Finite Group Databasebeta

← #7711411

Subgroups — #7711411

Order 512.

6978 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3544C212yesmaximal, complemented
2#59655C214yesmaximal, complemented
3#59652C214yesmaximal, complemented
4#46281C218yesmaximal, complemented
5#39487C214yesmaximal, complemented
6#39424C218yesmaximal, complemented
7#39381C214yesmaximal, complemented
8#39365C214yesmaximal, complemented
9#39359C214yesmaximal, complemented
10#37620C212yesmaximal, complemented
11#37190C214yesmaximal, complemented
12#36631C212yesmaximal, complemented
13#36424C218yesmaximal, complemented
14#35153C214yesmaximal, complemented
15C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
16#3544—232no—
17C2×#563C2218yes—
18C2×Q82C2212yes—
19C2×#555C2212yes—
20C2×#554C2214yes—
21C2×#552C2218yes—
22C2×#547C2218yes—
23C2×#544C2218yes—
24C2×#542C2218yes—
25C2×#537C2214yes—
26C2×#536C2218yes—
27C22×C4⋊Q8C2211yescharacteristic
28C2×#529C2214yes—
29C22×C4×Q8C2212yes—
30#2861C2214yes—
31#2860C2214yes—
32#2860C2214yes—
33#2859C2212yes—
34#2858C2212yes—
35#2857C2212yes—
36#2856C2212yes—
37#2854C2214yes—
38#2854C2214yes—
39#2854C2218yes—
40#2850C2214yes—
41#2823C2218yes—
42#2734C2214yes—
43#2733C2214yes—
44#2732C2218yes—
45#2731C2218yes—
46#2730C22116yes—
47#2729C22116yes—
48#2728C2218yes—
49#2727C2218yes—
50#2726C2218yes—
51#2725C22116yes—
52#2683C2218yes—
53#2682C2218yes—
54#2681C22116yes—
55#2629C2218yes—
56#2620C2218yes—
57#2618C2218yes—
58#2618C2218yes—
59#2582C2212yes—
60#2560C22116yes—
61#2558C22116yes—
62#2556C22116yes—
63#2554C22116yes—
64#2550C2218yes—
65#2540C2218yes—
66#2538C2214yes—
67#2537C2212yes—
68#2498C22116yes—
69#2492C2218yes—
70#2490C2218yes—
71#2453C2218yes—
72#2452C2218yes—
73#2441C2218yes—
74#2440C2218yes—
75#2430C22116yes—
76#2429C22116yes—
77#2429C2218yes—
78#2429C22116yes—
79#2428C2218yes—
80#2426C22116yes—
81#2426C22116yes—
82#2417C2218yes—
83#2416C2218yes—
84#2416C2218yes—
85#2415C2218yes—
86#2367C2218yes—
87#2363C2218yes—
88#2360C2218yes—
89#2352C2218yes—
90#2316C2214yes—
91C4×C4⋊Q8C2214yes—
92C4×C4⋊Q8C2214yes—
93C4×C4⋊Q8C2212yes—
94C4×C4⋊Q8C2214yes—
95C4×C4⋊Q8C2214yes—
96C4×#127C2212yes—
97C4×#124C2214yes—
98C4×#121C2214yes—
99C4×#121C2214yes—
100#2297C2218yes—
101C2×#391C2212yes—
102#2293C2214yes—
103#2293C2214yes—
104C2×#389C2214yes—
105C2×#384C2211yescharacteristic
106C2×C4×C4⋊C4C2214yes—
107C2×C43C2211yescharacteristic
108C22×#41C2318yes—
109C23×Q8C2312yes—
110#563—264no—
111Q82—216no—
112#555—216no—
113#554—232no—
114#552—264no—
115#547—264no—
116#544—264no—
117#542—264no—
118#537—416no—
119#536—264no—
120C2×C4⋊Q8D4116yescomplemented
121C2×C4⋊Q8C2318yes—
122C2×C4⋊Q8C2318yes—
123C2×C4⋊Q8C2318yes—
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2314yes—
126C2×C4⋊Q8C2312yes—
127C2×C4⋊Q8C2312yes—
128C2×#127C2312yes—
129#529—232no—
130C2×#126C23116yes—
131C2×#126C2318yes—
132C2×#125C2314yes—
133C2×#125C2318yes—
134C2×#125C23116yes—
135C2×#124C2314yes—
136C2×#124C2314yes—
137C2×#124C2318yes—
138C2×#124C2318yes—
139C2×#123C2318yes—
140C2×#123C23116yes—
141C2×#123C23116yes—
142C2×C22⋊Q8C23116yes—
143C2×C22⋊Q8C2314yes—
144C2×C22⋊Q8C2318yes—
145C2×C22⋊Q8C23116yes—
146C2×C22⋊Q8C23116yes—
147C2×C22⋊Q8C2318yes—
148C2×C22⋊Q8C23116yes—
149C2×#121C2314yes—
150C2×#121C2314yes—
151C2×#121C2318yes—
152C2×#121C2314yes—
153C2×#121C2318yes—
154C2×#121C2318yes—
155C2×#121C23116yes—
156C2×C4×Q8—48no—
157C2×C4×Q8C2314yes—
158C2×C4×Q8C2318yes—
159C2×C4×Q8C2318yes—
160C2×C4×Q8C2318yes—
161C2×C4×D4C2314yes—
162C2×C4×D4C2314yes—
163C2×C4×D4C2318yes—
164C2×C4×D4C2318yes—
165C2×C4×D4C2314yes—
166C2×#117C2318yes—
167C2×#117C2318yes—
168C2×#117C2318yes—
169C22×C4⋊C4C2314yes—
170C22×C4⋊C4C2318yes—
171C22×C4⋊C4C2312yes—
172C22×C4⋊C4C2314yes—
173C22×C4⋊C4C2314yes—
174C22×C42C2312yes—
175C22×C42C2311yescharacteristic
176C22×C42C2311yescharacteristic
177C22×C42C2314yes—
178C22×C42C2312yes—
179C22×C42C2311yescharacteristic
180#400C23116yes—
181#399C23116yes—
182#398C23116yes—
183#398C2318yes—
184#398C2318yes—
185#398C23116yes—
186#398C23116yes—
187#397C23116yes—
188#396C23116yes—
189#396C23116yes—
190#396C23116yes—
191#396C23116yes—
192#396C23116yes—
193#396C23116yes—
194#395C23116yes—
195#394C23116yes—
196#394C2318yes—
197#394C2318yes—
198#394C23116yes—
199#394C23116yes—
200#393C23116yes—
201#392C23116yes—
202#391C23116yes—
203#391C2314yes—
204#391C2318yes—
205#391C2314yes—
206#391C2314yes—
207#391C2318yes—
208#391C2314yes—
209#390C23116yes—
210#390C2314yes—
211#390C2314yes—
212#390C2314yes—
213#390C2314yes—
214#390C2314yes—
215#390C2314yes—
216#390C2314yes—
217#390C2314yes—
218#390C2314yes—
219#390C2314yes—
220#390C2318yes—
221#390C23116yes—
222#389C2318yes—
223#389C2318yes—
224#389C2318yes—
225#389C2318yes—
226#389C2314yes—
227#389C2314yes—
228#389C23116yes—
229#389C23116yes—
230#389C2318yes—
231#389C2314yes—
232#389C2314yes—
233#389C23116yes—
234#389C2314yes—
235#389C2314yes—
236#389C2314yes—
237#389C2314yes—
238#389C2318yes—
239#388C2318yes—
240#388C2318yes—
241#388C23116yes—
242#388C2318yes—
243#388C2318yes—
244#388C2318yes—
245#388C2318yes—
246#388C2318yes—
247#388C23116yes—
248#388C2318yes—
249#388C23116yes—
250#388C23116yes—
251#387C23116yes—
252#387C23116yes—
253#386C23116yes—
254#386C2318yes—
255#386C2318yes—
256#386C23116yes—
257#386C23116yes—
258#385C23116yes—
259#385C23116yes—
260#384C2312yes—
261#384C2312yes—
262#384C2312yes—
263#384C2314yes—
264#384C2312yes—
265#382C2314yes—
266#380C2314yes—
267#380C2318yes—
268#380C2314yes—
269#380C2318yes—
270#380C2318yes—
271C4×C4⋊C4C2318yes—
272C4×C4⋊C4C2318yes—
273C4×C4⋊C4C2318yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2314yes—
276C4×C4⋊C4C2314yes—
277C4×C22⋊C4C23116yes—
278C4×C22⋊C4C23116yes—
279C4×C22⋊C4C2318yes—
280C4×C22⋊C4C2318yes—
281C2×#95C2318yes—
282C43C2312yes—
283C43C2312yes—
284C43C2312yes—
285C43C2312yes—
286C2×#41—232no—
287C2×#41—416no—
288C2×#41—416no—
289C22×Q8—44no—
290C22×Q8—44no—
291C22×Q8C2×D4116yes—
292C22×Q8C2414yes—
293C22×Q8C2418yes—
294C22×Q8C2414yes—
295C22×Q8C2414yes—
296C22×D4C2414yes—
297C22×D4C2418yes—
298C22×D4C2414yes—
299C22×D4C2414yes—
300C22×D4C2418yes—
301C22×D4C2418yes—
302C23×C4C2418yes—
303C23×C4C2411yescharacteristic
304C23×C4C2412yes—
305C23×C4C2412yes—
306C23×C4C2414yes—
307C23×C4C2411yescharacteristic
308C23×C4C2414yes—
309C23×C4C24116yes—
310C23×C4C2411yescharacteristic
311C4⋊Q8—232no—
312C4⋊Q8—232no—
313C4⋊Q8—232no—
314C4⋊Q8—44no—
315C4⋊Q8C2×D4132yes—
316C4⋊Q8C2×D4116yes—
317C4⋊Q8C2×D4116yes—
318#127—44no—
319#126—264no—
320#126—232no—
321#125—232no—
322#125—264no—
323#125—216no—
324#124—232no—
325#124—232no—
326#124—216no—
327#124—216no—
328#123—232no—
329#123—264no—
330#123—264no—
331C22⋊Q8—264no—
332C22⋊Q8—264no—
333C22⋊Q8—216no—
334C22⋊Q8—232no—
335C22⋊Q8—264no—
336C22⋊Q8—432no—
337C22⋊Q8—416no—
338#121—216no—
339#121—216no—
340#121—232no—
341#121—432no—
342#121—416no—
343#121—48no—
344#121—416no—
345C4×Q8—232no—
346C4×Q8—216no—
347C4×Q8—416no—
348C4×Q8—416no—
349C4×D4—232no—
350C4×D4—232no—
351C4×D4—216no—
352C4×D4—216no—
353C4×D4—48no—
354#117—232no—
355#117—232no—
356#117—232no—
357C2×C4⋊C4—216no—
358C2×C4⋊C4—232no—
359C2×C4⋊C4—48no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4C2×D4132yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C24116yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C24116yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C24116yes—
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C24116yes—
374C2×C4⋊C4C2414yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2414yes—
377C2×C4⋊C4C2418yes—
378C2×C4⋊C4C2414yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2414yes—
381C2×C4⋊C4C24116yes—
382C2×C4⋊C4C2414yes—
383C2×C4⋊C4C2414yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C2418yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C2418yes—
388C2×C22⋊C4C24116yes—
389C2×C22⋊C4C2418yes—
390C2×C22⋊C4C24116yes—
391C2×C22⋊C4C2418yes—
392C2×C22⋊C4C2418yes—
393C2×C22⋊C4C2418yes—
394C2×C22⋊C4C2418yes—
395C2×C22⋊C4C2418yes—
396C2×C22⋊C4C2418yes—
397C2×C22⋊C4C2418yes—
398C2×C22⋊C4C24116yes—
399C2×C22⋊C4C24116yes—
400C2×C22⋊C4C2418yes—
401C2×C22⋊C4C24116yes—
402C2×C42—24no—
403C2×C42—28no—
404C2×C42—44no—
405C2×C42—48no—
406C2×C42—42no—
407C2×C42C2×D418yes—
408C2×C42C2418yes—
409C2×C42C2418yes—
410C2×C42C2418yes—
411C2×C42C2412yes—
412C2×C42C2412yes—
413C2×C42C2412yes—
414C2×C42C2412yes—
415C2×C42C2412yes—
416C2×C42C2412yes—
417C2×C42C2412yes—
418C2×C42C24116yes—
419C2×C42C2411yescharacteristic
420C2×C42C2412yes—
421C2×C42C2412yes—
422C2×C42C2412yes—
423C2×C42C2411yescharacteristic
424C2×C42C2412yes—
425C2×C42C2411yescharacteristic
426C2×C42C2411yescharacteristic
427C2×C42C2412yes—
428#95C24116yes—
429#95C24116yes—
430#95C2418yes—
431#95C2418yes—
432#95C24116yes—
433#95C2418yes—
434#95C2418yes—
435#95C2418yes—
436#95C24116yes—
437#95C2418yes—
438#95C2418yes—
439#95C2418yes—
440C24C2518yes—
441C24C2511yescharacteristic
442#41—864no—
443C2×Q8—28no—
444C2×Q8—216no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—432no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—48no—
453C2×Q8—48no—
454C2×Q8C22×D4116yes—
455C2×Q8C22×D4116yes—
456C2×D4—216no—
457C2×D4—216no—
458C2×D4—216no—
459C2×D4—28no—
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4—28no—
465C2×D4—44no—
466C2×D4—48no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—48no—
472C2×D4—416no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—216no—
484C22×C4—24no—
485C22×C4—216no—
486C22×C4—216no—
487C22×C4—232no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—48no—
491C22×C4—416no—
492C22×C4—416no—
493C22×C4C22×D418yes—
494C22×C4C22×D414yes—
495C22×C4C22×D4116yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4C2512yes—
498C22×C4C2518yes—
499C22×C4C2518yes—
500C22×C4C2518yes—
501C22×C4C2518yes—
502C22×C4C2518yes—
503C22×C4C2512yes—
504C22×C4C2511yescharacteristic
505C22×C4C2511yescharacteristic
506C22×C4C2512yes—
507C22×C4C2512yes—
508C22×C4C2512yes—
509C22×C4C2512yes—
510C4⋊C4—28no—
511C4⋊C4—232no—
512C4⋊C4—28no—
513C4⋊C4—232no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—232no—
519C4⋊C4—232no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—416no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4C22×D4132yes—
533C4⋊C4C22×D4132yes—
534C22⋊C4—216no—
535C22⋊C4—216no—
536C22⋊C4—232no—
537C22⋊C4—216no—
538C22⋊C4—232no—
539C22⋊C4—216no—
540C22⋊C4—216no—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—216no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—416no—
548C22⋊C4—416no—
549C22⋊C4—416no—
550C42—216no—
551C42—216no—
552C42—216no—
553C42—24no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—24no—
558C42—24no—
559C42—24no—
560C42—42no—
561C42—42no—
562C42—41no—
563C42—42no—
564C42—41no—
565C42—416no—
566C42C22×D418yes—
567C42C22×D414yes—
568C42C22×D414yes—
569C422−1+418yes—
570C23—22no—
571C23—22no—
572C23—28no—
573C23—28no—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—28no—
578C23—28no—
579C23C23×D412yes—
580C23C2×H(5,𝔽2)12yes—
581C23#58512yes—
582C23C2611yescharacteristic
583Q8—416no—
584Q8—416no—
585Q8—416no—
586Q8—816no—
587D4—432no—
588D4—416no—
589D4—88no—
590D4—88no—
591D4—816no—
592D4—168no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—232no—
610C2×C4—216no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—28no—
622C2×C4—216no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—432no—
628C2×C4—832no—
629C2×C4—88no—
630C2×C4C23×D4116yes—
631C2×C4C23×D414yes—
632C2×C4C23×D412yes—
633C2×C4C23×D412yes—
634C2×C4C23×D414yes—
635C2×C4C2×H(5,𝔽2)14yes—
636C2×C4C2×2−1+414yes—
637C2×C4C2×2−1+414yes—
638C2×C4C2×2−1+414yes—
639C2×C4#58514yes—
640C22—24no—
641C22—24no—
642C22—22no—
643C22—22no—
644C22—48no—
645C22—48no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—48no—
650C22—48no—
651C22#349614yes—
652C22C24×D411yescharacteristic
653C22C22×H(5,𝔽2)11yescharacteristic
654C22C22×2−1+412yes—
655C22C2×#58511yescharacteristic
656C22C2×#58512yes—
657C4—24no—
658C4—216no—
659C4—22no—
660C4—22no—
661C4—24no—
662C4—24no—
663C4—24no—
664C4—42no—
665C4—48no—
666C4—48no—
667C4—48no—
668C4—48no—
669C4#349718yes—
670C2—24no—
671C2—88no—
672C2#6275212yes—
673C2C2×#349611yescharacteristic
674C2C2×#349712yes—
675C2#6282211yescharacteristic
676C2#6283311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3544C2semidirect
2. #59655C2semidirect
3. #59652C2semidirect
4. #46281C2semidirect
5. #39487C2semidirect
6. #39424C2semidirect
7. #39381C2semidirect
8. #39365C2semidirect
9. #39359C2semidirect
10. #37620C2semidirect
11. #37190C2semidirect
12. #36631C2semidirect
13. #36424C2semidirect
14. #35153C2semidirect
15. C2×C4×C4⋊Q8C2semidirect
120. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#35442
#596552
#596522
#462812
#394872
#394242
#393812
#393652
#393592
#376202
#371902
#366312
#364242
#351532
C2×C4×C4⋊Q82