Finite Group Databasebeta

← #7688273

Subgroups — #7688273

Order 512.

7283 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C212yesmaximal, complemented
2#59823C213yesmaximal, complemented
3#59817C216yesmaximal, complemented
4#59136C216yesmaximal, complemented
5#44879C2112yesmaximal, complemented
6#39598C216yesmaximal, complemented
7#39370C216yesmaximal, complemented
8#36559C213yesmaximal, complemented
9Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
10#36466C212yesmaximal, complemented
11#36456C212yesmaximal, complemented
12D4×C4⋊Q8C213yesmaximal, complemented
13#35816C214yesmaximal, complemented
14#34929C213yesmaximal, complemented
15#34313C212yesmaximal
16C4×Q82C211yesmaximal, characteristic, complemented
17C4×#550C211yesmaximal, characteristic
18#3503—232no—
19C2×Q82C2212yes—
20C2×Q82C2211yescharacteristic
21C2×#557C2214yes—
22C2×#554C2216yes—
23C2×#554C2216yes—
24C2×#550C2211yescharacteristic
25C2×D4×Q8C22112yes—
26C2×D4×Q8C2216yes—
27C2×#544C22112yes—
28C2×#533C2216yes—
29C22×C4⋊Q8C2216yes—
30C2×#520C2214yes—
31C2×C4×#41C2212yes—
32#2868C2213yes—
33#2859C2216yes—
34#2857C2216yes—
35#2732C22112yes—
36#2729C22124yes—
37#2704C22124yes—
38#2683C22124yes—
39#2629C22112yes—
40#2584C22112yes—
41#2567C2216yes—
42#2566C22112yes—
43#2563C2216yes—
44#2556C22112yes—
45#2552C2216yes—
46#2551C2213yes—
47#2551C2216yes—
48#2551C2216yes—
49#2551C2216yes—
50#2550C22112yes—
51#2546C2216yes—
52#2546C2211yescharacteristic
53#2542C2213yes—
54#2538C2216yes—
55#2538C2213yes—
56#2538C22112yes—
57#2538C22112yes—
58#2538C2216yes—
59#2506C2216yes—
60#2492C2216yes—
61#2454C22112yes—
62#2452C22112yes—
63#2452C22112yes—
64#2452C22112yes—
65#2452C2216yes—
66#2452C22112yes—
67#2442C22112yes—
68#2440C22112yes—
69#2440C22112yes—
70#2439C22112yes—
71#2434C22112yes—
72#2429C2216yes—
73#2417C22112yes—
74#2409C22124yes—
75#2396C22112yes—
76#2369C2213yes—
77#2369C2216yes—
78#2366C2216yes—
79#2366C2213yes—
80#2363C2216yes—
81#2361C22112yes—
82#2354C2213yes—
83#2351C2214yes—
84Q8×C4⋊C4C2212yes—
85Q8×C4⋊C4C2211yescharacteristic
86Q8×C4⋊C4C2211yescharacteristic
87Q8×C4⋊C4C2213yes—
88Q8×C4⋊C4C2212yes—
89Q8×C4⋊C4C2212yes—
90D4×C4⋊C4C2216yes—
91D4×C4⋊C4C2216yes—
92D4×C4⋊C4C2216yes—
93#2341C2214yes—
94#2341C2212yes—
95#2324C22112yes—
96C4×C4⋊Q8C2213yes—
97C4×C4⋊Q8C2216yes—
98C4×C4⋊Q8C2213yes—
99C4×C4⋊Q8C2216yes—
100C4×#127C2216yes—
101C4×#127C2213yes—
102C4×C22⋊Q8C2216yes—
103C4×#121C2216yes—
104#2293C2216yes—
105C42×Q8C2212yes—
106C42×Q8C2211yescharacteristic
107C42×Q8C2213yes—
108C42×D4C2213yes—
109C22×#41C2312yes—
110C23×Q8C23112yes—
111Q82—216no—
112Q82D4116yescomplemented
113#557—232no—
114#554—248no—
115#554—248no—
116#550Q8116yescomplemented
117D4×Q8—296no—
118D4×Q8—424no—
119#544—296no—
120#533—248no—
121C2×C4⋊Q8—248no—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2316yes—
124C2×C4⋊Q8C2316yes—
125C2×C4⋊Q8C2316yes—
126C2×C4⋊Q8C23112yes—
127C2×C4⋊Q8C23112yes—
128C2×C4⋊Q8C2312yes—
129C2×C4⋊Q8C23112yes—
130C2×C4⋊Q8C2316yes—
131C2×C4⋊Q8C2316yes—
132C2×C4⋊Q8C2313yes—
133C2×C4⋊Q8C2316yes—
134C2×#127C2313yes—
135C2×#125C23112yes—
136C2×#125C23112yes—
137C2×#125C23112yes—
138C2×C22⋊Q8C23124yes—
139C2×C22⋊Q8C2316yes—
140C2×C22⋊Q8C23112yes—
141C2×C22⋊Q8C2316yes—
142C2×C22⋊Q8C23124yes—
143C2×C22⋊Q8C23112yes—
144C2×C22⋊Q8C23124yes—
145C2×C22⋊Q8C2316yes—
146C2×#121C2316yes—
147#520—232no—
148C4×#41—216no—
149C2×C4×Q8C2316yes—
150C2×C4×Q8C23112yes—
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2312yes—
153C2×C4×Q8C2314yes—
154C2×C4×Q8C2312yes—
155C2×C4×Q8C2311yescharacteristic
156C2×C4×Q8C2316yes—
157C2×C4×Q8C2313yes—
158C2×C4×Q8C2311yescharacteristic
159C2×C4×Q8C2312yes—
160C2×C4×Q8C2313yes—
161C2×C4×Q8C2311yescharacteristic
162C2×C4×D4C2313yes—
163C2×C4×D4C2316yes—
164C2×C4×D4C23112yes—
165C2×C4×D4C23112yes—
166C2×C4×D4C2316yes—
167C2×C4×D4C2313yes—
168C2×C4×D4C23112yes—
169C2×C4×D4C2316yes—
170C2×C4×D4C23112yes—
171C2×C4×D4C2313yes—
172C2×#117C23112yes—
173C2×#117C2316yes—
174C22×C4⋊C4C23112yes—
175C22×C4⋊C4C23112yes—
176C22×C42C2316yes—
177#400C23112yes—
178#399C23112yes—
179#398C23124yes—
180#398C23112yes—
181#398C23124yes—
182#398C2314yes—
183#396C23112yes—
184#396C23124yes—
185#396C23124yes—
186#396C23112yes—
187#396C23124yes—
188#395C23124yes—
189#395C23124yes—
190#395C23112yes—
191#393C23124yes—
192#393C23112yes—
193#391C2316yes—
194#391C23112yes—
195#391C2316yes—
196#391C23112yes—
197#391C23112yes—
198#391C2312yes—
199#390C2316yes—
200#390C23112yes—
201#390C2316yes—
202#390C23112yes—
203#390C2316yes—
204#390C2316yes—
205#390C23112yes—
206#390C2316yes—
207#390C2316yes—
208#389C2312yes—
209#389C23112yes—
210#389C2316yes—
211#389C2316yes—
212#389C23112yes—
213#389C2316yes—
214#389C2316yes—
215#389C2316yes—
216#389C2316yes—
217#389C2316yes—
218#389C23112yes—
219#389C23112yes—
220#389C23112yes—
221#389C2316yes—
222#389C23112yes—
223#389C23112yes—
224#389C2316yes—
225#389C23112yes—
226#389C23112yes—
227#389C2316yes—
228#389C2312yes—
229#388C23112yes—
230#388C23112yes—
231#388C23112yes—
232#387C23112yes—
233#386C23124yes—
234#386C23112yes—
235#385C23124yes—
236#385C23124yes—
237#385C23124yes—
238#385C23112yes—
239#384C2313yes—
240#384C2316yes—
241#384C2313yes—
242#384C2313yes—
243#384C2316yes—
244#384C2316yes—
245#384C2316yes—
246#384C2316yes—
247#382C23112yes—
248#380C2316yes—
249#380C23112yes—
250#380C23112yes—
251#380C2316yes—
252#380C23124yes—
253C4×C4⋊C4C2313yes—
254C4×C4⋊C4C2316yes—
255C4×C4⋊C4C2313yes—
256C4×C4⋊C4C2316yes—
257C4×C4⋊C4C2313yes—
258C4×C4⋊C4C2316yes—
259C4×C4⋊C4C2316yes—
260C4×C4⋊C4C23112yes—
261C4×C4⋊C4C2313yes—
262C4×C4⋊C4C2316yes—
263C4×C4⋊C4C2313yes—
264C4×C4⋊C4C2316yes—
265C4×C4⋊C4C2313yes—
266C4×C4⋊C4C2313yes—
267C4×C4⋊C4C2316yes—
268C4×C4⋊C4C2313yes—
269C4×C4⋊C4C2316yes—
270C4×C4⋊C4C2311yescharacteristic
271C4×C22⋊C4C23112yes—
272C4×C22⋊C4C2316yes—
273C4×C22⋊C4C2316yes—
274C4×C22⋊C4C2316yes—
275C43C2316yes—
276C43C2313yes—
277C2×#41—28no—
278C2×#41—28no—
279C2×#41C2×Q8116yescomplemented
280C22×Q8—248no—
281C22×Q8—424no—
282C22×Q8—812no—
283C22×Q8C2412yes—
284C22×Q8C24112yes—
285C22×Q8C2411yescharacteristic
286C22×Q8C24112yes—
287C22×Q8C2416yes—
288C22×Q8C2411yescharacteristic
289C22×D4C2413yes—
290C22×D4C2416yes—
291C22×D4C2416yes—
292C23×C4C2416yes—
293C23×C4C2416yes—
294C23×C4C2416yes—
295C23×C4C24124yes—
296C4⋊Q8—248no—
297C4⋊Q8—28no—
298C4⋊Q8—224no—
299C4⋊Q8—248no—
300C4⋊Q8—248no—
301C4⋊Q8—224no—
302C4⋊Q8—224no—
303C4⋊Q8—24no—
304C4⋊Q8—412no—
305C4⋊Q8—86no—
306C4⋊Q8C2×D4124yes—
307C4⋊Q8C2×D4148yes—
308#127C2×Q8124yescomplemented
309#125—248no—
310#125—248no—
311#125—248no—
312C22⋊Q8—224no—
313C22⋊Q8—296no—
314C22⋊Q8—248no—
315C22⋊Q8—224no—
316C22⋊Q8—296no—
317C22⋊Q8—448no—
318C22⋊Q8—424no—
319C22⋊Q8—412no—
320#121C2×Q8148yescomplemented
321C4×Q8—28no—
322C4×Q8—216no—
323C4×Q8—24no—
324C4×Q8—28no—
325C4×Q8—248no—
326C4×Q8—224no—
327C4×Q8—24no—
328C4×Q8—412no—
329C4×Q8—46no—
330C4×Q8C2×D418yes—
331C4×Q8C2×D4116yes—
332C4×Q8C2×D4124yes—
333C4×Q8C2×Q818yescomplemented
334C4×D4—224no—
335C4×D4—248no—
336C4×D4—212no—
337C4×D4—224no—
338C4×D4—248no—
339C4×D4—212no—
340C4×D4—424no—
341C4×D4—412no—
342C4×D4—812no—
343C4×D4C2×Q8124yescomplemented
344#117—224no—
345#117—248no—
346C2×C4⋊C4—248no—
347C2×C4⋊C4—248no—
348C2×C4⋊C4C24112yes—
349C2×C4⋊C4C2416yes—
350C2×C4⋊C4C24112yes—
351C2×C4⋊C4C24124yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2416yes—
354C2×C4⋊C4C2416yes—
355C2×C4⋊C4C2413yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C24112yes—
358C2×C4⋊C4C24112yes—
359C2×C4⋊C4C24112yes—
360C2×C4⋊C4C2413yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2413yes—
364C2×C4⋊C4C2416yes—
365C2×C4⋊C4C24112yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C24112yes—
369C2×C4⋊C4C2416yes—
370C2×C4⋊C4C24112yes—
371C2×C4⋊C4C2416yes—
372C2×C4⋊C4C2416yes—
373C2×C4⋊C4C24112yes—
374C2×C4⋊C4C24112yes—
375C2×C4⋊C4C2416yes—
376C2×C4⋊C4C2413yes—
377C2×C4⋊C4C2416yes—
378C2×C4⋊C4C2413yes—
379C2×C4⋊C4C2416yes—
380C2×C4⋊C4C2411yescharacteristic
381C2×C4⋊C4C2413yes—
382C2×C4⋊C4C2411yescharacteristic
383C2×C22⋊C4C24112yes—
384C2×C22⋊C4C24112yes—
385C2×C22⋊C4C24112yes—
386C2×C22⋊C4C2416yes—
387C2×C22⋊C4C2416yes—
388C2×C22⋊C4C24112yes—
389C2×C22⋊C4C24124yes—
390C2×C22⋊C4C24112yes—
391C2×C22⋊C4C24112yes—
392C2×C22⋊C4C2416yes—
393C2×C42—224no—
394C2×C42C2416yes—
395C2×C42C24112yes—
396C2×C42C2416yes—
397C2×C42C2413yes—
398C2×C42C2416yes—
399C2×C42C2413yes—
400C2×C42C2412yes—
401C2×C42C2416yes—
402C2×C42C2413yes—
403C2×C42C2412yes—
404C2×C42C2416yes—
405C2×C42C2416yes—
406C2×C42C2413yes—
407C2×C42C2416yes—
408C2×C42C2413yes—
409C2×C42C2413yes—
410C2×C42C2411yescharacteristic
411#95C2412yes—
412#95C2414yes—
413#95C2416yes—
414#95C2416yes—
415#95C24112yes—
416#95C2416yes—
417#95C2416yes—
418#95C2414yes—
419#95C24112yes—
420#95C2416yes—
421#95C2416yes—
422#95C24112yes—
423#95C24112yes—
424#95C24112yes—
425#95C24124yes—
426C24C2516yes—
427#41—264no—
428C2×Q8—24no—
429C2×Q8—224no—
430C2×Q8—224no—
431C2×Q8—224no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—46no—
437C2×Q8—46no—
438C2×Q8—412no—
439C2×Q8—412no—
440C2×Q8—412no—
441C2×Q8—896no—
442C2×Q8C22×D414yes—
443C2×Q8C22×D4124yes—
444C2×Q8C22×D414yes—
445C2×Q8C22×Q814yes—
446C2×Q82−1+414yes—
447C2×D4—212no—
448C2×D4—26no—
449C2×D4—26no—
450C2×D4—212no—
451C2×D4—46no—
452C2×D4—46no—
453C2×D4C22×Q8112yes—
454C2×D4C22×Q8124yes—
455C2×D4C22×Q8124yes—
456C22×C4—212no—
457C22×C4—212no—
458C22×C4—212no—
459C22×C4—212no—
460C22×C4—212no—
461C22×C4—212no—
462C22×C4—212no—
463C22×C4—248no—
464C22×C4—46no—
465C22×C4—424no—
466C22×C4—424no—
467C22×C4C22×Q8124yes—
468C22×C4C2512yes—
469C22×C4C2513yes—
470C22×C4C2516yes—
471C22×C4C2514yes—
472C22×C4C2516yes—
473C22×C4C25112yes—
474C22×C4C2513yes—
475C22×C4C2516yes—
476C22×C4C2512yes—
477C22×C4C2513yes—
478C22×C4C2512yes—
479C22×C4C2512yes—
480C22×C4C2511yescharacteristic
481C22×C4C2513yes—
482C22×C4C2511yescharacteristic
483C22×C4C2511yescharacteristic
484C4⋊C4—224no—
485C4⋊C4—212no—
486C4⋊C4—224no—
487C4⋊C4—248no—
488C4⋊C4—28no—
489C4⋊C4—212no—
490C4⋊C4—28no—
491C4⋊C4—224no—
492C4⋊C4—224no—
493C4⋊C4—224no—
494C4⋊C4—212no—
495C4⋊C4—224no—
496C4⋊C4—26no—
497C4⋊C4—24no—
498C4⋊C4—26no—
499C4⋊C4—212no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—26no—
504C4⋊C4—46no—
505C4⋊C4—412no—
506C4⋊C4—46no—
507C4⋊C4—46no—
508C4⋊C4—412no—
509C4⋊C4—412no—
510C4⋊C4—412no—
511C4⋊C4—46no—
512C4⋊C4C22×D4112yes—
513C4⋊C4C22×D4124yes—
514C4⋊C4C22×D4112yes—
515C4⋊C4C22×D4124yes—
516C4⋊C4C22×Q814yes—
517C4⋊C4C22×Q8112yes—
518C4⋊C4H(5,𝔽2)14yes—
519C22⋊C4—224no—
520C22⋊C4—224no—
521C22⋊C4—224no—
522C22⋊C4—212no—
523C22⋊C4—212no—
524C22⋊C4—412no—
525C22⋊C4—424no—
526C22⋊C4—412no—
527C22⋊C4—412no—
528C22⋊C4C22×Q8124yes—
529C42—24no—
530C42—212no—
531C42—212no—
532C42—224no—
533C42—26no—
534C42—212no—
535C42—26no—
536C42—24no—
537C42—212no—
538C42—26no—
539C42—46no—
540C42—46no—
541C42—43no—
542C42C22×D4112yes—
543C42C22×D4124yes—
544C42C22×Q8112yes—
545C422−1+414yes—
546C23—26no—
547C23—26no—
548C23—26no—
549C23—26no—
550C23—26no—
551C23—26no—
552C23C23×Q8112yes—
553C23C2611yescharacteristic
554Q8—28no—
555Q8—424no—
556Q8—448no—
557Q8—48no—
558Q8—824no—
559Q8D4×Q8116yescomplemented
560D4—248no—
561D4—224no—
562D4—424no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—23no—
566C2×C4—26no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—23no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—23no—
574C2×C4—26no—
575C2×C4—23no—
576C2×C4—22no—
577C2×C4—248no—
578C2×C4—24no—
579C2×C4—26no—
580C2×C4—212no—
581C2×C4—23no—
582C2×C4—26no—
583C2×C4—212no—
584C2×C4—22no—
585C2×C4—23no—
586C2×C4—26no—
587C2×C4—22no—
588C2×C4—26no—
589C2×C4—24no—
590C2×C4—23no—
591C2×C4—26no—
592C2×C4—24no—
593C2×C4—26no—
594C2×C4—22no—
595C2×C4—212no—
596C2×C4—424no—
597C2×C4—424no—
598C2×C4—848no—
599C2×C4C23×D416yes—
600C2×C4C23×D414yes—
601C2×C4C23×D4112yes—
602C2×C4C23×D416yes—
603C2×C4C23×D412yes—
604C2×C4C23×Q816yes—
605C2×C4C23×Q814yes—
606C2×C4C23×Q816yes—
607C2×C4C23×Q812yes—
608C2×C4C2×H(5,𝔽2)14yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+412yes—
611C2×C4C2×2−1+416yes—
612C2×C4C2×2−1+412yes—
613C2×C4C2×2−1+412yes—
614C2×C4#58512yes—
615C22—212no—
616C22—212no—
617C22—212no—
618C22—46no—
619C22—46no—
620C22—46no—
621C22—46no—
622C22C24×D411yescharacteristic
623C22C24×Q811yescharacteristic
624C22C22×H(5,𝔽2)11yescharacteristic
625C22C22×2−1+411yescharacteristic
626C22C22×2−1+411yescharacteristic
627C22C2×#58511yescharacteristic
628C22H(7,𝔽2)11yescharacteristic
629C4—24no—
630C4—26no—
631C4—212no—
632C4—26no—
633C4—24no—
634C4—22no—
635C4—24no—
636C4—412no—
637C4—46no—
638C4—44no—
639C4—43no—
640C4—46no—
641C4—42no—
642C4C2×D4×Q8112yes—
643C4#349714yes—
644C4#350314yescomplemented
645C2—412no—
646C2C22×D4×Q811yescharacteristic
647C2C2×#349711yescharacteristic
648C2C2×#350311yescharacteristic
649C2#6294311yescharacteristic
650C2D4×H(5,𝔽2)11yescharacteristic
651C2#6298911yescharacteristic
652C2#6303911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#550C22
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×#3503C2semidirect
2. #59823C2semidirect
3. #59817C2semidirect
4. #59136C2semidirect
5. #44879C2semidirect
6. #39598C2semidirect
7. #39370C2semidirect
8. #36559C2semidirect
9. Q8×C4⋊Q8C2semidirect
10. #36466C2semidirect
11. #36456C2semidirect
12. D4×C4⋊Q8C2semidirect
13. #35816C2semidirect
14. #34929C2semidirect
16. C4×Q82C2semidirect
112. Q82D4semidirect
116. #550Q8semidirect
279. C2×#41C2×Q8semidirect
308. #127C2×Q8semidirect
320. #121C2×Q8semidirect
333. C4×Q8C2×Q8semidirect
343. C4×D4C2×Q8semidirect
559. Q8D4×Q8semidirect
644. C4#3503semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
#598232
#598172
#591362
#448792
#395982
#393702
#365592
Q8×C4⋊Q82
#364662
#364562
D4×C4⋊Q82
#358162
#349292
#343132
C4×Q822
C4×#5502