Finite Group Databasebeta

← #30639120

Subgroups — #30639120

Order 16256.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861207C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic
5#22549637C211yesmaximal, characteristic
6#22549635C211yesmaximal, characteristic
7#419×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861207—22no—
11#16861207C414yescomplemented
12C4×Dic254—22no—
13#16861174—22no—
14#16861168C2211yescharacteristic
15#16861165—22no—
16C2×Dic127⋊C4—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861133—22no—
19#16861127—22no—
20C22×Q8×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#103×C127—22no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254C2×C412yes—
29C2×Dic254D412yes—
30C22×Dic127—21no—
31C22×Dic127D412yes—
32#14523232—21no—
33#14523232—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D412yes—
36Dic127⋊C4—24no—
37Dic127⋊C4—28no—
38C22⋊Dic127—21no—
39C4×Dic127—24no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127C2×C412yes—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127C2×C412yes—
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127—21no—
54C4⋊C4×C127D412yes—
55C22⋊C4×C127—21no—
56C42×C127—21no—
57Dic254—22no—
58Dic254—22no—
59Dic254—22no—
60Dic254—22no—
61C2×Dic127—24no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127C22⋊C414yes—
67C127⋊C8—41no—
68C23×C127C2×D411yescharacteristic
69Q8×C127—22no—
70Q8×C127—22no—
71Q8×C127—41no—
72Q8×C127—41no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C22×C411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C8×C127—41no—
84C22×C127—22no—
85C22×C127C2×C22⋊C411yescharacteristic
86C22×C127#12011yescharacteristic
87C22×C127#12311yescharacteristic
88C4×C127—42no—
89C4×C127—42no—
90C4×C127C4×D411yescharacteristic
91C4×C127C4×D411yescharacteristic
92C4×C127#12111yescharacteristic
93C4×C127#12111yescharacteristic
94Dic127—24no—
95Dic127—24no—
96Dic127—42no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127#38611yescharacteristic
100C2×C127#45612yes—
101#1886—1271nomaximal, Sylow
102C127#188611yescharacteristic, complemented, Sylow
103C2×C4⋊Q8—1271no—
104#519—1271no—
105#419—1271no—
106#419D12711yescharacteristic
107C2×#101—1271no—
108#339—1271no—
109#339—1271no—
110C22×Q8—1271no—
111C22×Q8C2×D12711yescharacteristic
112C2×M16—1271no—
113C2×M16C2×D12711yescharacteristic
114C4⋊Q8—1274no—
115C4⋊Q8—2542no—
116C4×Q8—2542no—
117C4×Q8—2542no—
118#117—1271no—
119#117—2542no—
120#117C2×D12711yescharacteristic
121C2×C4⋊C4—2542no—
122C2×C42—1271no—
123#103—22no—
124#103—2542no—
125#101—2542no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—1272no—
132C2×Q8—1272no—
133C2×Q8—2541no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8C4×D12712yes—
138C22×C4—21no—
139C22×C4—1272no—
140C22×C4—2541no—
141C22×C4C22×D12711yescharacteristic
142M16—21no—
143M16—21no—
144M16—2541no—
145M16—2541no—
146C2×C8—21no—
147C2×C8—2541no—
148C4⋊C4—1272no—
149C4⋊C4—2548no—
150C4⋊C4—2544no—
151C4⋊C4C22⋊D12712yes—
152C22⋊C4—21no—
153C22⋊C4—2541no—
154C42—21no—
155C42—1272no—
156C42—2544no—
157C42—2541no—
158C42—2541no—
159C23D4×D12711yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—2542no—
165Q8—2542no—
166Q8—2542no—
167Q8—2542no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—1274no—
173C2×C4—2542no—
174C2×C4—2544no—
175C2×C4—2542no—
176C2×C4—2542no—
177C2×C4—2542no—
178C2×C4C2×C4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4C2×C22⋊D12711yescharacteristic
182C2×C4#1452324711yescharacteristic
183C2×C4#1452324811yescharacteristic
184C8—41no—
185C8—5081no—
186C22—22no—
187C22C22⋊C4×D12711yescharacteristic
188C22#1686117711yescharacteristic
189C22#1686119111yescharacteristic
190C4—42no—
191C4—42no—
192C4—5082no—
193C4—2544no—
194C4—2544no—
195C4(C22⋊D127)⋊C411yescharacteristic
196C4C4×C22⋊D12711yescharacteristic
197C4#1686118011yescharacteristic
198C4#1686118211yescharacteristic
199C2—21no—
200C2—21no—
201C2#2254975211yescharacteristic
202C2#2255005712yes—
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1886
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #16861207C4semidirect
102. C127#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
#225503532
#225498852
C2×#168611272
#225496372
#225496352
#419×C1272
#1886127