Finite Group Databasebeta

← C2×#3370 #61312

Subgroups — C2×#3370 #61312

Order 256.

975 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559C211yesmaximal, characteristic, complemented
2C2×#554C211yesmaximal, characteristic, complemented
3#3370C2116yesmaximal, direct product factor
4C2×#490C211yesmaximal, characteristic, complemented
5C2×#489C211yesmaximal, characteristic, complemented
6C2×#488C211yesmaximal, characteristic, complemented
7C2×#479C211yesmaximal, characteristic, complemented
8C2×#478C211yesmaximal, characteristic, complemented
9C2×#474C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#463C211yesmaximal, characteristic, complemented
12C2×#459C211yesmaximal, characteristic
13C2×#446C211yesmaximal, characteristic, complemented
14C2×#442C211yesmaximal, characteristic, complemented
15C2×#440C211yesmaximal, characteristic, complemented
16C2×C4×D8C211yesmaximal, characteristic, complemented
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20#559—24no—
21#559C2218yescomplemented
22#554—24no—
23#554C2218yescomplemented
24C2×C4⋊Q8C2211yescharacteristic
25C2×#127C2211yescharacteristic
26C2×#125C2211yescharacteristic
27C2×#125C2211yescharacteristic
28C2×#124—21no—
29C2×#124C2211yescharacteristic
30C2×#124C2211yescharacteristic
31C2×#123—21no—
32C2×#123—21no—
33C2×C22⋊Q8—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8—21no—
36C2×C4×Q8C2211yescharacteristic
37C2×C4×Q8C2211yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic
43C2×C4×D4C2211yescharacteristic
44C22×C4⋊C4—21no—
45#490C2218yescomplemented
46#489C2218yescomplemented
47#488C2218yescomplemented
48#479C2218yescomplemented
49#478C2218yescomplemented
50#474C2218yescomplemented
51#463C2218yescomplemented
52#463C2218yescomplemented
53#459C2218yes—
54#446C2218yescomplemented
55#442C2218yescomplemented
56#440C2218yescomplemented
57C4×D8C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×C4⋊C8C2211yescharacteristic
62C2×C4⋊C8C2211yescharacteristic
63C2×C4⋊C8C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#102C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×#97C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8D411yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4—21no—
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C2×SD16—24no—
91C2×SD16C2314yes—
92C2×D8—24no—
93C2×D8—24no—
94C2×D8C2314yes—
95C2×D8C2314yes—
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C4⋊Q8—22no—
101C4⋊Q8C2314yes—
102#127—22no—
103#127C2314yes—
104#125—22no—
105#125—22no—
106#125C2314yes—
107#125C2314yes—
108#124—28no—
109#124—22no—
110#124—22no—
111#124C2314yes—
112#124C2314yes—
113#123—28no—
114#123—28no—
115C22⋊Q8—28no—
116C22⋊Q8—28no—
117C22⋊Q8—28no—
118C4×Q8—22no—
119C4×Q8D414yes—
120C4×Q8C2314yes—
121C4×Q8C2314yes—
122C4×D4—28no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4C2314yes—
129C4×D4C2314yes—
130C4×D4C2314yes—
131C4×D4C2314yes—
132C4×D4C2314yes—
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⋊C4—28no—
140C2×C4⋊C4D411yescharacteristic
141C2×C4⋊C4D411yescharacteristic
142C2×C4⋊C4D411yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C42—21no—
157C2×C42C2311yescharacteristic
158C2×C42C2311yescharacteristic
159C2×C42C2311yescharacteristic
160#107C2314yes—
161#107C2314yes—
162#106C2314yes—
163C4⋊C8C2314yes—
164C4⋊C8C2314yes—
165C4⋊C8C2314yes—
166#103C2314yes—
167#103C2314yes—
168#103C2314yes—
169#102C2314yes—
170#102C2314yes—
171#102C2314yes—
172#102C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176#97C2314yes—
177#97C2314yes—
178C4×C8C2314yes—
179C24—21no—
180C24—21no—
181C24—21no—
182C2×Q8—21no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—42no—
187C2×Q8—82no—
188C2×Q8C2×D414yes—
189C2×Q8C2412yes—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—44no—
201C2×D4—82no—
202C2×D4#4114yes—
203C2×D4C2412yes—
204C2×D4C2412yes—
205C2×D4C2412yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—44no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2411yescharacteristic
228SD16—28no—
229D8—28no—
230D8—28no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8C2412yes—
236C2×C8C2412yes—
237C2×C8C2412yes—
238C2×C8C2412yes—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4C2×D414yes—
250C4⋊C4C2×D414yes—
251C4⋊C4C2×D414yes—
252C4⋊C4C2412yes—
253C4⋊C4C2412yes—
254C4⋊C4C2412yes—
255C4⋊C4C2412yes—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C42—24no—
266C42C2×D412yes—
267C42C2×D412yes—
268C42C2×D412yes—
269C42C2412yes—
270C42C2412yes—
271C42C2412yes—
272C23—24no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23C22×D411yescharacteristic
285Q8—22no—
286Q8—22no—
287Q8—44no—
288Q8—42no—
289Q8—84no—
290D4—24no—
291D4—22no—
292D4—22no—
293D4—22no—
294D4—22no—
295D4—22no—
296D4—48no—
297D4—42no—
298D4—42no—
299D4—84no—
300D4C2×#4114yes—
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—22no—
312C2×C4—48no—
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×D412yes—
327C2×C4C22×D412yes—
328C2×C4C22×D411yescharacteristic
329C2×C4C2×#4112yes—
330C2×C42−1+412yes—
331C2×C4C2511yescharacteristic
332C8—22no—
333C8—22no—
334C8—22no—
335C8—22no—
336C22—24no—
337C22—48no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—84no—
347C22—84no—
348C22#54812yes—
349C22#57512yes—
350C22#57612yes—
351C22C23×D411yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4#54812yes—
364C4C22×#4111yescharacteristic
365C4C2×2−1+411yescharacteristic
366C2—44no—
367C2—82no—
368C2—82no—
369C2#337014yesdirect product factor
370C2C2×#54811yescharacteristic
371C2C2×#57511yescharacteristic
372C2C2×#57611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#459C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#559C2semidirect
2. C2×#554C2semidirect
3. #3370C2direct (N × H)
4. C2×#490C2semidirect
5. C2×#489C2semidirect
6. C2×#488C2semidirect
7. C2×#479C2semidirect
8. C2×#478C2semidirect
9. C2×#474C2semidirect
10. C2×#463C2semidirect
11. C2×#463C2semidirect
13. C2×#446C2semidirect
14. C2×#442C2semidirect
15. C2×#440C2semidirect
16. C2×C4×D8C2semidirect
21. #559C22semidirect
23. #554C22semidirect
45. #490C22semidirect
46. #489C22semidirect
47. #488C22semidirect
48. #479C22semidirect
49. #478C22semidirect
50. #474C22semidirect
51. #463C22semidirect
52. #463C22semidirect
54. #446C22semidirect
55. #442C22semidirect
56. #440C22semidirect
57. C4×D8C22semidirect
369. C2#3370direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#5542
#33702
C2×#4902
C2×#4892
C2×#4882
C2×#4792
C2×#4782
C2×#4742
C2×#4632
C2×#4592
C2×#4462
C2×#4422
C2×#4402
C2×C4×D82