Finite Group Databasebeta

← C2×#3371 #61314

Subgroups — C2×#3371 #61314

Order 256.

937 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559C211yesmaximal, characteristic
2C2×#557C211yesmaximal, characteristic, complemented
3#3371C2116yesmaximal, direct product factor
4C2×#490C211yesmaximal, characteristic
5C2×#489C211yesmaximal, characteristic, complemented
6C2×#488C211yesmaximal, characteristic, complemented
7C2×#479C211yesmaximal, characteristic, complemented
8C2×#478C211yesmaximal, characteristic, complemented
9C2×#476C211yesmaximal, characteristic, complemented
10C2×#464C211yesmaximal, characteristic, complemented
11C2×#464C211yesmaximal, characteristic, complemented
12C2×#460C211yesmaximal, characteristic
13C2×#446C211yesmaximal, characteristic, complemented
14C2×#441C211yesmaximal, characteristic, complemented
15C2×#440C211yesmaximal, characteristic, complemented
16C2×C4×SD16C211yesmaximal, characteristic, complemented
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20#559—24no—
21#559C2218yes—
22#557—24no—
23#557C2218yescomplemented
24C2×C4⋊Q8—21no—
25C2×C4⋊Q8C2211yescharacteristic
26C2×#127C2211yescharacteristic
27C2×#125—21no—
28C2×#125—21no—
29C2×#125C2211yescharacteristic
30C2×#125C2211yescharacteristic
31C2×#124—21no—
32C2×#124C2211yescharacteristic
33C2×#124C2211yescharacteristic
34C2×#123—21no—
35C2×#123—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39C2×C4×Q8C2211yescharacteristic
40C2×C4×Q8C2211yescharacteristic
41C2×C4×Q8C2211yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4C2211yescharacteristic
44C2×C4×D4C2211yescharacteristic
45#490C2218yes—
46#489C2218yescomplemented
47#488C2218yescomplemented
48#479C2218yescomplemented
49#478C2218yescomplemented
50#476C2218yescomplemented
51#464C2218yescomplemented
52#464C2218yescomplemented
53#460C2218yes—
54#446C2218yescomplemented
55#441C2218yescomplemented
56#440C2218yescomplemented
57C4×SD16C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#106C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×C4⋊C8C2211yescharacteristic
62C2×C4⋊C8C2211yescharacteristic
63C2×C4⋊C8C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
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×Q8C2311yescharacteristic
81C22×D4—21no—
82C22×D4—21no—
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C2×Q16—24no—
88C2×Q16C2314yes—
89C2×SD16—24no—
90C2×SD16—24no—
91C2×SD16C2314yes—
92C2×SD16C2314yes—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C4⋊Q8—28no—
98C4⋊Q8—22no—
99C4⋊Q8C2314yes—
100#127—22no—
101#127C2314yes—
102#125—28no—
103#125—28no—
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—
115C4×Q8—28no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8D414yes—
121C4×Q8C2314yes—
122C4×Q8C2314yes—
123C4×Q8C2314yes—
124C4×Q8C2314yes—
125C4×Q8C2314yes—
126C4×D4—28no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4C2314yes—
130C4×D4C2314yes—
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⋊C4—21no—
139C2×C4⋊C4—21no—
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×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42C2311yescharacteristic
158C2×C42C2311yescharacteristic
159C2×C42C2311yescharacteristic
160#107C2314yes—
161#106C2314yes—
162#106C2314yes—
163C4⋊C8C2314yes—
164C4⋊C8C2314yes—
165C4⋊C8C2314yes—
166#103C2314yes—
167#103C2314yes—
168#103C2314yes—
169#103C2314yes—
170#102C2314yes—
171#102C2314yes—
172#102C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176#97C2314yes—
177#97C2314yes—
178C4×C8C2314yes—
179C24—21no—
180C24—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8C2×D414yes—
190C2×Q8#4114yes—
191C2×Q8C2412yes—
192C2×Q8C2412yes—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—44no—
202C2×D4—82no—
203C2×D4C2412yes—
204C2×D4C2412yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2411yescharacteristic
222Q16—28no—
223SD16—28no—
224SD16—28no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8C2412yes—
230C2×C8C2412yes—
231C2×C8C2412yes—
232C2×C8C2412yes—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4C2×D414yes—
247C4⋊C4C2×D414yes—
248C4⋊C4C2×D414yes—
249C4⋊C4C2412yes—
250C4⋊C4C2412yes—
251C4⋊C4C2412yes—
252C4⋊C4C2412yes—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C42—24no—
260C42—24no—
261C42—24no—
262C42—24no—
263C42C2×D412yes—
264C42C2×D412yes—
265C42C2×D412yes—
266C42C2412yes—
267C42C2412yes—
268C42C2412yes—
269C23—22no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23C22×D411yescharacteristic
278Q8—24no—
279Q8—22no—
280Q8—22no—
281Q8—22no—
282Q8—44no—
283Q8—48no—
284Q8—42no—
285Q8C2×#4114yes—
286D4—22no—
287D4—22no—
288D4—22no—
289D4—22no—
290D4—48no—
291D4—42no—
292D4—42no—
293D4—84no—
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—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4C22×D412yes—
315C2×C4C22×D412yes—
316C2×C4C22×D412yes—
317C2×C4C22×D412yes—
318C2×C4C22×D412yes—
319C2×C4C22×D412yes—
320C2×C4C22×D411yescharacteristic
321C2×C4C2×#4112yes—
322C2×C42−1+412yes—
323C2×C4C2511yescharacteristic
324C8—22no—
325C8—22no—
326C8—22no—
327C8—22no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—84no—
335C22—84no—
336C22#54812yes—
337C22#57512yes—
338C22#57712yes—
339C22C23×D411yescharacteristic
340C4—24no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4#54812yes—
353C4C22×#4111yescharacteristic
354C4C2×2−1+411yescharacteristic
355C2—82no—
356C2—82no—
357C2#337114yesdirect product factor
358C2C2×#54811yescharacteristic
359C2C2×#57511yescharacteristic
360C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#127C22
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
2. C2×#557C2semidirect
3. #3371C2direct (N × H)
5. C2×#489C2semidirect
6. C2×#488C2semidirect
7. C2×#479C2semidirect
8. C2×#478C2semidirect
9. C2×#476C2semidirect
10. C2×#464C2semidirect
11. C2×#464C2semidirect
13. C2×#446C2semidirect
14. C2×#441C2semidirect
15. C2×#440C2semidirect
16. C2×C4×SD16C2semidirect
23. #557C22semidirect
46. #489C22semidirect
47. #488C22semidirect
48. #479C22semidirect
49. #478C22semidirect
50. #476C22semidirect
51. #464C22semidirect
52. #464C22semidirect
54. #446C22semidirect
55. #441C22semidirect
56. #440C22semidirect
57. C4×SD16C22semidirect
357. C2#3371direct (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×#5572
#33712
C2×#4902
C2×#4892
C2×#4882
C2×#4792
C2×#4782
C2×#4762
C2×#4642
C2×#4602
C2×#4462
C2×#4412
C2×#4402
C2×C4×SD162