Finite Group Databasebeta

← C2×#3349 #61241

Subgroups — C2×#3349 #61241

Order 256.

917 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic
2C2×#552C211yesmaximal, characteristic
3#3349C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#484C211yesmaximal, characteristic, complemented
6C2×#485C211yesmaximal, characteristic, complemented
7C2×#485C211yesmaximal, characteristic, complemented
8C2×#485C211yesmaximal, characteristic, complemented
9C2×#479C211yesmaximal, characteristic, complemented
10C2×#477C211yesmaximal, characteristic, complemented
11C2×#470C211yesmaximal, characteristic, complemented
12C2×#467C211yesmaximal, characteristic, complemented
13C2×#441C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic
15C2×#428C211yesmaximal, characteristic, complemented
16C2×#427C211yesmaximal, characteristic, complemented
17C22×Q16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#556—24no—
21#556C2218yes—
22#552—24no—
23#552C2218yes—
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#126—21no—
27C2×#126—21no—
28C2×#125—21no—
29C2×#125C2211yescharacteristic
30C2×#124—21no—
31C2×#123—21no—
32C2×C22⋊Q8C2211yescharacteristic
33C2×C22⋊Q8C2211yescharacteristic
34C2×C22⋊Q8C2211yescharacteristic
35C2×C22⋊Q8C2211yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×#117—21no—
41C2×#117C2211yescharacteristic
42C2×#117C2211yescharacteristic
43C2×#117C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2218yescomplemented
46#485C2218yescomplemented
47#485C2218yescomplemented
48#485C2218yescomplemented
49#484C2218yescomplemented
50#479C2218yescomplemented
51#477C2218yescomplemented
52#470C2218yescomplemented
53#467C2218yescomplemented
54#441C2218yescomplemented
55#435C2218yes—
56#428C2218yescomplemented
57#427C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#107C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#106C2211yescharacteristic
65C2×#106C2211yescharacteristic
66C2×#106C2211yescharacteristic
67C2×C4⋊C8C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#103C2211yescharacteristic
72C2×#103C2211yescharacteristic
73C2×#103C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4D411yescharacteristic
81C23×C4—21no—
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C2×Q16—24no—
85C2×Q16C2314yes—
86C2×M16—24no—
87C2×M16C2314yes—
88C22×C8—24no—
89C22×C8C2311yescharacteristic
90C22×C8C2314yes—
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C4⋊Q8—22no—
95C4⋊Q8C2314yes—
96#126—28no—
97#126—28no—
98#126—28no—
99#125—28no—
100#125—22no—
101#125C2314yes—
102#124—28no—
103#123—28no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8C2314yes—
109C22⋊Q8C2314yes—
110C22⋊Q8C2314yes—
111C22⋊Q8C2314yes—
112C4×Q8—28no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8C2314yes—
116C4×Q8C2314yes—
117C4×D4D414yes—
118C4×D4C2314yes—
119#117—22no—
120#117—22no—
121#117—22no—
122#117—28no—
123#117C2314yes—
124#117C2314yes—
125#117C2314yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4D411yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2314yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4D411yescharacteristic
153C2×C22⋊C4D411yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42C2311yescharacteristic
160#107C2314yes—
161#107C2314yes—
162#107C2314yes—
163#107C2314yes—
164#107C2314yes—
165#106C2314yes—
166#106C2314yes—
167#106C2314yes—
168#106C2314yes—
169C4⋊C8C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176C22⋊C8C2314yes—
177C22⋊C8C2314yes—
178#97C2314yes—
179C24C2×D411yescharacteristic
180C24C2×D411yescharacteristic
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—44no—
189C2×Q8C2412yes—
190C2×Q8C2412yes—
191C2×D4—42no—
192C2×D4C2×D414yes—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—44no—
207C22×C4C2×D412yes—
208C22×C4C2×D412yes—
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2412yes—
214C22×C4C2412yes—
215C22×C4C2411yescharacteristic
216Q16—28no—
217M16—28no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—28no—
222C2×C8#4114yes—
223C2×C8C2412yes—
224C2×C8C2412yes—
225C2×C8C2412yes—
226C2×C8C2412yes—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4C2×D414yes—
244C4⋊C4C2412yes—
245C4⋊C4C2412yes—
246C4⋊C4C2412yes—
247C4⋊C4C2412yes—
248C4⋊C4C2412yes—
249C4⋊C4C2412yes—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4C2×D414yes—
258C22⋊C4C2×D414yes—
259C42—24no—
260C42—24no—
261C42—24no—
262C42—24no—
263C42—24no—
264C42C2×D412yes—
265C42C2412yes—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23C22×D412yes—
273C23C22×D412yes—
274C23C22×D411yescharacteristic
275Q8—22no—
276Q8—22no—
277Q8—22no—
278Q8—22no—
279Q8—48no—
280Q8—42no—
281Q8—42no—
282D4—44no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—48no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4C22×D412yes—
311C2×C4C22×D412yes—
312C2×C4C22×D412yes—
313C2×C4C22×D412yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C2×#4112yes—
316C2×C42−1+412yes—
317C2×C4C2511yescharacteristic
318C8—22no—
319C8—22no—
320C8—22no—
321C8C2×#4114yes—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—44no—
329C22—44no—
330C22#54712yes—
331C22#57512yes—
332C22#57812yes—
333C22C23×D411yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4#54712yes—
347C4C22×#4111yescharacteristic
348C4C2×2−1+411yescharacteristic
349C2—42no—
350C2—42no—
351C2#334914yesdirect product factor
352C2C2×#54711yescharacteristic
353C2C2×#57511yescharacteristic
354C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC22×D4D4
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
3. #3349C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#484C2semidirect
6. C2×#485C2semidirect
7. C2×#485C2semidirect
8. C2×#485C2semidirect
9. C2×#479C2semidirect
10. C2×#477C2semidirect
11. C2×#470C2semidirect
12. C2×#467C2semidirect
13. C2×#441C2semidirect
15. C2×#428C2semidirect
16. C2×#427C2semidirect
45. #501C22semidirect
46. #485C22semidirect
47. #485C22semidirect
48. #485C22semidirect
49. #484C22semidirect
50. #479C22semidirect
51. #477C22semidirect
52. #470C22semidirect
53. #467C22semidirect
54. #441C22semidirect
56. #428C22semidirect
57. #427C22semidirect
351. C2#3349direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#5522
#33492
C2×#5012
C2×#4842
C2×#4852
C2×#4792
C2×#4772
C2×#4702
C2×#4672
C2×#4412
C2×#4352
C2×#4282
C2×#4272