Finite Group Databasebeta

← #21461804

Subgroups — #21461804

Order 7808.

404 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#6923C211yesmaximal, characteristic
2C2×#14519840C212yesmaximal, complemented
3D61⋊(C22⋊C8)C211yesmaximal, characteristic
4#16840635C212yesmaximal, complemented
5C22⋊Q8×D61C211yesmaximal, characteristic
6C23×#6923—21no—
7C2×D61⋊C8C2212yes—
8C2×C4⋊#6923—21no—
9C2×C4⋊#6923—24no—
10C2×C4⋊#6923C2212yes—
11#14519840—24no—
12#14519834—21no—
13C22⋊#92420—21no—
14C2×Q8×D61C2211yescharacteristic
15C22×C4×D61C2211yescharacteristic
16#14519757C411yescharacteristic, complemented
17#14519756C411yescharacteristic, complemented
18#14519755—21no—
19#14519754—21no—
20#14519753C411yescharacteristic, complemented
21C4⋊C4×D61—21no—
22C4⋊C4×D61C2211yescharacteristic
23C22⋊C4×D61—21no—
24C22⋊Q8×C61C411yescharacteristic, complemented
25C23×D61D411yescharacteristic
26C22×#6923—24no—
27C22×#6923—22no—
28C22×#6923—42no—
29C22×#6923—42no—
30C2×Q8×C61C2×C411yescharacteristic, complemented
31C22×C4×C61C2×C411yescharacteristic
32C22×Dic61D411yescharacteristic
33Q8×D61—22no—
34Q8×D61—22no—
35C2×C4×D61—22no—
36C2×C4×D61—22no—
37C2×C4×D61—21no—
38C2×C4×D61—21no—
39C2×C4×D61C2×C411yescharacteristic
40C2×C4×D61D411yescharacteristic
41C2×C4×D61D411yescharacteristic
42C2×C4×D61C2311yescharacteristic
43C2×Dic122—21no—
44C2×Dic122C2×C411yescharacteristic, complemented
45C2×#92420—22no—
46C4⋊#6923—24no—
47C4⋊#6923—22no—
48C4⋊#6923D414yes—
49D61⋊C8—22no—
50D61⋊C8—22no—
51C4⋊C4×C61—21no—
52C4⋊C4×C61C2×C411yescharacteristic, complemented
53C22⋊C4×C61—21no—
54C22⋊Dic61—21no—
55#11758198—21no—
56#11758198—21no—
57C4⋊Dic61—21no—
58C4⋊Dic61C2×C411yescharacteristic, complemented
59#11758196—21no—
60#11758196—21no—
61#11758196—21no—
62C23×C61C22⋊C411yescharacteristic
63C22×D61—21no—
64C22×D61—21no—
65C22×D61—21no—
66C22×D61—21no—
67C22×D61C22⋊C411yescharacteristic
68C22×D61SD1612yes—
69C22×D61Q1612yes—
70C22×D61C2×D411yescharacteristic
71C2×#6923—24no—
72C2×#6923—48no—
73C2×#6923—42no—
74C2×#6923—42no—
75Q8×C61—22no—
76C2×C4×C61—22no—
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61C22⋊C411yescharacteristic
80C2×C4×C61C22×C411yescharacteristic
81C2×Dic61—22no—
82C2×Dic61—21no—
83C2×Dic61—21no—
84C2×Dic61C22⋊C411yescharacteristic
85C2×Dic61C2×D411yescharacteristic
86C4×D61—21no—
87C4×D61—21no—
88C4×D61—22no—
89C4×D61—22no—
90C4×D61—41no—
91C4×D61—41no—
92C4×D61—41no—
93C4×D61—41no—
94C4×D61C2×D412yes—
95C4×D61#4112yes—
96Dic122—22no—
97Dic122—42no—
98#92420—42no—
99C22×C61—21no—
100C22×C61—21no—
101C22×C61#10312yescomplemented
102C22×C61C2×C22⋊C411yescharacteristic
103C2×D61—21no—
104C2×D61—21no—
105C2×D61—22no—
106C2×D61—22no—
107C2×D61—22no—
108C2×D61—22no—
109C2×D61#10312yescomplemented
110C2×D61#12011yescharacteristic
111C2×D61#12311yescharacteristic
112C2×D61C2×SD1611yescharacteristic
113C2×D61C2×Q1611yescharacteristic
114C2×D61AGL(1,ℤ/8)11yescharacteristic
115C2×D61#13711yescharacteristic
116#6923—44no—
117C4×C61—21no—
118C4×C61—41no—
119C4×C61—41no—
120C4×C61C4×D412yes—
121Dic61—21no—
122Dic61—41no—
123Dic61—41no—
124Dic61#12112yes—
125#1872—611nomaximal, Sylow
126C2×C61—22no—
127C2×C61#38611yescharacteristic
128C2×C61C2×#10311yescharacteristic
129C2×C61#41711yescharacteristic
130D61—22no—
131D61#45011yescharacteristic
132D61#45111yescharacteristic
133D61#48311yescharacteristic
134D61#48411yescharacteristic
135C2×C22⋊Q8—611no—
136C22×C4⋊C4—611no—
137C2×#103—612no—
138C2×C22⋊C8—611no—
139#340—612no—
140C61#187211yescharacteristic, complemented, Sylow
141C22×Q8—611no—
142C23×C4—611no—
143C23×C4—1221no—
144C22×C8—612no—
145C22⋊Q8—611no—
146C22⋊Q8—611no—
147C22⋊Q8—611no—
148C22⋊Q8—1221no—
149C22⋊Q8—1221no—
150C22⋊Q8#692311yescharacteristic, complemented
151C2×C4⋊C4—612no—
152C2×C4⋊C4—611no—
153C2×C4⋊C4—1221no—
154C2×C4⋊C4—1224no—
155C2×C4⋊C4—1221no—
156C2×C22⋊C4—1221no—
157#103—1224no—
158C22⋊C8—1221no—
159C22⋊C8—1221no—
160C24—611no—
161C2×Q8—611no—
162C2×Q8—1221no—
163C2×Q8—1222no—
164C2×Q8—1222no—
165C2×Q8C2×#692311yescharacteristic, complemented
166C22×C4—611no—
167C22×C4—611no—
168C22×C4—611no—
169C22×C4—611no—
170C22×C4—611no—
171C22×C4—1222no—
172C22×C4—1222no—
173C22×C4—1221no—
174C22×C4—1221no—
175C22×C4—1222no—
176C22×C4—1224no—
177C22×C4—2442no—
178C22×C4—2442no—
179C22×C4C2×#692311yescharacteristic
180C2×C8—1222no—
181C2×C8—1222no—
182C2×C8—1222no—
183C4⋊C4—21no—
184C4⋊C4—614no—
185C4⋊C4—611no—
186C4⋊C4—1221no—
187C4⋊C4—1221no—
188C4⋊C4—1221no—
189C4⋊C4—1222no—
190C4⋊C4—1224no—
191C4⋊C4—1221no—
192C4⋊C4C2×#692311yescharacteristic, complemented
193C22⋊C4—21no—
194C22⋊C4—1221no—
195C22⋊C4—1221no—
196C22⋊C4—1221no—
197C23—611no—
198C23—612no—
199C23—612no—
200C23—611no—
201C23—1221no—
202C23—1221no—
203C23—1221no—
204C23—1221no—
205C23C22⋊#692311yescharacteristic
206Q8—22no—
207Q8—1222no—
208Q8—2442no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—611no—
213C2×C4—612no—
214C2×C4—612no—
215C2×C4—611no—
216C2×C4—1222no—
217C2×C4—1221no—
218C2×C4—1221no—
219C2×C4—1222no—
220C2×C4—1222no—
221C2×C4—1224no—
222C2×C4—1221no—
223C2×C4—1221no—
224C2×C4—2441no—
225C2×C4—2441no—
226C2×C4—2441no—
227C2×C4—2441no—
228C2×C4—2442no—
229C2×C4—2442no—
230C2×C4—2448no—
231C2×C4C22⋊#692311yescharacteristic
232C2×C4C22×#692311yescharacteristic
233C8—2442no—
234C22—21no—
235C22—21no—
236C22—611no—
237C22—611no—
238C22—611no—
239C22—611no—
240C22—611no—
241C22—611no—
242C22—612no—
243C22—1222no—
244C22—1222no—
245C22—1222no—
246C22—1222no—
247C22—1221no—
248C22—1221no—
249C22#1451984012yescomplemented
250C22C2×C22⋊#692311yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—612no—
255C4—1221no—
256C4—2444no—
257C4—2441no—
258C4—2441no—
259C4D4×#692312yes—
260C2—22no—
261C2—611no—
262C2—611no—
263C2—611no—
264C2—611no—
265C2—1222no—
266C2#1684071011yescharacteristic
267C2C2×#1451984011yescharacteristic
268C2#1684075311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6923
commutatorC2×C4×C61C22×C4
FrattiniC2×C4C22×#6923
FittingC22⋊Q8×C61C4
socleC22×C61C2×C22⋊C4
hypercenterC22⋊Q8#6923
hypocenterC61#1872
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14519840C2semidirect
4. #16840635C2semidirect
16. #14519757C4semidirect
17. #14519756C4semidirect
20. #14519753C4semidirect
24. C22⋊Q8×C61C4semidirect
30. C2×Q8×C61C2×C4semidirect
44. C2×Dic122C2×C4semidirect
52. C4⋊C4×C61C2×C4semidirect
58. C4⋊Dic61C2×C4semidirect
101. C22×C61#103semidirect
109. C2×D61#103semidirect
140. C61#1872semidirect
150. C22⋊Q8#6923semidirect
165. C2×Q8C2×#6923semidirect
192. C4⋊C4C2×#6923semidirect
249. C22#14519840semidirect