Finite Group Databasebeta

← C2×#35723 #7629623

Subgroups — C2×#35723 #7629623

Order 512.

8438 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C22×#519C211yesmaximal, characteristic, complemented
3#35723C2132yesmaximal, direct product factor
4C2×#2499C213yesmaximal, complemented
5C2×#2485C213yesmaximal
6C2×#2421C213yesmaximal, complemented
7C2×#2446C213yesmaximal, complemented
8C2×#2446C213yesmaximal, complemented
9C2×#2423C213yesmaximal, complemented
10C2×#2625C213yesmaximal, complemented
11C2×#2395C213yesmaximal
12C2×#2351C211yesmaximal, characteristic, complemented
13C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
14C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
15C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
16C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
17C24×Q8C2211yescharacteristic
18C24×Q8C2211yescharacteristic
19C2×D4×Q8—216no—
20C2×D4×Q8C22116yes—
21C22×C4⋊Q8C2213yes—
22C22×C22⋊Q8C2213yes—
23C22×C22⋊Q8C2213yes—
24C2×#519—216no—
25C2×#519C22116yescomplemented
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2213yes—
32C22×C4×Q8C2213yes—
33C22×C4×D4C2213yes—
34C22×#117C2213yes—
35C22×#117C2213yes—
36#2625C22148yescomplemented
37#2499C22148yescomplemented
38#2485C22148yes—
39#2446C22148yescomplemented
40#2446C22148yescomplemented
41#2423C22148yescomplemented
42#2421C22148yescomplemented
43#2395C22148yes—
44C2×#400C2216yes—
45C2×#400C2213yes—
46C2×#399C2213yes—
47C2×#398C2216yes—
48C2×#398C2213yes—
49C2×#397C2213yes—
50C2×#397C2216yes—
51C2×#397C2213yes—
52C2×#397C2213yes—
53C2×#394C2213yes—
54C2×#394C2211yescharacteristic
55#2351C22116yescomplemented
56Q8×C4⋊C4C22116yescomplemented
57Q8×C22⋊C4C22116yescomplemented
58Q8×C22⋊C4C22116yescomplemented
59Q8×C22⋊C4C22116yescomplemented
60C2×#391C2213yes—
61C2×#391C2213yes—
62C2×#391C2213yes—
63C2×#391C2213yes—
64C2×#391C2213yes—
65C2×#391C2213yes—
66C2×#389C2213yes—
67C2×#389C2213yes—
68C2×#389C2213yes—
69C2×#389C2213yes—
70C2×#388C2216yes—
71C2×#387C2216yes—
72C2×#387C2216yes—
73C2×#387C2216yes—
74C2×#384C2213yes—
75C2×#382C2213yes—
76C2×#380C2213yes—
77C2×#380C2213yes—
78C2×#380C2213yes—
79C2×C4×C4⋊C4C2213yes—
80C2×C4×C4⋊C4C2213yes—
81C2×C4×C22⋊C4C2213yes—
82C2×C4×C22⋊C4C2213yes—
83C2×C4×C22⋊C4C2213yes—
84C23×Q8—28no—
85C23×Q8—28no—
86C23×Q8—28no—
87C23×Q8—28no—
88C23×Q8—28no—
89C23×Q8D4116yes—
90C23×Q8C2318yes—
91C23×Q8C2313yes—
92C23×Q8C2318yes—
93C23×Q8C2313yes—
94C23×Q8C2311yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4C2313yes—
97C24×C4C2313yes—
98D4×Q8—2128no—
99C2×C4⋊Q8—224no—
100C2×C4⋊Q8C23124yes—
101C2×C22⋊Q8—224no—
102C2×C22⋊Q8—224no—
103C2×C22⋊Q8C23124yes—
104C2×C22⋊Q8C23124yes—
105#519—2128no—
106C2×C4×Q8—224no—
107C2×C4×Q8—224no—
108C2×C4×Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8C2318yes—
114C2×C4×Q8C2318yes—
115C2×C4×Q8C2318yes—
116C2×C4×Q8C2318yes—
117C2×C4×Q8C2318yes—
118C2×C4×Q8C23124yes—
119C2×C4×Q8C23124yes—
120C2×C4×D4—224no—
121C2×C4×D4C23124yes—
122C2×#117—224no—
123C2×#117—224no—
124C2×#117C23124yes—
125C2×#117C23124yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2313yes—
128C22×C4⋊C4C2313yes—
129C22×C4⋊C4C2313yes—
130C22×C4⋊C4C2313yes—
131C22×C4⋊C4C2313yes—
132C22×C4⋊C4C2316yes—
133C22×C4⋊C4C2313yes—
134C22×C4⋊C4C2313yes—
135C22×C4⋊C4C2313yes—
136C22×C4⋊C4C2313yes—
137C22×C4⋊C4C2313yes—
138C22×C4⋊C4C2313yes—
139C22×C4⋊C4C2316yes—
140C22×C22⋊C4C2313yes—
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2311yescharacteristic
143C22×C22⋊C4C2311yescharacteristic
144C22×C22⋊C4C2313yes—
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2313yes—
147C22×C22⋊C4C2313yes—
148C22×C22⋊C4C2313yes—
149C22×C22⋊C4C2313yes—
150C22×C42C2313yes—
151C22×C42C2313yes—
152C22×C42C2316yes—
153C22×C42C2313yes—
154C22×C42C2313yes—
155C22×C42C2313yes—
156#400C23148yes—
157#400C23124yes—
158#399C23124yes—
159#398C23148yes—
160#398C23124yes—
161#397C23124yes—
162#397C23148yes—
163#397C23124yes—
164#397C23124yes—
165#394C2318yes—
166#394C23124yes—
167#391C23124yes—
168#391C23124yes—
169#391C23124yes—
170#391C23124yes—
171#391C23124yes—
172#391C23124yes—
173#389C23124yes—
174#389C23124yes—
175#389C23124yes—
176#389C23124yes—
177#388C23148yes—
178#387C23148yes—
179#387C23148yes—
180#387C23148yes—
181#384C23124yes—
182#382C23124yes—
183#380C23124yes—
184#380C23124yes—
185#380C23124yes—
186C4×C4⋊C4C23124yes—
187C4×C4⋊C4C23124yes—
188C4×C22⋊C4C23124yes—
189C4×C22⋊C4C23124yes—
190C4×C22⋊C4C23124yes—
191C2×#95C2311yescharacteristic
192C2×#95C2313yes—
193C2×#95C2313yes—
194C2×#95C2313yes—
195C2×#95C2313yes—
196C2×#95C2313yes—
197C2×#95C2313yes—
198C2×#95C2316yes—
199C2×#95C2313yes—
200C2×#95C2313yes—
201C2×#95C2316yes—
202C2×#95C2316yes—
203C2×#95C2313yes—
204C2×#95C2316yes—
205C25C2411yescharacteristic
206C25C2411yescharacteristic
207C22×Q8—2128no—
208C22×Q8—232no—
209C22×Q8—232no—
210C22×Q8—232no—
211C22×Q8—232no—
212C22×Q8—232no—
213C22×Q8—212no—
214C22×Q8—212no—
215C22×Q8—212no—
216C22×Q8—212no—
217C22×Q8—212no—
218C22×Q8—464no—
219C22×Q8C2×D4124yes—
220C22×Q8C2×D4164yes—
221C22×Q8C2×D418yes—
222C22×Q8#4118yes—
223C22×Q8#4118yes—
224C22×Q8C24112yes—
225C22×Q8C24112yes—
226C22×Q8C2414yes—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4C2414yes—
231C23×C4—212no—
232C23×C4—212no—
233C23×C4—212no—
234C23×C4—212no—
235C23×C4—212no—
236C23×C4C2×D4124yes—
237C23×C4C24112yes—
238C23×C4C2413yes—
239C23×C4C2411yescharacteristic
240C23×C4C2413yes—
241C23×C4C2413yes—
242C23×C4C2413yes—
243C23×C4C2411yescharacteristic
244C23×C4C2413yes—
245C23×C4C2411yescharacteristic
246C23×C4C2413yes—
247C23×C4C24112yes—
248C23×C4C2413yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2413yes—
252C4⋊Q8—296no—
253C22⋊Q8—296no—
254C22⋊Q8—296no—
255C4×Q8—296no—
256C4×Q8—296no—
257C4×Q8—232no—
258C4×Q8—232no—
259C4×Q8—232no—
260C4×Q8—232no—
261C4×Q8—232no—
262C4×D4—296no—
263#117—296no—
264#117—296no—
265C2×C4⋊C4—212no—
266C2×C4⋊C4—212no—
267C2×C4⋊C4—212no—
268C2×C4⋊C4—212no—
269C2×C4⋊C4—212no—
270C2×C4⋊C4—212no—
271C2×C4⋊C4—224no—
272C2×C4⋊C4—212no—
273C2×C4⋊C4—212no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—212no—
276C2×C4⋊C4—212no—
277C2×C4⋊C4—46no—
278C2×C4⋊C4—412no—
279C2×C4⋊C4C24112yes—
280C2×C4⋊C4C24112yes—
281C2×C4⋊C4C24112yes—
282C2×C4⋊C4C24112yes—
283C2×C4⋊C4C24112yes—
284C2×C4⋊C4C24112yes—
285C2×C4⋊C4C24112yes—
286C2×C4⋊C4C24112yes—
287C2×C4⋊C4C24124yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C24112yes—
290C2×C4⋊C4C24112yes—
291C2×C4⋊C4C24112yes—
292C2×C4⋊C4C24124yes—
293C2×C22⋊C4—212no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—212no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—212no—
298C2×C22⋊C4—212no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—46no—
302C2×C22⋊C4—46no—
303C2×C22⋊C4C2414yes—
304C2×C22⋊C4C24112yes—
305C2×C22⋊C4C24112yes—
306C2×C22⋊C4C24112yes—
307C2×C22⋊C4C2414yes—
308C2×C22⋊C4C2414yes—
309C2×C22⋊C4C2414yes—
310C2×C22⋊C4C24112yes—
311C2×C22⋊C4C24112yes—
312C2×C22⋊C4C24112yes—
313C2×C42—224no—
314C2×C42—212no—
315C2×C42—212no—
316C2×C42—212no—
317C2×C42—212no—
318C2×C42—212no—
319C2×C42C24112yes—
320C2×C42C24112yes—
321C2×C42C24124yes—
322C2×C42C24112yes—
323C2×C42C24112yes—
324C2×C42C24112yes—
325#95C24112yes—
326#95C24112yes—
327#95C24112yes—
328#95C24112yes—
329#95C24112yes—
330#95C24112yes—
331#95C2414yes—
332#95C24112yes—
333#95C24112yes—
334#95C24112yes—
335#95C24124yes—
336#95C24124yes—
337#95C24124yes—
338#95C24124yes—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24C22×D414yes—
352C242−1+414yes—
353C24C2512yes—
354C24C2512yes—
355C24C2511yescharacteristic
356C2×Q8—2256no—
357C2×Q8—296no—
358C2×Q8—224no—
359C2×Q8—224no—
360C2×Q8—224no—
361C2×Q8—224no—
362C2×Q8—224no—
363C2×Q8—4128no—
364C2×Q8—448no—
365C2×Q8#123164yes—
366C2×Q8C22×D4148yes—
367C2×Q8C22×D4116yes—
368C2×Q8C2×#41116yes—
369C2×Q8C2×#41116yes—
370C2×D4—232no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—26no—
377C22×C4—26no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—296no—
381C22×C4—224no—
382C22×C4—224no—
383C22×C4—26no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—26no—
387C22×C4—26no—
388C22×C4—26no—
389C22×C4—22no—
390C22×C4—26no—
391C22×C4—26no—
392C22×C4—26no—
393C22×C4—26no—
394C22×C4—22no—
395C22×C4—26no—
396C22×C4—22no—
397C22×C4—26no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—26no—
402C22×C4—26no—
403C22×C4—26no—
404C22×C4—26no—
405C22×C4—26no—
406C22×C4—26no—
407C22×C4—26no—
408C22×C4—26no—
409C22×C4—22no—
410C22×C4—224no—
411C22×C4—224no—
412C22×C4—224no—
413C22×C4—448no—
414C22×C4C22×D4112yes—
415C22×C4C22×D4148yes—
416C22×C4C22×D4112yes—
417C22×C4C2×#41112yes—
418C22×C4C2×#41112yes—
419C22×C42−1+414yes—
420C22×C42−1+414yes—
421C22×C4C2512yes—
422C22×C4C2516yes—
423C22×C4C2516yes—
424C22×C4C2516yes—
425C22×C4C2516yes—
426C22×C4C2512yes—
427C22×C4C2512yes—
428C22×C4C2516yes—
429C22×C4C2516yes—
430C22×C4C2516yes—
431C22×C4C2512yes—
432C22×C4C2512yes—
433C22×C4C2516yes—
434C4⋊C4—224no—
435C4⋊C4—224no—
436C4⋊C4—224no—
437C4⋊C4—224no—
438C4⋊C4—224no—
439C4⋊C4—224no—
440C4⋊C4—248no—
441C4⋊C4—224no—
442C4⋊C4—224no—
443C4⋊C4—28no—
444C4⋊C4—224no—
445C4⋊C4—224no—
446C4⋊C4—412no—
447C4⋊C4—424no—
448C22⋊C4—28no—
449C22⋊C4—224no—
450C22⋊C4—224no—
451C22⋊C4—28no—
452C22⋊C4—224no—
453C22⋊C4—224no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—412no—
457C22⋊C4—412no—
458C42—248no—
459C42—224no—
460C42—224no—
461C42—224no—
462C42—224no—
463C42—224no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23C23×D414yes—
491C23C23×D412yes—
492C23C22×#4112yes—
493C23C22×#4112yes—
494C23C2×2−1+414yes—
495C23C2×2−1+412yes—
496C23C2×2−1+412yes—
497C23C2×2−1+412yes—
498C23#58512yes—
499C23C2611yescharacteristic
500Q8—296no—
501Q8—448no—
502Q8C2×#123164yes—
503D4—232no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—26no—
507C2×C4—26no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—224no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—296no—
514C2×C4—26no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—26no—
518C2×C4—26no—
519C2×C4—26no—
520C2×C4—22no—
521C2×C4—26no—
522C2×C4—26no—
523C2×C4—26no—
524C2×C4—26no—
525C2×C4—22no—
526C2×C4—26no—
527C2×C4—22no—
528C2×C4—26no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—26no—
533C2×C4—26no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—26no—
537C2×C4—26no—
538C2×C4—26no—
539C2×C4—22no—
540C2×C4—26no—
541C2×C4—412no—
542C2×C4—412no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—412no—
546C2×C4—412no—
547C2×C4—412no—
548C2×C4—412no—
549C2×C4—44no—
550C2×C4—448no—
551C2×C4C2×#123148yes—
552C2×C4C23×D4112yes—
553C2×C4C23×D4112yes—
554C2×C4C22×#41112yes—
555C2×C4C22×#41112yes—
556C2×C4C2×2−1+414yes—
557C2×C4C2×2−1+414yes—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—416no—
572C22—416no—
573C22—42no—
574C22C22×#12314yes—
575C22#348614yes—
576C22#349714yes—
577C22#351614yes—
578C22#351814yes—
579C22#352314yes—
580C22#352314yes—
581C22C24×D411yescharacteristic
582C22C23×#4111yescharacteristic
583C22C23×#4111yescharacteristic
584C22C22×2−1+411yescharacteristic
585C22C22×2−1+411yescharacteristic
586C22C22×2−1+411yescharacteristic
587C22C2×#58511yescharacteristic
588C4—24no—
589C4—24no—
590C4—212no—
591C4—46no—
592C4—42no—
593C4—42no—
594C4—46no—
595C4—46no—
596C4—46no—
597C4—42no—
598C4—46no—
599C4—46no—
600C4C22×#123124yes—
601C2—44no—
602C2—44no—
603C2#3572318yesdirect product factor
604C2C23×#12311yescharacteristic
605C2C2×#348611yescharacteristic
606C2C2×#349711yescharacteristic
607C2C2×#351611yescharacteristic
608C2C2×#352311yescharacteristic
609C2C2×#352311yescharacteristic
610C2C2×#351811yescharacteristic
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
2. C22×#519C2semidirect
3. #35723C2direct (N × H)
4. C2×#2499C2semidirect
6. C2×#2421C2semidirect
7. C2×#2446C2semidirect
8. C2×#2446C2semidirect
9. C2×#2423C2semidirect
10. C2×#2625C2semidirect
12. C2×#2351C2semidirect
13. C2×Q8×C4⋊C4C2semidirect
14. C2×Q8×C22⋊C4C2semidirect
15. C2×Q8×C22⋊C4C2semidirect
16. C2×Q8×C22⋊C4C2semidirect
25. C2×#519C22semidirect
36. #2625C22semidirect
37. #2499C22semidirect
39. #2446C22semidirect
40. #2446C22semidirect
41. #2423C22semidirect
42. #2421C22semidirect
55. #2351C22semidirect
56. Q8×C4⋊C4C22semidirect
57. Q8×C22⋊C4C22semidirect
58. Q8×C22⋊C4C22semidirect
59. Q8×C22⋊C4C22semidirect
603. C2#35723direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C22×#5192
#357232
C2×#24992
C2×#24852
C2×#24212
C2×#24462
C2×#24232
C2×#26252
C2×#23952
C2×#23512
C2×Q8×C4⋊C42
C2×Q8×C22⋊C42