Finite Group Databasebeta

← C2×#3346 #61238

Subgroups — C2×#3346 #61238

Order 256.

971 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#553C211yesmaximal, characteristic
3#3346C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#484C211yesmaximal, characteristic, complemented
6C2×#481C211yesmaximal, characteristic, complemented
7C2×#485C211yesmaximal, characteristic, complemented
8C2×#480C211yesmaximal, characteristic, complemented
9C2×#479C211yesmaximal, characteristic, complemented
10C2×#474C211yesmaximal, characteristic, complemented
11C2×#468C211yesmaximal, characteristic, complemented
12C2×#465C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#428C211yesmaximal, characteristic, complemented
16C22×#107C211yesmaximal, characteristic, complemented
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#554—24no—
21#554C2218yescomplemented
22#553—24no—
23#553C2218yes—
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#125—21no—
27C2×#125C2211yescharacteristic
28C2×#123—21no—
29C2×C22⋊Q8—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8C2211yescharacteristic
32C2×C22⋊Q8C2211yescharacteristic
33C2×#121—21no—
34C2×#121C2211yescharacteristic
35C2×#121C2211yescharacteristic
36C2×C4×Q8C2211yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×#117C2211yescharacteristic
42C22×C4⋊C4C2211yescharacteristic
43C22×C4⋊C4C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2218yescomplemented
46#485C2218yescomplemented
47#484C2218yescomplemented
48#481C2218yescomplemented
49#480C2218yescomplemented
50#479C2218yescomplemented
51#474C2218yescomplemented
52#468C2218yescomplemented
53#465C2218yescomplemented
54#440C2218yescomplemented
55#435C2218yescomplemented
56#428C2218yescomplemented
57C2×#107C2211yescharacteristic
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#107C2211yescharacteristic
63C2×#107C2218yescomplemented
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×#97C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4—21no—
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×SD16—24no—
90C2×SD16C2314yes—
91C2×M16—24no—
92C2×M16C2314yes—
93C22×C8—24no—
94C22×C8C2311yescharacteristic
95C22×C8C2314yes—
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C4⋊Q8—22no—
100C4⋊Q8C2314yes—
101#126—28no—
102#125—28no—
103#125—22no—
104#125C2314yes—
105#123—28no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8—28no—
109C22⋊Q8—28no—
110C22⋊Q8C2314yes—
111C22⋊Q8C2314yes—
112#121—28no—
113#121—22no—
114#121—22no—
115#121C2314yes—
116#121C2314yes—
117C4×Q8—22no—
118C4×Q8C2314yes—
119C4×D4—28no—
120C4×D4—28no—
121C4×D4—22no—
122C4×D4D414yes—
123C4×D4C2314yes—
124C4×D4C2314yes—
125#117—22no—
126#117C2314yes—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4D411yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2314yes—
146C2×C4⋊C4C2314yes—
147C2×C4⋊C4C2314yes—
148C2×C4⋊C4C2311yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4D411yescharacteristic
156C2×C22⋊C4D411yescharacteristic
157C2×C42—21no—
158C2×C42—21no—
159C2×C42C2311yescharacteristic
160#107C2314yes—
161#107C2314yes—
162#107C2314yes—
163#107C2314yes—
164#107C2314yes—
165#107C2314yes—
166#106C2314yes—
167#106C2314yes—
168#106C2314yes—
169C4⋊C8C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176C22⋊C8C2314yes—
177C22⋊C8C2314yes—
178#97C2314yes—
179C24—21no—
180C24C2×D411yescharacteristic
181C24C2×D411yescharacteristic
182C2×Q8—21no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—82no—
187C2×Q8C2412yes—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—21no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—44no—
195C2×D4—82no—
196C2×D4C2×D414yes—
197C2×D4C2412yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4C2×D412yes—
218C22×C4C2×D412yes—
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2412yes—
224C22×C4C2412yes—
225C22×C4C2411yescharacteristic
226SD16—28no—
227M16—28no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—28no—
232C2×C8#4114yes—
233C2×C8C2412yes—
234C2×C8C2412yes—
235C2×C8C2412yes—
236C2×C8C2412yes—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4C2×D414yes—
253C4⋊C4C2412yes—
254C4⋊C4C2412yes—
255C4⋊C4C2412yes—
256C4⋊C4C2412yes—
257C4⋊C4C2412yes—
258C4⋊C4C2412yes—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4C2×D414yes—
266C22⋊C4C2×D414yes—
267C42—24no—
268C42—24no—
269C42C2×D412yes—
270C42C2412yes—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23#13714yes—
281C23C22×D412yes—
282C23C22×D412yes—
283C23C22×D411yescharacteristic
284Q8—22no—
285Q8—22no—
286Q8—42no—
287Q8—84no—
288D4—28no—
289D4—22no—
290D4—22no—
291D4—48no—
292D4—42no—
293D4—84no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
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—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4C22×D412yes—
323C2×C4C22×D412yes—
324C2×C4C22×D412yes—
325C2×C4C22×D412yes—
326C2×C4C22×D411yescharacteristic
327C2×C4C2×#4112yes—
328C2×C42−1+412yes—
329C2×C4C2511yescharacteristic
330C8—22no—
331C8—22no—
332C8—22no—
333C8C2×#4114yes—
334C22—28no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—44no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—84no—
345C22#54712yes—
346C22C2×#13714yes—
347C22C2×#13712yes—
348C22#57612yes—
349C22C23×D411yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4#54712yes—
362C4C22×#4111yescharacteristic
363C4C2×2−1+411yescharacteristic
364C2—24no—
365C2—42no—
366C2—82no—
367C2#334614yesdirect product factor
368C2C2×#54711yescharacteristic
369C2C22×#13711yescharacteristic
370C2C2×#57611yescharacteristic
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
1. C2×#554C2semidirect
3. #3346C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#484C2semidirect
6. C2×#481C2semidirect
7. C2×#485C2semidirect
8. C2×#480C2semidirect
9. C2×#479C2semidirect
10. C2×#474C2semidirect
11. C2×#468C2semidirect
12. C2×#465C2semidirect
13. C2×#440C2semidirect
14. C2×#435C2semidirect
15. C2×#428C2semidirect
16. C22×#107C2semidirect
21. #554C22semidirect
45. #501C22semidirect
46. #485C22semidirect
47. #484C22semidirect
48. #481C22semidirect
49. #480C22semidirect
50. #479C22semidirect
51. #474C22semidirect
52. #468C22semidirect
53. #465C22semidirect
54. #440C22semidirect
55. #435C22semidirect
56. #428C22semidirect
63. C2×#107C22semidirect
367. C2#3346direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5532
#33462
C2×#5012
C2×#4842
C2×#4812
C2×#4852
C2×#4802
C2×#4792
C2×#4742
C2×#4682
C2×#4652
C2×#4402
C2×#4352
C2×#4282
C22×#1072