Finite Group Databasebeta

← #30637058

Subgroups — #30637058

Order 16256.

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

normal subgroups only

This group has 312 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
1C2×C4⋊C4×D127C211yesmaximal, characteristic
2#22549838C212yesmaximal
3#22549602C211yesmaximal, characteristic, complemented
4#22549601C212yesmaximal, complemented
5#324×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C411yescharacteristic
7C22×C4×D127C411yescharacteristic
8C22×C4×D127C2211yescharacteristic
9C2×#14523231—22no—
10C2×C4⋊Dic127C411yescharacteristic
11C2×Dic127⋊C4C411yescharacteristic
12C2×Dic127⋊C4C2211yescharacteristic
13C4⋊C4×D127—22no—
14C4⋊C4×D127—22no—
15C22⋊(C127⋊C8)C2212yes—
16(C2×D127)⋊C8—22no—
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C23×D127C2×C411yescharacteristic
20C22×Dic127C2×C411yescharacteristic
21C22×Dic127C2×C411yescharacteristic
22C22×Dic127C2×C411yescharacteristic
23C2×C4×D127—21no—
24C2×C4×D127—21no—
25C2×C4×D127—22no—
26C2×C4×D127—21no—
27C2×C4×D127—21no—
28C2×C4×D127—22no—
29C2×C4×D127—42no—
30C2×C4×D127—42no—
31C2×C4×D127D412yes—
32C2×C4×D127D412yes—
33#14523231—44no—
34C2×C127⋊C8—22no—
35C4⋊Dic127—22no—
36Dic127⋊C4—22no—
37Dic127⋊C4—22no—
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127C2×C411yescharacteristic
40C22×C4×C127C2311yescharacteristic
41C2×C8×C127—22no—
42C4⋊C4×C127—22no—
43C22×D127—21no—
44C22×D127—21no—
45C22×D127—22no—
46C22×D127—22no—
47C22×D127C22⋊C411yescharacteristic
48C22×D127C22⋊C411yescharacteristic
49C2×Dic127—21no—
50C2×Dic127—21no—
51C2×Dic127—21no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127C22⋊C412yes—
55C2×Dic127M1612yescomplemented
56C4×D127—24no—
57C4×D127—24no—
58C4×D127—24no—
59C4×D127—42no—
60C4×D127—42no—
61C4×D127—42no—
62C127⋊C8—42no—
63C23×C127C22×C411yescharacteristic
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127M1612yescomplemented
70C2×C4×C127C2×D412yes—
71C8×C127—42no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127C2×C22⋊C411yescharacteristic
75C22×C127C2×M1611yescharacteristic
76C22×C127C2×M1611yescharacteristic
77C2×D127—24no—
78C2×D127—24no—
79C2×D127—21no—
80C2×D127—21no—
81C2×D127—21no—
82C2×D127—21no—
83C2×D127#9912yes—
84C2×D127#10112yes—
85C4×C127—22no—
86C4×C127—22no—
87C4×C127—41no—
88Dic127—22no—
89Dic127—22no—
90Dic127—41no—
91C2×C127—21no—
92C2×C127—21no—
93C2×C127#40711yescharacteristic
94C2×C127C2×#9911yescharacteristic
95C2×C127C2×#10111yescharacteristic
96D127—22no—
97D127—22no—
98#1463—1271nomaximal, Sylow
99C127#146311yescharacteristic, complemented, Sylow
100C22×C4⋊C4—1271no—
101#407—1272no—
102#324—1271no—
103#324—1272no—
104#324D12711yescharacteristic, complemented
105C23×C4—1271no—
106C23×C4—1271no—
107C23×C4—1271no—
108C2×M16—2542no—
109C2×C4⋊C4—1271no—
110C2×C4⋊C4—1271no—
111C2×C4⋊C4—1271no—
112C2×C4⋊C4—2542no—
113C2×C4⋊C4—2542no—
114C2×C4⋊C4C2×D12711yescharacteristic
115C22⋊C8—1272no—
116C22⋊C8—2542no—
117C22⋊C8C2×D12712yes—
118C24—1271no—
119C22×C4—1271no—
120C22×C4—1271no—
121C22×C4—1272no—
122C22×C4—1272no—
123C22×C4—1271no—
124C22×C4—5082no—
125C22×C4—5082no—
126C22×C4—2542no—
127C22×C4—2541no—
128C22×C4—2541no—
129C22×C4—2541no—
130C22×C4—2541no—
131C22×C4—2542no—
132C22×C4C4×D12711yescharacteristic
133C22×C4C4×D12711yescharacteristic
134C22×C4C22×D12711yescharacteristic
135M16—5084no—
136C2×C8—22no—
137C2×C8—2542no—
138C4⋊C4—22no—
139C4⋊C4—2542no—
140C4⋊C4—2542no—
141C4⋊C4—2542no—
142C23—1271no—
143C23—1271no—
144C23—2542no—
145C23—2542no—
146C23—2541no—
147C23—2541no—
148C23C2×C4×D12711yescharacteristic
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—1272no—
155C2×C4—1272no—
156C2×C4—5082no—
157C2×C4—5082no—
158C2×C4—5082no—
159C2×C4—5081no—
160C2×C4—5081no—
161C2×C4—2544no—
162C2×C4—2544no—
163C2×C4—2544no—
164C2×C4—2541no—
165C2×C4—2541no—
166C2×C4—2541no—
167C2×C4#1452323112yescomplemented
168C2×C4D4×D12712yes—
169C8—42no—
170C8—5082no—
171C22—21no—
172C22—21no—
173C22—1272no—
174C22—1272no—
175C22—2544no—
176C22—2544no—
177C22—2541no—
178C22—2541no—
179C22—2541no—
180C22—2541no—
181C22C22⋊C4×D12711yescharacteristic
182C22M16×D12711yescharacteristic
183C22C2×#1452323111yescharacteristic
184C4—22no—
185C4—22no—
186C4—41no—
187C4—5081no—
188C4—2542no—
189C4—2542no—
190C2—21no—
191C2—21no—
192C2—2542no—
193C2—2542no—
194C2#2254983811yescharacteristic
195C2#99×D12711yescharacteristic
196C2#101×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#324D127
hypocenterC127#1463
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. #22549602C2semidirect
4. #22549601C2semidirect
5. #324×C127C2semidirect
55. C2×Dic127M16semidirect
69. C2×C4×C127M16semidirect
99. C127#1463semidirect
104. #324D127semidirect
167. C2×C4#14523231semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
#225498382
#225496022
#225496012
#324×C1272
#1463127