⋊C4C24116yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C24116yes—
391C2×C22⋊C4—216no—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—48no—
394C2×C22⋊C4C2414yes—
395C2×C22⋊C4C2414yes—
396C2×C22⋊C4C2414yes—
397C2×C22⋊C4C2414yes—
398C2×C22⋊C4C2418yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2412yes—
401C2×C22⋊C4C2412yes—
402C2×C22⋊C4C2414yes—
403C2×C22⋊C4C2412yes—
404C2×C22⋊C4C2414yes—
405C2×C22⋊C4C2414yes—
406C2×C22⋊C4C2418yes—
407C2×C22⋊C4C24116yes—
408C2×C22⋊C4C2418yes—
409C2×C22⋊C4C24116yes—
410C2×C22⋊C4C2418yes—
411C2×C42—24no—
412C2×C42—28no—
413C2×C42—48no—
414C2×C42#4118yes—
415C2×C42C2414yes—
416C2×C42C2414yes—
417C2×C42C2412yes—
418C2×C42C2412yes—
419C2×C42C2412yes—
420C2×C42C2412yes—
421C2×C42C2412yes—
422C2×C42C2414yes—
423C2×C42C2418yes—
424C2×C42C24116yes—
425C2×C42C2418yes—
426C2×C42C24116yes—
427C2×C42C2411yescharacteristic
428C2×C42C2412yes—
429C2×C42C2411yescharacteristic
430#95C2418yes—
431#95C2418yes—
432#95C24116yes—
433#95C24116yes—
434#95C2414yes—
435#95C2418yes—
436#95C24116yes—
437#95C2414yes—
438#95C2418yes—
439#95C24116yes—
440#95C24116yes—
441#95C24116yes—
442#95C2418yes—
443#95C24116yes—
444#95C24116yes—
445#95C2418yes—
446#95C2414yes—
447#95C2414yes—
448#95C2418yes—
449#95C24116yes—
450#95C2418yes—
451#95C24116yes—
452C24—22no—
453C24—22no—
454C24—44no—
455C24C2×#4114yes—
456C24C2512yes—
457C24C2512yes—
458C24C2511yescharacteristic
459#41—416no—
460C2×Q8—22no—
461C2×Q8—28no—
462C2×Q8—28no—
463C2×Q8—22no—
464C2×Q8—28no—
465C2×Q8—28no—
466C2×Q8—28no—
467C2×Q8—28no—
468C2×Q8—22no—
469C2×Q8—432no—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—28no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—48no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—232no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—28no—
488C22×C4—24no—
489C22×C4—28no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—28no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4C2×#4114yes—
502C22×C4C2×#4118yes—
503C22×C4C2×#4118yes—
504C22×C4C2×#4118yes—
505C22×C4C2×#4114yes—
506C22×C4C2×#4114yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C4C2514yes—
509C22×C4C2514yes—
510C22×C4C2514yes—
511C22×C4C2518yes—
512C22×C4C2518yes—
513C22×C4C2518yes—
514C22×C4C2518yes—
515C22×C4C2512yes—
516C22×C4C2514yes—
517C22×C4C2512yes—
518C22×C4C2512yes—
519C22×C4C2512yes—
520C22×C4C2511yescharacteristic
521C22×C4C2511yescharacteristic
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—216no—
527C4⋊C4—216no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—24no—
531C4⋊C4—28no—
532C4⋊C4—24no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—24no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—416no—
543C4⋊C4—416no—
544C4⋊C4—48no—
545C4⋊C4—416no—
546C4⋊C4—416no—
547C22⋊C4—216no—
548C22⋊C4—216no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—28no—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—24no—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—28no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—416no—
563C22⋊C4—416no—
564C42—24no—
565C42—24no—
566C42—24no—
567C42—28no—
568C42—24no—
569C42—28no—
570C42—28no—
571C42—24no—
572C42—48no—
573C42—48no—
574C42—416no—
575C42—416no—
576C42C2×#4114yes—
577C42C2×#4114yes—
578C42C2×#4118yes—
579C23—24no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—24no—
584C23—28no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—28no—
589C23—48no—
590C23—84no—
591C23—84no—
592C23C22×#4114yes—
593C23C22×#4112yes—
594C23C22×#4112yes—
595C23C2×H(5,𝔽2)12yes—
596C23C2611yescharacteristic
597Q8—416no—
598Q8—416no—
599Q8—44no—
600D4—48no—
601D4—44no—
602D4—88no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—216no—
609C2×C4—216no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—216no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—48no—
634C2×C4—416no—
635C2×C4—48no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4—816no—
640C2×C4#553116yes—
641C2×C4C22×#4114yes—
642C2×C4C22×#4114yes—
643C2×C4C22×#4114yes—
644C2×C4C22×#4112yes—
645C2×C4C22×#4112yes—
646C2×C4C22×#4112yes—
647C2×C4C22×#4112yes—
648C2×C4C2×H(5,𝔽2)12yes—
649C2×C4C2×H(5,𝔽2)12yes—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—88no—
663C22C2×#55314yes—
664C22C23×#4111yescharacteristic
665C22C23×#4111yescharacteristic
666C22C22×H(5,𝔽2)11yescharacteristic
667C222−1+614yes—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—48no—
674C4—44no—
675C4—44no—
676C4—48no—
677C4—48no—
678C4—42no—
679C4—48no—
680C4C2×#55314yes—
681C4C2×#55314yes—
682C2—24no—
683C2—44no—
684C2—82no—
685C2C22×#55311yescharacteristic
686C2#6286012yes—
687C2#6286012yes—
688C2#6288812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #44462C2semidirect
2. #43511C2semidirect
3. #39822C2semidirect
4. #39819C2semidirect
5. #39553C2semidirect
6. #37947C2semidirect
7. #35673C2semidirect
8. #35630C2semidirect
9. #35339C2semidirect
10. #35316C2semidirect
12. C2×#2621C2semidirect
13. #34209C2semidirect
14. #34203C2semidirect
15. #34189C2semidirect
17. #34052C2semidirect
26. #2864C22semidirect
28. #2857C22semidirect
30. #2802C22semidirect
33. #2777C22semidirect
34. #2776C22semidirect
37. #2739C22semidirect
38. #2647C22semidirect
43. #2621C22semidirect
45. #2606C22semidirect
47. #2594C22semidirect
48. #2590C22semidirect
51. #2570C22semidirect
53. #2547C22semidirect
54. #2546C22semidirect
56. #2522C22semidirect
57. #2514C22semidirect
58. #2514C22semidirect
59. #2511C22semidirect
60. #2511C22semidirect
64. #2448C22semidirect
65. #2446C22semidirect
66. #2423C22semidirect
67. #2421C22semidirect
74. #2365C22semidirect
77. #2357C22semidirect
78. #2356C22semidirect
83. #2327C22semidirect
95. #2314C22semidirect
99. C4×C4⋊Q8C22semidirect
110. #2283C22semidirect

Maximal subgroups

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

subgroupindex
#444622
#435112
#398222
#398192
#395532
#379472
#356732
#356302
#353392
#353162
#353062
C2×#26212
#342092
#342032
#341892
#340652
#340522