Finite Group Databasebeta

← C2×#3243 #60857

Subgroups — C2×#3243 #60857

Order 256.

871 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic, complemented
2C2×#542C211yesmaximal, characteristic
3#3243C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#500C211yesmaximal, characteristic, complemented
6C2×#489C211yesmaximal, characteristic, complemented
7C2×#488C211yesmaximal, characteristic, complemented
8C2×#484C211yesmaximal, characteristic, complemented
9C2×#485C211yesmaximal, characteristic, complemented
10C2×#482C211yesmaximal, characteristic
11C2×#480C211yesmaximal, characteristic
12C2×#479C211yesmaximal, characteristic, complemented
13C2×#477C211yesmaximal, characteristic, complemented
14C2×#478C211yesmaximal, characteristic, complemented
15C2×#474C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17#544—24no—
18#544C2218yescomplemented
19#542—24no—
20#542C2218yes—
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#125—21no—
27C2×#125C2211yescharacteristic
28C2×#125C2211yescharacteristic
29C2×#124—21no—
30C2×#124C2211yescharacteristic
31C2×#123—21no—
32C2×C22⋊Q8—21no—
33C2×C22⋊Q8—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8C2211yescharacteristic
36C2×#121C2211yescharacteristic
37C2×C4×Q8C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39C2×#117C2211yescharacteristic
40C2×#117C2211yescharacteristic
41C22×C4⋊C4C2211yescharacteristic
42#501C2218yescomplemented
43#500C2218yescomplemented
44#489C2218yescomplemented
45#488C2218yescomplemented
46#485C2218yescomplemented
47#484C2218yescomplemented
48#482C2218yes—
49#480C2218yes—
50#479C2218yescomplemented
51#478C2218yescomplemented
52#477C2218yescomplemented
53#474C2218yescomplemented
54#433C2218yescomplemented
55C2×#107C2211yescharacteristic
56C2×#107C2211yescharacteristic
57C2×#107C2211yescharacteristic
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#106C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×C4⋊C8C2211yescharacteristic
64C2×C4⋊C8C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×Q8C2311yescharacteristic
81C22×D4—21no—
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C4⋊Q8—28no—
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94C4⋊Q8C2314yes—
95C4⋊Q8C2314yes—
96C4⋊Q8C2314yes—
97#126—28no—
98#125—28no—
99#125—22no—
100#125—22no—
101#125C2314yes—
102#125C2314yes—
103#124—28no—
104#124—22no—
105#124C2314yes—
106#123—28no—
107C22⋊Q8—22no—
108C22⋊Q8—28no—
109C22⋊Q8—28no—
110C22⋊Q8—28no—
111C22⋊Q8C2314yes—
112#121—22no—
113#121C2314yes—
114C4×Q8—22no—
115C4×Q8C2314yes—
116C4×D4—22no—
117C4×D4C2314yes—
118#117—22no—
119#117D414yes—
120#117C2314yes—
121#117C2314yes—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4D411yescharacteristic
134C2×C4⋊C4D411yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2314yes—
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4D411yescharacteristic
149C2×C22⋊C4D411yescharacteristic
150C2×C42—21no—
151C2×C42—21no—
152C2×C42C2311yescharacteristic
153C2×C42C2311yescharacteristic
154#107C2314yes—
155#107C2314yes—
156#107C2314yes—
157#107C2314yes—
158#107C2314yes—
159#107C2314yes—
160#106C2314yes—
161#106C2314yes—
162C4⋊C8C2314yes—
163C4⋊C8C2314yes—
164#103C2314yes—
165#103C2314yes—
166#103C2314yes—
167#103C2314yes—
168#102C2314yes—
169#102C2314yes—
170#102C2314yes—
171#102C2314yes—
172C22⋊C8C2314yes—
173C22⋊C8C2314yes—
174#97C2314yes—
175C4×C8C2314yes—
176C24—21no—
177C24C2×D411yescharacteristic
178C2×Q8—21no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—24no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8C2412yes—
188C2×D4—24no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—82no—
193C2×D4C2412yes—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4C2×D412yes—
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2412yes—
214C22×C4C2411yescharacteristic
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8C2412yes—
220C2×C8C2412yes—
221C2×C8C2412yes—
222C2×C8C2412yes—
223C4⋊C4—24no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4C2×D414yes—
240C4⋊C4C2×D414yes—
241C4⋊C4C2412yes—
242C4⋊C4C2412yes—
243C4⋊C4C2412yes—
244C4⋊C4C2412yes—
245C4⋊C4C2412yes—
246C4⋊C4C2412yes—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4C2×D414yes—
254C22⋊C4C2×D414yes—
255C42—24no—
256C42—24no—
257C42C2×D412yes—
258C42C2×D412yes—
259C42C2412yes—
260C42C2412yes—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C22×D412yes—
269C23C22×D411yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—42no—
273Q8—84no—
274Q8—84no—
275Q8—84no—
276D4—22no—
277D4—22no—
278D4—42no—
279D4—84no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4C22×D412yes—
305C2×C4C22×D412yes—
306C2×C4C22×D412yes—
307C2×C4C22×D412yes—
308C2×C4C22×D412yes—
309C2×C4C22×D411yescharacteristic
310C2×C42−1+412yes—
311C2×C42−1+412yes—
312C2×C4C2511yescharacteristic
313C8—22no—
314C8—22no—
315C8—22no—
316C8—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—84no—
325C22#53612yes—
326C22#57612yes—
327C22#57812yes—
328C22C23×D411yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4#53612yes—
342C4C2×2−1+411yescharacteristic
343C4C2×2−1+411yescharacteristic
344C2—42no—
345C2—82no—
346C2#324314yesdirect product factor
347C2C2×#53611yescharacteristic
348C2C2×#57611yescharacteristic
349C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
3. #3243C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#500C2semidirect
6. C2×#489C2semidirect
7. C2×#488C2semidirect
8. C2×#484C2semidirect
9. C2×#485C2semidirect
12. C2×#479C2semidirect
13. C2×#477C2semidirect
14. C2×#478C2semidirect
15. C2×#474C2semidirect
16. C2×#433C2semidirect
18. #544C22semidirect
42. #501C22semidirect
43. #500C22semidirect
44. #489C22semidirect
45. #488C22semidirect
46. #485C22semidirect
47. #484C22semidirect
50. #479C22semidirect
51. #478C22semidirect
52. #477C22semidirect
53. #474C22semidirect
54. #433C22semidirect
346. C2#3243direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#5422
#32432
C2×#5012
C2×#5002
C2×#4892
C2×#4882
C2×#4842
C2×#4852
C2×#4822
C2×#4802
C2×#4792
C2×#4772
C2×#4782
C2×#4742
C2×#4332