Finite Group Databasebeta

← C2×#3515 #62779

Subgroups — C2×#3515 #62779

Order 256.

3571 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3515C2132yesmaximal, direct product factor
2C2×#557C211yesmaximal, characteristic, complemented
3C2×#554C211yesmaximal, characteristic, complemented
4C2×#554C211yesmaximal, characteristic, complemented
5C2×#553C212yesmaximal, complemented
6C2×#552C212yesmaximal, complemented
7C2×#552C214yesmaximal, complemented
8C2×#552C212yesmaximal, complemented
9C2×#547C211yesmaximal, characteristic, complemented
10C2×#547C212yesmaximal, complemented
11C2×#541C212yesmaximal, complemented
12C2×#539C214yesmaximal, complemented
13C2×#533C211yesmaximal, characteristic, complemented
14C2×#529C211yesmaximal, characteristic, complemented
15C2×#529C212yesmaximal, complemented
16C22×#125C212yesmaximal, complemented
17C2×#520C212yesmaximal, complemented
18C2×C4×#41C211yesmaximal, characteristic, complemented
19C22×#41C2211yescharacteristic
20C22×#41C2212yes—
21#557C22116yescomplemented
22#554C22116yescomplemented
23#554C22116yescomplemented
24#553C22132yescomplemented
25#552C22132yescomplemented
26#552C22164yescomplemented
27#552C22132yescomplemented
28#547C22116yescomplemented
29#547C22132yescomplemented
30#541C22132yescomplemented
31#539C22164yescomplemented
32#533C22116yescomplemented
33C2×C4⋊Q8C2211yescharacteristic
34C2×C4⋊Q8C2211yescharacteristic
35C2×C4⋊Q8C2212yes—
36#529C22132yescomplemented
37#529C22116yescomplemented
38C2×#126C2214yes—
39C2×#126C2212yes—
40C2×#126C2214yes—
41C2×#126C2214yes—
42C2×#125C2211yescharacteristic
43C2×#125C2212yes—
44C2×#125C2214yes—
45C2×#125C2214yes—
46C2×#125C2212yes—
47C2×#125C2212yes—
48C2×#125C2214yes—
49C2×#125C2214yes—
50C2×#125C2212yes—
51C2×#125C22132yescomplemented
52C2×#124C2211yescharacteristic
53C2×#123C2212yes—
54C2×#123C2214yes—
55C2×#123C2214yes—
56C2×#123C2214yes—
57C2×#123C2214yes—
58C2×C22⋊Q8C2211yescharacteristic
59C2×C22⋊Q8C2212yes—
60C2×C22⋊Q8C2212yes—
61C2×C22⋊Q8C2212yes—
62C2×C22⋊Q8C2214yes—
63C2×C22⋊Q8C2214yes—
64C2×C22⋊Q8C2212yes—
65C2×C22⋊Q8C2212yes—
66C2×#121C2212yes—
67C2×#121C2211yescharacteristic
68C2×#121C2214yes—
69#520C22132yescomplemented
70C4×#41C22116yescomplemented
71C2×C4×Q8C2211yescharacteristic
72C2×C4×Q8C2211yescharacteristic
73C2×C4×Q8C2212yes—
74C2×C4×Q8C2212yes—
75C2×C4×Q8C2212yes—
76C2×C4×Q8C2211yescharacteristic
77C2×C4×Q8C2212yes—
78C2×C4×D4C2211yescharacteristic
79C2×C4×D4C2212yes—
80C2×C4×D4C2211yescharacteristic
81C2×C4×D4C2212yes—
82C2×C4×D4C2214yes—
83C2×C4×D4C2212yes—
84C2×C4×D4C2214yes—
85C2×C4×D4C2212yes—
86C2×C4×D4C2211yescharacteristic
87C2×C4×D4C2212yes—
88C2×#117C2211yescharacteristic
89C2×#117C2212yes—
90C2×#117C2212yes—
91C2×#117C2214yes—
92C2×#117C2214yes—
93C2×#117C2212yes—
94C22×C4⋊C4C2214yes—
95C22×C4⋊C4C2212yes—
96C22×C4⋊C4C2212yes—
97C22×C4⋊C4C2214yes—
98C22×C4⋊C4C2212yes—
99C22×C42C2212yes—
100C22×C42C2211yescharacteristic
101C2×#41—28no—
102C2×#41—216no—
103C2×#41C2318yes—
104C2×#41C23116yes—
105C22×Q8C2311yescharacteristic
106C22×Q8C2311yescharacteristic
107C22×Q8C2312yes—
108C22×Q8C2312yes—
109C22×D4C2311yescharacteristic
110C22×D4C2312yes—
111C22×D4C2314yes—
112C22×D4C2312yes—
113C22×D4C2311yescharacteristic
114C23×C4C2314yes—
115C23×C4C2312yes—
116C23×C4C2312yes—
117C23×C4C2312yes—
118C23×C4C2311yescharacteristic
119C23×C4C2311yescharacteristic
120C23×C4C2314yes—
121C23×C4C2312yes—
122C23×C4C2314yes—
123C23×C4C2311yescharacteristic
124C4⋊Q8C2318yes—
125C4⋊Q8C2318yes—
126C4⋊Q8C23116yes—
127#126C23132yes—
128#126C23116yes—
129#126C23132yes—
130#126C23132yes—
131#125C2318yes—
132#125C23116yes—
133#125C23132yes—
134#125C23132yes—
135#125C23116yes—
136#125C23116yes—
137#125C23132yes—
138#125C23132yes—
139#125C23116yes—
140#124C2318yes—
141#123C23116yes—
142#123C23132yes—
143#123C23132yes—
144#123C23132yes—
145#123C23132yes—
146C22⋊Q8C2318yes—
147C22⋊Q8C23116yes—
148C22⋊Q8C23116yes—
149C22⋊Q8C23116yes—
150C22⋊Q8C23132yes—
151C22⋊Q8C23132yes—
152C22⋊Q8C23116yes—
153C22⋊Q8C23116yes—
154#121C23116yes—
155#121C2318yes—
156#121C23132yes—
157C4×Q8C2318yes—
158C4×Q8C2318yes—
159C4×Q8C23116yes—
160C4×Q8C23116yes—
161C4×Q8C23116yes—
162C4×Q8C2318yes—
163C4×Q8C23116yes—
164C4×D4C2318yes—
165C4×D4C23116yes—
166C4×D4C2318yes—
167C4×D4C23116yes—
168C4×D4C23132yes—
169C4×D4C23116yes—
170C4×D4C23132yes—
171C4×D4C23116yes—
172C4×D4C2318yes—
173C4×D4C23116yes—
174#117C2318yes—
175#117C23116yes—
176#117C23116yes—
177#117C23132yes—
178#117C23132yes—
179#117C23116yes—
180C2×C4⋊C4C2311yescharacteristic
181C2×C4⋊C4C2314yes—
182C2×C4⋊C4C2312yes—
183C2×C4⋊C4C2311yescharacteristic
184C2×C4⋊C4C2312yes—
185C2×C4⋊C4C2311yescharacteristic
186C2×C4⋊C4C2312yes—
187C2×C4⋊C4C2312yes—
188C2×C4⋊C4C2314yes—
189C2×C4⋊C4C2312yes—
190C2×C4⋊C4C2314yes—
191C2×C4⋊C4C2314yes—
192C2×C4⋊C4C23132yes—
193C2×C4⋊C4C23116yes—
194C2×C4⋊C4C2314yes—
195C2×C4⋊C4C2314yes—
196C2×C4⋊C4C2314yes—
197C2×C4⋊C4C2314yes—
198C2×C4⋊C4C23116yes—
199C2×C4⋊C4C23132yes—
200C2×C4⋊C4C23116yes—
201C2×C4⋊C4C2312yes—
202C2×C4⋊C4C2312yes—
203C2×C4⋊C4C2311yescharacteristic
204C2×C4⋊C4C2312yes—
205C2×C4⋊C4C2314yes—
206C2×C4⋊C4C2312yes—
207C2×C4⋊C4C2312yes—
208C2×C4⋊C4C2314yes—
209C2×C22⋊C4C2312yes—
210C2×C22⋊C4C2312yes—
211C2×C22⋊C4C2312yes—
212C2×C22⋊C4C2314yes—
213C2×C22⋊C4C2312yes—
214C2×C22⋊C4C2312yes—
215C2×C22⋊C4C2312yes—
216C2×C22⋊C4C2314yes—
217C2×C22⋊C4C2314yes—
218C2×C22⋊C4C2314yes—
219C2×C22⋊C4C2312yes—
220C2×C22⋊C4C2312yes—
221C2×C42C2312yes—
222C2×C42C2311yescharacteristic
223C2×C42C2314yes—
224C2×C42C2314yes—
225C2×C42C2311yescharacteristic
226C2×C42C2312yes—
227C2×C42C2312yes—
228C2×C42C2311yescharacteristic
229C2×C42C2312yes—
230C2×C42C23116yes—
231C2×C42C2318yes—
232C2×C42C2311yescharacteristic
233C24C2411yescharacteristic
234C24C2412yes—
235C24C2412yes—
236#41—232no—
237#41—264no—
238C2×Q8—24no—
239C2×Q8—28no—
240C2×Q8—24no—
241C2×Q8—44no—
242C2×Q8C2414yes—
243C2×Q8C2414yes—
244C2×Q8C2418yes—
245C2×Q8C2418yes—
246C2×D4—28no—
247C2×D4—216no—
248C2×D4—24no—
249C2×D4—44no—
250C2×D4#4118yes—
251C2×D4C2414yes—
252C2×D4C2418yes—
253C2×D4C24116yes—
254C2×D4C2418yes—
255C2×D4C2414yes—
256C22×C4—216no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—44no—
263C22×C4—48no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4C2418yes—
267C22×C4C2418yes—
268C22×C4C2414yes—
269C22×C4C2414yes—
270C22×C4C24116yes—
271C22×C4C2418yes—
272C22×C4C2411yescharacteristic
273C22×C4C2411yescharacteristic
274C22×C4C2412yes—
275C22×C4C2411yescharacteristic
276C22×C4C2412yes—
277C22×C4C2412yes—
278C22×C4C2411yescharacteristic
279C22×C4C2414yes—
280C22×C4C2412yes—
281C22×C4C2414yes—
282C22×C4C2412yes—
283C22×C4C2412yes—
284C22×C4C2418yes—
285C22×C4C2414yes—
286C22×C4C24116yes—
287C22×C4C24116yes—
288C22×C4C2411yescharacteristic
289C22×C4C2411yescharacteristic
290C4⋊C4C2414yes—
291C4⋊C4C24116yes—
292C4⋊C4C2418yes—
293C4⋊C4C2414yes—
294C4⋊C4C2418yes—
295C4⋊C4C2414yes—
296C4⋊C4C2418yes—
297C4⋊C4C2418yes—
298C4⋊C4C24116yes—
299C4⋊C4C2418yes—
300C4⋊C4C24116yes—
301C4⋊C4C24116yes—
302C4⋊C4C24116yes—
303C4⋊C4C24116yes—
304C4⋊C4C24116yes—
305C4⋊C4C24116yes—
306C4⋊C4C2418yes—
307C4⋊C4C2418yes—
308C4⋊C4C2414yes—
309C4⋊C4C2418yes—
310C4⋊C4C24116yes—
311C4⋊C4C2418yes—
312C4⋊C4C2418yes—
313C4⋊C4C24116yes—
314C22⋊C4C2418yes—
315C22⋊C4C2418yes—
316C22⋊C4C2418yes—
317C22⋊C4C24116yes—
318C22⋊C4C2418yes—
319C22⋊C4C2418yes—
320C22⋊C4C2418yes—
321C22⋊C4C24116yes—
322C22⋊C4C24116yes—
323C22⋊C4C24116yes—
324C22⋊C4C2418yes—
325C22⋊C4C2418yes—
326C42C2414yes—
327C42C2418yes—
328C42C2414yes—
329C42C2414yes—
330C42C2418yes—
331C42C2414yes—
332C42C2418yes—
333C42C24116yes—
334C42C24116yes—
335C42C2418yes—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—24no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23C2×#4118yes—
345C23C2512yes—
346C23C2514yes—
347C23C2514yes—
348C23C2511yescharacteristic
349Q8—28no—
350Q8—216no—
351Q8—28no—
352Q8—48no—
353D4—216no—
354D4—232no—
355D4—28no—
356D4—48no—
357D4C2×#41116yes—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—216no—
363C2×C4—232no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—28no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—216no—
374C2×C4—216no—
375C2×C4—28no—
376C2×C4—48no—
377C2×C4—416no—
378C2×C4—44no—
379C2×C4—416no—
380C2×C4C2×#4114yes—
381C2×C42−1+414yes—
382C2×C4C2512yes—
383C2×C4C2512yes—
384C2×C4C2512yes—
385C2×C4C2514yes—
386C2×C4C2514yes—
387C2×C4C2514yes—
388C2×C4C2512yes—
389C2×C4C2514yes—
390C2×C4C2518yes—
391C2×C4C2518yes—
392C2×C4C2514yes—
393C2×C4C2514yes—
394C2×C4C2512yes—
395C2×C4C2512yes—
396C22—24no—
397C22—24no—
398C22—216no—
399C22—22no—
400C22—24no—
401C22—22no—
402C22—24no—
403C22—24no—
404C22—22no—
405C22—44no—
406C22—48no—
407C22C22×#4118yes—
408C22C22×#4112yes—
409C22C2×2−1+412yes—
410C22#58512yes—
411C22C2611yescharacteristic
412C4—22no—
413C4—24no—
414C4—24no—
415C4—28no—
416C4—22no—
417C4—24no—
418C4—24no—
419C4—22no—
420C4—24no—
421C4—24no—
422C4—28no—
423C4—22no—
424C4C22×#4114yes—
425C4C2×2−1+414yes—
426C2—28no—
427C2—42no—
428C2—44no—
429C2#351514yesdirect product factor
430C2C23×#4111yescharacteristic
431C2C22×2−1+411yescharacteristic
432C2C2×#58511yescharacteristic
subgroupG/H
center, socleC23C25
commutator, Frattini, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3515C2direct (N × H)
2. C2×#557C2semidirect
3. C2×#554C2semidirect
4. C2×#554C2semidirect
5. C2×#553C2semidirect
6. C2×#552C2semidirect
7. C2×#552C2semidirect
8. C2×#552C2semidirect
9. C2×#547C2semidirect
10. C2×#547C2semidirect
11. C2×#541C2semidirect
12. C2×#539C2semidirect
13. C2×#533C2semidirect
14. C2×#529C2semidirect
15. C2×#529C2semidirect
16. C22×#125C2semidirect
17. C2×#520C2semidirect
18. C2×C4×#41C2semidirect
21. #557C22semidirect
22. #554C22semidirect
23. #554C22semidirect
24. #553C22semidirect
25. #552C22semidirect
26. #552C22semidirect
27. #552C22semidirect
28. #547C22semidirect
29. #547C22semidirect
30. #541C22semidirect
31. #539C22semidirect
32. #533C22semidirect
36. #529C22semidirect
37. #529C22semidirect
51. C2×#125C22semidirect
69. #520C22semidirect
70. C4×#41C22semidirect
429. C2#3515direct (N × H)

Maximal subgroups

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

subgroupindex
#35152
C2×#5572
C2×#5542
C2×#5532
C2×#5522
C2×#5472
C2×#5412
C2×#5392
C2×#5332
C2×#5292
C22×#1252
C2×#5202
C2×C4×#412