Finite Group Databasebeta

← #7688198

Subgroups — #7688198

Order 512.

6973 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3518C212yesmaximal, complemented
2#59821C213yesmaximal, complemented
3#59244C216yesmaximal, complemented
4#59065C216yesmaximal, complemented
5#45287C2112yesmaximal, complemented
6#39351C216yesmaximal, complemented
7#38922C216yesmaximal, complemented
8#36576C212yesmaximal, complemented
9Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
10#36456C212yesmaximal, complemented
11#36187C213yesmaximal, complemented
12#36134C213yesmaximal, complemented
13#35801C214yesmaximal, complemented
14#34932C213yesmaximal, complemented
15#34182C212yesmaximal, complemented
16C4×D4×Q8C211yesmaximal, characteristic, complemented
17C4×#550C211yesmaximal, characteristic, complemented
18#3518—232no—
19C2×Q82C2212yes—
20C2×#556C22112yes—
21C2×#556C2216yes—
22C2×#555C2216yes—
23C2×#550C2211yescharacteristic
24C2×D4×Q8C2211yescharacteristic
25C2×#548C2214yes—
26C2×#542C22112yes—
27C2×#533C2216yes—
28C2×#529C2216yes—
29C2×#519C2214yes—
30C2×C4×#41C2212yes—
31C22×C4×Q8C2212yes—
32#2868C2213yes—
33#2857C2216yes—
34#2856C2216yes—
35#2727C22112yes—
36#2723C22124yes—
37#2715C22124yes—
38#2672C22124yes—
39#2624C22112yes—
40#2567C2216yes—
41#2563C2216yes—
42#2553C22112yes—
43#2552C2213yes—
44#2551C2216yes—
45#2551C2216yes—
46#2548C2216yes—
47#2547C2211yescharacteristic
48#2542C2216yes—
49#2542C2213yes—
50#2538C2216yes—
51#2538C2216yes—
52#2537C2216yes—
53#2537C2213yes—
54#2512C22112yes—
55#2500C22112yes—
56#2496C2216yes—
57#2491C22112yes—
58#2491C22112yes—
59#2489C2216yes—
60#2485C22112yes—
61#2469C22124yes—
62#2452C22112yes—
63#2448C22112yes—
64#2448C22112yes—
65#2441C22112yes—
66#2439C22112yes—
67#2429C22112yes—
68#2427C2216yes—
69#2426C22112yes—
70#2423C22112yes—
71#2422C2216yes—
72#2421C22112yes—
73#2416C22112yes—
74#2415C22112yes—
75#2395C22112yes—
76#2366C2213yes—
77#2365C2216yes—
78#2365C2213yes—
79#2363C2216yes—
80#2360C2216yes—
81#2354C2216yes—
82#2354C2216yes—
83#2354C2213yes—
84#2351C2214yes—
85#2351C22112yes—
86Q8×C4⋊C4C2216yes—
87Q8×C4⋊C4C2211yescharacteristic
88Q8×C4⋊C4C2216yes—
89Q8×C4⋊C4C2211yescharacteristic
90Q8×C4⋊C4C2212yes—
91Q8×C4⋊C4C2212yes—
92D4×C4⋊C4C2213yes—
93#2341C2214yes—
94#2341C2212yes—
95Q8×C22⋊C4C2212yes—
96#2316C22112yes—
97C4×C4⋊Q8C2213yes—
98C4×C4⋊Q8C2213yes—
99C4×#127C2213yes—
100C4×C22⋊Q8C2216yes—
101C4×C22⋊Q8C2216yes—
102C4×#121C2216yes—
103C4×#121C2216yes—
104#2285C2216yes—
105#2285C2216yes—
106C42×Q8C2211yescharacteristic
107C42×D4C2213yes—
108C42×D4C2213yes—
109C22×#41C2312yes—
110C23×Q8C2312yes—
111Q82—216no—
112#556—248no—
113#556—296no—
114#555—248no—
115#550—28no—
116D4×Q8D4116yescomplemented
117#548—232no—
118#542—296no—
119#533—248no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2316yes—
122C2×C4⋊Q8C23112yes—
123C2×C4⋊Q8C2316yes—
124C2×C4⋊Q8C2316yes—
125C2×C4⋊Q8C2313yes—
126C2×#127C2313yes—
127#529—248no—
128C2×#126C23112yes—
129C2×#126C23124yes—
130C2×#125C2316yes—
131C2×#124C2312yes—
132C2×#124C23124yes—
133C2×#124C23112yes—
134C2×#123C23112yes—
135C2×C22⋊Q8C2316yes—
136C2×C22⋊Q8C23112yes—
137C2×C22⋊Q8C23112yes—
138C2×C22⋊Q8C2316yes—
139C2×C22⋊Q8C23112yes—
140C2×C22⋊Q8C2316yes—
141C2×#121C23112yes—
142C2×#121C2316yes—
143C2×#121C2316yes—
144C2×#121C2316yes—
145C2×#121C23112yes—
146#519—232no—
147C4×#41—216no—
148C2×C4×Q8—216no—
149C2×C4×Q8C2316yes—
150C2×C4×Q8C2316yes—
151C2×C4×Q8C2311yescharacteristic
152C2×C4×Q8C2312yes—
153C2×C4×Q8C2314yes—
154C2×C4×Q8C2312yes—
155C2×C4×Q8C23112yes—
156C2×C4×Q8C23112yes—
157C2×C4×Q8C2316yes—
158C2×C4×Q8C23112yes—
159C2×C4×Q8C2311yescharacteristic
160C2×C4×Q8C2311yescharacteristic
161C2×C4×D4C2313yes—
162C2×C4×D4C2316yes—
163C2×C4×D4C2313yes—
164C2×C4×D4C2316yes—
165C2×C4×D4C23112yes—
166C2×C4×D4C2311yescharacteristic
167C2×C4×D4C2316yes—
168C2×C4×D4C2313yes—
169C2×C4×D4C2313yes—
170C2×C4×D4C2313yes—
171C2×#117C23112yes—
172C2×#117C23112yes—
173C2×#117C23112yes—
174C2×#117C2316yes—
175C22×C4⋊C4C2316yes—
176C22×C42C2316yes—
177C22×C42C2316yes—
178#400C23112yes—
179#399C23124yes—
180#399C23112yes—
181#398C23112yes—
182#397C23124yes—
183#397C2314yes—
184#397C23112yes—
185#397C23112yes—
186#396C23124yes—
187#396C23124yes—
188#395C23124yes—
189#394C23112yes—
190#394C23124yes—
191#394C23112yes—
192#394C23124yes—
193#393C23124yes—
194#391C2316yes—
195#391C2316yes—
196#391C2316yes—
197#391C23112yes—
198#391C2312yes—
199#390C2316yes—
200#390C2316yes—
201#390C2316yes—
202#390C23112yes—
203#390C2316yes—
204#390C23112yes—
205#390C23112yes—
206#390C23112yes—
207#390C2316yes—
208#389C2316yes—
209#389C2316yes—
210#389C2316yes—
211#389C2316yes—
212#389C23112yes—
213#389C23112yes—
214#389C2316yes—
215#389C23112yes—
216#389C23124yes—
217#389C23124yes—
218#389C23112yes—
219#389C2316yes—
220#389C2316yes—
221#389C2316yes—
222#388C23112yes—
223#388C23124yes—
224#388C23124yes—
225#388C23124yes—
226#388C23112yes—
227#387C23112yes—
228#387C23124yes—
229#387C23112yes—
230#386C23112yes—
231#386C23112yes—
232#385C23112yes—
233#385C23112yes—
234#384C2313yes—
235#384C2313yes—
236#384C2313yes—
237#384C2316yes—
238#384C2316yes—
239#384C2316yes—
240#384C2316yes—
241#384C2316yes—
242#382C23112yes—
243#382C23112yes—
244#380C2316yes—
245#380C23112yes—
246#380C23112yes—
247#380C23112yes—
248#380C2316yes—
249#380C23112yes—
250#380C2316yes—
251#380C2312yes—
252#380C2316yes—
253#380C2312yes—
254C4×C4⋊C4C2313yes—
255C4×C4⋊C4C2316yes—
256C4×C4⋊C4C2316yes—
257C4×C4⋊C4C2316yes—
258C4×C4⋊C4C2313yes—
259C4×C4⋊C4C2313yes—
260C4×C4⋊C4C2313yes—
261C4×C4⋊C4C23112yes—
262C4×C4⋊C4C23112yes—
263C4×C4⋊C4C23112yes—
264C4×C4⋊C4C2313yes—
265C4×C4⋊C4C2313yes—
266C4×C4⋊C4C2313yes—
267C4×C4⋊C4C2313yes—
268C4×C4⋊C4C2311yescharacteristic
269C4×C22⋊C4C2316yes—
270C4×C22⋊C4C2316yes—
271C4×C22⋊C4C23112yes—
272C4×C22⋊C4C2316yes—
273C4×C22⋊C4C2316yes—
274C4×C22⋊C4C2316yes—
275C4×C22⋊C4C2316yes—
276C43C2313yes—
277C2×#41—28no—
278C2×#41—28no—
279C2×#41—28no—
280C22×Q8—28no—
281C22×Q8—28no—
282C22×Q8C2×D4116yes—
283C22×Q8C2412yes—
284C22×Q8C2416yes—
285C22×Q8C24112yes—
286C22×Q8C2411yescharacteristic
287C22×D4C2413yes—
288C22×D4C2416yes—
289C22×D4C2411yescharacteristic
290C22×D4C24112yes—
291C22×D4C2416yes—
292C23×C4C2416yes—
293C23×C4C2416yes—
294C23×C4C2412yes—
295C23×C4C2416yes—
296C23×C4C2416yes—
297C23×C4C2416yes—
298C4⋊Q8—248no—
299C4⋊Q8—224no—
300C4⋊Q8—24no—
301C4⋊Q8—412no—
302C4⋊Q8—86no—
303C4⋊Q8C2×D4124yes—
304#127—212no—
305#126—296no—
306#126—248no—
307#125—224no—
308#124—248no—
309#124—28no—
310#124—296no—
311#123—248no—
312C22⋊Q8—248no—
313C22⋊Q8—224no—
314C22⋊Q8—248no—
315C22⋊Q8—412no—
316C22⋊Q8—812no—
317C22⋊Q8C2×D4148yes—
318#121—224no—
319#121—248no—
320#121—224no—
321#121—412no—
322#121—812no—
323C4×Q8—28no—
324C4×Q8—248no—
325C4×Q8—216no—
326C4×Q8—248no—
327C4×Q8—28no—
328C4×Q8—24no—
329C4×Q8—248no—
330C4×Q8—224no—
331C4×Q8—224no—
332C4×Q8—224no—
333C4×Q8C2×D418yes—
334C4×Q8#4118yes—
335C4×D4—248no—
336C4×D4—224no—
337C4×D4—224no—
338C4×D4—212no—
339C4×D4—24no—
340C4×D4—212no—
341C4×D4—46no—
342C4×D4—46no—
343C4×D4—412no—
344C4×D4C2×D4124yes—
345#117—248no—
346#117—248no—
347#117—224no—
348#117—248no—
349C2×C4⋊C4—224no—
350C2×C4⋊C4C2416yes—
351C2×C4⋊C4C24112yes—
352C2×C4⋊C4C2416yes—
353C2×C4⋊C4C2416yes—
354C2×C4⋊C4C24112yes—
355C2×C4⋊C4C24112yes—
356C2×C4⋊C4C24112yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C24112yes—
360C2×C4⋊C4C2413yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2413yes—
363C2×C4⋊C4C24112yes—
364C2×C4⋊C4C2413yes—
365C2×C4⋊C4C2416yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2416yes—
369C2×C4⋊C4C24112yes—
370C2×C4⋊C4C24112yes—
371C2×C4⋊C4C2416yes—
372C2×C4⋊C4C24112yes—
373C2×C4⋊C4C2416yes—
374C2×C4⋊C4C24112yes—
375C2×C4⋊C4C2413yes—
376C2×C4⋊C4C2413yes—
377C2×C4⋊C4C2416yes—
378C2×C4⋊C4C2413yes—
379C2×C22⋊C4C24112yes—
380C2×C22⋊C4C2414yes—
381C2×C22⋊C4C24112yes—
382C2×C22⋊C4C24124yes—
383C2×C22⋊C4C2414yes—
384C2×C22⋊C4C2416yes—
385C2×C22⋊C4C24112yes—
386C2×C22⋊C4C2412yes—
387C2×C22⋊C4C2416yes—
388C2×C22⋊C4C2416yes—
389C2×C22⋊C4C2416yes—
390C2×C22⋊C4C24112yes—
391C2×C22⋊C4C24124yes—
392C2×C22⋊C4C24112yes—
393C2×C22⋊C4C24112yes—
394C2×C22⋊C4C2416yes—
395C2×C42—224no—
396C2×C42—224no—
397C2×C42C2416yes—
398C2×C42C24124yes—
399C2×C42C2416yes—
400C2×C42C24112yes—
401C2×C42C2413yes—
402C2×C42C2413yes—
403C2×C42C2416yes—
404C2×C42C2416yes—
405C2×C42C2416yes—
406C2×C42C2413yes—
407C2×C42C2412yes—
408C2×C42C2413yes—
409C2×C42C2413yes—
410C2×C42C2413yes—
411C2×C42C2411yescharacteristic
412#95C2412yes—
413#95C2414yes—
414#95C2416yes—
415#95C2416yes—
416#95C2416yes—
417#95C2416yes—
418#95C2416yes—
419#95C24112yes—
420#95C24112yes—
421#95C2414yes—
422#95C2416yes—
423#95C24124yes—
424#95C24112yes—
425#95C24112yes—
426#95C24112yes—
427C24C2516yes—
428C24C2512yes—
429#41—432no—
430C2×Q8—264no—
431C2×Q8—24no—
432C2×Q8—224no—
433C2×Q8—212no—
434C2×Q8—212no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—412no—
438C2×Q8—412no—
439C2×Q8C22×D4124yes—
440C2×Q8C22×D414yes—
441C2×Q8C2×#4114yes—
442C2×Q82−1+414yes—
443C2×D4—224no—
444C2×D4—212no—
445C2×D4—26no—
446C2×D4—212no—
447C2×D4—212no—
448C2×D4—22no—
449C2×D4—26no—
450C2×D4—212no—
451C2×D4—22no—
452C2×D4—26no—
453C2×D4—46no—
454C2×D4—412no—
455C2×D4—46no—
456C2×D4—412no—
457C2×D4C22×D414yes—
458C22×C4—212no—
459C22×C4—212no—
460C22×C4—212no—
461C22×C4—212no—
462C22×C4—24no—
463C22×C4—212no—
464C22×C4—212no—
465C22×C4—24no—
466C22×C4—212no—
467C22×C4—212no—
468C22×C4—24no—
469C22×C4—212no—
470C22×C4—212no—
471C22×C4—46no—
472C22×C4—46no—
473C22×C4—46no—
474C22×C4—46no—
475C22×C4C22×D4124yes—
476C22×C4C2512yes—
477C22×C4C2513yes—
478C22×C4C2512yes—
479C22×C4C2516yes—
480C22×C4C2516yes—
481C22×C4C2514yes—
482C22×C4C2512yes—
483C22×C4C25112yes—
484C22×C4C2516yes—
485C22×C4C2513yes—
486C22×C4C2513yes—
487C22×C4C2511yescharacteristic
488C22×C4C2513yes—
489C22×C4C2511yescharacteristic
490C22×C4C2511yescharacteristic
491C4⋊C4—212no—
492C4⋊C4—212no—
493C4⋊C4—224no—
494C4⋊C4—224no—
495C4⋊C4—224no—
496C4⋊C4—224no—
497C4⋊C4—224no—
498C4⋊C4—224no—
499C4⋊C4—212no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—26no—
503C4⋊C4—26no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—26no—
508C4⋊C4—212no—
509C4⋊C4—46no—
510C4⋊C4—412no—
511C4⋊C4—412no—
512C4⋊C4—46no—
513C4⋊C4—412no—
514C4⋊C4—412no—
515C4⋊C4—46no—
516C4⋊C4C22×D4112yes—
517C4⋊C4C22×D4112yes—
518C4⋊C4C22×D4124yes—
519C4⋊C4C2×#41112yes—
520C22⋊C4—224no—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—224no—
524C22⋊C4—248no—
525C22⋊C4—224no—
526C22⋊C4—212no—
527C22⋊C4—24no—
528C22⋊C4—212no—
529C22⋊C4—46no—
530C22⋊C4—412no—
531C22⋊C4—46no—
532C22⋊C4—412no—
533C22⋊C4—424no—
534C22⋊C4—412no—
535C22⋊C4C22×D4124yes—
536C42—24no—
537C42—212no—
538C42—212no—
539C42—248no—
540C42—224no—
541C42—212no—
542C42—212no—
543C42—212no—
544C42—26no—
545C42—26no—
546C42—26no—
547C42—43no—
548C42C22×D4112yes—
549C42C2×#41112yes—
550C422−1+414yes—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—26no—
558C23—26no—
559C23—26no—
560C23—26no—
561C23—26no—
562C23—26no—
563C23—26no—
564C23C23×D414yes—
565C23C2611yescharacteristic
566Q8—248no—
567Q8—48no—
568Q8—824no—
569Q8#547116yes—
570D4—28no—
571D4—424no—
572D4—412no—
573D4—812no—
574D4—824no—
575C2×C4—26no—
576C2×C4—21no—
577C2×C4—26no—
578C2×C4—23no—
579C2×C4—23no—
580C2×C4—21no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—248no—
584C2×C4—26no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—212no—
589C2×C4—22no—
590C2×C4—26no—
591C2×C4—22no—
592C2×C4—23no—
593C2×C4—26no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—23no—
598C2×C4—23no—
599C2×C4—22no—
600C2×C4—26no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—26no—
604C2×C4—212no—
605C2×C4—26no—
606C2×C4—212no—
607C2×C4—22no—
608C2×C4—424no—
609C2×C4—424no—
610C2×C4—48no—
611C2×C4—812no—
612C2×C4—812no—
613C2×C4C23×D4112yes—
614C2×C4C23×D416yes—
615C2×C4C23×D416yes—
616C2×C4C23×D412yes—
617C2×C4C22×#4116yes—
618C2×C4C22×#4116yes—
619C2×C4C22×#4112yes—
620C2×C4C2×2−1+412yes—
621C2×C4C2×2−1+416yes—
622C2×C4C2×2−1+412yes—
623C2×C4C2×2−1+412yes—
624C2×C4#58512yes—
625C2×C4#58512yes—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—46no—
634C22—46no—
635C22—46no—
636C22—46no—
637C22—46no—
638C22—46no—
639C22—46no—
640C22C24×D411yescharacteristic
641C22C23×#4111yescharacteristic
642C22C22×2−1+411yescharacteristic
643C22C22×2−1+411yescharacteristic
644C22C2×#58511yescharacteristic
645C22C2×#58511yescharacteristic
646C22H(7,𝔽2)11yescharacteristic
647C4—26no—
648C4—26no—
649C4—22no—
650C4—212no—
651C4—46no—
652C4—42no—
653C4—412no—
654C4—43no—
655C4—46no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4C2×#547112yes—
660C4#349714yes—
661C4#351814yescomplemented
662C2—44no—
663C2—86no—
664C2C22×#54711yescharacteristic
665C2C2×#349711yescharacteristic
666C2C2×#351811yescharacteristic
667C2#6294311yescharacteristic
668C2#6299411yescharacteristic
669C2#6300311yescharacteristic
670C2#6303411yescharacteristic
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×#3518C2semidirect
2. #59821C2semidirect
3. #59244C2semidirect
4. #59065C2semidirect
5. #45287C2semidirect
6. #39351C2semidirect
7. #38922C2semidirect
8. #36576C2semidirect
9. Q8×C4⋊Q8C2semidirect
10. #36456C2semidirect
11. #36187C2semidirect
12. #36134C2semidirect
13. #35801C2semidirect
14. #34932C2semidirect
15. #34182C2semidirect
16. C4×D4×Q8C2semidirect
17. C4×#550C2semidirect
116. D4×Q8D4semidirect
661. C4#3518semidirect

Maximal subgroups

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

subgroupindex
C2×#35182
#598212
#592442
#590652
#452872
#393512
#389222
#365762
Q8×C4⋊Q82
#364562
#361872
#361342
#358012
#349322
#341822
C4×D4×Q82
C4×#5502