Finite Group Databasebeta

← #30668714

Subgroups — #30668714

Order 16768.

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

normal subgroups only

This group has 484 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×#16862225C211yesmaximal, characteristic
2#22553216C212yesmaximal, complemented
3#22553206C212yesmaximal
4#22553200C212yesmaximal
5#22553195C212yesmaximal, complemented
6#22553194C212yesmaximal, complemented
7#22553163C212yesmaximal, complemented
8C4×C22⋊Dic131C211yesmaximal, characteristic
9#381×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11#16862225C418yescomplemented
12C2×C4⋊Dic131C2211yescharacteristic
13C2×Dic131⋊C4C2212yes—
14C2×Dic131⋊C4C2212yes—
15C2×Dic131⋊C4C2212yes—
16C2×C22⋊Dic131C2212yes—
17C2×C22⋊Dic131C2212yes—
18C2×C4×Dic131C2212yes—
19C2×C4×Dic131C2211yescharacteristic
20C2×C4×Dic131C2211yescharacteristic
21C2×C4×Dic131C2212yes—
22C131⋊#95C2212yes—
23C131⋊#95C2212yes—
24C131⋊#95C2212yes—
25C131⋊#95C2212yes—
26C131⋊#95C2212yes—
27C23×C4×C131C2211yescharacteristic
28C2×C22⋊C4×C131C2211yescharacteristic
29C2×C22⋊C4×C131C2211yescharacteristic
30#95×C131C2212yes—
31#95×C131C2212yes—
32C2×Dic262—41no—
33C2×Dic262—41no—
34C2×Dic262C2×C414yes—
35C22×Dic131C2312yes—
36C22×Dic131C2312yes—
37C22×Dic131C2312yes—
38C22×Dic131C2312yes—
39C4⋊Dic131C2×C414yes—
40Dic131⋊C4—24no—
41Dic131⋊C4—42no—
42Dic131⋊C4C2×C418yes—
43C22⋊Dic131—24no—
44C22⋊Dic131C2×C418yes—
45C4×Dic131—24no—
46C4×Dic131—41no—
47C4×Dic131—41no—
48C4×Dic131—42no—
49C24×C131C2311yescharacteristic
50C22×C4×C131—41no—
51C22×C4×C131—41no—
52C22×C4×C131C2×C414yes—
53C22×C4×C131C2312yes—
54C22×C4×C131C2312yes—
55C22×C4×C131C2311yescharacteristic
56C22×C4×C131C2311yescharacteristic
57C22⋊C4×C131—41no—
58C22⋊C4×C131—41no—
59Dic262—44no—
60C2×Dic131—22no—
61C2×Dic131—22no—
62C2×Dic131—22no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131—22no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131C22×C414yes—
70C2×Dic131C22×C414yes—
71C2×Dic131#4114yes—
72C23×C131—22no—
73C23×C131—21no—
74C23×C131—21no—
75C23×C131—22no—
76C23×C131C22×C412yes—
77C23×C131C2411yescharacteristic
78C2×C4×C131—21no—
79C2×C4×C131—21no—
80C2×C4×C131—21no—
81C2×C4×C131—21no—
82C2×C4×C131—22no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—22no—
87C2×C4×C131—22no—
88C2×C4×C131—44no—
89C2×C4×C131C22×C412yes—
90C2×C4×C131C22×C412yes—
91C22×C131—22no—
92C22×C131—22no—
93C22×C131—22no—
94C22×C131—42no—
95C22×C131—42no—
96C22×C131C23×C411yescharacteristic
97C22×C131C2×#4112yes—
98C22×C131H(5,𝔽2)11yescharacteristic
99C22×C131H(5,𝔽2)11yescharacteristic
100C22×C1312−1+412yes—
101C4×C131—22no—
102C4×C131—22no—
103C4×C131—42no—
104C4×C131—42no—
105Dic131—24no—
106Dic131—24no—
107Dic131—24no—
108Dic131—42no—
109C2×C131—42no—
110C2×C131C4×#4111yescharacteristic
111C2×C131#51811yescharacteristic
112C2×C131#51911yescharacteristic
113C2×C131#53912yes—
114C2×C131#54212yes—
115C131#233211yescharacteristic, complemented, Sylow
116#2332—1311nomaximal, Sylow
117C2×C22⋊Q8—1311no—
118#391—1312no—
119#388—1312no—
120#388—1312no—
121#387—1312no—
122#387—1312no—
123#381D13111yescharacteristic
124C4×C4⋊C4—1312no—
125C4×C22⋊C4—1311no—
126C22×Q8—1311no—
127C23×C4C2×D13111yescharacteristic
128C22⋊Q8—1318no—
129C2×C4⋊C4—1312no—
130C2×C4⋊C4—1312no—
131C2×C4⋊C4—1312no—
132C2×C4⋊C4—1311no—
133C2×C22⋊C4—1312no—
134C2×C22⋊C4—1312no—
135C2×C22⋊C4C2×D13111yescharacteristic
136C2×C22⋊C4C2×D13111yescharacteristic
137C2×C42—1312no—
138C2×C42—1311no—
139C2×C42—1311no—
140C2×C42—1312no—
141#95—1312no—
142#95—1312no—
143#95—1312no—
144#95—1312no—
145#95—1312no—
146#95C2×D13112yes—
147#95C2×D13112yes—
148C24C22×D13111yescharacteristic
149C2×Q8—5241no—
150C2×Q8—5241no—
151C2×Q8—1314no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—1312no—
155C22×C4—1312no—
156C22×C4—1312no—
157C22×C4—1312no—
158C22×C4C4×D13114yes—
159C22×C4C22×D13112yes—
160C22×C4C22×D13112yes—
161C22×C4C22×D13111yescharacteristic
162C22×C4C22×D13111yescharacteristic
163C4⋊C4—2624no—
164C4⋊C4—5242no—
165C4⋊C4—1318no—
166C4⋊C4—1314no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—2624no—
170C22⋊C4—1318no—
171C42—2624no—
172C42—5241no—
173C42—5241no—
174C42—5242no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C2×C4×D13112yes—
180C23C23×D13111yescharacteristic
181Q8—5244no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—2622no—
194C2×C4—2622no—
195C2×C4—2622no—
196C2×C4—2622no—
197C2×C4—2622no—
198C2×C4—2622no—
199C2×C4—2622no—
200C2×C4—2622no—
201C2×C4—2622no—
202C2×C4—1314no—
203C2×C4—1314no—
204C2×C4—1314no—
205C2×C4C2×C4×D13112yes—
206C2×C4C2×C4×D13112yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C22×C4×D13111yescharacteristic
213C22#41×D13112yes—
214C22#1686229111yescharacteristic
215C22#1686229111yescharacteristic
216C22#1686229312yes—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—42no—
221C4—2624no—
222C4—2624no—
223C4—2624no—
224C4—5242no—
225C2—42no—
226C2#2255378011yescharacteristic
227C2#2255378511yescharacteristic
228C2#2255378911yescharacteristic
229C2#2255390512yes—
230C2#2255393012yes—
subgroupG/H
center, FrattiniC23C23×D131
commutatorC22×C131C23×C4
Fitting#381×C131C2
socle, cosocleC23×C131C24
prime radicalC24×C131C23
hypercenter#381D131
hypocenterC131#2332
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #22553216C2semidirect
5. #22553195C2semidirect
6. #22553194C2semidirect
7. #22553163C2semidirect
11. #16862225C4semidirect
115. C131#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#168622252
#225532162
#225532062
#225532002
#225531952
#225531942
#225531632
C4×C22⋊Dic1312
#381×C1312
#2332131