Finite Group Databasebeta

← #30641446

Subgroups — #30641446

Order 16256.

516 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 205 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×#16861207C211yesmaximal, characteristic
2C2×C4×Dic254C211yesmaximal, characteristic
3#22549783C211yesmaximal, characteristic
4#22549781C211yesmaximal, characteristic
5#22549760C214yesmaximal
6#22549757C214yesmaximal
7#22549741C211yesmaximal, characteristic
8#22549722C211yesmaximal, characteristic
9C2×C4×Q8×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861207C418yescomplemented
12C4×Dic254—24no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4C2212yes—
15C2×Dic127⋊C4C2214yes—
16C2×C4×Dic127C2211yescharacteristic
17C2×C4×Dic127C2212yes—
18C2×C4×Dic127C2211yescharacteristic
19C127⋊#95C2212yes—
20C127⋊#95C2214yes—
21C127⋊#95C2214yes—
22C127⋊#95C2214yes—
23C127⋊#95C2212yes—
24C22×Q8×C127C2211yescharacteristic
25C4×Q8×C127—24no—
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2212yes—
28C2×C42×C127C2212yes—
29C2×C42×C127C2211yescharacteristic
30C2×Dic254—22no—
31C2×Dic254—22no—
32C2×Dic254C2×C414yes—
33C22×Dic127C2312yes—
34C22×Dic127C2312yes—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C4⋊Dic127—22no—
38Dic127⋊C4—24no—
39Dic127⋊C4C2×C4116yes—
40C4×Dic127—24no—
41C4×Dic127—22no—
42C4×Dic127C2×C414yes—
43C2×Q8×C127—22no—
44C2×Q8×C127—22no—
45C2×Q8×C127C2×C414yes—
46C22×C4×C127C2312yes—
47C22×C4×C127C2312yes—
48C22×C4×C127C2311yescharacteristic
49C22×C4×C127C2311yescharacteristic
50C22×C4×C127C2311yescharacteristic
51C4⋊C4×C127—24no—
52C4⋊C4×C127—22no—
53C42×C127—24no—
54C42×C127D414yes—
55Dic254—28no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—24no—
64C2×Dic127—22no—
65C2×Dic127C22×C414yes—
66C2×Dic127C22×C414yes—
67C23×C127C2411yescharacteristic
68Q8×C127—28no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—24no—
75C2×C4×C127C22×C414yes—
76C2×C4×C127C22×C412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127#4112yes—
81C22×C127C23×C411yescharacteristic
82C22×C127C22×D411yescharacteristic
83C22×C127C2×#4111yescharacteristic
84C22×C1272−1+412yes—
85C22×C1272−1+412yes—
86C4×C127—24no—
87C4×C127—22no—
88C4×C127—22no—
89C4×C127—42no—
90C4×C127C4×D414yes—
91Dic127—24no—
92Dic127—24no—
93Dic127—42no—
94Dic127—42no—
95C2×C127C2×C4×D411yescharacteristic
96C2×C127#51912yes—
97C2×C127#53611yescharacteristic
98C2×C127#53611yescharacteristic
99C2×C127#54112yes—
100#2327—1271nomaximal, Sylow
101C127#232711yescharacteristic, complemented, Sylow
102C2×C4⋊Q8—1271no—
103C2×C4×Q8—1271no—
104C2×C4×Q8D12711yescharacteristic
105#391—1271no—
106#391—1271no—
107#387—1274no—
108#387—1274no—
109#382—1271no—
110#376—1271no—
111C22×Q8—1271no—
112C22×Q8C2×D12711yescharacteristic
113C4⋊Q8—1278no—
114C4×Q8—24no—
115C4×Q8—2544no—
116C2×C4⋊C4—1274no—
117C2×C4⋊C4—1271no—
118C2×C4⋊C4—1272no—
119C2×C4⋊C4C2×D12711yescharacteristic
120C2×C4⋊C4C2×D12712yes—
121C2×C42—1271no—
122C2×C42—1272no—
123C2×C42—1271no—
124C2×C42C2×D12712yes—
125C2×C42C2×D12711yescharacteristic
126#95—1272no—
127#95—1272no—
128#95—1274no—
129#95—1274no—
130#95—1274no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—1274no—
134C2×Q8—2542no—
135C2×Q8—2542no—
136C2×Q8C4×D12714yes—
137C22×C4—1272no—
138C22×C4—1272no—
139C22×C4—1272no—
140C22×C4—1272no—
141C22×C4C22×D12712yes—
142C22×C4C22×D12712yes—
143C22×C4C22×D12711yescharacteristic
144C22×C4C22×D12711yescharacteristic
145C22×C4C22×D12711yescharacteristic
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—12716no—
149C4⋊C4—2544no—
150C4⋊C4—2542no—
151C42—24no—
152C42—1274no—
153C42—2544no—
154C42—2542no—
155C42C22⋊D12714yes—
156C23C23×D12711yescharacteristic
157Q8—28no—
158Q8—2548no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—1274no—
166C2×C4—1274no—
167C2×C4—2542no—
168C2×C4—2542no—
169C2×C4—2542no—
170C2×C4—2544no—
171C2×C4—2542no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4—2542no—
175C2×C4—2542no—
176C2×C4C2×C4×D12714yes—
177C2×C4C2×C4×D12712yes—
178C2×C4C2×C22⋊D12712yes—
179C2×C4C2×C22⋊D12712yes—
180C2×C4C2×C22⋊D12712yes—
181C2×C4#1452324812yes—
182C22C22×C4×D12711yescharacteristic
183C22C22×C22⋊D12711yescharacteristic
184C22C2×#1452324811yescharacteristic
185C22#1686125612yes—
186C22#1686125712yes—
187C4—24no—
188C4—22no—
189C4—22no—
190C4—42no—
191C4—5082no—
192C4—5082no—
193C4—2544no—
194C4—2544no—
195C4C4×C22⋊D12714yes—
196C2C2×C4×C22⋊D12711yescharacteristic
197C2#2255035212yes—
198C2#2255044711yescharacteristic
199C2#2255045011yescharacteristic
200C2#2255048412yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×C4×Q8×C127C2
socle, prime radical, cosocleC23×C127C24
hypercenterC2×C4×Q8D127
hypocenterC127#2327
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
11. #16861207C4semidirect
101. C127#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×C4×Dic2542
#225497832
#225497812
#225497602
#225497572
#225497412
#225497222
C2×C4×Q8×C1272
#2327127