Finite Group Databasebeta

← #30664458

Subgroups — #30664458

Order 16768.

240 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#22553864C211yesmaximal, characteristic
2#22553362C211yesmaximal, characteristic
3#22553163C211yesmaximal, characteristic
4#22553050C211yesmaximal, characteristic
5#22553049C211yesmaximal, characteristic
6#22553048C211yesmaximal, characteristic
7#328×C131C211yesmaximal, characteristic
8#16862245C411yescharacteristic, complemented
9#16862244C411yescharacteristic, complemented
10#16862236—21no—
11#16862225C2211yescharacteristic
12#16862223—21no—
13Q8×Dic131C411yescharacteristic, complemented
14#16862211—21no—
15#16862211C411yescharacteristic, complemented
16#16862202—21no—
17C2×Dic131⋊C4—21no—
18C2×C4×Dic131—21no—
19C2×C4×Dic131C2211yescharacteristic
20Dic131⋊C8—21no—
21C22⋊(C131⋊C8)C2211yescharacteristic
22C8×Dic131—21no—
23C131⋊#95C2211yescharacteristic
24C22⋊Q8×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C2×Dic262—21no—
28C2×Dic262C2×C411yescharacteristic
29C22×Dic131—21no—
30C22×Dic131D411yescharacteristic
31C22×Dic131D411yescharacteristic
32C2×C131⋊C8—21no—
33C4⋊Dic131—21no—
34C4⋊Dic131C2×C411yescharacteristic
35Dic131⋊C4—24no—
36Dic131⋊C4—21no—
37Dic131⋊C4—21no—
38Dic131⋊C4—21no—
39Dic131⋊C4—21no—
40Dic131⋊C4—21no—
41C22⋊Dic131—21no—
42C4×Dic131—21no—
43C4×Dic131—21no—
44C4×Dic131—42no—
45C4×Dic131C2×C411yescharacteristic
46C4×Dic131C2×C411yescharacteristic
47C4×Dic131D411yescharacteristic
48C4×Dic131D411yescharacteristic
49C2×Q8×C131C2×C411yescharacteristic
50C22×C4×C131—21no—
51C22×C4×C131C2311yescharacteristic
52C2×C8×C131—21no—
53C4⋊C4×C131—21no—
54C4⋊C4×C131C2×C411yescharacteristic
55C22⋊C4×C131—21no—
56Dic262—42no—
57Dic262—41no—
58C2×Dic131—22no—
59C2×Dic131—22no—
60C2×Dic131—21no—
61C2×Dic131—21no—
62C2×Dic131—41no—
63C2×Dic131—41no—
64C2×Dic131C22⋊C411yescharacteristic
65C2×Dic131C22⋊C411yescharacteristic
66C2×Dic131C22⋊C411yescharacteristic
67C2×Dic131C22⋊C411yescharacteristic
68C2×Dic131SD1612yes—
69C2×Dic131Q1612yes—
70C131⋊C8—41no—
71C23×C131C2×D411yescharacteristic
72Q8×C131—41no—
73C2×C4×C131—22no—
74C2×C4×C131—21no—
75C2×C4×C131—21no—
76C2×C4×C131—41no—
77C2×C4×C131—41no—
78C2×C4×C131—41no—
79C2×C4×C131C22×C411yescharacteristic
80C2×C4×C131C2×D411yescharacteristic
81C8×C131—41no—
82C22×C131—21no—
83C22×C131—21no—
84C22×C131C2×C22⋊C411yescharacteristic
85C22×C131C2×SD1611yescharacteristic
86C22×C131C2×Q1611yescharacteristic
87C4×C131—21no—
88C4×C131—21no—
89C4×C131—42no—
90C4×C131—41no—
91C4×C131—41no—
92Dic131—21no—
93Dic131—42no—
94Dic131—41no—
95Dic131—41no—
96Dic131#10312yes—
97Dic131#10312yes—
98Dic131#10412yes—
99C2×C131—21no—
100C2×C131—21no—
101C2×C131#41011yescharacteristic
102C2×C131C2×#10311yescharacteristic
103C2×C131C2×#10411yescharacteristic
104C131#150311yescharacteristic, complemented, Sylow
105#1503—1311nomaximal, Sylow
106#533—1311no—
107#431—1311no—
108C4×C4⋊C4—1311no—
109#328—1311no—
110#328—1311no—
111#328—1311no—
112#328D13111yescharacteristic
113C4⋊Q8—1311no—
114C4⋊Q8—1311no—
115#125—2621no—
116C22⋊Q8—2621no—
117C22⋊Q8—1311no—
118C22⋊Q8C2×D13111yescharacteristic
119C4×Q8—2621no—
120C4×Q8—1311no—
121C4×Q8—1311no—
122#117—2621no—
123C2×C4⋊C4—2621no—
124C2×C42—2621no—
125C2×C42—1311no—
126C4⋊C8—2621no—
127C22⋊C8—1311no—
128C22⋊C8C2×D13111yescharacteristic
129C4×C8—2621no—
130#95—1311no—
131#95C2×D13111yescharacteristic
132C2×Q8—2621no—
133C2×Q8—1311no—
134C2×Q8C4×D13111yescharacteristic
135C22×C4—21no—
136C22×C4—2621no—
137C22×C4—1311no—
138C22×C4—1311no—
139C22×C4C22×D13111yescharacteristic
140C2×C8—21no—
141C2×C8—2621no—
142C4⋊C4—21no—
143C4⋊C4—2624no—
144C4⋊C4—2621no—
145C4⋊C4—2621no—
146C4⋊C4—2621no—
147C4⋊C4—2621no—
148C4⋊C4—2621no—
149C4⋊C4—2621no—
150C4⋊C4—1311no—
151C4⋊C4C4×D13111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2621no—
154C42—2621no—
155C42—2621no—
156C42—5242no—
157C42—1311no—
158C42—1311no—
159C42—1311no—
160C42—1311no—
161C23D4×D13111yescharacteristic
162Q8—41no—
163Q8—5241no—
164Q8—5242no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—2622no—
172C2×C4—2622no—
173C2×C4—2621no—
174C2×C4—2621no—
175C2×C4—5241no—
176C2×C4—5241no—
177C2×C4—1311no—
178C2×C4—1312no—
179C2×C4—1312no—
180C2×C4—1311no—
181C2×C4—1311no—
182C2×C4—1311no—
183C2×C4C2×C4×D13111yescharacteristic
184C2×C4D4×D13111yescharacteristic
185C8—41no—
186C8—5241no—
187C22—21no—
188C22—21no—
189C22C22⋊C4×D13111yescharacteristic
190C22SD16×D13111yescharacteristic
191C22Q16×D13111yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—2621no—
198C4—5242no—
199C4—5241no—
200C4—5241no—
201C4—1312no—
202C4—1312no—
203C4—1312no—
204C2—21no—
205C2—21no—
206C2#2255328411yescharacteristic
207C2#103×D13111yescharacteristic
208C2#104×D13111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#328×C131C2
socleC22×C131C2×C22⋊C4
prime radicalC23×C131C2×D4
cosocleC22×C4×C131C23
hypercenter#328D131
hypocenterC131#1503
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
8. #16862245C4semidirect
9. #16862244C4semidirect
13. Q8×Dic131C4semidirect
15. #16862211C4semidirect
104. C131#1503semidirect

Maximal subgroups

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

subgroupindex
#225538642
#225533622
#225531632
#225530502
#225530492
#225530482
#328×C1312
#1503131