Finite Group Databasebeta

← #30638577

Subgroups — #30638577

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
2C2×#16861136C211yesmaximal, characteristic
3#22549832C211yesmaximal, characteristic, complemented
4C2×#16861119C211yesmaximal, characteristic, complemented
5#22549641C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#416×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216C414yescomplemented
10C2×#14523232—21no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8—21no—
14C22×C127⋊C8C2211yescharacteristic
15C2×C4×Dic127C2211yescharacteristic
16#16861136C414yescomplemented
17#16861120—21no—
18#16861119—22no—
19C4×C127⋊C8—21no—
20C2×#41×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#103×C127C412yescomplemented
24#102×C127C412yescomplemented
25C22×Dic127—21no—
26#14523232—24no—
27#14523232—21no—
28#14523232—21no—
29#14523232C2×C412yes—
30#14523232C2×C412yes—
31C2×C127⋊C8—24no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8C2×C412yes—
35C2×C127⋊C8D411yescharacteristic
36C2×C127⋊C8D411yescharacteristic
37C2×C127⋊C8D412yes—
38C4×Dic127—21no—
39C4×Dic127C2×C412yes—
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×C127C2311yescharacteristic
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127C2×C412yes—
52C4⋊C4×C127C2×C412yes—
53C22⋊C4×C127—21no—
54C42×C127—21no—
55C2×Dic127—22no—
56C2×Dic127—42no—
57C127⋊C8—22no—
58C127⋊C8—22no—
59C127⋊C8C22⋊C412yes—
60C127⋊C8C22⋊C412yes—
61C23×C127—21no—
62C23×C127#4111yescharacteristic
63Q8×C127—21no—
64Q8×C127C4212yes—
65D4×C127—22no—
66D4×C127—22no—
67D4×C127—21no—
68D4×C127C4212yes—
69C2×C4×C127—21no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
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×#104—1271no—
102#416D12711yescharacteristic
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—1271no—
112C2×M16—2541no—
113C22×C8—1271no—
114C22×C8—2541no—
115C22×C8C2×D12711yescharacteristic
116#117C2×D12711yescharacteristic
117C2×C42—1271no—
118#104—1274no—
119#103Dic12712yescomplemented
120#102Dic12712yescomplemented
121#97—2541no—
122#97—2542no—
123C4×C8—2541no—
124#41—21no—
125#41—21no—
126#41C4×D12712yes—
127#41C4×D12712yes—
128C2×Q8C2×Dic12711yescharacteristic
129C2×D4—21no—
130C2×D4C2×Dic12711yescharacteristic
131C22×C4—21no—
132C22×C4—2541no—
133C22×C4C22×D12711yescharacteristic
134M16—1272no—
135M16—1272no—
136M16—2544no—
137M16—2541no—
138M16—2541no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—1272no—
142C2×C8—1272no—
143C2×C8—1271no—
144C2×C8—1271no—
145C2×C8—2544no—
146C2×C8—2541no—
147C2×C8—2541no—
148C2×C8C2×Dic12712yes—
149C4⋊C4C2×Dic12712yes—
150C22⋊C4—21no—
151C42—21no—
152C42—1272no—
153C42—2541no—
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—5082no—
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#2254999812yes—
subgroupG/H
centerC2×C4D4×D127
commutatorC4×C127C2×C42
FrattiniC22×C4C22×D127
Fitting#416×C127C2
socleC22×C127C4×D4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#416D127
hypocenterC127#1765
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
3. #22549832C2semidirect
4. C2×#16861119C2semidirect
5. #22549641C2semidirect
6. #22549635C2semidirect
9. #16861216C4semidirect
16. #16861136C4semidirect
23. #103×C127C4semidirect
24. #102×C127C4semidirect
99. C127#1765semidirect
119. #103Dic127semidirect
120. #102Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
C2×#168611362
#225498322
C2×#168611192
#225496412
#225496352
#416×C1272
#1765127