Finite Group Databasebeta

← C2×#47544 #7700907

Subgroups — C2×#47544 #7700907

Order 512.

4718 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2789C211yesmaximal, characteristic
2C2×#2776C212yesmaximal
3C2×#2769C211yesmaximal, characteristic
4C2×#2766C212yesmaximal
5C2×#2762C212yesmaximal
6C2×#2734C212yesmaximal
7C2×#2713C212yesmaximal
8#47544C2132yesmaximal, direct product factor
9C2×#2833C211yesmaximal, characteristic
10C2×#2546C211yesmaximal, characteristic
11C2×#2581C211yesmaximal, characteristic
12C2×#2605C212yesmaximal
13C2×#2590C212yesmaximal
14C2×#2586C211yesmaximal, characteristic
15C2×#2533C212yesmaximal
16C2×#2528C212yesmaximal
17C2×#2453C212yesmaximal
18C2×#2462C212yesmaximal
19C2×#2331C212yesmaximal
20C2×#2327C211yesmaximal, characteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28#2833C22116yes—
29#2789C22116yes—
30#2776C22132yes—
31#2769C22116yes—
32#2766C22132yes—
33#2762C22132yes—
34#2734C22132yes—
35#2713C22132yes—
36#2605C22132yes—
37#2590C22132yes—
38#2586C22116yes—
39#2581C22116yes—
40#2546C22116yes—
41#2533C22132yes—
42#2528C22132yes—
43#2462C22132yes—
44#2453C22132yes—
45C2×#401C2212yes—
46C2×#401C2212yes—
47C2×#400C2212yes—
48C2×#400C2212yes—
49C2×#400C2212yes—
50C2×#400C2212yes—
51C2×#400C2211yescharacteristic
52C2×#400C2212yes—
53C2×#400C2211yescharacteristic
54C2×#400C2212yes—
55C2×#400C2212yes—
56C2×#400C2212yes—
57C2×#398C2211yescharacteristic
58C2×#398C2211yescharacteristic
59C2×#398C2211yescharacteristic
60C2×#398C2212yes—
61C2×#398C2212yes—
62C2×#398C2211yescharacteristic
63C2×#398C2212yes—
64C2×#398C2212yes—
65C2×#398C2212yes—
66C2×#398C2212yes—
67C2×#395C2211yescharacteristic
68C2×#395C2212yes—
69C2×#395C2211yescharacteristic
70C2×#395C2212yes—
71C2×#395C2212yes—
72C2×#395C2212yes—
73#2331C22132yes—
74#2327C22116yes—
75C2×#391C2212yes—
76C2×#391C2211yescharacteristic
77C2×#391C2212yes—
78C2×#391C2211yescharacteristic
79C2×#391C2212yes—
80C2×#391C2212yes—
81C2×#391C2212yes—
82C2×#391C2212yes—
83C2×#389C2212yes—
84C2×#389C2212yes—
85C2×#389C2212yes—
86C2×#389C2212yes—
87C2×#389C2212yes—
88C2×#389C2212yes—
89C2×#389C2212yes—
90C2×#389C2212yes—
91C2×#389C2212yes—
92C2×#387C2212yes—
93C2×#387C2212yes—
94C2×#387C2212yes—
95C2×#387C2212yes—
96C2×#387C2212yes—
97C2×#387C2212yes—
98C2×#387C2212yes—
99C2×#387C2212yes—
100C2×#387C2212yes—
101C2×#387C2212yes—
102C2×#387C2212yes—
103C2×#387C2212yes—
104C2×#387C2212yes—
105C2×#387C2212yes—
106C2×#387C2212yes—
107C2×#387C2212yes—
108C2×#387C2212yes—
109C2×#387C2212yes—
110C2×#387C2212yes—
111C2×#387C2212yes—
112C2×#387C2212yes—
113C2×#387C2212yes—
114C2×#387C2212yes—
115C2×#387C2212yes—
116C2×#383C2212yes—
117C2×#383C2212yes—
118C2×#382C2211yescharacteristic
119C2×#382C2212yes—
120C2×#382C2211yescharacteristic
121C2×C4×C4⋊C4C2212yes—
122C2×C4×C4⋊C4C2212yes—
123C2×C4×C4⋊C4C2212yes—
124C2×C4×C4⋊C4C2211yescharacteristic
125C2×#376C2211yescharacteristic
126C2×#376C2212yes—
127C23×Q8C2311yescharacteristic
128C23×Q8C2311yescharacteristic
129C23×Q8C2312yes—
130C23×Q8C2311yescharacteristic
131C2×C4⋊Q8—28no—
132C2×C4⋊Q8—216no—
133C2×C4⋊Q8—44no—
134C2×C4⋊Q8C2318yes—
135C2×C4⋊Q8C23116yes—
136C2×C4⋊Q8C2318yes—
137C2×#125—48no—
138C2×#125C23116yes—
139C2×C4×Q8—44no—
140C2×C4×Q8—44no—
141C2×C4×Q8—44no—
142C2×C4×Q8C2318yes—
143C2×C4×Q8C2318yes—
144C2×C4×Q8C2318yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2312yes—
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2312yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2312yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2311yescharacteristic
166C22×C4⋊C4C2311yescharacteristic
167C22×C4⋊C4C2311yescharacteristic
168C22×C4⋊C4C2311yescharacteristic
169C22×C4⋊C4C2311yescharacteristic
170C22×C4⋊C4C2311yescharacteristic
171C22×C4⋊C4C2312yes—
172C22×C4⋊C4C2312yes—
173C22×C4⋊C4C2312yes—
174C22×C4⋊C4C2311yescharacteristic
175C22×C42C2311yescharacteristic
176C22×C42C2311yescharacteristic
177C22×C42C2311yescharacteristic
178C22×C42C2312yes—
179C22×C42C2311yescharacteristic
180C22×C42C2312yes—
181C22×C42C2312yes—
182C22×C42C2312yes—
183C22×C42C2312yes—
184C22×C42C2312yes—
185C22×C42C2311yescharacteristic
186C22×C42C2312yes—
187#401C23116yes—
188#401C23116yes—
189#400C2318yes—
190#400C23116yes—
191#400C23116yes—
192#400C2318yes—
193#400C23116yes—
194#400C23116yes—
195#400C23116yes—
196#400C23116yes—
197#400C23116yes—
198#400C23116yes—
199#398C23116yes—
200#398C2318yes—
201#398C2318yes—
202#398C2318yes—
203#398C23116yes—
204#398C23116yes—
205#398C23116yes—
206#398C23116yes—
207#398C23116yes—
208#398C2318yes—
209#395C23116yes—
210#395C2318yes—
211#395C2318yes—
212#395C23116yes—
213#395C23116yes—
214#395C23116yes—
215#391C2318yes—
216#391C23116yes—
217#391C2318yes—
218#391C23116yes—
219#391C23116yes—
220#391C23116yes—
221#391C23116yes—
222#391C23116yes—
223#389C23116yes—
224#389C23116yes—
225#389C23116yes—
226#389C23116yes—
227#389C23116yes—
228#389C23116yes—
229#389C23116yes—
230#389C23116yes—
231#389C23116yes—
232#387C23116yes—
233#387C23116yes—
234#387C23116yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C23116yes—
241#387C23116yes—
242#387C23116yes—
243#387C23116yes—
244#387C23116yes—
245#387C23116yes—
246#387C23116yes—
247#387C23116yes—
248#387C23116yes—
249#387C23116yes—
250#387C23116yes—
251#387C23116yes—
252#387C23116yes—
253#387C23116yes—
254#387C23116yes—
255#387C23116yes—
256#383C23116yes—
257#383C23116yes—
258#382C2318yes—
259#382C2318yes—
260#382C23116yes—
261C4×C4⋊C4C23116yes—
262C4×C4⋊C4C23116yes—
263C4×C4⋊C4C23116yes—
264C4×C4⋊C4C2318yes—
265#376C2318yes—
266#376C23116yes—
267C2×#95C2312yes—
268C2×#95C2312yes—
269C2×#95C2312yes—
270C2×#95C2312yes—
271C2×#95C2312yes—
272C2×#95C2312yes—
273C2×#95C2312yes—
274C2×#95C2312yes—
275C2×#95C2312yes—
276C2×#95C2311yescharacteristic
277C2×#95C2311yescharacteristic
278C2×#95C2312yes—
279C2×#95C2312yes—
280C2×#95C2312yes—
281C2×#95C2311yescharacteristic
282C2×#95C2311yescharacteristic
283C2×#95C2312yes—
284C2×#95C2312yes—
285C2×#95C2312yes—
286C2×#95C2312yes—
287C2×#95C2312yes—
288C2×#95C2312yes—
289C2×#95C2312yes—
290C2×#95C2312yes—
291C2×#95C2312yes—
292C2×#95C2312yes—
293C2×#95C2312yes—
294C2×#95C2312yes—
295C2×#95C2312yes—
296C2×#95C2312yes—
297C2×#95C2312yes—
298C2×#95C2312yes—
299C2×#95C2312yes—
300C2×#95C2312yes—
301C2×#95C2312yes—
302C2×#95C2312yes—
303C2×#95C2312yes—
304C2×#95C2312yes—
305C2×#95C2312yes—
306C2×#95C2312yes—
307C2×#95C2312yes—
308C22×Q8—24no—
309C22×Q8—28no—
310C22×Q8—28no—
311C22×Q8—42no—
312C22×Q8—42no—
313C22×Q8—42no—
314C22×Q8—42no—
315C22×Q8—42no—
316C22×Q8—44no—
317C22×Q8—44no—
318C22×Q8#4118yes—
319C22×Q8C2414yes—
320C22×Q8C2414yes—
321C22×Q8C2418yes—
322C22×Q8C2414yes—
323C23×C4C2412yes—
324C23×C4C2412yes—
325C23×C4C2412yes—
326C23×C4C2411yescharacteristic
327C23×C4C2412yes—
328C23×C4C2412yes—
329C23×C4C2412yes—
330C23×C4C2412yes—
331C23×C4C2412yes—
332C23×C4C2411yescharacteristic
333C23×C4C2412yes—
334C23×C4C2411yescharacteristic
335C23×C4C2412yes—
336C23×C4C2411yescharacteristic
337C23×C4C2411yescharacteristic
338C23×C4C2411yescharacteristic
339C23×C4C2411yescharacteristic
340C23×C4C2412yes—
341C23×C4C2412yes—
342C4⋊Q8—232no—
343C4⋊Q8—264no—
344C4⋊Q8—416no—
345#125—432no—
346C4×Q8—416no—
347C4×Q8—416no—
348C4×Q8—416no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—28no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4—28no—
357C2×C4⋊C4—28no—
358C2×C4⋊C4—28no—
359C2×C4⋊C4—28no—
360C2×C4⋊C4—28no—
361C2×C4⋊C4—28no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—28no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4—44no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—44no—
375C2×C4⋊C4—44no—
376C2×C4⋊C4—44no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C2418yes—
383C2×C4⋊C4C2418yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C2418yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C2418yes—
388C2×C4⋊C4C2418yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2418yes—
392C2×C4⋊C4C2418yes—
393C2×C4⋊C4C2418yes—
394C2×C4⋊C4C2418yes—
395C2×C4⋊C4C2414yes—
396C2×C4⋊C4C2414yes—
397C2×C4⋊C4C2414yes—
398C2×C4⋊C4C2414yes—
399C2×C4⋊C4C2414yes—
400C2×C4⋊C4C2418yes—
401C2×C4⋊C4C2418yes—
402C2×C4⋊C4C2418yes—
403C2×C4⋊C4C2414yes—
404C2×C4⋊C4C2414yes—
405C2×C4⋊C4C2418yes—
406C2×C4⋊C4C2418yes—
407C2×C4⋊C4C2418yes—
408C2×C4⋊C4C2418yes—
409C2×C42—28no—
410C2×C42—24no—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—28no—
415C2×C42—28no—
416C2×C42—44no—
417C2×C42—44no—
418C2×C42—44no—
419C2×C42—42no—
420C2×C42—44no—
421C2×C42C2414yes—
422C2×C42C2414yes—
423C2×C42C2414yes—
424C2×C42C2418yes—
425C2×C42C2414yes—
426C2×C42C2418yes—
427C2×C42C2418yes—
428C2×C42C2418yes—
429C2×C42C2418yes—
430C2×C42C2418yes—
431C2×C42C2414yes—
432C2×C42C2418yes—
433#95C2418yes—
434#95C2418yes—
435#95C2418yes—
436#95C2418yes—
437#95C2418yes—
438#95C2418yes—
439#95C2418yes—
440#95C2414yes—
441#95C2418yes—
442#95C2418yes—
443#95C2418yes—
444#95C2418yes—
445#95C2418yes—
446#95C2414yes—
447#95C2414yes—
448#95C2418yes—
449#95C2418yes—
450#95C2418yes—
451#95C2418yes—
452#95C2418yes—
453#95C2418yes—
454#95C2414yes—
455#95C2418yes—
456#95C2418yes—
457#95C2418yes—
458#95C2418yes—
459#95C2418yes—
460#95C2418yes—
461#95C2418yes—
462#95C2418yes—
463#95C2418yes—
464#95C2418yes—
465#95C2418yes—
466#95C2418yes—
467#95C2418yes—
468#95C2418yes—
469#95C2418yes—
470#95C2418yes—
471#95C2418yes—
472#95C2418yes—
473#95C2418yes—
474C24C2511yescharacteristic
475C2×Q8—28no—
476C2×Q8—216no—
477C2×Q8—216no—
478C2×Q8—44no—
479C2×Q8—44no—
480C2×Q8—44no—
481C2×Q8—416no—
482C2×Q8—44no—
483C2×Q8—44no—
484C2×Q8—48no—
485C2×Q8—48no—
486C2×Q8—416no—
487C2×Q8—416no—
488C2×Q8—816no—
489C2×Q8C2×#41116yes—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—24no—
521C22×C4—24no—
522C22×C4—24no—
523C22×C4—24no—
524C22×C4—24no—
525C22×C4—24no—
526C22×C4—24no—
527C22×C4—24no—
528C22×C4—24no—
529C22×C4—24no—
530C22×C4—24no—
531C22×C4—24no—
532C22×C4—24no—
533C22×C4—24no—
534C22×C4—24no—
535C22×C4—24no—
536C22×C4—24no—
537C22×C4C2×#4114yes—
538C22×C4C2×#4114yes—
539C22×C4C2×#4114yes—
540C22×C4H(5,𝔽2)14yes—
541C22×C42−1+418yes—
542C22×C42−1+418yes—
543C22×C4C2514yes—
544C22×C4C2514yes—
545C22×C4C2514yes—
546C22×C4C2514yes—
547C22×C4C2514yes—
548C22×C4C2512yes—
549C22×C4C2512yes—
550C22×C4C2514yes—
551C22×C4C2512yes—
552C22×C4C2514yes—
553C22×C4C2514yes—
554C22×C4C2514yes—
555C22×C4C2514yes—
556C22×C4C2512yes—
557C22×C4C2512yes—
558C22×C4C2512yes—
559C22×C4C2512yes—
560C22×C4C2514yes—
561C22×C4C2514yes—
562C4⋊C4—216no—
563C4⋊C4—216no—
564C4⋊C4—216no—
565C4⋊C4—216no—
566C4⋊C4—216no—
567C4⋊C4—216no—
568C4⋊C4—216no—
569C4⋊C4—216no—
570C4⋊C4—216no—
571C4⋊C4—216no—
572C4⋊C4—216no—
573C4⋊C4—216no—
574C4⋊C4—216no—
575C4⋊C4—216no—
576C4⋊C4—216no—
577C4⋊C4—216no—
578C4⋊C4—48no—
579C4⋊C4—48no—
580C4⋊C4—48no—
581C4⋊C4—48no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—44no—
587C4⋊C4—48no—
588C4⋊C4—48no—
589C4⋊C4—48no—
590C4⋊C4—44no—
591C4⋊C4—44no—
592C42—216no—
593C42—28no—
594C42—28no—
595C42—28no—
596C42—28no—
597C42—216no—
598C42—216no—
599C42—44no—
600C42—48no—
601C42—48no—
602C42—48no—
603C42—48no—
604C23C22×#4112yes—
605C23C2×H(5,𝔽2)12yes—
606C23C2×2−1+414yes—
607C23C2×2−1+414yes—
608C23#58512yes—
609C23C2611yescharacteristic
610Q8—416no—
611Q8—416no—
612Q8—416no—
613Q8—816no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—216no—
629C2×C4—216no—
630C2×C4—28no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—48no—
671C2×C4—48no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—48no—
676C2×C4—48no—
677C2×C4—48no—
678C2×C4—48no—
679C2×C4—48no—
680C2×C4C22×#4114yes—
681C2×C4C22×#4114yes—
682C2×C4C22×#4114yes—
683C2×C4C2×H(5,𝔽2)14yes—
684C2×C4C2×2−1+418yes—
685C2×C4C2×2−1+418yes—
686C22#350514yes—
687C22#352318yes—
688C22#353814yes—
689C22#354414yes—
690C22#355518yes—
691C22C23×#4111yescharacteristic
692C22C22×H(5,𝔽2)11yescharacteristic
693C22C22×2−1+412yes—
694C22C22×2−1+412yes—
695C22C2×#58511yescharacteristic
696C4—28no—
697C4—24no—
698C4—24no—
699C4—24no—
700C4—24no—
701C4—28no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—44no—
715C2#4754418yesdirect product factor
716C2C2×#353811yescharacteristic
717C2C2×#354411yescharacteristic
718C2C2×#355512yes—
719C2C2×#350511yescharacteristic
720C2C2×#352312yes—
subgroupG/H
center, socle, prime radicalC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #47544C2direct (N × H)
715. C2#47544direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27892
C2×#27762
C2×#27692
C2×#27662
C2×#27622
C2×#27342
C2×#27132
#475442
C2×#28332
C2×#25462
C2×#25812
C2×#26052
C2×#25902
C2×#25862
C2×#25332
C2×#25282
C2×#24532
C2×#24622
C2×#23312
C2×#23272