Finite Group Databasebeta

← #30670913

Subgroups — #30670913

Order 16768.

676 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 177 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×#16862245C211yesmaximal, characteristic, complemented
2C2×#16862224C214yesmaximal, complemented
3C2×#16862208C211yesmaximal, characteristic
4#22553247C214yesmaximal, complemented
5#22553214C211yesmaximal, characteristic
6#22553213C211yesmaximal, characteristic
7#22553181C211yesmaximal, characteristic, complemented
8C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
9#384×C131C211yesmaximal, characteristic, complemented
10C22×Dic262C2212yes—
11C22×D524C2211yescharacteristic
12C22×C4×D131C2212yes—
13#16862245—24no—
14#16862224—216no—
15#16862208—24no—
16C2×C4⋊Dic131C2211yescharacteristic
17C2×Dic131⋊C4C2214yes—
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2214yes—
21C2×(C2×D131)⋊C4C2214yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×C4×Dic131C2211yescharacteristic
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C4⋊C4×C131C2211yescharacteristic
28C2×C4⋊C4×C131C2214yes—
29C2×C42×C131C2211yescharacteristic
30C23×D131C2312yes—
31C2×Dic262—28no—
32C2×Dic262—24no—
33C2×D524—22no—
34C2×D524—22no—
35C2×D524Q814yescomplemented
36C22×Dic131C2312yes—
37C22×Dic131C2311yescharacteristic
38C22×Dic131C2312yes—
39C22×Dic131C2311yescharacteristic
40C2×C4×D131—24no—
41C2×C4×D131—28no—
42C4⋊Dic131—22no—
43Dic131⋊C4—28no—
44Dic131⋊C4—24no—
45Dic131⋊C4D418yescomplemented
46(C2×D131)⋊C4—24no—
47(C2×D131)⋊C4—28no—
48(C2×D131)⋊C4—28no—
49C4×Dic131—22no—
50C4×Dic131D414yes—
51C22×C4×C131C2312yes—
52C22×C4×C131C2312yes—
53C22×C4×C131C2312yes—
54C22×C4×C131C2311yescharacteristic
55C4⋊C4×C131—22no—
56C4⋊C4×C131—28no—
57C4⋊C4×C131—22no—
58C42×C131—22no—
59C22×D131—24no—
60C22×D131—22no—
61C22×D131—22no—
62C22×D131—24no—
63C22×D131C2×Q814yes—
64Dic262—48no—
65D524—28no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—24no—
70C2×Dic131—22no—
71C2×Dic131C2×D412yes—
72C2×Dic131C2×D414yes—
73C2×Dic131C2×D412yes—
74C2×Dic131C2×D414yes—
75C4×D131—48no—
76C23×C131C2411yescharacteristic
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—22no—
80C2×C4×C131—24no—
81C2×C4×C131—24no—
82C2×C4×C131—22no—
83C2×C4×C131C2×D414yes—
84C2×C4×C131C2×D412yes—
85C2×C4×C131C2×Q812yes—
86C2×C4×C131#4112yes—
87C22×C131C22×D411yescharacteristic
88C22×C131C22×D412yes—
89C22×C131C22×Q811yescharacteristic
90C22×C131C2×#4111yescharacteristic
91C22×C1312−1+412yes—
92C2×D131—24no—
93C2×D131—28no—
94C2×D131—42no—
95C2×D131—42no—
96C2×D131—44no—
97C4×C131—24no—
98C4×C131—42no—
99C4×C131—42no—
100C4×C131C22⋊Q814yescomplemented
101Dic131—22no—
102Dic131—24no—
103Dic131—42no—
104Dic131#12714yes—
105C2×C131C2×C22⋊Q811yescharacteristic
106C2×C131C2×#12711yescharacteristic
107C2×C131#53611yescharacteristic
108C2×C131#54712yes—
109C2×C131D4×Q812yes—
110D131—44no—
111C131#254211yescharacteristic, complemented, Sylow
112#2542—1311nomaximal, Sylow
113C2×C4⋊Q8—1311no—
114C2×C22⋊Q8—1314no—
115C2×C4×D4—1311no—
116#399—1314no—
117#390—1311no—
118#390—1311no—
119#384—1311no—
120#384D13111yescharacteristic, complemented
121C4×C4⋊C4—1311no—
122C22×Q8—1312no—
123C22×D4—1311no—
124C23×C4—1312no—
125C4⋊Q8—2624no—
126C22⋊Q8—26216no—
127C4×D4—2624no—
128C2×C4⋊C4—1314no—
129C2×C4⋊C4—1311no—
130C2×C4⋊C4—1312no—
131C2×C4⋊C4—1312no—
132C2×C4⋊C4C2×D13111yescharacteristic
133C2×C4⋊C4C2×D13114yes—
134C2×C4⋊C4C2×D13111yescharacteristic
135C2×C22⋊C4—1312no—
136C2×C22⋊C4—1314no—
137C2×C22⋊C4—1314no—
138C2×C42—1311no—
139C2×C42—1311no—
140C2×C42C2×D13111yescharacteristic
141#95—1312no—
142C24—1312no—
143C2×Q8—2628no—
144C2×Q8—2624no—
145C2×D4—2622no—
146C2×D4—2622no—
147C2×D4—1314no—
148C22×C4—2624no—
149C22×C4—2628no—
150C22×C4—1312no—
151C22×C4—1311no—
152C22×C4—1312no—
153C22×C4—1311no—
154C22×C4C22×D13112yes—
155C22×C4C22×D13112yes—
156C22×C4C22×D13112yes—
157C22×C4C22×D13111yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—2628no—
162C4⋊C4—2624no—
163C4⋊C4—2622no—
164C4⋊C4—1318no—
165C22⋊C4—2624no—
166C22⋊C4—2628no—
167C22⋊C4—2628no—
168C42—22no—
169C42—2622no—
170C42—1314no—
171C23—2624no—
172C23—2622no—
173C23—2622no—
174C23—2624no—
175C23—1314no—
176C23C23×D13111yescharacteristic
177Q8—5248no—
178D4—2628no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—2622no—
186C2×C4—2622no—
187C2×C4—2622no—
188C2×C4—2624no—
189C2×C4—2622no—
190C2×C4—5248no—
191C2×C4—1312no—
192C2×C4—1314no—
193C2×C4—1312no—
194C2×C4—1314no—
195C2×C4D4×D13114yes—
196C2×C4D4×D13112yes—
197C2×C4Q8×D13112yes—
198C2×C4#1452351312yes—
199C22—2628no—
200C22—2624no—
201C22—5242no—
202C22—5244no—
203C22—5242no—
204C22C2×D4×D13111yescharacteristic
205C22C2×D4×D13112yes—
206C22C2×Q8×D13111yescharacteristic
207C22C2×#1452351311yescharacteristic
208C22#1686229412yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—2622no—
213C4—2624no—
214C4—5242no—
215C4—1314no—
216C4#1686222414yescomplemented
217C2—5244no—
218C2C2×#1686222411yescharacteristic
219C2#127×D13111yescharacteristic
220C2#2255388711yescharacteristic
221C2#2255396312yes—
222C2#2255397712yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#384×C131C2
prime radicalC22×D524C22
hypercenter#384D131
hypocenterC131#2542
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#16862245C2semidirect
2. C2×#16862224C2semidirect
4. #22553247C2semidirect
7. #22553181C2semidirect
8. C4×Dic131⋊C4C2semidirect
9. #384×C131C2semidirect
35. C2×D524Q8semidirect
45. Dic131⋊C4D4semidirect
100. C4×C131C22⋊Q8semidirect
111. C131#2542semidirect
120. #384D131semidirect
216. C4#16862224semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622242
C2×#168622082
#225532472
#225532142
#225532132
#225531812
C4×Dic131⋊C42
#384×C1312
#2542131