Finite Group Databasebeta

← C2×#3241 #60855

Subgroups — C2×#3241 #60855

Order 256.

903 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic, complemented
2C2×#540C211yesmaximal, characteristic
3#3241C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#499C211yesmaximal, characteristic, complemented
6C2×#490C211yesmaximal, characteristic, complemented
7C2×#486C211yesmaximal, characteristic, complemented
8C2×#481C211yesmaximal, characteristic, complemented
9C2×#482C211yesmaximal, characteristic, complemented
10C2×#482C211yesmaximal, characteristic, complemented
11C2×#480C211yesmaximal, characteristic, complemented
12C2×#478C211yesmaximal, characteristic, complemented
13C2×#478C211yesmaximal, characteristic, complemented
14C2×#476C211yesmaximal, characteristic, complemented
15C2×#474C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17#544—24no—
18#544C2218yescomplemented
19#540—24no—
20#540C2218yes—
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×#127C2211yescharacteristic
25C2×#125—21no—
26C2×#125C2211yescharacteristic
27C2×#125C2211yescharacteristic
28C2×#125C2211yescharacteristic
29C2×#123—21no—
30C2×#123—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8—21no—
33C2×#121—21no—
34C2×#121—21no—
35C2×#121C2211yescharacteristic
36C2×#121C2211yescharacteristic
37C2×C4×D4C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39C2×#117C2211yescharacteristic
40C2×#117C2211yescharacteristic
41C22×C4⋊C4C2211yescharacteristic
42#501C2218yescomplemented
43#499C2218yescomplemented
44#490C2218yescomplemented
45#486C2218yescomplemented
46#482C2218yescomplemented
47#482C2218yescomplemented
48#481C2218yescomplemented
49#480C2218yescomplemented
50#478C2218yescomplemented
51#478C2218yescomplemented
52#476C2218yescomplemented
53#474C2218yescomplemented
54#433C2218yescomplemented
55C2×#107C2211yescharacteristic
56C2×#107C2211yescharacteristic
57C2×#107C2211yescharacteristic
58C2×#107C2211yescharacteristic
59C2×#106C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×#106C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×C4⋊C8C2211yescharacteristic
64C2×C4⋊C8C2211yescharacteristic
65C2×#102C2211yescharacteristic
66C2×#102C2211yescharacteristic
67C2×#102C2211yescharacteristic
68C2×#102C2211yescharacteristic
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×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C4⋊Q8—28no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8C2314yes—
97C4⋊Q8C2314yes—
98#127—22no—
99#127C2314yes—
100#125—28no—
101#125—22no—
102#125—22no—
103#125—22no—
104#125C2314yes—
105#125C2314yes—
106#125C2314yes—
107#123—28no—
108#123—28no—
109C22⋊Q8—28no—
110C22⋊Q8—28no—
111#121—22no—
112#121—22no—
113#121—28no—
114#121—28no—
115#121C2314yes—
116#121C2314yes—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4C2314yes—
120C4×D4C2314yes—
121#117—22no—
122#117D414yes—
123#117C2314yes—
124#117C2314yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
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⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4D411yescharacteristic
136C2×C4⋊C4D411yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2314yes—
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4D411yescharacteristic
150C2×C22⋊C4D411yescharacteristic
151C2×C42—21no—
152C2×C42C2311yescharacteristic
153C2×C42C2311yescharacteristic
154#107C2314yes—
155#107C2314yes—
156#107C2314yes—
157#107C2314yes—
158#106C2314yes—
159#106C2314yes—
160#106C2314yes—
161#106C2314yes—
162C4⋊C8C2314yes—
163C4⋊C8C2314yes—
164#102C2314yes—
165#102C2314yes—
166#102C2314yes—
167#102C2314yes—
168#102C2314yes—
169#102C2314yes—
170#102C2314yes—
171#102C2314yes—
172C22⋊C8C2314yes—
173C22⋊C8C2314yes—
174#97C2314yes—
175C4×C8C2314yes—
176C24—21no—
177C24—21no—
178C24C2×D411yescharacteristic
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—82no—
194C2×D4—82no—
195C2×D4—82no—
196C2×D4—82no—
197C2×D4C2412yes—
198C2×D4C2412yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4C2×D412yes—
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2412yes—
220C22×C4C2411yescharacteristic
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8C2412yes—
226C2×C8C2412yes—
227C2×C8C2412yes—
228C2×C8C2412yes—
229C4⋊C4—24no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4C2×D414yes—
245C4⋊C4C2×D414yes—
246C4⋊C4C2412yes—
247C4⋊C4C2412yes—
248C4⋊C4C2412yes—
249C4⋊C4C2412yes—
250C4⋊C4C2412yes—
251C4⋊C4C2412yes—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4C2×D414yes—
258C22⋊C4C2×D414yes—
259C42—24no—
260C42C2×D412yes—
261C42C2×D412yes—
262C42C2412yes—
263C42C2412yes—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23C22×D412yes—
276C23C22×D411yescharacteristic
277Q8—84no—
278Q8—84no—
279D4—22no—
280D4—22no—
281D4—22no—
282D4—22no—
283D4—42no—
284D4—42no—
285D4—84no—
286D4—84no—
287D4—84no—
288D4—84no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4C22×D412yes—
313C2×C4C22×D412yes—
314C2×C4C22×D412yes—
315C2×C4C22×D412yes—
316C2×C4C22×D412yes—
317C2×C4C22×D411yescharacteristic
318C2×C42−1+412yes—
319C2×C42−1+412yes—
320C2×C4C2511yescharacteristic
321C8—22no—
322C8—22no—
323C8—22no—
324C8—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—44no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—84no—
336C22—84no—
337C22#53612yes—
338C22#57612yes—
339C22#57712yes—
340C22C23×D411yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4#53612yes—
353C4C2×2−1+411yescharacteristic
354C4C2×2−1+411yescharacteristic
355C2—42no—
356C2—82no—
357C2—82no—
358C2#324114yesdirect product factor
359C2C2×#53611yescharacteristic
360C2C2×#57611yescharacteristic
361C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#540C2
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. #3241C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#499C2semidirect
6. C2×#490C2semidirect
7. C2×#486C2semidirect
8. C2×#481C2semidirect
9. C2×#482C2semidirect
10. C2×#482C2semidirect
11. C2×#480C2semidirect
12. C2×#478C2semidirect
13. C2×#478C2semidirect
14. C2×#476C2semidirect
15. C2×#474C2semidirect
16. C2×#433C2semidirect
18. #544C22semidirect
42. #501C22semidirect
43. #499C22semidirect
44. #490C22semidirect
45. #486C22semidirect
46. #482C22semidirect
47. #482C22semidirect
48. #481C22semidirect
49. #480C22semidirect
50. #478C22semidirect
51. #478C22semidirect
52. #476C22semidirect
53. #474C22semidirect
54. #433C22semidirect
358. C2#3241direct (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×#5402
#32412
C2×#5012
C2×#4992
C2×#4902
C2×#4862
C2×#4812
C2×#4822
C2×#4802
C2×#4782
C2×#4762
C2×#4742
C2×#4332