Finite Group Databasebeta

← #21449462

Subgroups — #21449462

Order 7808.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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×Q8×D61C211yesmaximal, characteristic
2C2×#14519777C211yesmaximal, characteristic, complemented
3C2×#14519775C211yesmaximal, characteristic, complemented
4C2×#14519757C211yesmaximal, characteristic
5#16839683C214yesmaximal, complemented
6#16839667C211yesmaximal, characteristic, complemented
7#16839636C214yesmaximal
8C4×#11758198C211yesmaximal, characteristic
9C61×#391C211yesmaximal, characteristic, complemented
10C22×Dic122C2212yes—
11C22×Dic122C2211yescharacteristic
12C22×Dic122C2211yescharacteristic
13C2×Q8×D61—24no—
14C2×Q8×D61—24no—
15C2×Q8×D61—24no—
16C22×C4×D61C2212yes—
17C22×C4×D61C2211yescharacteristic
18#14519777—24no—
19#14519775—24no—
20#14519757—24no—
21C2×C4⋊Dic61C2212yes—
22C2×C4⋊Dic61C2214yes—
23C2×#11758196C2214yes—
24C2×#11758198C2212yes—
25C2×#11758198C2212yes—
26C2×C4×Dic61C2211yescharacteristic
27#14519684C2212yes—
28#14519684C2214yes—
29C22×Q8×C61C2211yescharacteristic
30C2×C4⋊C4×C61C2211yescharacteristic
31C2×C42×C61C2211yescharacteristic
32#95×C61C2214yes—
33C23×D61C2311yescharacteristic
34C2×Q8×C61—22no—
35C2×Q8×C61—22no—
36C2×Q8×C61D414yes—
37C22×C4×C61C2312yes—
38C22×C4×C61C2312yes—
39C22×C4×C61C2312yes—
40C22×C4×C61C2311yescharacteristic
41C22×Dic61C2312yes—
42C22×Dic61C2311yescharacteristic
43C22×Dic61C2312yes—
44C22×Dic61C2312yes—
45Q8×D61—232no—
46C2×C4×D61—24no—
47C2×C4×D61—24no—
48C2×C4×D61—22no—
49C2×C4×D61—22no—
50C2×C4×D61—24no—
51C2×C4×D61D414yes—
52C2×Dic122—24no—
53C2×Dic122—22no—
54C2×Dic122—22no—
55C2×Dic122—24no—
56C2×Dic122—24no—
57C2×Dic122—22no—
58C2×Dic122—22no—
59C2×Dic122—22no—
60C2×Dic122D414yes—
61C4⋊C4×C61—22no—
62C42×C61—22no—
63#11758198—24no—
64#11758198Q818yescomplemented
65C4⋊Dic61—24no—
66C4⋊Dic61—28no—
67#11758196—28no—
68C4×Dic61—22no—
69C23×C61C2411yescharacteristic
70C22×D61—21no—
71C22×D61—21no—
72C22×D61—22no—
73C22×D61C2×D412yes—
74C22×D61C2×Q814yes—
75Q8×C61—28no—
76C2×C4×C61—22no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—24no—
82C2×C4×C61C2×D414yes—
83C2×C4×C61C2×D412yes—
84C2×C4×C61C2×D412yes—
85C2×C4×C61C2×D412yes—
86C2×C4×C61C2×Q814yescomplemented
87C2×Dic61—22no—
88C2×Dic61—22no—
89C2×Dic61—21no—
90C2×Dic61—21no—
91C2×Dic61—22no—
92C2×Dic61—22no—
93C2×Dic61—22no—
94C2×Dic61—22no—
95C2×Dic61—22no—
96C2×Dic61C2×D412yes—
97C2×Dic61C2×D414yes—
98C2×Dic61C2×Q814yescomplemented
99C4×D61—216no—
100C4×D61—28no—
101Dic122—216no—
102Dic122—28no—
103Dic122—44no—
104C22×C61C22×D411yescharacteristic
105C22×C61C22×D411yescharacteristic
106C22×C61C22×D411yescharacteristic
107C22×C61C22×Q812yes—
108C22×C612−1+412yes—
109C2×D61—22no—
110C2×D61—22no—
111C2×D61—24no—
112C2×D61—24no—
113C2×D61C4⋊Q814yescomplemented
114C4×C61—24no—
115C4×C61—24no—
116C4×C61—42no—
117C4×C61#12014yes—
118Dic61—22no—
119Dic61—24no—
120Dic61—24no—
121Dic61—42no—
122#2434—611nomaximal, Sylow
123C2×C61C2×#12011yescharacteristic
124C2×C61C2×C4⋊Q811yescharacteristic
125C2×C61#53611yescharacteristic
126C2×C61D4×Q812yes—
127C2×C61D4×Q812yes—
128D61—24no—
129C23×Q8—611no—
130C2×C4⋊Q8—611no—
131C2×C4⋊Q8—611no—
132C2×C22⋊Q8—611no—
133#395—614no—
134#391—611no—
135#391D6111yescharacteristic, complemented
136#385—614no—
137C4×C22⋊C4—611no—
138C61#243411yescharacteristic, complemented, Sylow
139C22×Q8—611no—
140C22×Q8—611no—
141C22×Q8—612no—
142C22×Q8—1224no—
143C22×Q8—1224no—
144C22×Q8—1224no—
145C22×Q8C2×D6111yescharacteristic
146C23×C4—611no—
147C23×C4—612no—
148C4⋊Q8—1224no—
149C4⋊Q8—1224no—
150C22⋊Q8—1224no—
151C2×C4⋊C4—612no—
152C2×C4⋊C4—614no—
153C2×C4⋊C4—614no—
154C2×C4⋊C4C2×D6111yescharacteristic
155C2×C22⋊C4—612no—
156C2×C22⋊C4—612no—
157C2×C42—611no—
158C2×C42C2×D6111yescharacteristic
159#95—612no—
160#95—614no—
161#95C2×D6114yes—
162C24—611no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—614no—
166C2×Q8—1224no—
167C2×Q8—1224no—
168C2×Q8—12232no—
169C2×Q8—1224no—
170C2×Q8—1222no—
171C2×Q8—1222no—
172C2×Q8—1222no—
173C2×Q8—1222no—
174C2×Q8—1222no—
175C2×Q8D24414yes—
176C22×C4—612no—
177C22×C4—611no—
178C22×C4—612no—
179C22×C4—614no—
180C22×C4—612no—
181C22×C4—1224no—
182C22×C4—1224no—
183C22×C4—1224no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4C22×D6112yes—
187C22×C4C22×D6112yes—
188C22×C4C22×D6112yes—
189C22×C4C22×D6111yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—1224no—
192C4⋊C4—1228no—
193C4⋊C4—1228no—
194C22⋊C4—618no—
195C22⋊C4—1224no—
196C42—22no—
197C42—1222no—
198C23—612no—
199C23—614no—
200C23—1221no—
201C23—1221no—
202C23—1222no—
203C23C23×D6111yescharacteristic
204Q8—28no—
205Q8—12216no—
206Q8—1228no—
207Q8—2444no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—612no—
215C2×C4—614no—
216C2×C4—614no—
217C2×C4—12216no—
218C2×C4—1222no—
219C2×C4—1221no—
220C2×C4—1222no—
221C2×C4—1221no—
222C2×C4—1222no—
223C2×C4—1222no—
224C2×C4—1228no—
225C2×C4—1222no—
226C2×C4—1222no—
227C2×C4—1222no—
228C2×C4C2×D24414yes—
229C2×C4C2×D24412yes—
230C2×C4D4×D6112yes—
231C2×C4D4×D6112yes—
232C2×C4Q8×D6114yescomplemented
233C22—614no—
234C22—1222no—
235C22—1222no—
236C22—1224no—
237C22—1224no—
238C22C2×D4×D6111yescharacteristic
239C22C2×D4×D6111yescharacteristic
240C22C22×D24411yescharacteristic
241C22C2×Q8×D6112yes—
242C22#1451982412yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—1222no—
247C4—1224no—
248C4—1224no—
249C4—2442no—
250C4#1451974714yes—
251C2—1224no—
252C2C2×#1451974711yescharacteristic
253C2C4⋊Q8×D6111yescharacteristic
254C2#1684033211yescharacteristic
255C2#1684042512yes—
256C2Q8×D24412yes—
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC61×#391C2
hypercenter#391D61
hypocenterC61#2434
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#14519777C2semidirect
3. C2×#14519775C2semidirect
5. #16839683C2semidirect
6. #16839667C2semidirect
9. C61×#391C2semidirect
64. #11758198Q8semidirect
86. C2×C4×C61C2×Q8semidirect
98. C2×Dic61C2×Q8semidirect
113. C2×D61C4⋊Q8semidirect
135. #391D61semidirect
138. C61#2434semidirect
232. C2×C4Q8×D61semidirect