Finite Group Databasebeta

← C2×#42813 #7641438

Subgroups — C2×#42813 #7641438

Order 512.

7128 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
1C22×Q82C211yesmaximal, characteristic, complemented
2C22×#552C211yesmaximal, characteristic
3C2×#2713C214yesmaximal, complemented
4C2×#2674C214yesmaximal
5#42813C2132yesmaximal, direct product factor
6C2×#2821C212yesmaximal, complemented
7C2×#2547C211yesmaximal, characteristic
8C2×#2584C211yesmaximal, characteristic, complemented
9C2×#2599C212yesmaximal, complemented
10C2×#2439C212yesmaximal, complemented
11C2×#2421C212yesmaximal, complemented
12C2×#2423C212yesmaximal, complemented
13C2×#2440C212yesmaximal, complemented
14C2×#2621C212yesmaximal, complemented
15C2×#2660C211yesmaximal, characteristic, complemented
16C2×#2637C212yesmaximal, complemented
17C2×#2398C211yesmaximal, characteristic
18C2×#2320C211yesmaximal, characteristic, complemented
19C2×Q82—216no—
20C2×Q82C22116yescomplemented
21C2×#552—216no—
22C2×#552C22116yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2212yes—
25C22×C4⋊Q8C2212yes—
26C22×C4⋊Q8C2214yes—
27C22×#126C2212yes—
28C22×#125C2211yescharacteristic
29C22×#125C2212yes—
30C22×#124C2212yes—
31C22×#123C2212yes—
32C22×C22⋊Q8C2211yescharacteristic
33C22×C22⋊Q8C2211yescharacteristic
34C22×C22⋊Q8C2214yes—
35C22×C4×Q8C2212yes—
36C22×C4×Q8C2211yescharacteristic
37C22×C4×Q8C2212yes—
38C22×C4×Q8C2211yescharacteristic
39C22×C4×D4C2211yescharacteristic
40C22×#117C2212yes—
41C22×#117C2211yescharacteristic
42C23×C4⋊C4C2211yescharacteristic
43#2821C22132yescomplemented
44#2713C22164yescomplemented
45#2674C22164yes—
46#2660C22116yescomplemented
47#2637C22132yescomplemented
48#2621C22132yescomplemented
49#2599C22132yescomplemented
50#2584C22116yescomplemented
51#2547C22116yes—
52#2440C22132yescomplemented
53#2439C22132yescomplemented
54#2423C22132yescomplemented
55#2421C22132yescomplemented
56#2398C22116yes—
57C2×#400C2214yes—
58C2×#400C2212yes—
59C2×#399C2214yes—
60C2×#398C2212yes—
61C2×#398C2214yes—
62C2×#398C2212yes—
63C2×#398C2212yes—
64C2×#398C2214yes—
65C2×#397C2212yes—
66C2×#397C2212yes—
67C2×#397C2211yescharacteristic
68C2×#397C2214yes—
69C2×#397C2211yescharacteristic
70C2×#396C2214yes—
71C2×#395C2212yes—
72C2×#395C2214yes—
73C2×#395C2214yes—
74C2×#394C2212yes—
75C2×#393C2211yescharacteristic
76C2×#393C2211yescharacteristic
77C2×#393C2214yes—
78#2320C22116yescomplemented
79C2×#391C2212yes—
80C2×#391C2214yes—
81C2×#391C2212yes—
82C2×#389C2212yes—
83C2×#389C2212yes—
84C2×#389C2214yes—
85C2×#389C2212yes—
86C2×#388C2214yes—
87C2×#388C2214yes—
88C2×#387C2214yes—
89C2×#387C2214yes—
90C2×#387C2214yes—
91C2×#387C2214yes—
92C2×#385C2214yes—
93C2×#384C2212yes—
94C2×#383C2212yes—
95C2×#382C2212yes—
96C2×#382C2212yes—
97C2×#380C2212yes—
98C2×#380C2212yes—
99C2×C4×C4⋊C4C2211yescharacteristic
100C2×C4×C22⋊C4C2212yes—
101C2×C4×C22⋊C4C2212yes—
102C2×#376C2211yescharacteristic
103C23×Q8C2311yescharacteristic
104C23×Q8C2311yescharacteristic
105C23×Q8C2314yes—
106C23×Q8C2312yes—
107C23×D4C2311yescharacteristic
108C24×C4C2311yescharacteristic
109C24×C4C2311yescharacteristic
110C24×C4C2312yes—
111Q82—2128no—
112#552—2128no—
113C2×C4⋊Q8—216no—
114C2×C4⋊Q8—232no—
115C2×C4⋊Q8—216no—
116C2×C4⋊Q8—28no—
117C2×C4⋊Q8C23116yes—
118C2×C4⋊Q8C2318yes—
119C2×C4⋊Q8C23132yes—
120C2×C4⋊Q8C23116yes—
121C2×#126—216no—
122C2×#126C23116yes—
123C2×#125—216no—
124C2×#125—28no—
125C2×#125C2318yes—
126C2×#125C23116yes—
127C2×#124—48no—
128C2×#124C23116yes—
129C2×#123—216no—
130C2×#123C23116yes—
131C2×C22⋊Q8—28no—
132C2×C22⋊Q8—28no—
133C2×C22⋊Q8—416no—
134C2×C22⋊Q8C2318yes—
135C2×C22⋊Q8C2318yes—
136C2×C22⋊Q8C23132yes—
137C2×C4×Q8—216no—
138C2×C4×Q8—216no—
139C2×C4×Q8—28no—
140C2×C4×Q8D4116yes—
141C2×C4×Q8C2318yes—
142C2×C4×Q8C23116yes—
143C2×C4×Q8C23116yes—
144C2×C4×Q8C2318yes—
145C2×C4×D4—28no—
146C2×C4×D4C2318yes—
147C2×#117—28no—
148C2×#117—216no—
149C2×#117C23116yes—
150C2×#117C2318yes—
151C22×C4⋊C4—28no—
152C22×C4⋊C4C2312yes—
153C22×C4⋊C4C2312yes—
154C22×C4⋊C4C2312yes—
155C22×C4⋊C4C2314yes—
156C22×C4⋊C4C2312yes—
157C22×C4⋊C4C2314yes—
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2314yes—
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2311yescharacteristic
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2311yescharacteristic
165C22×C4⋊C4C2318yes—
166C22×C4⋊C4C2311yescharacteristic
167C22×C4⋊C4C2312yes—
168C22×C4⋊C4C2312yes—
169C22×C4⋊C4C2314yes—
170C22×C4⋊C4C2314yes—
171C22×C4⋊C4C2314yes—
172C22×C4⋊C4C2314yes—
173C22×C4⋊C4C2311yescharacteristic
174C22×C4⋊C4C2312yes—
175C22×C22⋊C4C2312yes—
176C22×C22⋊C4C2312yes—
177C22×C22⋊C4C2312yes—
178C22×C22⋊C4C2311yescharacteristic
179C22×C22⋊C4C2311yescharacteristic
180C22×C22⋊C4C2312yes—
181C22×C22⋊C4C2312yes—
182C22×C22⋊C4C2314yes—
183C22×C22⋊C4C2312yes—
184C22×C22⋊C4C2312yes—
185C22×C22⋊C4C2314yes—
186C22×C42C2312yes—
187C22×C42C2312yes—
188C22×C42C2314yes—
189C22×C42C2312yes—
190C22×C42C2312yes—
191C22×C42C2312yes—
192C22×C42C2311yescharacteristic
193#400C23132yes—
194#400C23116yes—
195#399C23132yes—
196#398C23116yes—
197#398C23116yes—
198#398C23116yes—
199#398C23132yes—
200#398C23132yes—
201#397C2318yes—
202#397C23116yes—
203#397C23116yes—
204#397C2318yes—
205#397C23132yes—
206#396C23132yes—
207#395C23132yes—
208#395C23116yes—
209#395C23132yes—
210#394C23116yes—
211#393C23132yes—
212#393C2318yes—
213#393C2318yes—
214#391C23116yes—
215#391C23132yes—
216#391C23116yes—
217#389C23116yes—
218#389C23132yes—
219#389C23116yes—
220#389C23116yes—
221#388C23132yes—
222#388C23132yes—
223#387C23132yes—
224#387C23132yes—
225#387C23132yes—
226#387C23132yes—
227#385C23132yes—
228#384C23116yes—
229#383C23116yes—
230#382C23116yes—
231#382C23116yes—
232#380C23116yes—
233#380C23116yes—
234C4×C4⋊C4C2318yes—
235C4×C22⋊C4C23116yes—
236C4×C22⋊C4C23116yes—
237#376C2318yes—
238C2×#95C2314yes—
239C2×#95C2314yes—
240C2×#95C2311yescharacteristic
241C2×#95C2311yescharacteristic
242C2×#95C2314yes—
243C2×#95C2312yes—
244C2×#95C2312yes—
245C2×#95C2312yes—
246C2×#95C2312yes—
247C2×#95C2314yes—
248C2×#95C2312yes—
249C2×#95C2314yes—
250C2×#95C2314yes—
251C2×#95C2312yes—
252C2×#95C2314yes—
253C2×#95C2314yes—
254C2×#95C2314yes—
255C25C2411yescharacteristic
256C25C2411yescharacteristic
257C22×Q8—24no—
258C22×Q8—216no—
259C22×Q8—28no—
260C22×Q8—28no—
261C22×Q8—28no—
262C22×Q8—48no—
263C22×Q8—44no—
264C22×Q8—48no—
265C22×Q8—44no—
266C22×Q8C2×D418yes—
267C22×Q8C2414yes—
268C22×Q8C2414yes—
269C22×Q8C24116yes—
270C22×Q8C2418yes—
271C22×D4—24no—
272C22×D4—24no—
273C22×D4—42no—
274C22×D4C2414yes—
275C23×C4—28no—
276C23×C4—28no—
277C23×C4—24no—
278C23×C4—24no—
279C23×C4—28no—
280C23×C4—28no—
281C23×C4—44no—
282C23×C4C2414yes—
283C23×C4C2414yes—
284C23×C4C2414yes—
285C23×C4C2412yes—
286C23×C4C2412yes—
287C23×C4C2414yes—
288C23×C4C2412yes—
289C23×C4C2412yes—
290C23×C4C2412yes—
291C23×C4C2411yescharacteristic
292C23×C4C2411yescharacteristic
293C23×C4C2418yes—
294C23×C4C2412yes—
295C23×C4C2411yescharacteristic
296C23×C4C2412yes—
297C23×C4C2411yescharacteristic
298C23×C4C2412yes—
299C23×C4C2411yescharacteristic
300C4⋊Q8—264no—
301C4⋊Q8—2128no—
302C4⋊Q8—264no—
303C4⋊Q8—232no—
304#126—264no—
305#125—264no—
306#125—232no—
307#124—432no—
308#123—264no—
309C22⋊Q8—232no—
310C22⋊Q8—232no—
311C22⋊Q8—464no—
312C4×Q8—264no—
313C4×Q8—264no—
314C4×Q8—232no—
315C4×Q8C2×D4164yes—
316C4×D4—232no—
317#117—232no—
318#117—264no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—216no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—216no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—216no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—232no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4—48no—
340C2×C4⋊C4C2×D418yes—
341C2×C4⋊C4C2×D4116yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2418yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2418yes—
346C2×C4⋊C4C2418yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2418yes—
349C2×C4⋊C4C24116yes—
350C2×C4⋊C4C2418yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2418yes—
353C2×C4⋊C4C24116yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2418yes—
356C2×C4⋊C4C2418yes—
357C2×C4⋊C4C24116yes—
358C2×C4⋊C4C24116yes—
359C2×C4⋊C4C24116yes—
360C2×C4⋊C4C24116yes—
361C2×C4⋊C4C24116yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2418yes—
364C2×C22⋊C4—28no—
365C2×C22⋊C4—216no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—48no—
375C2×C22⋊C4C2414yes—
376C2×C22⋊C4C2418yes—
377C2×C22⋊C4C2418yes—
378C2×C22⋊C4C2414yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2418yes—
381C2×C22⋊C4C24116yes—
382C2×C22⋊C4C2418yes—
383C2×C22⋊C4C2418yes—
384C2×C22⋊C4C24116yes—
385C2×C22⋊C4C2418yes—
386C2×C42—28no—
387C2×C42—216no—
388C2×C42—28no—
389C2×C42—28no—
390C2×C42—44no—
391C2×C42C2×D4116yes—
392C2×C42C2×D418yes—
393C2×C42C2418yes—
394C2×C42C2418yes—
395C2×C42C24116yes—
396C2×C42C2418yes—
397C2×C42C2418yes—
398C2×C42C2418yes—
399C2×C42C2414yes—
400#95C2414yes—
401#95C2414yes—
402#95C2418yes—
403#95C2418yes—
404#95C2418yes—
405#95C2418yes—
406#95C2418yes—
407#95C2418yes—
408#95C24116yes—
409#95C24116yes—
410#95C24116yes—
411#95C24116yes—
412#95C24116yes—
413#95C24116yes—
414#95C24116yes—
415#95C24116yes—
416#95C24116yes—
417C24—22no—
418C24—24no—
419C24—24no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—24no—
424C24—24no—
425C24—22no—
426C24H(5,𝔽2)14yes—
427C24C2512yes—
428C24C2512yes—
429C24C2511yescharacteristic
430C2×Q8—232no—
431C2×Q8—232no—
432C2×Q8—28no—
433C2×Q8—232no—
434C2×Q8—216no—
435C2×Q8—216no—
436C2×Q8—216no—
437C2×Q8—416no—
438C2×Q8—48no—
439C2×Q8—416no—
440C2×Q8—48no—
441C2×Q8—832no—
442C2×Q8—816no—
443C2×Q8C22×D4116yes—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—44no—
447C2×D4—416no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—28no—
481C22×C4—216no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—216no—
485C22×C4—216no—
486C22×C4—48no—
487C22×C4—432no—
488C22×C4—416no—
489C22×C4—416no—
490C22×C4C22×D418yes—
491C22×C4C22×D418yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C42−1+418yes—
496C22×C42−1+418yes—
497C22×C4C2514yes—
498C22×C4C2512yes—
499C22×C4C2518yes—
500C22×C4C2514yes—
501C22×C4C2512yes—
502C22×C4C2514yes—
503C22×C4C2518yes—
504C22×C4C2514yes—
505C22×C4C2514yes—
506C22×C4C2514yes—
507C22×C4C2512yes—
508C22×C4C2512yes—
509C22×C4C2514yes—
510C22×C4C2514yes—
511C22×C4C2512yes—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—28no—
518C4⋊C4—232no—
519C4⋊C4—216no—
520C4⋊C4—232no—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—232no—
527C4⋊C4—216no—
528C4⋊C4—416no—
529C4⋊C4—416no—
530C4⋊C4—416no—
531C4⋊C4—416no—
532C4⋊C4C22×D4116yes—
533C4⋊C4C22×D4132yes—
534C22⋊C4—216no—
535C22⋊C4—232no—
536C22⋊C4—28no—
537C22⋊C4—216no—
538C22⋊C4—216no—
539C22⋊C4—28no—
540C22⋊C4—216no—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—48no—
544C22⋊C4—416no—
545C42—216no—
546C42—232no—
547C42—216no—
548C42—216no—
549C42—48no—
550C42C22×D4132yes—
551C42C22×D4116yes—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—22no—
556C23—28no—
557C23—216no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—44no—
564C23—44no—
565C23—48no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—48no—
570C23—48no—
571C23C23×D412yes—
572C23C2×H(5,𝔽2)14yes—
573C23C2×H(5,𝔽2)12yes—
574C23C2×2−1+414yes—
575C23C2×2−1+414yes—
576C23#58512yes—
577C23C2611yescharacteristic
578Q8—232no—
579Q8—232no—
580Q8—832no—
581Q8—816no—
582D4—416no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—216no—
592C2×C4—28no—
593C2×C4—216no—
594C2×C4—216no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—432no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—416no—
626C2×C4—48no—
627C2×C4—416no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—416no—
631C2×C4—416no—
632C2×C4#536116yes—
633C2×C4C23×D418yes—
634C2×C4C23×D418yes—
635C2×C4C23×D414yes—
636C2×C4C23×D414yes—
637C2×C4C23×D414yes—
638C2×C4C2×2−1+418yes—
639C2×C4C2×2−1+418yes—
640C22—28no—
641C22—24no—
642C22—44no—
643C22—416no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—88no—
652C22C2×#53614yes—
653C22#348614yes—
654C22#348814yes—
655C22#354418yes—
656C22#355518yes—
657C22C24×D411yescharacteristic
658C22C22×H(5,𝔽2)11yescharacteristic
659C22C22×2−1+412yes—
660C22C22×2−1+412yes—
661C22C2×#58511yescharacteristic
662C4—24no—
663C4—28no—
664C4—28no—
665C4—24no—
666C4—28no—
667C4—48no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—48no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4C2×#53618yes—
677C2—44no—
678C2—82no—
679C2#4281318yesdirect product factor
680C2C22×#53611yescharacteristic
681C2C2×#354412yes—
682C2C2×#355512yes—
683C2C2×#348611yescharacteristic
684C2C2×#348811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×Q82C2semidirect
3. C2×#2713C2semidirect
5. #42813C2direct (N × H)
6. C2×#2821C2semidirect
8. C2×#2584C2semidirect
9. C2×#2599C2semidirect
10. C2×#2439C2semidirect
11. C2×#2421C2semidirect
12. C2×#2423C2semidirect
13. C2×#2440C2semidirect
14. C2×#2621C2semidirect
15. C2×#2660C2semidirect
16. C2×#2637C2semidirect
18. C2×#2320C2semidirect
20. C2×Q82C22semidirect
43. #2821C22semidirect
44. #2713C22semidirect
46. #2660C22semidirect
47. #2637C22semidirect
48. #2621C22semidirect
49. #2599C22semidirect
50. #2584C22semidirect
52. #2440C22semidirect
53. #2439C22semidirect
54. #2423C22semidirect
55. #2421C22semidirect
78. #2320C22semidirect
679. C2#42813direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q822
C22×#5522
C2×#27132
C2×#26742
#428132
C2×#28212
C2×#25472
C2×#25842
C2×#25992
C2×#24392
C2×#24212
C2×#24232
C2×#24402
C2×#26212
C2×#26602
C2×#26372
C2×#23982
C2×#23202