Finite Group Databasebeta

← C2×#3552 #62745

Subgroups — C2×#3552 #62745

Order 256.

3543 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3552C2132yesmaximal, direct product factor
2C2×#563C214yesmaximal, complemented
3C2×#557C211yesmaximal, characteristic, complemented
4C2×#556C211yesmaximal, characteristic, complemented
5C2×#554C212yesmaximal, complemented
6C2×#553C212yesmaximal, complemented
7C2×D4×Q8C211yesmaximal, characteristic, complemented
8C2×D4×Q8C212yesmaximal, complemented
9C2×#548C212yesmaximal, complemented
10C2×#547C211yesmaximal, characteristic, complemented
11C2×#544C212yesmaximal, complemented
12C2×#542C212yesmaximal, complemented
13C2×#541C212yesmaximal, complemented
14C2×#539C214yesmaximal, complemented
15C2×#536C212yesmaximal, complemented
16C22×C4⋊Q8C211yesmaximal, characteristic, complemented
17C2×#529C211yesmaximal, characteristic, complemented
18C2×#529C211yesmaximal, characteristic, complemented
19C22×#41C2212yes—
20C23×Q8C2212yes—
21C23×Q8C2212yes—
22#563C22164yescomplemented
23#557C22116yescomplemented
24#556C22116yescomplemented
25#554C22132yescomplemented
26#553C22132yescomplemented
27D4×Q8C22116yescomplemented
28D4×Q8C22132yescomplemented
29#548C22132yescomplemented
30#547C22116yescomplemented
31#544C22132yescomplemented
32#542C22132yescomplemented
33#541C22132yescomplemented
34#539C22164yescomplemented
35#536C22132yescomplemented
36C2×C4⋊Q8C2212yes—
37C2×C4⋊Q8C2212yes—
38C2×C4⋊Q8C2211yescharacteristic
39C2×C4⋊Q8C2211yescharacteristic
40C2×C4⋊Q8C2212yes—
41C2×C4⋊Q8C2212yes—
42C2×C4⋊Q8C2212yes—
43C2×C4⋊Q8C2214yes—
44C2×C4⋊Q8C22116yescomplemented
45#529C22116yescomplemented
46#529C22116yescomplemented
47C2×#126C2212yes—
48C2×#126C2212yes—
49C2×#126C2214yes—
50C2×#126C2214yes—
51C2×#126C2214yes—
52C2×#125C2211yescharacteristic
53C2×#125C2211yescharacteristic
54C2×#125C2214yes—
55C2×#125C2214yes—
56C2×#125C2214yes—
57C2×#125C2212yes—
58C2×#124C2211yescharacteristic
59C2×#124C2211yescharacteristic
60C2×#124C2212yes—
61C2×#124C2214yes—
62C2×#123C2212yes—
63C2×#123C2212yes—
64C2×#123C2214yes—
65C2×#123C2214yes—
66C2×#123C2214yes—
67C2×C22⋊Q8C2211yescharacteristic
68C2×C22⋊Q8C2211yescharacteristic
69C2×C22⋊Q8C2214yes—
70C2×C22⋊Q8C2214yes—
71C2×C22⋊Q8C2214yes—
72C2×C22⋊Q8C2214yes—
73C2×C22⋊Q8C2212yes—
74C2×C22⋊Q8C2212yes—
75C2×C22⋊Q8C2214yes—
76C2×C22⋊Q8C2212yes—
77C2×C22⋊Q8C2214yes—
78C2×#121C2211yescharacteristic
79C2×#121C2211yescharacteristic
80C2×#121C2214yes—
81C2×#121C2212yes—
82C2×C4×Q8C2211yescharacteristic
83C2×C4×Q8C2211yescharacteristic
84C2×C4×Q8C2212yes—
85C2×C4×Q8C2212yes—
86C2×C4×Q8C2212yes—
87C2×C4×D4C2211yescharacteristic
88C2×C4×D4C2211yescharacteristic
89C2×C4×D4C2212yes—
90C2×C4×D4C2212yes—
91C2×C4×D4C2214yes—
92C2×C4×D4C2214yes—
93C2×C4×D4C2212yes—
94C2×#117C2212yes—
95C2×#117C2212yes—
96C2×#117C2212yes—
97C22×C4⋊C4C2212yes—
98C22×C4⋊C4C2212yes—
99C22×C4⋊C4C2212yes—
100C22×C42C2211yescharacteristic
101C2×#41—48no—
102C2×#41C23116yes—
103C22×Q8—48no—
104C22×Q8—48no—
105C22×Q8C2311yescharacteristic
106C22×Q8C2311yescharacteristic
107C22×Q8C2312yes—
108C22×Q8C2312yes—
109C22×Q8C2314yes—
110C22×Q8C2314yes—
111C22×Q8C23116yes—
112C22×Q8C23116yes—
113C22×D4C2312yes—
114C22×D4C2312yes—
115C22×D4C2311yescharacteristic
116C22×D4C2311yescharacteristic
117C22×D4C2314yes—
118C23×C4C2311yescharacteristic
119C23×C4C2311yescharacteristic
120C23×C4C2312yes—
121C23×C4C2312yes—
122C23×C4C2311yescharacteristic
123C23×C4C2314yes—
124C23×C4C2314yes—
125C23×C4C2314yes—
126C4⋊Q8C23116yes—
127C4⋊Q8C23116yes—
128C4⋊Q8C2318yes—
129C4⋊Q8C2318yes—
130C4⋊Q8C23116yes—
131C4⋊Q8C23116yes—
132C4⋊Q8C23116yes—
133C4⋊Q8C23132yes—
134#126C23116yes—
135#126C23116yes—
136#126C23132yes—
137#126C23132yes—
138#126C23132yes—
139#125C2318yes—
140#125C2318yes—
141#125C23132yes—
142#125C23132yes—
143#125C23132yes—
144#125C23116yes—
145#124C2318yes—
146#124C2318yes—
147#124C23116yes—
148#124C23132yes—
149#123C23116yes—
150#123C23116yes—
151#123C23132yes—
152#123C23132yes—
153#123C23132yes—
154C22⋊Q8C2318yes—
155C22⋊Q8C2318yes—
156C22⋊Q8C23132yes—
157C22⋊Q8C23132yes—
158C22⋊Q8C23132yes—
159C22⋊Q8C23132yes—
160C22⋊Q8C23116yes—
161C22⋊Q8C23116yes—
162C22⋊Q8C23132yes—
163C22⋊Q8C23116yes—
164C22⋊Q8C23132yes—
165#121C2318yes—
166#121C2318yes—
167#121C23132yes—
168#121C23116yes—
169C4×Q8C2318yes—
170C4×Q8C2318yes—
171C4×Q8C23116yes—
172C4×Q8C23116yes—
173C4×Q8C23116yes—
174C4×D4C2318yes—
175C4×D4C2318yes—
176C4×D4C23116yes—
177C4×D4C23116yes—
178C4×D4C23132yes—
179C4×D4C23132yes—
180C4×D4C23116yes—
181#117C23116yes—
182#117C23116yes—
183#117C23116yes—
184C2×C4⋊C4C2311yescharacteristic
185C2×C4⋊C4C2312yes—
186C2×C4⋊C4C2312yes—
187C2×C4⋊C4C2314yes—
188C2×C4⋊C4C2312yes—
189C2×C4⋊C4C2314yes—
190C2×C4⋊C4C2314yes—
191C2×C4⋊C4C2312yes—
192C2×C4⋊C4C23116yes—
193C2×C4⋊C4C23116yes—
194C2×C4⋊C4C23116yes—
195C2×C4⋊C4C2311yescharacteristic
196C2×C4⋊C4C2312yes—
197C2×C4⋊C4C2312yes—
198C2×C4⋊C4C2312yes—
199C2×C4⋊C4C2312yes—
200C2×C4⋊C4C2311yescharacteristic
201C2×C4⋊C4C2314yes—
202C2×C4⋊C4C2312yes—
203C2×C4⋊C4C2314yes—
204C2×C4⋊C4C2312yes—
205C2×C4⋊C4C2311yescharacteristic
206C2×C4⋊C4C2314yes—
207C2×C4⋊C4C2314yes—
208C2×C4⋊C4C2314yes—
209C2×C4⋊C4C2312yes—
210C2×C4⋊C4C2312yes—
211C2×C22⋊C4C2314yes—
212C2×C22⋊C4C2312yes—
213C2×C22⋊C4C2312yes—
214C2×C22⋊C4C2312yes—
215C2×C22⋊C4C2312yes—
216C2×C22⋊C4C2312yes—
217C2×C22⋊C4C2312yes—
218C2×C22⋊C4C2314yes—
219C2×C22⋊C4C2314yes—
220C2×C22⋊C4C2312yes—
221C2×C22⋊C4C2312yes—
222C2×C22⋊C4C2314yes—
223C2×C22⋊C4C2312yes—
224C2×C22⋊C4C2312yes—
225C2×C42C2311yescharacteristic
226C2×C42C2311yescharacteristic
227C2×C42C2312yes—
228C2×C42C2312yes—
229C2×C42C2314yes—
230C2×C42C2311yescharacteristic
231C2×C42C2314yes—
232C2×C42C2311yescharacteristic
233C2×C42C2318yes—
234C24C2412yes—
235C24C2412yes—
236C24C2411yescharacteristic
237#41—432no—
238C2×Q8—28no—
239C2×Q8—24no—
240C2×Q8—28no—
241C2×Q8—24no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—432no—
245C2×Q8—432no—
246C2×Q8C2414yes—
247C2×Q8C2414yes—
248C2×Q8C2418yes—
249C2×Q8C2418yes—
250C2×Q8C24116yes—
251C2×Q8C24116yes—
252C2×D4—28no—
253C2×D4—28no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—48no—
257C2×D4C2418yes—
258C2×D4C2418yes—
259C2×D4C2414yes—
260C2×D4C2414yes—
261C2×D4C24116yes—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4C2414yes—
271C22×C4C2414yes—
272C22×C4C2414yes—
273C22×C4C2418yes—
274C22×C4C2418yes—
275C22×C4C2412yes—
276C22×C4C2412yes—
277C22×C4C2412yes—
278C22×C4C2411yescharacteristic
279C22×C4C2411yescharacteristic
280C22×C4C2412yes—
281C22×C4C2411yescharacteristic
282C22×C4C2411yescharacteristic
283C22×C4C2412yes—
284C22×C4C2414yes—
285C22×C4C2414yes—
286C22×C4C2412yes—
287C22×C4C24116yes—
288C22×C4C24116yes—
289C22×C4C24116yes—
290C22×C4C2411yescharacteristic
291C22×C4C2411yescharacteristic
292C4⋊C4C2414yes—
293C4⋊C4C2418yes—
294C4⋊C4C2418yes—
295C4⋊C4C24116yes—
296C4⋊C4C2418yes—
297C4⋊C4C24116yes—
298C4⋊C4C24116yes—
299C4⋊C4C2418yes—
300C4⋊C4C2414yes—
301C4⋊C4C2418yes—
302C4⋊C4C2418yes—
303C4⋊C4C2418yes—
304C4⋊C4C2418yes—
305C4⋊C4C2414yes—
306C4⋊C4C24116yes—
307C4⋊C4C2418yes—
308C4⋊C4C24116yes—
309C4⋊C4C2418yes—
310C4⋊C4C2414yes—
311C4⋊C4C24116yes—
312C4⋊C4C24116yes—
313C4⋊C4C24116yes—
314C4⋊C4C2418yes—
315C4⋊C4C2418yes—
316C22⋊C4C24116yes—
317C22⋊C4C2418yes—
318C22⋊C4C2418yes—
319C22⋊C4C2418yes—
320C22⋊C4C2418yes—
321C22⋊C4C2418yes—
322C22⋊C4C2418yes—
323C22⋊C4C24116yes—
324C22⋊C4C24116yes—
325C22⋊C4C2418yes—
326C22⋊C4C2418yes—
327C22⋊C4C24116yes—
328C22⋊C4C2418yes—
329C22⋊C4C2418yes—
330C42C2414yes—
331C42C2414yes—
332C42C2414yes—
333C42C2414yes—
334C42C2418yes—
335C42C2418yes—
336C42C24116yes—
337C42C24116yes—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C232−1+414yes—
347C23C2514yes—
348C23C2514yes—
349C23C2512yes—
350C23C2511yescharacteristic
351Q8—216no—
352Q8—28no—
353Q8—216no—
354Q8—28no—
355Q8—416no—
356Q8—416no—
357D4—216no—
358D4—216no—
359D4—28no—
360D4—28no—
361D4—416no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—28no—
376C2×C4—216no—
377C2×C4—28no—
378C2×C4—216no—
379C2×C4—416no—
380C2×C4—416no—
381C2×C4—416no—
382C2×C4H(5,𝔽2)14yes—
383C2×C42−1+414yes—
384C2×C4C2514yes—
385C2×C4C2514yes—
386C2×C4C2514yes—
387C2×C4C2512yes—
388C2×C4C2512yes—
389C2×C4C2512yes—
390C2×C4C2514yes—
391C2×C4C2512yes—
392C2×C4C2514yes—
393C2×C4C2518yes—
394C2×C4C2518yes—
395C2×C4C2514yes—
396C2×C4C2512yes—
397C2×C4C2512yes—
398C22—22no—
399C22—22no—
400C22—28no—
401C22—24no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—24no—
407C22—48no—
408C22—48no—
409C22C2×H(5,𝔽2)12yes—
410C22C2×2−1+414yes—
411C22C2×2−1+412yes—
412C22C2×2−1+412yes—
413C22C2611yescharacteristic
414C4—24no—
415C4—22no—
416C4—28no—
417C4—28no—
418C4—24no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—24no—
423C4—24no—
424C4—24no—
425C4—24no—
426C4C2×H(5,𝔽2)14yes—
427C4C2×2−1+414yes—
428C2—24no—
429C2—44no—
430C2—44no—
431C2#355214yesdirect product factor
432C2C22×H(5,𝔽2)11yescharacteristic
433C2C22×2−1+411yescharacteristic
434C2C22×2−1+411yescharacteristic
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. #3552C2direct (N × H)
2. C2×#563C2semidirect
3. C2×#557C2semidirect
4. C2×#556C2semidirect
5. C2×#554C2semidirect
6. C2×#553C2semidirect
7. C2×D4×Q8C2semidirect
8. C2×D4×Q8C2semidirect
9. C2×#548C2semidirect
10. C2×#547C2semidirect
11. C2×#544C2semidirect
12. C2×#542C2semidirect
13. C2×#541C2semidirect
14. C2×#539C2semidirect
15. C2×#536C2semidirect
16. C22×C4⋊Q8C2semidirect
17. C2×#529C2semidirect
18. C2×#529C2semidirect
22. #563C22semidirect
23. #557C22semidirect
24. #556C22semidirect
25. #554C22semidirect
26. #553C22semidirect
27. D4×Q8C22semidirect
28. D4×Q8C22semidirect
29. #548C22semidirect
30. #547C22semidirect
31. #544C22semidirect
32. #542C22semidirect
33. #541C22semidirect
34. #539C22semidirect
35. #536C22semidirect
44. C2×C4⋊Q8C22semidirect
45. #529C22semidirect
46. #529C22semidirect
431. C2#3552direct (N × H)

Maximal subgroups

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

subgroupindex
#35522
C2×#5632
C2×#5572
C2×#5562
C2×#5542
C2×#5532
C2×D4×Q82
C2×#5482
C2×#5472
C2×#5442
C2×#5422
C2×#5412
C2×#5392
C2×#5362
C22×C4⋊Q82
C2×#5292