Finite Group Databasebeta

← C2×#58752 #8104982

Subgroups — C2×#58752 #8104982

Order 512.

4386 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#58752C2132yesmaximal, direct product factor
2C2×#2794C211yesmaximal, characteristic
3C2×#2792C211yesmaximal, characteristic
4C2×#2774C212yesmaximal
5C2×#2769C212yesmaximal
6C2×#2767C212yesmaximal
7C2×#2758C212yesmaximal
8C2×#2758C212yesmaximal
9C2×#2719C212yesmaximal
10C2×#2713C212yesmaximal
11C2×#2843C211yesmaximal, characteristic
12C2×#2844C212yesmaximal
13C2×#2571C211yesmaximal, characteristic
14C2×#2567C211yesmaximal, characteristic
15C2×#2581C211yesmaximal, characteristic
16C2×#2524C212yesmaximal
17C2×#2520C212yesmaximal
18C2×#2462C212yesmaximal
19C2×#2634C211yesmaximal, characteristic
20C2×#2666C212yesmaximal
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×#125C2212yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#2844C22132yes—
27#2843C22116yes—
28#2794C22116yes—
29#2792C22116yes—
30#2774C22132yes—
31#2769C22132yes—
32#2767C22132yes—
33#2758C22132yes—
34#2758C22132yes—
35#2719C22132yes—
36#2713C22132yes—
37#2666C22132yes—
38#2634C22116yes—
39#2581C22116yes—
40#2571C22116yes—
41#2567C22116yes—
42#2524C22132yes—
43#2520C22132yes—
44#2462C22132yes—
45C2×#400C2212yes—
46C2×#400C2211yescharacteristic
47C2×#400C2212yes—
48C2×#400C2212yes—
49C2×#400C2212yes—
50C2×#400C2212yes—
51C2×#400C2211yescharacteristic
52C2×#400C2211yescharacteristic
53C2×#400C2212yes—
54C2×#400C2212yes—
55C2×#400C2212yes—
56C2×#400C2212yes—
57C2×#400C2211yescharacteristic
58C2×#400C2212yes—
59C2×#400C2212yes—
60C2×#400C2212yes—
61C2×#400C2212yes—
62C2×#398C2211yescharacteristic
63C2×#398C2212yes—
64C2×#398C2211yescharacteristic
65C2×#398C2212yes—
66C2×#398C2212yes—
67C2×#398C2212yes—
68C2×#398C2212yes—
69C2×#398C2212yes—
70C2×#398C2212yes—
71C2×#398C2212yes—
72C2×#395C2212yes—
73C2×#395C2211yescharacteristic
74C2×#395C2212yes—
75C2×#395C2212yes—
76C2×#395C2212yes—
77C2×#395C2211yescharacteristic
78C2×#395C2212yes—
79C2×#395C2212yes—
80C2×#395C2212yes—
81C2×#391C2212yes—
82C2×#391C2212yes—
83C2×#391C2211yescharacteristic
84C2×#391C2211yescharacteristic
85C2×#391C2212yes—
86C2×#389C2212yes—
87C2×#389C2212yes—
88C2×#389C2212yes—
89C2×#389C2212yes—
90C2×#389C2212yes—
91C2×#389C2212yes—
92C2×#389C2212yes—
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×#384C2211yescharacteristic
114C2×#384C2211yescharacteristic
115C2×#383C2212yes—
116C2×#383C2211yescharacteristic
117C2×#383C2212yes—
118C2×#383C2212yes—
119C2×#382C2212yes—
120C2×#382C2212yes—
121C2×#382C2212yes—
122C2×#382C2212yes—
123C2×#382C2211yescharacteristic
124C2×C4×C4⋊C4C2212yes—
125C2×C4×C4⋊C4C2211yescharacteristic
126C2×C4×C4⋊C4C2212yes—
127C23×Q8C2311yescharacteristic
128C23×Q8C2311yescharacteristic
129C23×Q8C2312yes—
130C2×C4⋊Q8—44no—
131C2×C4⋊Q8C2318yes—
132C2×#125—44no—
133C2×#125—48no—
134C2×#125C2318yes—
135C2×#125C23116yes—
136C2×C4×Q8—44no—
137C2×C4×Q8—44no—
138C2×C4×Q8C2318yes—
139C2×C4×Q8C2318yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2312yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2312yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2311yescharacteristic
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2311yescharacteristic
166C22×C4⋊C4C2312yes—
167C22×C4⋊C4C2311yescharacteristic
168C22×C4⋊C4C2312yes—
169C22×C4⋊C4C2311yescharacteristic
170C22×C4⋊C4C2312yes—
171C22×C4⋊C4C2311yescharacteristic
172C22×C42C2311yescharacteristic
173C22×C42C2311yescharacteristic
174C22×C42C2311yescharacteristic
175C22×C42C2311yescharacteristic
176C22×C42C2312yes—
177C22×C42C2312yes—
178C22×C42C2311yescharacteristic
179C22×C42C2312yes—
180C22×C42C2312yes—
181C22×C42C2312yes—
182#400C2318yes—
183#400C2318yes—
184#400C23116yes—
185#400C2318yes—
186#400C23116yes—
187#400C23116yes—
188#400C23116yes—
189#400C23116yes—
190#400C23116yes—
191#400C23116yes—
192#400C23116yes—
193#400C2318yes—
194#400C23116yes—
195#400C23116yes—
196#400C23116yes—
197#400C23116yes—
198#400C23116yes—
199#398C23116yes—
200#398C23116yes—
201#398C23116yes—
202#398C2318yes—
203#398C2318yes—
204#398C23116yes—
205#398C23116yes—
206#398C23116yes—
207#398C23116yes—
208#398C23116yes—
209#395C23116yes—
210#395C23116yes—
211#395C2318yes—
212#395C2318yes—
213#395C23116yes—
214#395C23116yes—
215#395C23116yes—
216#395C23116yes—
217#395C23116yes—
218#391C2318yes—
219#391C2318yes—
220#391C23116yes—
221#391C23116yes—
222#391C23116yes—
223#389C23116yes—
224#389C23116yes—
225#389C23116yes—
226#389C23116yes—
227#389C23116yes—
228#389C23116yes—
229#389C23116yes—
230#387C23116yes—
231#387C23116yes—
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#384C2318yes—
251#384C2318yes—
252#383C23116yes—
253#383C2318yes—
254#383C23116yes—
255#383C23116yes—
256#382C23116yes—
257#382C23116yes—
258#382C2318yes—
259#382C23116yes—
260#382C23116yes—
261C4×C4⋊C4C2318yes—
262C4×C4⋊C4C23116yes—
263C4×C4⋊C4C23116yes—
264C2×#95C2312yes—
265C2×#95C2312yes—
266C2×#95C2312yes—
267C2×#95C2312yes—
268C2×#95C2312yes—
269C2×#95C2312yes—
270C2×#95C2312yes—
271C2×#95C2312yes—
272C2×#95C2312yes—
273C2×#95C2312yes—
274C2×#95C2311yescharacteristic
275C2×#95C2312yes—
276C2×#95C2311yescharacteristic
277C2×#95C2312yes—
278C2×#95C2312yes—
279C2×#95C2312yes—
280C2×#95C2312yes—
281C2×#95C2312yes—
282C2×#95C2312yes—
283C2×#95C2312yes—
284C2×#95C2312yes—
285C2×#95C2312yes—
286C2×#95C2312yes—
287C2×#95C2312yes—
288C2×#95C2311yescharacteristic
289C2×#95C2312yes—
290C2×#95C2312yes—
291C2×#95C2312yes—
292C2×#95C2312yes—
293C2×#95C2312yes—
294C2×#95C2311yescharacteristic
295C2×#95C2312yes—
296C2×#95C2312yes—
297C2×#95C2312yes—
298C2×#95C2312yes—
299C2×#95C2312yes—
300C2×#95C2312yes—
301C2×#95C2312yes—
302C2×#95C2312yes—
303C2×#95C2312yes—
304C2×#95C2312yes—
305C2×#95C2312yes—
306C22×Q8—24no—
307C22×Q8—24no—
308C22×Q8—42no—
309C22×Q8—42no—
310C22×Q8—44no—
311C22×Q8—44no—
312C22×Q8—44no—
313C22×Q8—44no—
314C22×Q8C2414yes—
315C22×Q8C2414yes—
316C22×Q8C2418yes—
317C23×C4C2411yescharacteristic
318C23×C4C2411yescharacteristic
319C23×C4C2412yes—
320C23×C4C2412yes—
321C23×C4C2412yes—
322C23×C4C2412yes—
323C23×C4C2412yes—
324C23×C4C2411yescharacteristic
325C23×C4C2412yes—
326C23×C4C2412yes—
327C23×C4C2412yes—
328C23×C4C2412yes—
329C23×C4C2412yes—
330C23×C4C2412yes—
331C23×C4C2411yescharacteristic
332C23×C4C2411yescharacteristic
333C23×C4C2412yes—
334C23×C4C2411yescharacteristic
335C23×C4C2411yescharacteristic
336C4⋊Q8—416no—
337#125—416no—
338#125—432no—
339C4×Q8—416no—
340C4×Q8—416no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—28no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—28no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—28no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—42no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4—44no—
369C2×C4⋊C4—44no—
370C2×C4⋊C4—44no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C2418yes—
377C2×C4⋊C4C2418yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C2414yes—
383C2×C4⋊C4C2414yes—
384C2×C4⋊C4C2414yes—
385C2×C4⋊C4C2414yes—
386C2×C4⋊C4C2418yes—
387C2×C4⋊C4C2418yes—
388C2×C4⋊C4C2414yes—
389C2×C4⋊C4C2418yes—
390C2×C4⋊C4C2418yes—
391C2×C4⋊C4C2414yes—
392C2×C4⋊C4C2414yes—
393C2×C4⋊C4C2414yes—
394C2×C4⋊C4C2418yes—
395C2×C4⋊C4C2418yes—
396C2×C4⋊C4C2418yes—
397C2×C4⋊C4C2418yes—
398C2×C4⋊C4C2418yes—
399C2×C4⋊C4C2418yes—
400C2×C4⋊C4C2418yes—
401C2×C4⋊C4C2418yes—
402C2×C4⋊C4C2418yes—
403C2×C4⋊C4C2418yes—
404C2×C4⋊C4C2418yes—
405C2×C42—24no—
406C2×C42—24no—
407C2×C42—24no—
408C2×C42—24no—
409C2×C42—28no—
410C2×C42—28no—
411C2×C42—24no—
412C2×C42—44no—
413C2×C42—44no—
414C2×C42—44no—
415C2×C42C2414yes—
416C2×C42C2414yes—
417C2×C42C2414yes—
418C2×C42C2414yes—
419C2×C42C2418yes—
420C2×C42C2418yes—
421C2×C42C2414yes—
422C2×C42C2418yes—
423C2×C42C2418yes—
424C2×C42C2418yes—
425#95C2418yes—
426#95C2418yes—
427#95C2418yes—
428#95C2418yes—
429#95C2418yes—
430#95C2414yes—
431#95C2414yes—
432#95C2418yes—
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#95C2418yes—
447#95C2418yes—
448#95C2418yes—
449#95C2418yes—
450#95C2418yes—
451#95C2418yes—
452#95C2418yes—
453#95C2418yes—
454#95C2418yes—
455#95C2418yes—
456#95C2418yes—
457#95C2414yes—
458#95C2418yes—
459#95C2418yes—
460#95C2418yes—
461#95C2418yes—
462#95C2418yes—
463#95C2418yes—
464#95C2418yes—
465#95C2418yes—
466#95C2418yes—
467C24C2511yescharacteristic
468C2×Q8—28no—
469C2×Q8—28no—
470C2×Q8—44no—
471C2×Q8—44no—
472C2×Q8—48no—
473C2×Q8—48no—
474C2×Q8—48no—
475C2×Q8—48no—
476C2×Q8—416no—
477C2×Q8—416no—
478C2×Q8—816no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—22no—
513C22×C4—22no—
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×C4H(5,𝔽2)14yes—
529C22×C42−1+418yes—
530C22×C42−1+414yes—
531C22×C42−1+414yes—
532C22×C4C2514yes—
533C22×C4C2514yes—
534C22×C4C2514yes—
535C22×C4C2514yes—
536C22×C4C2514yes—
537C22×C4C2512yes—
538C22×C4C2514yes—
539C22×C4C2514yes—
540C22×C4C2514yes—
541C22×C4C2512yes—
542C22×C4C2512yes—
543C22×C4C2514yes—
544C22×C4C2514yes—
545C22×C4C2514yes—
546C22×C4C2512yes—
547C22×C4C2512yes—
548C22×C4C2514yes—
549C22×C4C2512yes—
550C22×C4C2512yes—
551C4⋊C4—216no—
552C4⋊C4—216no—
553C4⋊C4—216no—
554C4⋊C4—216no—
555C4⋊C4—216no—
556C4⋊C4—216no—
557C4⋊C4—216no—
558C4⋊C4—216no—
559C4⋊C4—216no—
560C4⋊C4—48no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—48no—
564C4⋊C4—48no—
565C4⋊C4—48no—
566C4⋊C4—48no—
567C4⋊C4—48no—
568C4⋊C4—48no—
569C4⋊C4—48no—
570C4⋊C4—48no—
571C4⋊C4—44no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—48no—
578C4⋊C4—48no—
579C4⋊C4—48no—
580C4⋊C4—48no—
581C4⋊C4—44no—
582C4⋊C4—44no—
583C42—28no—
584C42—28no—
585C42—28no—
586C42—28no—
587C42—216no—
588C42—216no—
589C42—28no—
590C42—48no—
591C42—48no—
592C42—48no—
593C23C2×H(5,𝔽2)12yes—
594C23C2×2−1+412yes—
595C23C2×2−1+412yes—
596C23C2×2−1+414yes—
597C23#58514yes—
598C23C2611yescharacteristic
599Q8—416no—
600Q8—416no—
601Q8—816no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—216no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
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—48no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—48no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—44no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—48no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4C2×H(5,𝔽2)14yes—
671C2×C4C2×2−1+418yes—
672C2×C4C2×2−1+414yes—
673C2×C4C2×2−1+414yes—
674C22#353814yes—
675C22#354014yes—
676C22#354414yes—
677C22#355518yes—
678C22#356318yes—
679C22C22×H(5,𝔽2)11yescharacteristic
680C22C22×2−1+411yescharacteristic
681C22C22×2−1+411yescharacteristic
682C22C22×2−1+412yes—
683C22C2×#58512yes—
684C4—24no—
685C4—24no—
686C4—28no—
687C4—24no—
688C4—44no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—44no—
703C2#5875218yesdirect product factor
704C2C2×#353811yescharacteristic
705C2C2×#354011yescharacteristic
706C2C2×#356312yes—
707C2C2×#354411yescharacteristic
708C2C2×#355512yes—
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
1. #58752C2direct (N × H)
703. C2#58752direct (N × H)

Maximal subgroups

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

subgroupindex
#587522
C2×#27942
C2×#27922
C2×#27742
C2×#27692
C2×#27672
C2×#27582
C2×#27192
C2×#27132
C2×#28432
C2×#28442
C2×#25712
C2×#25672
C2×#25812
C2×#25242
C2×#25202
C2×#24622
C2×#26342
C2×#26662