Finite Group Databasebeta

← C2×#3308 #61129

Subgroups — C2×#3308 #61129

Order 256.

1187 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic, complemented
3#3308C2116yesmaximal, direct product factor
4C2×#489C211yesmaximal, characteristic, complemented
5C2×#484C211yesmaximal, characteristic, complemented
6C2×#480C211yesmaximal, characteristic, complemented
7C2×#477C211yesmaximal, characteristic, complemented
8C2×#468C211yesmaximal, characteristic, complemented
9C2×#466C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#449C211yesmaximal, characteristic, complemented
12C2×#448C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#417C211yesmaximal, characteristic, complemented
16C22×#103C211yesmaximal, characteristic, complemented
17C22×#41—21no—
18C22×#41C2211yescharacteristic
19C23×Q8C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×D8C2211yescharacteristic
23C22×M16C2211yescharacteristic
24C23×C8C2211yescharacteristic
25#548—24no—
26#548C2218yescomplemented
27#547—24no—
28#547C2218yescomplemented
29C2×C4⋊Q8C2211yescharacteristic
30C2×#124—21no—
31C2×#124C2211yescharacteristic
32C2×#123—21no—
33C2×#123—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8C2211yescharacteristic
37C2×#121—21no—
38C2×#121C2211yescharacteristic
39C2×#121C2211yescharacteristic
40C2×#121C2211yescharacteristic
41C2×C4×Q8C2211yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4C2211yescharacteristic
44C2×C4×D4C2211yescharacteristic
45C22×C4⋊C4C2211yescharacteristic
46C22×C4⋊C4C2211yescharacteristic
47#489C2218yescomplemented
48#484C2218yescomplemented
49#480C2218yescomplemented
50#477C2218yescomplemented
51#468C2218yescomplemented
52#466C2218yescomplemented
53#463C2218yescomplemented
54#449C2218yescomplemented
55#448C2218yescomplemented
56#440C2218yescomplemented
57#435C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×C4⋊C8C2211yescharacteristic
62#417C2218yescomplemented
63C2×#103C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2218yescomplemented
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C2×#41—28no—
78C2×#41—22no—
79C2×#41—24no—
80C2×#41—42no—
81C2×#41—48no—
82C2×#41C2314yes—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—42no—
88C22×Q8C2311yescharacteristic
89C22×Q8C2314yes—
90C22×Q8C2311yescharacteristic
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—21no—
95C22×D4D411yescharacteristic
96C22×D4C2311yescharacteristic
97C22×D4C2311yescharacteristic
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2311yescharacteristic
104C23×C4C2311yescharacteristic
105C2×SD16—24no—
106C2×SD16—42no—
107C2×SD16C2314yes—
108C2×SD16C2314yes—
109C2×D8—42no—
110C2×D8C2314yes—
111C2×M16—42no—
112C2×M16C2314yes—
113C22×C8—42no—
114C22×C8C2311yescharacteristic
115C22×C8C2311yescharacteristic
116C22×C8C2311yescharacteristic
117C22×C8C2311yescharacteristic
118C22×C8C2314yes—
119C4⋊Q8—22no—
120C4⋊Q8C2314yes—
121#124—28no—
122#124—22no—
123#124C2314yes—
124#123—28no—
125#123—28no—
126C22⋊Q8—22no—
127C22⋊Q8—28no—
128C22⋊Q8—28no—
129C22⋊Q8C2314yes—
130#121—28no—
131#121—22no—
132#121—22no—
133#121—22no—
134#121C2314yes—
135#121C2314yes—
136#121C2314yes—
137C4×Q8—22no—
138C4×Q8C2314yes—
139C4×D4—28no—
140C4×D4—22no—
141C4×D4D414yes—
142C4×D4C2314yes—
143C4×D4C2314yes—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4D411yescharacteristic
152C2×C4⋊C4C2311yescharacteristic
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2311yescharacteristic
155C2×C4⋊C4C2314yes—
156C2×C4⋊C4C2314yes—
157C2×C4⋊C4C2311yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4D411yescharacteristic
166C2×C22⋊C4D411yescharacteristic
167C2×C42—21no—
168C2×C42C2311yescharacteristic
169#107C2314yes—
170#107C2314yes—
171#107C2314yes—
172C4⋊C8C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176#103C2314yes—
177#103C2314yes—
178#103C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182#102C2314yes—
183C22⋊C8C2314yes—
184C22⋊C8C2314yes—
185#97C2314yes—
186C24—21no—
187C24—21no—
188C24C2×D411yescharacteristic
189C24C2×D411yescharacteristic
190#41—28no—
191#41—44no—
192#41—48no—
193#41—432no—
194C2×Q8—21no—
195C2×Q8—24no—
196C2×Q8—21no—
197C2×Q8—24no—
198C2×Q8—22no—
199C2×Q8—28no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—48no—
205C2×Q8—44no—
206C2×Q8#4114yescomplemented
207C2×Q8C2412yes—
208C2×Q8C2412yes—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4C2×D414yes—
225C2×D4C2412yes—
226C2×D4C2412yes—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4C2×D412yes—
248C22×C4C2×D412yes—
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2412yes—
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256SD16—28no—
257SD16—44no—
258D8—44no—
259M16—44no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—44no—
265C2×C8C2412yes—
266C2×C8C2412yes—
267C2×C8C2412yes—
268C2×C8C2412yes—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—21no—
278C4⋊C4C2×D414yes—
279C4⋊C4C2412yes—
280C4⋊C4C2412yes—
281C4⋊C4C2412yes—
282C4⋊C4C2412yes—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4C2×D414yes—
291C22⋊C4C2×D414yes—
292C42—24no—
293C42C2×D412yes—
294C42C2412yes—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23AGL(1,ℤ/8)14yes—
309C23C22×D412yes—
310C23C22×D412yes—
311C23C22×D411yescharacteristic
312Q8—24no—
313Q8—22no—
314Q8—22no—
315Q8—22no—
316Q8—48no—
317Q8—48no—
318Q8—42no—
319Q8C2×#4114yescomplemented
320D4—28no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—48no—
326D4—48no—
327D4—48no—
328D4—48no—
329D4—42no—
330D4—42no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—42no—
358C2×C4C22×D412yes—
359C2×C4C22×D412yes—
360C2×C4C22×D412yes—
361C2×C4C22×D412yes—
362C2×C4C22×D411yescharacteristic
363C2×C4C2×#4112yes—
364C2×C4H(5,𝔽2)12yes—
365C2×C4C2511yescharacteristic
366C8—22no—
367C8—22no—
368C8—22no—
369C8—22no—
370C22—22no—
371C22—28no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—44no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—84no—
384C22—84no—
385C22#54612yes—
386C22C2×AGL(1,ℤ/8)14yes—
387C22C2×AGL(1,ℤ/8)12yes—
388C22#57812yes—
389C22C23×D411yescharacteristic
390C4—22no—
391C4—22no—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4#54612yes—
401C4C22×#4111yescharacteristic
402C4C2×H(5,𝔽2)11yescharacteristic
403C2—24no—
404C2—42no—
405C2—82no—
406C2—82no—
407C2#330814yesdirect product factor
408C2C2×#54611yescharacteristic
409C2C22×AGL(1,ℤ/8)11yescharacteristic
410C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#548C2semidirect
2. C2×#547C2semidirect
3. #3308C2direct (N × H)
4. C2×#489C2semidirect
5. C2×#484C2semidirect
6. C2×#480C2semidirect
7. C2×#477C2semidirect
8. C2×#468C2semidirect
9. C2×#466C2semidirect
10. C2×#463C2semidirect
11. C2×#449C2semidirect
12. C2×#448C2semidirect
13. C2×#440C2semidirect
14. C2×#435C2semidirect
15. C2×#417C2semidirect
16. C22×#103C2semidirect
26. #548C22semidirect
28. #547C22semidirect
47. #489C22semidirect
48. #484C22semidirect
49. #480C22semidirect
50. #477C22semidirect
51. #468C22semidirect
52. #466C22semidirect
53. #463C22semidirect
54. #449C22semidirect
55. #448C22semidirect
56. #440C22semidirect
57. #435C22semidirect
62. #417C22semidirect
69. C2×#103C22semidirect
206. C2×Q8#41semidirect
319. Q8C2×#41semidirect
407. C2#3308direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#5472
#33082
C2×#4892
C2×#4842
C2×#4802
C2×#4772
C2×#4682
C2×#4662
C2×#4632
C2×#4492
C2×#4482
C2×#4402
C2×#4352
C2×#4172
C22×#1032