Finite Group Databasebeta

← #30648860

Subgroups — #30648860

Order 16256.

380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550433C211yesmaximal, characteristic
2#22550206C212yesmaximal
3#22550161C212yesmaximal, complemented
4#22550086C212yesmaximal, complemented
5#22550032C212yesmaximal
6#22549981C214yesmaximal, complemented
7#22549933C211yesmaximal, characteristic
8C2×C4×Q8×C127C211yesmaximal, characteristic
9C2×#14523238C2214yes—
10#16861209C2211yescharacteristic
11#16861209C2211yescharacteristic
12#16861201—21no—
13#16861188—21no—
14#16861188C2212yes—
15C4×Dic254—21no—
16C4×Dic254C2212yes—
17#16861168—21no—
18#16861140C2214yes—
19C4⋊(C127⋊C8)C2212yes—
20#16861133C2214yes—
21#16861132C2214yes—
22C22⋊(C127⋊C8)C2212yes—
23#16861120C2212yes—
24C22×Q8×C127C2211yescharacteristic
25C4×Q8×C127—21no—
26C4×Q8×C127—21no—
27C4×Q8×C127C2212yes—
28C4×Q8×C127C2212yes—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127C2211yescharacteristic
31C2×C42×C127—21no—
32C2×C42×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254C2312yes—
35#14523238—28no—
36C2×C127⋊C8C2314yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127C2312yes—
41Dic127⋊C4—22no—
42Dic127⋊C4—22no—
43C22⋊Dic127—22no—
44C4×Dic127—22no—
45C2×Q8×C127—24no—
46C2×Q8×C127—21no—
47C2×Q8×C127C2312yes—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127D411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127—22no—
54C4⋊C4×C127—22no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127C2312yes—
57C42×C127—22no—
58C42×C127—22no—
59C42×C127D411yescharacteristic
60C42×C127D411yescharacteristic
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63Dic254—24no—
64Dic254—42no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C127⋊C8—24no—
68C23×C127C2×D411yescharacteristic
69Q8×C127—24no—
70Q8×C127—22no—
71Q8×C127#4114yes—
72C2×C4×C127—24no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—42no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2411yescharacteristic
85C22×C127—21no—
86C22×C127—21no—
87C22×C127#13712yes—
88C22×C127C22×D411yescharacteristic
89C4×C127—24no—
90C4×C127—21no—
91C4×C127—21no—
92C4×C127—21no—
93C4×C127—21no—
94C4×C127—42no—
95C4×C127#13712yes—
96C4×C127C2×#4112yes—
97Dic127—42no—
98Dic127—42no—
99C2×C127—22no—
100C2×C127#52511yescharacteristic
101C2×C127C2×#13711yescharacteristic
102C2×C127C2×#13711yescharacteristic
103#3113—1271nomaximal, Sylow
104C127#311311yescharacteristic, complemented, Sylow
105#533—1271no—
106C2×C4×Q8D12711yescharacteristic
107#483—1272no—
108#475—1272no—
109#462—1272no—
110#451—1272no—
111#441—1274no—
112#432—1271no—
113C22×Q8C2×D12711yescharacteristic
114C2×Q16—1274no—
115C4⋊Q8—1271no—
116C4⋊Q8—1271no—
117#125—2541no—
118C22⋊Q8—1272no—
119C22⋊Q8—2541no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—1272no—
123C4×Q8—2541no—
124C4×Q8C2×D12712yes—
125C4×Q8C2×D12712yes—
126#117—2541no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2×D12711yescharacteristic
129C2×C42—21no—
130C2×C42C2×D12711yescharacteristic
131#106—1274no—
132C4⋊C8—1272no—
133#103—1274no—
134#103—1274no—
135C22⋊C8—1272no—
136#97—1272no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8—1272no—
140C2×Q8—2541no—
141C2×Q8C22×D12712yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4C22⋊D12711yescharacteristic
145C22×C4C22⋊D12711yescharacteristic
146C22×C4C22×D12711yescharacteristic
147Q16—2548no—
148C2×C8—1274no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—1272no—
153C4⋊C4—2542no—
154C4⋊C4—2542no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4C22×D12712yes—
159C22⋊C4—2542no—
160C42—22no—
161C42—22no—
162C42—2542no—
163C42C22⋊D12711yescharacteristic
164C42C22⋊D12711yescharacteristic
165C42C22×D12711yescharacteristic
166C42C22×D12711yescharacteristic
167C23C2×C22⋊D12711yescharacteristic
168Q8—24no—
169Q8—22no—
170Q8—5082no—
171Q8—2544no—
172Q8#1452324814yes—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—2542no—
181C2×C4—2542no—
182C2×C4C2×C22⋊D12711yescharacteristic
183C2×C4C2×C22⋊D12711yescharacteristic
184C2×C4C2×C22⋊D12711yescharacteristic
185C2×C4C2×C22⋊D12711yescharacteristic
186C2×C4C2×C22⋊D12711yescharacteristic
187C2×C4C23×D12711yescharacteristic
188C8—2544no—
189C22—21no—
190C22—21no—
191C22#1686123912yes—
192C22C22×C22⋊D12711yescharacteristic
193C4—24no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—5082no—
200C4—5082no—
201C4#1686124012yes—
202C4C2×#1452324812yes—
203C2—22no—
204C2#2255038711yescharacteristic
205C2C2×#1686123911yescharacteristic
206C2C2×#1686124011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×Q8×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenterC2×C4×Q8D127
hypocenterC127#3113
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #22550161C2semidirect
4. #22550086C2semidirect
6. #22549981C2semidirect
104. C127#3113semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225502062
#225501612
#225500862
#225500322
#225499812
#225499332
C2×C4×Q8×C1272
#3113127