Finite Group Databasebeta

← #21451549

Subgroups — #21451549

Order 7808.

816 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758222C211yesmaximal, characteristic
2C2×#14519753C212yesmaximal, complemented
3#16839682C212yesmaximal, complemented
4#16839679C212yesmaximal, complemented
5#16839677C214yesmaximal, complemented
6#16839622C211yesmaximal, characteristic, complemented
7#16839619C212yesmaximal, complemented
8C2×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
9C2×#11758222—24no—
10C2×#11758222—28no—
11C22×Dic122C2211yescharacteristic
12C22×D244C2211yescharacteristic
13C22×C22⋊D61C2212yes—
14C22×C4×D61C2212yes—
15#14519753—28no—
16C2×C4⋊Dic61C2212yes—
17C2×#11758196C2214yes—
18C2×C22⋊Dic61C2212yes—
19C2×#11758198C2214yes—
20C2×#11758198C2214yes—
21#14519684C2214yes—
22#14519684C2212yes—
23C22×Q8×C61C2211yescharacteristic
24C23×C4×C61C2211yescharacteristic
25C22⋊Q8×C61—24no—
26C2×C4⋊C4×C61C2212yes—
27C2×C4⋊C4×C61C2211yescharacteristic
28C2×C22⋊C4×C61C2212yes—
29C24×C61C2311yescharacteristic
30C23×D61C2312yes—
31C2×Q8×C61—22no—
32C2×Q8×C61—24no—
33C22×C4×C61—24no—
34C22×C4×C61D414yes—
35C22×C4×C61C2312yes—
36C22×C4×C61C2312yes—
37C22×C4×C61C2311yescharacteristic
38C22×C4×C61C2311yescharacteristic
39C2×C22⋊D61—24no—
40C2×C22⋊D61—24no—
41C2×C22⋊D61—24no—
42C22×Dic61C2314yes—
43C22×Dic61C2312yes—
44#11758222—232no—
45C2×D244—24no—
46C2×D244—22no—
47C2×C4×D61—24no—
48C2×C4×D61—24no—
49C2×C4×D61D418yes—
50C2×Dic122—22no—
51C2×Dic122—24no—
52C4⋊C4×C61—22no—
53C4⋊C4×C61—24no—
54C22⋊C4×C61—24no—
55C22⋊Dic61—42no—
56#11758198—28no—
57#11758198—44no—
58C4⋊Dic61—24no—
59#11758196—28no—
60C23×C61—22no—
61C23×C61—21no—
62C23×C61—22no—
63C23×C61—21no—
64C23×C61C2×D412yes—
65C23×C61C2411yescharacteristic
66C22×D61—22no—
67C22×D61—22no—
68C22×D61—22no—
69C22×D61—22no—
70C22×D61—22no—
71C22×D61—22no—
72C22×D61C2×D414yes—
73Q8×C61—44no—
74C2×C4×C61—22no—
75C2×C4×C61—28no—
76C2×C4×C61—24no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61C2×D412yes—
82C2×C4×C61C2×D414yes—
83C2×C4×C61C2×D412yes—
84C22⋊D61—216no—
85C2×Dic61—22no—
86C2×Dic61—22no—
87C2×Dic61—24no—
88C2×Dic61—24no—
89C2×Dic61—24no—
90C2×Dic61C2×D414yes—
91D244—28no—
92C4×D61—216no—
93Dic122—28no—
94C22×C61—22no—
95C22×C61—22no—
96C22×C61—22no—
97C22×C61—42no—
98C22×C61—42no—
99C22×C61C22×D412yes—
100C22×C61C22×D411yescharacteristic
101C22×C61H(5,𝔽2)11yescharacteristic
102C22×C612−1+411yescharacteristic
103C22×C612−1+412yes—
104C2×D61—24no—
105C2×D61—24no—
106C2×D61—24no—
107C2×D61—42no—
108C2×D61—42no—
109C2×D61—42no—
110C2×D61—42no—
111C4×C61—22no—
112C4×C61—42no—
113C4×C61—42no—
114C4×C61#12014yes—
115Dic61—24no—
116Dic61—44no—
117#2616—611nomaximal, Sylow
118C2×C61—42no—
119C2×C61C2×#12011yescharacteristic
120C2×C61#53612yes—
121C2×C61#53611yescharacteristic
122C2×C61#53711yescharacteristic
123C2×C61#53712yes—
124D61—44no—
125C22×#41—611no—
126C2×C22⋊Q8—612no—
127C2×C22⋊Q8D6111yescharacteristic, complemented
128#395—612no—
129#394—612no—
130#394—614no—
131#380—611no—
132#380—612no—
133C61#261611yescharacteristic, complemented, Sylow
134C2×#41—1228no—
135C2×#41—1224no—
136C22×Q8—611no—
137C22×Q8C2×D6111yescharacteristic
138C22×D4—612no—
139C22×D4—611no—
140C23×C4—612no—
141C23×C4C2×D6111yescharacteristic
142C22⋊Q8—24no—
143C22⋊Q8—1228no—
144C2×C4⋊C4—612no—
145C2×C4⋊C4—614no—
146C2×C4⋊C4C2×D6111yescharacteristic
147C2×C4⋊C4C2×D6112yes—
148C2×C22⋊C4—614no—
149C2×C22⋊C4—612no—
150C2×C22⋊C4—614no—
151C2×C22⋊C4C2×D6112yes—
152#95—614no—
153#95—612no—
154C24—612no—
155C24C22×D6111yescharacteristic
156#41—12232no—
157C2×Q8—24no—
158C2×Q8—22no—
159C2×Q8—1224no—
160C2×Q8—1222no—
161C2×D4—1224no—
162C2×D4—1224no—
163C2×D4—1224no—
164C2×D4—1224no—
165C2×D4—1222no—
166C22×C4—24no—
167C22×C4—612no—
168C22×C4—618no—
169C22×C4—614no—
170C22×C4—1224no—
171C22×C4—1224no—
172C22×C4C22⋊D6114yes—
173C22×C4C22×D6112yes—
174C22×C4C22×D6112yes—
175C22×C4C22×D6111yescharacteristic
176C22×C4C22×D6111yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—1228no—
180C4⋊C4—1224no—
181C22⋊C4—24no—
182C22⋊C4—1228no—
183C22⋊C4—2444no—
184C22⋊C4—2442no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—22no—
189C23—614no—
190C23—1222no—
191C23—1222no—
192C23—1222no—
193C23—1222no—
194C23—1222no—
195C23—1222no—
196C23C2×C22⋊D6112yes—
197C23C23×D6111yescharacteristic
198Q8—44no—
199Q8—1228no—
200D4—1228no—
201D4—12216no—
202C2×C4—22no—
203C2×C4—28no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—614no—
210C2×C4—1222no—
211C2×C4—1222no—
212C2×C4—12216no—
213C2×C4—1224no—
214C2×C4—1224no—
215C2×C4—1224no—
216C2×C4D4×D6114yes—
217C2×C4C2×C22⋊D6112yes—
218C2×C4C2×C22⋊D6112yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—42no—
223C22—42no—
224C22—1224no—
225C22—1224no—
226C22—1224no—
227C22—2442no—
228C22—2442no—
229C22—2442no—
230C22—2442no—
231C22C2×D4×D6112yes—
232C22C22×C22⋊D6111yescharacteristic
233C22#1451982311yescharacteristic
234C22#1451982411yescharacteristic
235C22#1451982512yes—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—1224no—
240C4—2444no—
241C4#1451974514yes—
242C2—42no—
243C2—2444no—
244C2C2×#1451974511yescharacteristic
245C2#1684033512yes—
246C2#1684033611yescharacteristic
247C2#1684034011yescharacteristic
248C2#1684034212yes—
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×C22⋊Q8×C61C2
hypercenterC2×C22⋊Q8D61
hypocenterC61#2616
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14519753C2semidirect
3. #16839682C2semidirect
4. #16839679C2semidirect
5. #16839677C2semidirect
6. #16839622C2semidirect
7. #16839619C2semidirect
8. C2×C22⋊Q8×C61C2semidirect
127. C2×C22⋊Q8D61semidirect
133. C61#2616semidirect