Finite Group Databasebeta

← #7630387

Subgroups — #7630387

Order 512.

5556 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C213yesmaximal, complemented
2#36593C211yesmaximal, characteristic, complemented
3Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
4#36456C212yesmaximal
5#36447C212yesmaximal, complemented
6#36445C212yesmaximal
7#36407C216yesmaximal, complemented
8#36288C216yesmaximal, complemented
9#36287C216yesmaximal, complemented
10#36194C213yesmaximal, complemented
11#36134C213yesmaximal, complemented
12#34978C216yesmaximal, complemented
13#34941C216yesmaximal, complemented
14#34929C216yesmaximal, complemented
15C4×Q82C212yesmaximal, complemented
16C4×#557C212yesmaximal, complemented
17#34129C212yesmaximal
18C4×#533C213yesmaximal, complemented
19C42×#41C211yesmaximal, characteristic
20C2×Q82C2212yes—
21C2×#557C2212yes—
22C2×#533C2213yes—
23C2×#533C2216yes—
24C2×C4×#41C2211yescharacteristic
25C2×C4×#41C2211yescharacteristic
26C2×C4×#41C2211yescharacteristic
27#2868C2213yes—
28#2868C2213yes—
29#2862C2216yes—
30#2859C2216yes—
31#2859C2216yes—
32#2857C2216yes—
33#2857C2216yes—
34#2857C2216yes—
35#2857C2216yes—
36#2612C22124yes—
37#2574C22124yes—
38#2571C2216yes—
39#2567C2216yes—
40#2563C2216yes—
41#2556C22124yes—
42#2552C2216yes—
43#2552C2216yes—
44#2552C2216yes—
45#2551C2216yes—
46#2551C2216yes—
47#2546C2216yes—
48#2542C2216yes—
49#2538C2216yes—
50#2538C2216yes—
51#2508C22112yes—
52#2506C22112yes—
53#2497C22112yes—
54#2492C22112yes—
55#2454C22112yes—
56#2452C22112yes—
57#2441C22112yes—
58#2441C22112yes—
59#2439C22112yes—
60#2439C22112yes—
61#2429C22112yes—
62#2423C22112yes—
63#2396C22124yes—
64#2370C22112yes—
65#2369C22112yes—
66#2369C2216yes—
67#2365C2216yes—
68#2355C2216yes—
69#2354C2216yes—
70#2354C2216yes—
71#2351C2214yes—
72Q8×C4⋊C4C2216yes—
73Q8×C4⋊C4C2212yes—
74Q8×C4⋊C4C2212yes—
75Q8×C4⋊C4C2214yes—
76Q8×C4⋊C4C2212yes—
77Q8×C4⋊C4C2212yes—
78#2346C22112yes—
79#2341C2214yes—
80C4×C4⋊Q8C2213yes—
81C4×C4⋊Q8C2216yes—
82C4×C4⋊Q8C2216yes—
83C4×C4⋊Q8C2216yes—
84C4×C4⋊Q8C2213yes—
85C4×C4⋊Q8C2213yes—
86C4×C4⋊Q8C22112yes—
87C4×C4⋊Q8C2211yescharacteristic
88C4×#125C2216yes—
89C4×#125C22112yes—
90C4×#125C2216yes—
91C4×C22⋊Q8C2216yes—
92C4×C22⋊Q8C2216yes—
93C4×C22⋊Q8C2216yes—
94C4×C22⋊Q8C2216yes—
95#2296C2216yes—
96#2295C2216yes—
97#2295C2216yes—
98#2293C22112yes—
99#2285C2216yes—
100#2285C2216yes—
101#2285C2213yes—
102C42×Q8C2214yes—
103C42×Q8C2211yescharacteristic
104C42×Q8C2216yes—
105C42×Q8C2211yescharacteristic
106C42×Q8C2211yescharacteristic
107C42×Q8C2216yes—
108C42×Q8C2211yescharacteristic
109C42×D4C2213yes—
110C42×D4C2213yes—
111C42×D4C2213yes—
112C42×D4C2213yes—
113C4×#117C2216yes—
114C4×#117C2213yes—
115C4×#117C2213yes—
116C4×#117C2213yes—
117C2×C43C2213yes—
118C22×#41C2311yescharacteristic
119Q82—216no—
120#557—216no—
121#533—224no—
122#533—424no—
123C2×C4⋊Q8C2313yes—
124C2×C4⋊Q8C2316yes—
125C2×C4⋊Q8C2316yes—
126C2×C4⋊Q8C23112yes—
127C2×C4⋊Q8C2316yes—
128C2×C4⋊Q8C2316yes—
129C2×C4⋊Q8C2316yes—
130C2×C4⋊Q8C2311yescharacteristic
131C2×#125C2316yes—
132C2×#125C23112yes—
133C2×#125C23112yes—
134C2×C22⋊Q8C2316yes—
135C2×C22⋊Q8C2316yes—
136C2×C22⋊Q8C23112yes—
137C2×C22⋊Q8C23112yes—
138C4×#41—28no—
139C4×#41—28no—
140C4×#41Q8116yescomplemented
141C2×C4×Q8C2316yes—
142C2×C4×Q8C2316yes—
143C2×C4×Q8C2316yes—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2314yes—
146C2×C4×Q8C2314yes—
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2316yes—
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2316yes—
151C2×C4×Q8C23112yes—
152C2×C4×Q8C2311yescharacteristic
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2311yescharacteristic
155C2×C4×Q8C2311yescharacteristic
156C2×C4×D4C2313yes—
157C2×C4×D4C2313yes—
158C2×C4×D4C2313yes—
159C2×C4×D4C2313yes—
160C2×C4×D4C2313yes—
161C2×C4×D4C2313yes—
162C2×#117C2316yes—
163C2×#117C2313yes—
164C2×#117C2313yes—
165C2×#117C2316yes—
166C2×#117C23112yes—
167C2×#117C2313yes—
168C22×C42C2313yes—
169C22×C42C2313yes—
170C22×C42C2313yes—
171C22×C42C2313yes—
172C22×C42C2313yes—
173C22×C42C2313yes—
174C22×C42C2313yes—
175#400C23124yes—
176#399C23124yes—
177#398C23124yes—
178#398C23124yes—
179#396C23124yes—
180#396C23124yes—
181#395C23124yes—
182#393C23124yes—
183#391C2316yes—
184#391C23112yes—
185#391C2316yes—
186#391C23112yes—
187#391C23112yes—
188#391C23124yes—
189#391C2316yes—
190#391C2312yes—
191#390C23112yes—
192#390C23112yes—
193#389C2312yes—
194#389C2316yes—
195#389C2316yes—
196#389C2316yes—
197#389C23112yes—
198#389C23112yes—
199#389C23112yes—
200#389C23112yes—
201#389C2316yes—
202#389C23112yes—
203#389C23112yes—
204#389C23112yes—
205#389C23112yes—
206#389C23112yes—
207#389C23124yes—
208#389C23112yes—
209#389C2316yes—
210#389C2316yes—
211#389C2312yes—
212#389C23112yes—
213#389C23112yes—
214#388C23124yes—
215#388C23124yes—
216#387C23112yes—
217#387C23124yes—
218#387C23124yes—
219#387C23112yes—
220#387C23112yes—
221#387C23112yes—
222#385C23112yes—
223#385C23112yes—
224#384C2311yescharacteristic
225#384C2313yes—
226#384C2316yes—
227#384C2316yes—
228#384C2316yes—
229#384C23112yes—
230#384C2316yes—
231#384C2316yes—
232#384C2316yes—
233#384C2316yes—
234#382C2316yes—
235#382C23112yes—
236#382C23112yes—
237#382C23112yes—
238#380C2316yes—
239#380C23112yes—
240#380C23112yes—
241#380C23112yes—
242#380C23112yes—
243#380C2316yes—
244C4×C4⋊C4C2313yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C23112yes—
247C4×C4⋊C4C23112yes—
248C4×C4⋊C4C2316yes—
249C4×C4⋊C4C23112yes—
250C4×C4⋊C4C23112yes—
251C4×C4⋊C4C2316yes—
252C4×C4⋊C4C2316yes—
253C4×C4⋊C4C23112yes—
254C4×C4⋊C4C23112yes—
255C4×C4⋊C4C2313yes—
256C4×C4⋊C4C2316yes—
257C4×C4⋊C4C2316yes—
258C4×C4⋊C4C2316yes—
259C4×C4⋊C4C2312yes—
260C4×C4⋊C4C2313yes—
261C4×C4⋊C4C2313yes—
262C4×C4⋊C4C23112yes—
263C4×C4⋊C4C23112yes—
264C4×C4⋊C4C23112yes—
265C4×C4⋊C4C2313yes—
266C4×C4⋊C4C2313yes—
267C4×C4⋊C4C2316yes—
268C4×C4⋊C4C2313yes—
269C4×C4⋊C4C2316yes—
270C4×C4⋊C4C2313yes—
271C4×C4⋊C4C23112yes—
272C4×C4⋊C4C2313yes—
273C4×C4⋊C4C2313yes—
274C4×C4⋊C4C2313yes—
275C4×C4⋊C4C2313yes—
276C4×C22⋊C4C2316yes—
277C4×C22⋊C4C23112yes—
278C4×C22⋊C4C2316yes—
279C4×C22⋊C4C23112yes—
280C4×C22⋊C4C2316yes—
281C4×C22⋊C4C2316yes—
282C4×C22⋊C4C2316yes—
283C4×C22⋊C4C2316yes—
284#376C23112yes—
285#376C2316yes—
286C43C2313yes—
287C43C2313yes—
288C43C23112yes—
289C43C2311yescharacteristic
290C43C2313yes—
291C43C2313yes—
292C43C2313yes—
293C43C2313yes—
294C43C2313yes—
295C2×#41—24no—
296C2×#41—24no—
297C2×#41C2×Q818yes—
298C22×Q8C2416yes—
299C22×Q8C2412yes—
300C22×Q8C24112yes—
301C22×Q8C2411yescharacteristic
302C22×D4C2413yes—
303C23×C4C2413yes—
304C23×C4C2413yes—
305C23×C4C2413yes—
306C23×C4C2413yes—
307C23×C4C2413yes—
308C23×C4C2413yes—
309C23×C4C2413yes—
310C4⋊Q8—248no—
311C4⋊Q8—212no—
312C4⋊Q8—224no—
313C4⋊Q8—224no—
314C4⋊Q8—412no—
315C4⋊Q8—412no—
316C4⋊Q8—412no—
317C4⋊Q8#4118yescomplemented
318#125—224no—
319#125—248no—
320#125—424no—
321C22⋊Q8—224no—
322C22⋊Q8—224no—
323C22⋊Q8—424no—
324C22⋊Q8—424no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—224no—
328C4×Q8—216no—
329C4×Q8—216no—
330C4×Q8—224no—
331C4×Q8—224no—
332C4×Q8—224no—
333C4×Q8—224no—
334C4×Q8—28no—
335C4×Q8—424no—
336C4×Q8C2×Q818yes—
337C4×Q8C2×Q818yes—
338C4×Q8#4118yes—
339C4×Q8#4118yes—
340C4×D4—212no—
341C4×D4—212no—
342C4×D4—212no—
343C4×D4—212no—
344C4×D4C2×Q8124yes—
345C4×D4C2×Q8124yes—
346#117—212no—
347#117—212no—
348#117—224no—
349#117—424no—
350#117—412no—
351#117C2×Q8124yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2413yes—
354C2×C4⋊C4C2413yes—
355C2×C4⋊C4C2416yes—
356C2×C4⋊C4C2416yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C24112yes—
359C2×C4⋊C4C2416yes—
360C2×C4⋊C4C24112yes—
361C2×C4⋊C4C24112yes—
362C2×C4⋊C4C2416yes—
363C2×C4⋊C4C24112yes—
364C2×C4⋊C4C24112yes—
365C2×C4⋊C4C2416yes—
366C2×C4⋊C4C24112yes—
367C2×C4⋊C4C24112yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C24112yes—
371C2×C4⋊C4C24112yes—
372C2×C4⋊C4C24112yes—
373C2×C4⋊C4C24124yes—
374C2×C4⋊C4C24112yes—
375C2×C4⋊C4C24112yes—
376C2×C4⋊C4C2413yes—
377C2×C4⋊C4C2413yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2413yes—
380C2×C4⋊C4C2413yes—
381C2×C4⋊C4C2412yes—
382C2×C22⋊C4C2416yes—
383C2×C22⋊C4C2416yes—
384C2×C22⋊C4C24112yes—
385C2×C22⋊C4C24124yes—
386C2×C22⋊C4C24112yes—
387C2×C22⋊C4C2416yes—
388C2×C42—212no—
389C2×C42—212no—
390C2×C42—212no—
391C2×C42—212no—
392C2×C42—212no—
393C2×C42—46no—
394C2×C42C2×Q8124yes—
395C2×C42C2413yes—
396C2×C42C2413yes—
397C2×C42C24112yes—
398C2×C42C2413yes—
399C2×C42C2413yes—
400C2×C42C24112yes—
401C2×C42C24112yes—
402C2×C42C2413yes—
403C2×C42C2413yes—
404C2×C42C2413yes—
405C2×C42C2413yes—
406C2×C42C2413yes—
407C2×C42C2413yes—
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2413yes—
412C2×C42C2413yes—
413C2×C42C24112yes—
414C2×C42C2413yes—
415C2×C42C2414yes—
416C2×C42C2413yes—
417C2×C42C24112yes—
418C2×C42C2413yes—
419C2×C42C2413yes—
420C2×C42C2413yes—
421C2×C42C2413yes—
422C2×C42C2413yes—
423C2×C42C2411yescharacteristic
424C2×C42C2413yes—
425C2×C42C2413yes—
426C2×C42C2411yescharacteristic
427C2×C42C2411yescharacteristic
428C2×C42C2411yescharacteristic
429#95C2414yes—
430#95C2416yes—
431#95C2416yes—
432#95C2416yes—
433#95C2416yes—
434#95C24112yes—
435#95C24112yes—
436#95C24112yes—
437#95C2414yes—
438#95C24112yes—
439#95C24112yes—
440#95C24124yes—
441#95C24112yes—
442#95C24112yes—
443#95C24112yes—
444#95C24124yes—
445C24C2513yes—
446#41—232no—
447C2×Q8—24no—
448C2×Q8—212no—
449C2×Q8—212no—
450C2×Q8—224no—
451C2×Q8—212no—
452C2×Q8—24no—
453C2×Q8—412no—
454C2×Q8—412no—
455C2×Q8C22×Q814yes—
456C2×Q8C2×#4118yes—
457C2×Q8C2×#4114yes—
458C2×Q8C2×#4114yes—
459C2×D4—26no—
460C2×D4—26no—
461C2×D4C22×Q8112yes—
462C22×C4—26no—
463C22×C4—26no—
464C22×C4—26no—
465C22×C4—26no—
466C22×C4—26no—
467C22×C4—26no—
468C22×C4—26no—
469C22×C4—26no—
470C22×C4—26no—
471C22×C4—26no—
472C22×C4—26no—
473C22×C4—26no—
474C22×C4—26no—
475C22×C4—26no—
476C22×C4—26no—
477C22×C4—26no—
478C22×C4—43no—
479C22×C4—43no—
480C22×C4C22×Q8112yes—
481C22×C4C22×Q8112yes—
482C22×C4C22×Q8112yes—
483C22×C4C2513yes—
484C22×C4C2514yes—
485C22×C4C25112yes—
486C22×C4C2513yes—
487C22×C4C25112yes—
488C22×C4C2511yescharacteristic
489C22×C4C2513yes—
490C22×C4C2513yes—
491C22×C4C2514yes—
492C22×C4C2513yes—
493C22×C4C2513yes—
494C22×C4C2511yescharacteristic
495C22×C4C2511yescharacteristic
496C22×C4C2513yes—
497C22×C4C2511yescharacteristic
498C22×C4C2511yescharacteristic
499C22×C4C2511yescharacteristic
500C22×C4C2511yescharacteristic
501C4⋊C4—26no—
502C4⋊C4—26no—
503C4⋊C4—212no—
504C4⋊C4—24no—
505C4⋊C4—224no—
506C4⋊C4—224no—
507C4⋊C4—212no—
508C4⋊C4—224no—
509C4⋊C4—224no—
510C4⋊C4—224no—
511C4⋊C4—224no—
512C4⋊C4—212no—
513C4⋊C4—224no—
514C4⋊C4—224no—
515C4⋊C4—212no—
516C4⋊C4—28no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—212no—
520C4⋊C4—412no—
521C4⋊C4—412no—
522C4⋊C4—424no—
523C4⋊C4—412no—
524C4⋊C4—412no—
525C4⋊C4C22×Q8112yes—
526C4⋊C4C22×Q8112yes—
527C4⋊C4C2×#4118yes—
528C4⋊C4C2×#41112yes—
529C4⋊C4C2×#41112yes—
530C4⋊C4C2×#4118yes—
531C22⋊C4—212no—
532C22⋊C4—212no—
533C22⋊C4—224no—
534C22⋊C4—412no—
535C22⋊C4—424no—
536C22⋊C4C22×Q8124yes—
537C42—22no—
538C42—22no—
539C42—26no—
540C42—26no—
541C42—26no—
542C42—26no—
543C42—26no—
544C42—26no—
545C42—26no—
546C42—224no—
547C42—224no—
548C42—224no—
549C42—26no—
550C42—26no—
551C42—26no—
552C42—224no—
553C42—26no—
554C42—26no—
555C42—26no—
556C42—28no—
557C42—22no—
558C42—26no—
559C42—43no—
560C42—43no—
561C42—412no—
562C42C22×Q814yes—
563C42C22×Q8112yes—
564C42C22×Q8112yes—
565C42C22×Q8112yes—
566C42C2×#4114yes—
567C42C2×#41112yes—
568C42C2×#41112yes—
569C42H(5,𝔽2)14yes—
570C422−1+414yes—
571C23—23no—
572C23—23no—
573C23—23no—
574C23—23no—
575C23—23no—
576C23—23no—
577C23C23×Q816yes—
578C23C2611yescharacteristic
579Q8—216no—
580Q8—424no—
581Q8—824no—
582Q8#533116yescomplemented
583D4—224no—
584C2×C4—23no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—23no—
591C2×C4—23no—
592C2×C4—23no—
593C2×C4—23no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—23no—
598C2×C4—23no—
599C2×C4—23no—
600C2×C4—224no—
601C2×C4—224no—
602C2×C4—224no—
603C2×C4—24no—
604C2×C4—23no—
605C2×C4—23no—
606C2×C4—23no—
607C2×C4—23no—
608C2×C4—212no—
609C2×C4—212no—
610C2×C4—212no—
611C2×C4—23no—
612C2×C4—212no—
613C2×C4—212no—
614C2×C4—23no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—212no—
618C2×C4—412no—
619C2×C4—412no—
620C2×C4—412no—
621C2×C4—412no—
622C2×C4C23×Q812yes—
623C2×C4C23×Q816yes—
624C2×C4C23×Q816yes—
625C2×C4C23×Q816yes—
626C2×C4C23×Q812yes—
627C2×C4C23×Q812yes—
628C2×C4C22×#4118yes—
629C2×C4C22×#4116yes—
630C2×C4C22×#4116yes—
631C2×C4C22×#4116yes—
632C2×C4C22×#4116yes—
633C2×C4C22×#4112yes—
634C2×C4C22×#4112yes—
635C2×C4C22×#4112yes—
636C2×C4C22×#4112yes—
637C2×C4C22×#4112yes—
638C2×C4C2×H(5,𝔽2)12yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4C2×2−1+412yes—
642C2×C4C2×2−1+412yes—
643C2×C4C2×2−1+412yes—
644C2×C4#58512yes—
645C22—26no—
646C22—26no—
647C22—26no—
648C22—43no—
649C22—43no—
650C22—43no—
651C22—43no—
652C22C24×Q811yescharacteristic
653C22C23×#4111yescharacteristic
654C22C23×#4111yescharacteristic
655C22C23×#4111yescharacteristic
656C22C22×H(5,𝔽2)11yescharacteristic
657C22C22×2−1+411yescharacteristic
658C22C2×#58511yescharacteristic
659C4—26no—
660C4—22no—
661C4—26no—
662C4—22no—
663C4—26no—
664C4—22no—
665C4—26no—
666C4—28no—
667C4—412no—
668C4—44no—
669C4—43no—
670C4—43no—
671C4—412no—
672C4C2×#533112yes—
673C4Q8×#4114yes—
674C4#348214yes—
675C4#348314yes—
676C4#350314yes—
677C2—46no—
678C2C22×#53311yescharacteristic
679C2C2×Q8×#4111yescharacteristic
680C2C2×#348211yescharacteristic
681C2C2×#348311yescharacteristic
682C2C2×#350311yescharacteristic
683C2#6280011yescharacteristic
684C2#6280511yescharacteristic
subgroupG/H
centerC22×C4C25
commutator, Frattini, socle, cosocleC23C26
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
2. #36593C2semidirect
3. Q8×C4⋊Q8C2semidirect
5. #36447C2semidirect
7. #36407C2semidirect
8. #36288C2semidirect
9. #36287C2semidirect
10. #36194C2semidirect
11. #36134C2semidirect
12. #34978C2semidirect
13. #34941C2semidirect
14. #34929C2semidirect
15. C4×Q82C2semidirect
16. C4×#557C2semidirect
18. C4×#533C2semidirect
140. C4×#41Q8semidirect
317. C4⋊Q8#41semidirect
582. Q8#533semidirect

Maximal subgroups

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

subgroupindex
#366452
#365932
Q8×C4⋊Q82
#364562
#364472
#364452
#364072
#362882
#362872
#361942
#361342
#349782
#349412
#349292
C4×Q822
C4×#5572
#341292
C4×#5332
C42×#412