Finite Group Databasebeta

← C2×#13067756 #14729649

Subgroups — C2×#13067756 #14729649

Order 2816.

1010 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622705C211yesmaximal, characteristic
2C2×#10622634C212yesmaximal
3#13067756C218yesmaximal, direct product factor
4C2×#10622184C211yesmaximal, characteristic
5C2×Q8⋊(C11⋊C8)C212yesmaximal
6C2×Q82×C11C211yesmaximal, characteristic
7Q82×C11—22no—
8Q82×C11—24no—
9Q82×C11C2214yes—
10C2×C11×C4⋊Q8—22no—
11C2×C11×C4⋊Q8—22no—
12C2×C11×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8×C11—22no—
14C2×C4×Q8×C11C2212yes—
15C22×#4326—22no—
16C2×#66922C2211yescharacteristic
17C2×#66899—22no—
18#10622705—22no—
19#10622705C2214yes—
20#10622634—24no—
21#10622634C2218yes—
22C2×#66871—22no—
23C2×C4⋊(C11⋊C8)C2212yes—
24C2×C4×C11⋊C8C2211yescharacteristic
25#10622184C2214yes—
26Q8⋊(C11⋊C8)C2218yes—
27C22×Q8×C11—22no—
28C22×Q8×C11D412yes—
29C22×Q8×C11D412yes—
30C22×Dic22—21no—
31C11×C4⋊Q8—28no—
32C11×C4⋊Q8—28no—
33C11×C4⋊Q8—28no—
34C11×C4⋊Q8—28no—
35C11×C4⋊Q8—22no—
36C11×C4⋊Q8D412yes—
37C11×C4⋊Q8C2312yes—
38C4×Q8×C11—28no—
39C4×Q8×C11—28no—
40C4×Q8×C11—22no—
41C4×Q8×C11—22no—
42C4×Q8×C11D414yes—
43C4×Q8×C11C2314yes—
44C2×C11×C4⋊C4—24no—
45C2×C11×C4⋊C4—22no—
46C2×C11×C4⋊C4—22no—
47C2×C11×C4⋊C4D412yes—
48C2×C42×C11—22no—
49C2×C42×C11—22no—
50C2×C42×C11C2311yescharacteristic
51C2×#4326—28no—
52C2×#4326—48no—
53C2×C4⋊Dic11—22no—
54C22×C11⋊C8—21no—
55C22×C11⋊C8—22no—
56#66922—22no—
57#66922—21no—
58#66922C2312yes—
59#66899—28no—
60#66871—28no—
61C4⋊(C11⋊C8)—22no—
62C4⋊(C11⋊C8)C2314yes—
63C4×C11⋊C8—21no—
64C4×C11⋊C8C2312yes—
65C2×#1633—111nomaximal, Sylow
66C2×Q8×C11—28no—
67C2×Q8×C11—22no—
68C2×Q8×C11—22no—
69C2×Q8×C11—24no—
70C2×Q8×C11—44no—
71C2×Q8×C11—84no—
72C2×Q8×C11Q1618yescomplemented
73C2×Q8×C11C2×D414yes—
74C2×Q8×C11C2×D414yes—
75C22×C4×C11—22no—
76C22×C4×C11—22no—
77C22×C4×C11—22no—
78C22×C4×C11C2×D411yescharacteristic
79C22×C4×C11C2×D412yes—
80C22×Dic11—41no—
81C2×Dic22—22no—
82C2×Dic22—44no—
83C2×Dic22—41no—
84C11×C4⋊C4—216no—
85C11×C4⋊C4—24no—
86C11×C4⋊C4—24no—
87C11×C4⋊C4—22no—
88C11×C4⋊C4—24no—
89C11×C4⋊C4—24no—
90C11×C4⋊C4C2×D414yes—
91C42×C11—28no—
92C42×C11—24no—
93C42×C11—24no—
94C42×C11C2×D411yescharacteristic
95C42×C11C2×D412yes—
96C42×C11C2411yescharacteristic
97#4326—416no—
98C4⋊Dic11—24no—
99C4⋊Dic11—42no—
100C2×C11⋊C8—24no—
101C2×C11⋊C8—22no—
102C2×C11⋊C8—24no—
103C2×C11⋊C8—44no—
104C2×Q82D1111yescharacteristic
105C2×#494—111no—
106C2×#477—112no—
107#1633—118no—
108C2×#333—111no—
109C2×Q8⋊C8—112no—
110C23×C11#12011yescharacteristic
111Q8×C11—24no—
112Q8×C11—28no—
113Q8×C11—44no—
114Q8×C11—82no—
115Q8×C11—88no—
116Q8×C11C2×Q1618yescomplemented
117C2×C4×C11—24no—
118C2×C4×C11—28no—
119C2×C4×C11—22no—
120C2×C4×C11—24no—
121C2×C4×C11—24no—
122C2×C4×C11—44no—
123C2×C4×C11—44no—
124C2×C4×C11#12011yescharacteristic
125C2×C4×C11#12012yes—
126C2×C4×C11C2×Q1614yes—
127C2×C4×C11#13714yes—
128C2×C4×C11C22×D411yescharacteristic
129C2×C4×C11C22×D412yes—
130C2×Dic11—42no—
131C2×Dic11—82no—
132Dic22—44no—
133Dic22—82no—
134C11⋊C8—24no—
135C11⋊C8—44no—
136C22×Q16—222no—
137Q82—22no—
138Q82—24no—
139Q82C2×D1114yes—
140C2×C4⋊Q8—22no—
141C2×C4⋊Q8—22no—
142C2×C4⋊Q8—111no—
143C2×C4⋊Q8C2×D1111yescharacteristic
144C2×C4×Q8—22no—
145C2×C4×Q8C2×D1112yes—
146#494—114no—
147#494—222no—
148#477—118no—
149#477—224no—
150C2×#107—222no—
151C2×C4⋊C8—112no—
152C2×#103—222no—
153C2×C4×C8—111no—
154#333—114no—
155Q8⋊C8—118no—
156C22×C11#45114yes—
157C22×C11#45312yes—
158C22×C11C2×#12011yescharacteristic
159C4×C11—28no—
160C4×C11—21no—
161C4×C11—21no—
162C4×C11—44no—
163C4×C11—44no—
164C4×C11#45114yes—
165C4×C11C22×Q1612yes—
166C4×C11C2×#13712yes—
167Dic11—82no—
168C22×Q8—22no—
169C22×Q8—221no—
170C22×Q8C22⋊D1112yes—
171C22×Q8C22⋊D1112yes—
172C2×Q16—228no—
173C2×Q16—448no—
174C22×C8—221no—
175C22×C8—222no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179C4⋊Q8—28no—
180C4⋊Q8—22no—
181C4⋊Q8—112no—
182C4⋊Q8—222no—
183C4⋊Q8—221no—
184C4⋊Q8C22⋊D1112yes—
185C4⋊Q8C22×D1112yes—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8C22⋊D1114yes—
191C4×Q8C22×D1114yes—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—222no—
196C2×C4⋊C4C22⋊D1112yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42C22×D1111yescharacteristic
200#107—228no—
201C4⋊C8—114no—
202C4⋊C8—222no—
203#103—228no—
204C4×C8—112no—
205C4×C8—221no—
206C2×C11#163314yescomplemented
207C2×C11C2×#45112yes—
208C2×C11C2×#45311yescharacteristic
209C2×Q8—28no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—44no—
214C2×Q8—84no—
215C2×Q8—222no—
216C2×Q8—444no—
217C2×Q8—441no—
218C2×Q8#432618yescomplemented
219C2×Q8C2×C22⋊D1114yes—
220C2×Q8C2×C22⋊D1114yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—441no—
225C22×C4C2×C22⋊D1111yescharacteristic
226C22×C4C2×C22⋊D1112yes—
227Q16—4416no—
228C2×C8—224no—
229C2×C8—222no—
230C2×C8—224no—
231C2×C8—444no—
232C4⋊C4—216no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—224no—
239C4⋊C4—442no—
240C4⋊C4C2×C22⋊D1114yes—
241C42—28no—
242C42—24no—
243C42—24no—
244C42C2×C22⋊D1111yescharacteristic
245C42C2×C22⋊D1112yes—
246C42C23×D1111yescharacteristic
247C11C2×#163311yescharacteristic, complemented, Sylow
248C23#6700611yescharacteristic
249Q8—24no—
250Q8—28no—
251Q8—44no—
252Q8—82no—
253Q8—88no—
254Q8—444no—
255Q8—882no—
256Q8C2×#432618yescomplemented
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—442no—
265C2×C4—882no—
266C2×C4#6699514yes—
267C2×C4C2×#432614yes—
268C2×C4#6700611yescharacteristic
269C2×C4#6700612yes—
270C2×C4C22×C22⋊D1111yescharacteristic
271C2×C4C22×C22⋊D1112yes—
272C8—224no—
273C8—444no—
274C22#1062284114yes—
275C22#1062285412yes—
276C22C2×#6700611yescharacteristic
277C4—28no—
278C4—21no—
279C4—21no—
280C4—44no—
281C4—44no—
282C4—882no—
283C4#1062284114yes—
284C4C2×#6699512yes—
285C4C22×#432612yes—
286C2#1306775614yesdirect product factor
287C2C2×#1062284112yes—
288C2C2×#1062285411yescharacteristic
subgroupG/H
centerC23#67006
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingC2×Q82×C11C2
socle, prime radicalC23×C11#120
hypercenterC2×Q82D11
hypocenterC11C2×#1633
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #13067756C2direct (N × H)
72. C2×Q8×C11Q16semidirect
116. Q8×C11C2×Q16semidirect
206. C2×C11#1633semidirect
218. C2×Q8#4326semidirect
247. C11C2×#1633semidirect
256. Q8C2×#4326semidirect
286. C2#13067756direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106227052
C2×#106226342
#130677562
C2×#106221842
C2×Q8⋊(C11⋊C8)2
C2×Q82×C112
C2×#163311