Finite Group Databasebeta

← #30638575

Subgroups — #30638575

Order 16256.

292 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16861216C211yesmaximal, characteristic
2#22549872C211yesmaximal, characteristic
3#22549834C211yesmaximal, characteristic, complemented
4C2×#16861120C211yesmaximal, characteristic, complemented
5#22549640C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216C414yescomplemented
10C2×#14523232—21no—
11C2×#14523232C2211yescharacteristic
12C22×C127⋊C8—21no—
13C22×C127⋊C8C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861133C412yescomplemented
17#16861130C412yescomplemented
18#16861120—22no—
19#16861119—21no—
20C8×Dic127—21no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C2×C42×C127C2211yescharacteristic
24#104×C127C414yescomplemented
25#14523232—24no—
26#14523232—21no—
27#14523232C2×C412yes—
28C2×C127⋊C8—24no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8C2×C412yes—
33C2×C127⋊C8C2×C412yes—
34C2×C127⋊C8D411yescharacteristic
35C2×C127⋊C8D411yescharacteristic
36C2×C127⋊C8D412yes—
37C4⋊Dic127C2×C412yes—
38C22⋊Dic127—21no—
39C4×Dic127—21no—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127C2×C412yes—
43#41×C127C2×C412yes—
44C2×Q8×C127C2×C411yescharacteristic
45C2×D4×C127—21no—
46C2×D4×C127C2×C411yescharacteristic
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50M16×C127—21no—
51M16×C127C2×C412yes—
52C2×C8×C127—21no—
53C42×C127—21no—
54C42×C127C2×C412yes—
55C2×Dic127—22no—
56C127⋊C8—22no—
57C127⋊C8—22no—
58C127⋊C8C22⋊C412yes—
59C127⋊C8C22⋊C412yes—
60C23×C127—21no—
61C23×C127#4111yescharacteristic
62Q8×C127—21no—
63Q8×C127C4212yes—
64D4×C127—22no—
65D4×C127—22no—
66D4×C127—21no—
67D4×C127C4212yes—
68C2×C4×C127—21no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—42no—
73C2×C4×C127C22×C411yescharacteristic
74C2×C4×C127C22×C411yescharacteristic
75C2×C4×C127C22×C411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C8×C127—22no—
80C22×C127—22no—
81C22×C127—22no—
82C22×C127—42no—
83C22×C127#11711yescharacteristic
84C22×C127C4×D411yescharacteristic
85C22×C127C4×D411yescharacteristic
86C4×C127—22no—
87C4×C127—42no—
88C4×C127C2×C4211yescharacteristic
89C4×C127C2×C22⋊C411yescharacteristic
90C4×C127C4×D411yescharacteristic
91C4×C127C4×D411yescharacteristic
92Dic127—42no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127—42no—
96C2×C127C4×C22⋊C411yescharacteristic
97C2×C127#44412yes—
98#1765—1271nomaximal, Sylow
99C127#176511yescharacteristic, complemented, Sylow
100C2×#131—1271no—
101C2×#104D12711yescharacteristic
102#416—1271no—
103#405—1271no—
104C2×#97—1271no—
105#341—1271no—
106#339—1271no—
107C2×#41C2×D12711yescharacteristic
108#131—1274no—
109#131—2542no—
110C2×M16—1271no—
111C2×M16—2541no—
112C2×M16C2×D12711yescharacteristic
113C22×C8—1271no—
114C22×C8—1271no—
115C22×C8—2541no—
116#117—1271no—
117C2×C42C2×D12711yescharacteristic
118#104Dic12714yescomplemented
119#103—1272no—
120#102—1272no—
121#97—2542no—
122#97—2541no—
123C4×C8—2541no—
124#41—21no—
125#41—21no—
126#41C4×D12712yes—
127#41C2×Dic12712yes—
128C2×Q8C4×D12711yescharacteristic
129C2×D4—21no—
130C2×D4C4×D12711yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4C22×D12711yescharacteristic
134M16—21no—
135M16—1272no—
136M16—2544no—
137M16—2541no—
138M16C2×Dic12712yes—
139C2×C8—21no—
140C2×C8—1272no—
141C2×C8—1272no—
142C2×C8—1271no—
143C2×C8—1271no—
144C2×C8—1272no—
145C2×C8—2544no—
146C2×C8—2541no—
147C2×C8—2541no—
148C2×C8—2541no—
149C4⋊C4—1272no—
150C22⋊C4—2541no—
151C42—21no—
152C42—2541no—
153C42C2×Dic12712yes—
154C23—21no—
155C23#1452324711yescharacteristic
156Q8—21no—
157Q8C4×Dic12712yes—
158D4—22no—
159D4—22no—
160D4—21no—
161D4C4×Dic12712yes—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—2542no—
168C2×C4C2×C4×D12711yescharacteristic
169C2×C4C2×C4×D12711yescharacteristic
170C2×C4C22×Dic12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C2×C4D4×D12711yescharacteristic
173C2×C4#1452324711yescharacteristic
174C8—22no—
175C8—1272no—
176C8—1272no—
177C8—2542no—
178C8—2542no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22#1686116511yescharacteristic
183C22(C22⋊D127)⋊C411yescharacteristic
184C22D4×Dic12711yescharacteristic
185C4—22no—
186C4—42no—
187C4—5082no—
188C4C2×C4×Dic12711yescharacteristic
189C4C22⋊C4×D12711yescharacteristic
190C4(C22⋊D127)⋊C411yescharacteristic
191C4D4×Dic12711yescharacteristic
192C2—21no—
193C2—21no—
194C2—42no—
195C2C22⋊C4×Dic12711yescharacteristic
196C2#2254999512yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC4×C127C2×C42
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127C4×D4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1765
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
3. #22549834C2semidirect
4. C2×#16861120C2semidirect
5. #22549640C2semidirect
6. #22549636C2semidirect
9. #16861216C4semidirect
16. #16861133C4semidirect
17. #16861130C4semidirect
24. #104×C127C4semidirect
99. C127#1765semidirect
118. #104Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
#225498722
#225498342
C2×#168611202
#225496402
#225496362
C2×#104×C1272
#1765127