Finite Group Databasebeta

← C2×#39510 #7636817

Subgroups — C2×#39510 #7636817

Order 512.

4878 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2800C214yesmaximal
2C2×#2776C214yesmaximal, complemented
3#39510C2132yesmaximal, direct product factor
4C2×#2859C211yesmaximal, characteristic, complemented
5C2×#2562C212yesmaximal
6C2×#2594C214yesmaximal, complemented
7C2×#2494C212yesmaximal, complemented
8C2×#2525C214yesmaximal, complemented
9C2×#2495C212yesmaximal, complemented
10C2×#2629C211yesmaximal, characteristic, complemented
11C2×#2629C211yesmaximal, characteristic, complemented
12C2×#2370C212yesmaximal, complemented
13C2×#2314C211yesmaximal, characteristic, complemented
14C2×#2314C211yesmaximal, characteristic, complemented
15C2×C4×#125C211yesmaximal, characteristic, complemented
16C2×#2293C211yesmaximal, characteristic
17C22×#125C2212yes—
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2859C22116yescomplemented
23#2800C22164yes—
24#2776C22164yescomplemented
25#2629C22116yescomplemented
26#2629C22116yescomplemented
27#2594C22164yescomplemented
28#2562C22132yes—
29#2525C22164yescomplemented
30#2495C22132yescomplemented
31#2494C22132yescomplemented
32C2×#401C2214yes—
33C2×#400C2214yes—
34C2×#400C2214yes—
35C2×#398C2214yes—
36C2×#398C2214yes—
37C2×#397C2214yes—
38C2×#397C2214yes—
39C2×#396C2214yes—
40C2×#396C2214yes—
41C2×#394C2214yes—
42#2370C22132yescomplemented
43#2314C22116yescomplemented
44#2314C22116yescomplemented
45C4×#125C22116yescomplemented
46#2293C22116yes—
47C2×#390C2212yes—
48C2×#389C2214yes—
49C2×#389C2214yes—
50C2×#389C2214yes—
51C2×#389C2212yes—
52C2×#389C2212yes—
53C2×#389C2212yes—
54C2×#389C2212yes—
55C2×#388C2214yes—
56C2×#388C2214yes—
57C2×#388C2214yes—
58C2×#388C2214yes—
59C2×#388C2214yes—
60C2×#388C2212yes—
61C2×#388C2214yes—
62C2×#388C2212yes—
63C2×#387C2214yes—
64C2×#387C2214yes—
65C2×#387C2214yes—
66C2×#387C2214yes—
67C2×#387C2214yes—
68C2×#387C2214yes—
69C2×#384C2211yescharacteristic
70C2×#384C2211yescharacteristic
71C2×#383C2214yes—
72C2×#382C2212yes—
73C2×#382C2211yescharacteristic
74C2×#380C2212yes—
75C2×#380C2212yes—
76C2×#380C2212yes—
77C2×#380C2212yes—
78C2×C4×C4⋊C4C2214yes—
79C2×C4×C4⋊C4C2214yes—
80C2×C4×C4⋊C4C2212yes—
81C2×C4×C4⋊C4C2211yescharacteristic
82C2×C4×C4⋊C4C2212yes—
83C2×C4×C4⋊C4C2212yes—
84C2×C4×C22⋊C4C2211yescharacteristic
85C2×C4×C22⋊C4C2211yescharacteristic
86C2×C43C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#125—28no—
91C2×#125—216no—
92C2×#125C2318yes—
93C2×#125C23116yes—
94C2×C4×D4—28no—
95C2×C4×D4C2318yes—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117C2318yes—
99C2×#117C2318yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2314yes—
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#401C23132yes—
137#400C23132yes—
138#400C23132yes—
139#398C23132yes—
140#398C23132yes—
141#397C23132yes—
142#397C23132yes—
143#396C23132yes—
144#396C23132yes—
145#394C23132yes—
146#390C23116yes—
147#389C23116yes—
148#389C23132yes—
149#389C23132yes—
150#389C23116yes—
151#389C23116yes—
152#389C23132yes—
153#389C23116yes—
154#388C23132yes—
155#388C23132yes—
156#388C23132yes—
157#388C23132yes—
158#388C23116yes—
159#388C23132yes—
160#388C23132yes—
161#388C23116yes—
162#387C23132yes—
163#387C23132yes—
164#387C23132yes—
165#387C23132yes—
166#387C23132yes—
167#387C23132yes—
168#384C2318yes—
169#384C2318yes—
170#383C23132yes—
171#382C2318yes—
172#382C23116yes—
173#380C23116yes—
174#380C23116yes—
175#380C23116yes—
176#380C23116yes—
177C4×C4⋊C4C23116yes—
178C4×C4⋊C4C23132yes—
179C4×C4⋊C4C23116yes—
180C4×C4⋊C4C23116yes—
181C4×C4⋊C4C23132yes—
182C4×C4⋊C4C2318yes—
183C4×C22⋊C4C2318yes—
184C4×C22⋊C4C2318yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2314yes—
201C43C2318yes—
202C25C2411yescharacteristic
203C25C2411yescharacteristic
204C22×D4—24no—
205C22×D4—44no—
206C22×D4C2414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#125—232no—
229#125—264no—
230C4×D4—232no—
231#117—232no—
232#117—232no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4C2418yes—
250C2×C4⋊C4C2418yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2418yes—
253C2×C4⋊C4C2418yes—
254C2×C4⋊C4C2418yes—
255C2×C4⋊C4C24116yes—
256C2×C4⋊C4C24116yes—
257C2×C4⋊C4C24116yes—
258C2×C4⋊C4C2418yes—
259C2×C4⋊C4C24116yes—
260C2×C4⋊C4C2418yes—
261C2×C4⋊C4C24116yes—
262C2×C4⋊C4C24116yes—
263C2×C4⋊C4C24116yes—
264C2×C4⋊C4C24116yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4C2418yes—
276C2×C22⋊C4C2418yes—
277C2×C22⋊C4C2414yes—
278C2×C22⋊C4C2414yes—
279C2×C22⋊C4C2418yes—
280C2×C22⋊C4C2418yes—
281C2×C22⋊C4C24116yes—
282C2×C22⋊C4C24116yes—
283C2×C22⋊C4C24116yes—
284C2×C22⋊C4C24116yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—44no—
289C2×C42—48no—
290C2×C42—44no—
291C2×C42—48no—
292C2×C42—48no—
293C2×C42#4118yes—
294C2×C42#4118yes—
295C2×C42C2414yes—
296C2×C42C2414yes—
297C2×C42C2414yes—
298C2×C42C2418yes—
299C2×C42C2418yes—
300C2×C42C24116yes—
301C2×C42C24116yes—
302C2×C42C24116yes—
303C2×C42C2414yes—
304C2×C42C2414yes—
305#95C2418yes—
306#95C2418yes—
307#95C2418yes—
308#95C24116yes—
309#95C24116yes—
310#95C24116yes—
311#95C24116yes—
312#95C24116yes—
313#95C24116yes—
314#95C24116yes—
315#95C24116yes—
316#95C24116yes—
317#95C24116yes—
318#95C24116yes—
319#95C24116yes—
320#95C24116yes—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24C2512yes—
332C24C2512yes—
333C24C2511yescharacteristic
334C2×D4—28no—
335C2×D4—48no—
336C2×D4—416no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4C2×#4114yes—
377C22×C4C2×#4118yes—
378C22×C4C2×#4114yes—
379C22×C4C2×#4114yes—
380C22×C4C2×#4114yes—
381C22×C42−1+414yes—
382C22×C4C2512yes—
383C22×C4C2512yes—
384C22×C4C2514yes—
385C22×C4C2518yes—
386C22×C4C2514yes—
387C22×C4C2514yes—
388C22×C4C2518yes—
389C22×C4C2518yes—
390C22×C4C2518yes—
391C22×C4C2512yes—
392C22×C4C2512yes—
393C22×C4C2514yes—
394C22×C4C2512yes—
395C4⋊C4—216no—
396C4⋊C4—216no—
397C4⋊C4—28no—
398C4⋊C4—232no—
399C4⋊C4—232no—
400C4⋊C4—232no—
401C4⋊C4—216no—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—416no—
406C4⋊C4—416no—
407C4⋊C4—48no—
408C4⋊C4—416no—
409C4⋊C4—416no—
410C4⋊C4—416no—
411C22⋊C4—216no—
412C22⋊C4—216no—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—48no—
416C22⋊C4—416no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C22⋊C4—416no—
421C42—28no—
422C42—28no—
423C42—28no—
424C42—48no—
425C42—416no—
426C42—48no—
427C42—416no—
428C42—416no—
429C42C2×#41116yes—
430C42C2×#41116yes—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—48no—
446C23—48no—
447C23—44no—
448C23—44no—
449C23—48no—
450C23—48no—
451C23C22×#4112yes—
452C23C22×#4112yes—
453C23C2×2−1+412yes—
454C23#58514yes—
455C23#58514yes—
456C23C2611yescharacteristic
457D4—416no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—216no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—416no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—416no—
498C2×C4—416no—
499C2×C4—48no—
500C2×C4—416no—
501C2×C4—416no—
502C2×C4—416no—
503C2×C4#556116yes—
504C2×C4C22×#4118yes—
505C2×C4C22×#4114yes—
506C2×C4C22×#4114yes—
507C2×C4C22×#4114yes—
508C2×C4C22×#4114yes—
509C2×C4C2×2−1+414yes—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—88no—
521C22—88no—
522C22#345514yes—
523C22C2×#55614yes—
524C22#351214yes—
525C22#351218yes—
526C22#356518yes—
527C22C23×#4111yescharacteristic
528C22C23×#4111yescharacteristic
529C22C22×2−1+411yescharacteristic
530C22C2×#58512yes—
531C22C2×#58512yes—
532C4—24no—
533C4—28no—
534C4—24no—
535C4—44no—
536C4—48no—
537C4—44no—
538C4—44no—
539C4—48no—
540C4—48no—
541C4—48no—
542C4—42no—
543C4—42no—
544C4C2×#55618yes—
545C2—82no—
546C2—82no—
547C2#3951018yesdirect product factor
548C2C2×#345511yescharacteristic
549C2C22×#55611yescharacteristic
550C2C2×#356512yes—
551C2C2×#351212yes—
552C2C2×#351211yescharacteristic
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. C2×#2776C2semidirect
3. #39510C2direct (N × H)
4. C2×#2859C2semidirect
6. C2×#2594C2semidirect
7. C2×#2494C2semidirect
8. C2×#2525C2semidirect
9. C2×#2495C2semidirect
10. C2×#2629C2semidirect
11. C2×#2629C2semidirect
12. C2×#2370C2semidirect
13. C2×#2314C2semidirect
14. C2×#2314C2semidirect
15. C2×C4×#125C2semidirect
22. #2859C22semidirect
24. #2776C22semidirect
25. #2629C22semidirect
26. #2629C22semidirect
27. #2594C22semidirect
29. #2525C22semidirect
30. #2495C22semidirect
31. #2494C22semidirect
42. #2370C22semidirect
43. #2314C22semidirect
44. #2314C22semidirect
45. C4×#125C22semidirect
547. C2#39510direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28002
C2×#27762
#395102
C2×#28592
C2×#25622
C2×#25942
C2×#24942
C2×#25252
C2×#24952
C2×#26292
C2×#23702
C2×#23142
C2×C4×#1252
C2×#22932