Finite Group Databasebeta

← #21458678

Subgroups — #21458678

Order 7808.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic122C211yesmaximal, characteristic, complemented
2#16840427C211yesmaximal, characteristic
3#16840159C212yesmaximal, complemented
4#16840158C211yesmaximal, characteristic, complemented
5#16840053C211yesmaximal, characteristic
6#16840051C212yesmaximal
7#16840045C212yesmaximal, complemented
8#16839886C211yesmaximal, characteristic, complemented
9C4×#11758201C211yesmaximal, characteristic
10#16839847C212yesmaximal
11C61×#475C211yesmaximal, characteristic, complemented
12C2×Q8×D61—21no—
13C2×#11758201C2211yescharacteristic
14#14519777—21no—
15#14519777C2211yescharacteristic
16#14519776—22no—
17#14519776C2212yes—
18#14519776C2212yes—
19#14519755—22no—
20#14519753—21no—
21Q8×Dic61C2212yes—
22C4×Dic122—21no—
23C4×Dic122C2211yescharacteristic
24C4×D244C2211yescharacteristic
25#14519739C2212yes—
26#14519709C2212yes—
27#14519708C2212yes—
28#14519708C2212yes—
29#14519708C2211yescharacteristic
30#14519706C2212yes—
31#14519701C2211yescharacteristic
32#14519699C2212yes—
33#14519697C2211yescharacteristic
34C4×#92418C2211yescharacteristic
35C8×Dic61C2212yes—
36C4⋊Q8×C61C2211yescharacteristic
37C4×Q8×C61C2211yescharacteristic
38#106×C61C2212yes—
39C4⋊C8×C61C2211yescharacteristic
40#103×C61C2212yes—
41C2×Q8×C61—21no—
42C2×Q8×C61C2311yescharacteristic
43Q8×D61—44no—
44C2×D244C2311yescharacteristic
45C2×C4×D61—21no—
46C2×C4×D61—22no—
47C2×Dic122—22no—
48C2×Dic122—21no—
49C2×Dic122—22no—
50C2×Dic122D411yescharacteristic
51C2×C8×C61C2312yes—
52C4⋊C4×C61—21no—
53C4⋊C4×C61—21no—
54C4⋊C4×C61C2312yes—
55C4⋊C4×C61C2311yescharacteristic
56C42×C61—21no—
57C42×C61C2311yescharacteristic
58#11758201Q814yescomplemented
59#11758198—21no—
60#11758198—22no—
61C4⋊Dic61—21no—
62C4⋊Dic61—22no—
63C4⋊Dic61D411yescharacteristic
64C4⋊Dic61C2312yes—
65C4⋊Dic61C2311yescharacteristic
66#11758196—22no—
67#11758196—22no—
68#11758196D412yes—
69C4×Dic61—22no—
70C4×Dic61C2312yes—
71C2×#92418C2311yescharacteristic
72C2×#92418C2311yescharacteristic
73C22×D61—21no—
74Q8×C61—21no—
75Q8×C61—42no—
76Q8×C61C2×Q812yescomplemented
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61—22no—
80C2×C4×C61C2×D411yescharacteristic
81C2×C4×C61C2×D411yescharacteristic
82C2×C4×C61C2411yescharacteristic
83C2×Dic61—22no—
84C2×Dic61—21no—
85C2×Dic61C2×D412yes—
86C2×Dic61C2×D412yes—
87D244—21no—
88D244C2×Q812yes—
89C4×D61—42no—
90C4×D61—44no—
91Dic122—44no—
92Dic122—82no—
93Dic122—81no—
94Dic122SD1614yescomplemented
95C8×C61—22no—
96#92418—21no—
97#92418C2×Q812yescomplemented
98C22×C61C22×D411yescharacteristic
99C2×D61—22no—
100C2×D61—41no—
101C2×D61—41no—
102C4×C61—22no—
103C4×C61—21no—
104C4×C61—42no—
105C4×C61—41no—
106C4×C61C2×SD1612yes—
107C4×C61C22×Q811yescharacteristic
108C4×C612−1+411yescharacteristic
109Dic61—22no—
110Dic61—41no—
111Dic61—42no—
112Dic61C2×SD1614yes—
113Q8×SD16—611nomaximal, Sylow
114C2×C61D4×Q811yescharacteristic
115C2×C61C22×SD1611yescharacteristic
116C2×C61#57711yescharacteristic
117D61—42no—
118Q82—611no—
119D4×Q8—611no—
120#498—611no—
121#498—612no—
122#476—612no—
123#476—611no—
124#475—612no—
125#475D6111yescharacteristic, complemented
126C8×Q8—611no—
127C4×SD16—611no—
128C4×SD16—612no—
129C61Q8×SD1611yescharacteristic, complemented, Sylow
130C22×Q8—1221no—
131C2×SD16—611no—
132C4⋊Q8—612no—
133C4⋊Q8—612no—
134C4⋊Q8—611no—
135C4⋊Q8—1221no—
136C4⋊Q8—1222no—
137C4⋊Q8C2×D6111yescharacteristic
138C22⋊Q8—1222no—
139C22⋊Q8—1221no—
140C4×Q8—611no—
141C4×Q8—612no—
142C4×Q8—1221no—
143C4×Q8C2×D6111yescharacteristic
144C4×D4—612no—
145C4×D4—611no—
146#106—612no—
147#106—612no—
148#106—611no—
149#106—612no—
150#106C2×D6112yes—
151C4⋊C8—612no—
152C4⋊C8C2×D6111yescharacteristic
153#103—611no—
154#103C2×D6112yes—
155#102—612no—
156#102—611no—
157C4×C8—612no—
158C4×C8—611no—
159C2×Q8—21no—
160C2×Q8—611no—
161C2×Q8—1222no—
162C2×Q8—1222no—
163C2×Q8—1221no—
164C2×Q8—2444no—
165C2×Q8C22×D6111yescharacteristic
166C2×D4—611no—
167C22×C4—1222no—
168C22×C4—1221no—
169SD16—614no—
170C2×C8—611no—
171C2×C8—611no—
172C2×C8C22×D6112yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—612no—
176C4⋊C4—611no—
177C4⋊C4—612no—
178C4⋊C4—611no—
179C4⋊C4—1222no—
180C4⋊C4—1222no—
181C4⋊C4—1222no—
182C4⋊C4—1221no—
183C4⋊C4C22×D6111yescharacteristic
184C4⋊C4C22×D6112yes—
185C22⋊C4—1222no—
186C22⋊C4—1221no—
187C42—21no—
188C42—612no—
189C42—1222no—
190C42C22×D6111yescharacteristic
191C23—1221no—
192Q8—21no—
193Q8—42no—
194Q8—614no—
195Q8—4881no—
196Q8—4882no—
197Q8—2444no—
198Q8Q8×D6112yescomplemented
199D4—612no—
200D4—1221no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—612no—
205C2×C4—612no—
206C2×C4—1222no—
207C2×C4—1221no—
208C2×C4—2444no—
209C2×C4—2442no—
210C2×C4D4×D6111yescharacteristic
211C2×C4D4×D6111yescharacteristic
212C2×C4C23×D6111yescharacteristic
213C8—22no—
214C8—612no—
215C8—1221no—
216C22—1222no—
217C22—2441no—
218C22—2441no—
219C22C2×D4×D6111yescharacteristic
220C4—21no—
221C4—22no—
222C4—42no—
223C4—41no—
224C4—614no—
225C4—1222no—
226C4—2441no—
227C4—2442no—
228C4SD16×D6112yes—
229C4C2×Q8×D6111yescharacteristic
230C4#1451982411yescharacteristic
231C2—2442no—
232C2#1684042511yescharacteristic
233C2C2×SD16×D6111yescharacteristic
234C2#1684057111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#475C2
socleC22×C61C22×D4
hypercenter#475D61
hypocenterC61Q8×SD16
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×Dic122C2semidirect
3. #16840159C2semidirect
4. #16840158C2semidirect
7. #16840045C2semidirect
8. #16839886C2semidirect
11. C61×#475C2semidirect
58. #11758201Q8semidirect
76. Q8×C61C2×Q8semidirect
94. Dic122SD16semidirect
97. #92418C2×Q8semidirect
125. #475D61semidirect
129. C61Q8×SD16semidirect
198. Q8Q8×D61semidirect