Finite Group Databasebeta

← C2×#3351 #61244

Subgroups — C2×#3351 #61244

Order 256.

949 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic
2C2×#552C211yesmaximal, characteristic, complemented
3#3351C2116yesmaximal, direct product factor
4C2×#499C211yesmaximal, characteristic, complemented
5C2×#481C211yesmaximal, characteristic, complemented
6C2×#485C211yesmaximal, characteristic, complemented
7C2×#482C211yesmaximal, characteristic, complemented
8C2×#483C211yesmaximal, characteristic, complemented
9C2×#479C211yesmaximal, characteristic, complemented
10C2×#478C211yesmaximal, characteristic, complemented
11C2×#468C211yesmaximal, characteristic, complemented
12C2×#468C211yesmaximal, characteristic, complemented
13C2×C4×SD16C211yesmaximal, characteristic, complemented
14C2×#436C211yesmaximal, characteristic, complemented
15C2×#428C211yesmaximal, characteristic, complemented
16C2×#428C211yesmaximal, characteristic, complemented
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20#553—24no—
21#553C2218yes—
22#552—24no—
23#552C2218yescomplemented
24C2×#126—21no—
25C2×#126—21no—
26C2×#125—21no—
27C2×#125C2211yescharacteristic
28C2×#125C2211yescharacteristic
29C2×#123—21no—
30C2×#123—21no—
31C2×C22⋊Q8C2211yescharacteristic
32C2×C22⋊Q8C2211yescharacteristic
33C2×#121—21no—
34C2×#121C2211yescharacteristic
35C2×#121C2211yescharacteristic
36C2×C4×Q8C2211yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×#117—21no—
41C2×#117C2211yescharacteristic
42C2×#117C2211yescharacteristic
43C22×C4⋊C4C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#499C2218yescomplemented
46#485C2218yescomplemented
47#483C2218yescomplemented
48#482C2218yescomplemented
49#481C2218yescomplemented
50#479C2218yescomplemented
51#478C2218yescomplemented
52#468C2218yescomplemented
53#468C2218yescomplemented
54C4×SD16C2218yescomplemented
55#436C2218yescomplemented
56#428C2218yescomplemented
57#428C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#106C2211yescharacteristic
65C2×#106C2211yescharacteristic
66C2×#106C2211yescharacteristic
67C2×C4⋊C8C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8C2311yescharacteristic
78C22×D4—21no—
79C22×D4—21no—
80C22×D4D411yescharacteristic
81C22×D4C2311yescharacteristic
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C2×SD16—24no—
88C2×SD16C2314yes—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16C2314yes—
92C2×M16C2314yes—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97#126—28no—
98#126—28no—
99#125—28no—
100#125—22no—
101#125—22no—
102#125C2314yes—
103#125C2314yes—
104#123—28no—
105#123—28no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8C2314yes—
109C22⋊Q8C2314yes—
110#121—28no—
111#121—22no—
112#121—22no—
113#121C2314yes—
114#121C2314yes—
115C4×Q8—22no—
116C4×Q8C2314yes—
117C4×D4—28no—
118C4×D4—22no—
119C4×D4D414yes—
120C4×D4C2314yes—
121C4×D4C2314yes—
122#117—22no—
123#117—22no—
124#117—28no—
125#117C2314yes—
126#117C2314yes—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
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⋊C4—21no—
138C2×C4⋊C4D411yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2314yes—
144C2×C4⋊C4C2314yes—
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4D411yescharacteristic
155C2×C22⋊C4D411yescharacteristic
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42C2311yescharacteristic
160#107C2314yes—
161#107C2314yes—
162#107C2314yes—
163#107C2314yes—
164#106C2314yes—
165#106C2314yes—
166#106C2314yes—
167#106C2314yes—
168#106C2314yes—
169C4⋊C8C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176C22⋊C8C2314yes—
177C22⋊C8C2314yes—
178C4×C8C2314yes—
179C24—21no—
180C24C2×D411yescharacteristic
181C24C2×D411yescharacteristic
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8C2412yes—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—82no—
194C2×D4C2×D414yes—
195C2×D4C2412yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—44no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4C2×D412yes—
214C22×C4C2×D412yes—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2412yes—
220C22×C4C2412yes—
221C22×C4C2411yescharacteristic
222SD16—28no—
223M16—28no—
224M16—28no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8#4114yes—
229C2×C8C2412yes—
230C2×C8C2412yes—
231C2×C8C2412yes—
232C2×C8C2412yes—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4C2×D414yes—
249C4⋊C4C2412yes—
250C4⋊C4C2412yes—
251C4⋊C4C2412yes—
252C4⋊C4C2412yes—
253C4⋊C4C2412yes—
254C4⋊C4C2412yes—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4C2×D414yes—
263C22⋊C4C2×D414yes—
264C42—24no—
265C42—24no—
266C42—24no—
267C42C2×D412yes—
268C42C2412yes—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23C22×D412yes—
280C23C22×D412yes—
281C23C22×D411yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285D4—22no—
286D4—22no—
287D4—44no—
288D4—48no—
289D4—42no—
290D4—84no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—48no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4C22×D412yes—
319C2×C4C22×D412yes—
320C2×C4C22×D412yes—
321C2×C4C22×D412yes—
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4112yes—
324C2×C42−1+412yes—
325C2×C4C2511yescharacteristic
326C8—22no—
327C8—22no—
328C8—22no—
329C8C2×#4114yes—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—44no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—84no—
342C22#54712yes—
343C22#57512yes—
344C22#57712yes—
345C22C23×D411yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4#54712yes—
358C4C22×#4111yescharacteristic
359C4C2×2−1+411yescharacteristic
360C2—42no—
361C2—42no—
362C2—82no—
363C2#335114yesdirect product factor
364C2C2×#54711yescharacteristic
365C2C2×#57511yescharacteristic
366C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#553C2
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
2. C2×#552C2semidirect
3. #3351C2direct (N × H)
4. C2×#499C2semidirect
5. C2×#481C2semidirect
6. C2×#485C2semidirect
7. C2×#482C2semidirect
8. C2×#483C2semidirect
9. C2×#479C2semidirect
10. C2×#478C2semidirect
11. C2×#468C2semidirect
12. C2×#468C2semidirect
13. C2×C4×SD16C2semidirect
14. C2×#436C2semidirect
15. C2×#428C2semidirect
16. C2×#428C2semidirect
23. #552C22semidirect
45. #499C22semidirect
46. #485C22semidirect
47. #483C22semidirect
48. #482C22semidirect
49. #481C22semidirect
50. #479C22semidirect
51. #478C22semidirect
52. #468C22semidirect
53. #468C22semidirect
54. C4×SD16C22semidirect
55. #436C22semidirect
56. #428C22semidirect
57. #428C22semidirect
363. C2#3351direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#5522
#33512
C2×#4992
C2×#4812
C2×#4852
C2×#4822
C2×#4832
C2×#4792
C2×#4782
C2×#4682
C2×C4×SD162
C2×#4362
C2×#4282