Finite Group Databasebeta

← #396×D127 #30641949

Subgroups — #396×D127 #30641949

Order 16256.

908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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×D127C213yesmaximal, complemented
2C2×C22⋊C4×D127C213yesmaximal
3#22549800C211yesmaximal, characteristic, complemented
4#22549799C213yesmaximal, complemented
5#22549798C213yesmaximal, complemented
6#95×D127C211yesmaximal, characteristic, complemented
7#396×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C22×C4×D127C2213yes—
10C22×C4×D127C2213yes—
11C2×C4⋊Dic127C2213yescomplemented
12C2×Dic127⋊C4C2213yescomplemented
13C2×Dic127⋊C4C2213yescomplemented
14C4⋊C4×D127—212no—
15C2×C22⋊Dic127C2213yes—
16C2×(C2×D127)⋊C4C2213yes—
17C2×(C2×D127)⋊C4C2213yes—
18C22⋊C4×D127—212no—
19C127⋊#95C2213yescomplemented
20C2×C4⋊C4×C127C2213yescomplemented
21C2×C22⋊C4×C127C2213yes—
22#95×C127C2211yescharacteristic, complemented
23C23×D127—26no—
24C23×D127—26no—
25C23×D127Q814yes—
26C23×D127C2311yescharacteristic
27C23×D127C2311yescharacteristic
28C22×Dic127C2313yes—
29C22×Dic127C2313yes—
30C2×C4×D127—26no—
31C2×C4×D127—26no—
32C2×C4×D127—26no—
33C2×C4×D127—26no—
34C2×C4×D127—26no—
35C2×C4×D127D4112yes—
36C4⋊Dic127—26no—
37Dic127⋊C4—26no—
38Dic127⋊C4—26no—
39C22⋊Dic127—26no—
40(C2×D127)⋊C4—26no—
41(C2×D127)⋊C4—26no—
42C24×C127C2311yescharacteristic
43C22×C4×C127C2313yes—
44C22×C4×C127C2313yes—
45C4⋊C4×C127—26no—
46C22⋊C4×C127—26no—
47C22×D127—23no—
48C22×D127—23no—
49C22×D127—224no—
50C22×D127—412no—
51C22×D127—44no—
52C22×D127C2×D416yes—
53C22×D127C2×Q812yes—
54C22×D127C2×Q812yes—
55C22×D127#4116yes—
56C2×Dic127—23no—
57C2×Dic127—23no—
58C2×Dic127—23no—
59C2×Dic127—23no—
60C2×Dic127—23no—
61C2×Dic127C2×D416yes—
62C4×D127—224no—
63C4×D127—412no—
64C23×C127—23no—
65C23×C127—23no—
66C23×C127C2×Q812yes—
67C23×C127C2411yescharacteristic
68C2×C4×C127—23no—
69C2×C4×C127—23no—
70C2×C4×C127—23no—
71C2×C4×C127—23no—
72C2×C4×C127—23no—
73C2×C4×C127C2×D416yes—
74C22×C127—26no—
75C22×C127—43no—
76C22×C127—41no—
77C22×C127C22×D413yes—
78C22×C127C22×Q811yescharacteristic
79C22×C127C2×#4113yes—
80C2×D127—26no—
81C2×D127—43no—
82C2×D127—41no—
83C2×D127—416no—
84C2×D127#121112yes—
85C2×D127C22⋊Q8112yes—
86C2×D127#12614yes—
87C4×C127—26no—
88C4×C127—43no—
89Dic127—26no—
90Dic127—43no—
91C2×C127—42no—
92C2×C127C2×#12113yes—
93C2×C127C2×C22⋊Q813yes—
94C2×C127C2×#12611yescharacteristic
95D127—42no—
96D127#39618yesdirect product factor
97C2×#396—1271nomaximal, Sylow
98C127C2×#39611yescharacteristic, complemented, Sylow
99C22×C4⋊C4—1273no—
100C22×C22⋊C4—1273no—
101#396—1273no—
102#396—1273no—
103#396—1271no—
104#396D12711yescharacteristic, direct product factor
105C2×#95—1271no—
106C25—1271no—
107C23×C4—1273no—
108C23×C4—1273no—
109C2×C4⋊C4—1273no—
110C2×C4⋊C4—1273no—
111C2×C4⋊C4—1273no—
112C2×C4⋊C4—25412no—
113C2×C4⋊C4C2×D12713yescomplemented
114C2×C22⋊C4—1273no—
115C2×C22⋊C4—1273no—
116C2×C22⋊C4—1273no—
117C2×C22⋊C4—25412no—
118C2×C22⋊C4C2×D12713yes—
119#95—1273no—
120#95C2×D12711yescharacteristic, complemented
121C24—1271no—
122C24—1274no—
123C24—1271no—
124C24—2546no—
125C24—2546no—
126C24C22×D12711yescharacteristic
127C22×C4—1273no—
128C22×C4—12712no—
129C22×C4—1273no—
130C22×C4—2546no—
131C22×C4—2546no—
132C22×C4—2546no—
133C22×C4—2546no—
134C22×C4—2546no—
135C22×C4C22×D12713yes—
136C22×C4C22×D12713yes—
137C4⋊C4—26no—
138C4⋊C4—2546no—
139C4⋊C4—2546no—
140C4⋊C4—2546no—
141C22⋊C4—26no—
142C22⋊C4—2546no—
143C22⋊C4—2546no—
144C22⋊C4—2546no—
145C23—23no—
146C23—23no—
147C23—1276no—
148C23—1272no—
149C23—1276no—
150C23—1272no—
151C23—50812no—
152C23—5084no—
153C23—25424no—
154C23—2543no—
155C23—2543no—
156C23Q8×D12712yes—
157C23C23×D12711yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—1276no—
164C2×C4—50812no—
165C2×C4—25424no—
166C2×C4—2543no—
167C2×C4—2543no—
168C2×C4—2543no—
169C2×C4—2543no—
170C2×C4—2543no—
171C2×C4D4×D12716yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—12712no—
176C22—12712no—
177C22—1274no—
178C22—50816no—
179C22—5083no—
180C22—5081no—
181C22—2546no—
182C22C2×D4×D12713yes—
183C22C2×Q8×D12711yescharacteristic
184C22#41×D12713yes—
185C4—26no—
186C4—43no—
187C4—5083no—
188C4—2546no—
189C2—42no—
190C2—1278no—
191C2—5082no—
192C2#121×D12713yes—
193C2C22⋊Q8×D12713yes—
194C2#126×D12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#396×C127C2
prime radicalC24×D127C22
hypercenter#396D127
hypocenterC127C2×#396
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D127C2semidirect
3. #22549800C2semidirect
4. #22549799C2semidirect
5. #22549798C2semidirect
6. #95×D127C2semidirect
7. #396×C127C2semidirect
11. C2×C4⋊Dic127C22semidirect
12. C2×Dic127⋊C4C22semidirect
13. C2×Dic127⋊C4C22semidirect
19. C127⋊#95C22semidirect
20. C2×C4⋊C4×C127C22semidirect
22. #95×C127C22semidirect
96. D127#396direct (N × H)
98. C127C2×#396semidirect
104. #396D127direct (N × H)
113. C2×C4⋊C4C2×D127semidirect
120. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
C2×C22⋊C4×D1272
#225498002
#225497992
#225497982
#95×D1272
#396×C1272
C2×#396127