Finite Group Databasebeta

← #30664294

Subgroups — #30664294

Order 16768.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16862204C211yesmaximal, characteristic
2#22553366C212yesmaximal, complemented
3#22553038C211yesmaximal, characteristic, complemented
4#22553035C212yesmaximal
5#324×C131C211yesmaximal, characteristic, complemented
6C22×C4×D131C411yescharacteristic
7#16862204—22no—
8#16862204—22no—
9C2×C4⋊Dic131C411yescharacteristic
10C2×(C2×D131)⋊C4C411yescharacteristic
11C2×(C2×D131)⋊C4C2211yescharacteristic
12C2×C4×Dic131C411yescharacteristic
13C2×C4×Dic131C2211yescharacteristic
14Dic131⋊C8—22no—
15C22⋊(C131⋊C8)C2212yes—
16#16862156—22no—
17C2×C4⋊C4×C131C2211yescharacteristic
18C22⋊C8×C131C2212yes—
19C23×D131C2×C411yescharacteristic
20C22×Dic131C2×C411yescharacteristic
21C22×Dic131C2×C411yescharacteristic
22C22×Dic131C2×C411yescharacteristic
23C2×C4×D131—22no—
24C2×C4×D131—21no—
25C2×C4×D131—21no—
26C2×C4×D131—22no—
27C2×C131⋊C8—22no—
28C4⋊Dic131—22no—
29(C2×D131)⋊C4—22no—
30(C2×D131)⋊C4—22no—
31C4×Dic131—21no—
32C4×Dic131—21no—
33C4×Dic131D412yes—
34C4×Dic131D412yes—
35C22×C4×C131C2×C411yescharacteristic
36C22×C4×C131C2×C411yescharacteristic
37C22×C4×C131C2311yescharacteristic
38C2×C8×C131—22no—
39C4⋊C4×C131—22no—
40C22×D131—21no—
41C22×D131—21no—
42C22×D131—41no—
43C22×D131—41no—
44C22×D131M1612yes—
45C2×Dic131—21no—
46C2×Dic131—21no—
47C2×Dic131—21no—
48C2×Dic131—41no—
49C2×Dic131—41no—
50C2×Dic131C22⋊C411yescharacteristic
51C2×Dic131C22⋊C411yescharacteristic
52C2×Dic131C22⋊C412yes—
53C4×D131—24no—
54C4×D131—24no—
55C131⋊C8—42no—
56C23×C131C22×C411yescharacteristic
57C2×C4×C131—21no—
58C2×C4×C131—21no—
59C2×C4×C131—21no—
60C2×C4×C131—41no—
61C2×C4×C131—41no—
62C2×C4×C131M1612yes—
63C2×C4×C131C2×D412yes—
64C8×C131—42no—
65C22×C131—21no—
66C22×C131—21no—
67C22×C131C2×C22⋊C411yescharacteristic
68C22×C131C2×M1611yescharacteristic
69C22×C131C2×M1611yescharacteristic
70C2×D131—22no—
71C2×D131—42no—
72C2×D131—41no—
73C2×D131—41no—
74C2×D131—42no—
75C4×C131—22no—
76C4×C131—22no—
77C4×C131—41no—
78Dic131—21no—
79Dic131—21no—
80Dic131—22no—
81Dic131—41no—
82C2×C131—21no—
83C2×C131—21no—
84C2×C131#40711yescharacteristic
85C2×C131#41011yescharacteristic
86C2×C131#41311yescharacteristic
87D131—42no—
88C131#146411yescharacteristic, complemented, Sylow
89#1464—1311nomaximal, Sylow
90C2×#117—1311no—
91#432—1312no—
92#324—1311no—
93#324D13111yescharacteristic, complemented
94#323—1312no—
95C23×C4—1311no—
96#117—2622no—
97#117—2622no—
98C2×C4⋊C4—1311no—
99C2×C4⋊C4C2×D13111yescharacteristic
100C2×C22⋊C4—1311no—
101C2×C22⋊C4—1311no—
102C2×C42—1311no—
103C2×C42—1311no—
104C4⋊C8—2622no—
105C22⋊C8—1312no—
106C22⋊C8C2×D13112yes—
107#97—2622no—
108C24—1311no—
109C22×C4—2622no—
110C22×C4—2621no—
111C22×C4—2621no—
112C22×C4—2622no—
113C22×C4—1311no—
114C22×C4—1311no—
115C22×C4—1311no—
116C22×C4C4×D13111yescharacteristic
117C22×C4C4×D13111yescharacteristic
118C22×C4C22×D13111yescharacteristic
119C2×C8—22no—
120C2×C8—2622no—
121C4⋊C4—22no—
122C4⋊C4—2622no—
123C22⋊C4—2622no—
124C22⋊C4—2622no—
125C42—2621no—
126C42—2621no—
127C42—1312no—
128C42—1312no—
129C23—2621no—
130C23—2621no—
131C23—5241no—
132C23—5241no—
133C23—1312no—
134C23C2×C4×D13111yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—2624no—
141C2×C4—2624no—
142C2×C4—2621no—
143C2×C4—2621no—
144C2×C4—2621no—
145C2×C4—5241no—
146C2×C4—5241no—
147C2×C4—1311no—
148C2×C4—1311no—
149C2×C4—1312no—
150C2×C4#1452349612yes—
151C2×C4D4×D13112yes—
152C8—42no—
153C8—5242no—
154C22—21no—
155C22—21no—
156C22—2622no—
157C22—5242no—
158C22—5242no—
159C22—5241no—
160C22—5241no—
161C22C22⋊C4×D13111yescharacteristic
162C22M16×D13111yescharacteristic
163C22C2×#1452349611yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—2621no—
168C4—2621no—
169C4—2622no—
170C4—5241no—
171C2—21no—
172C2—21no—
173C2—5242no—
174C2#2255327411yescharacteristic
175C2#2255328411yescharacteristic
176C2#2255329511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC23×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#324×C131C2
socleC22×C131C2×C22⋊C4
prime radicalC23×D131C2×C4
cosocleC22×C4×C131C23
hypercenter#324D131
hypocenterC131#1464
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22553366C2semidirect
3. #22553038C2semidirect
5. #324×C131C2semidirect
88. C131#1464semidirect
93. #324D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622042
#225533662
#225530382
#225530352
#324×C1312
#1464131