Finite Group Databasebeta

← #30637318

Subgroups — #30637318

Order 16256.

234 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C4×#14523232C211yesmaximal, characteristic, complemented
2C2×#16861120C211yesmaximal, characteristic, complemented
3#22549618C212yesmaximal, complemented
4#22549616C212yesmaximal, complemented
5#529×C127C211yesmaximal, characteristic
6C2×#14523232—21no—
7C22×C127⋊C8—21no—
8#16861120—21no—
9#16861120C2212yes—
10#16861120C2212yes—
11C4×C127⋊C8—21no—
12#126×C127—21no—
13#125×C127C2211yescharacteristic
14#124×C127C2211yescharacteristic
15#123×C127—21no—
16C22⋊Q8×C127C411yescharacteristic
17#121×C127C411yescharacteristic
18C4×Q8×C127C411yescharacteristic
19C4×D4×C127—21no—
20C4×D4×C127C411yescharacteristic
21#117×C127—21no—
22C2×C42×C127C2211yescharacteristic
23#14523232—24no—
24C2×C127⋊C8—24no—
25C2×C127⋊C8—22no—
26C2×C127⋊C8—22no—
27C2×Q8×C127C2×C411yescharacteristic
28C2×D4×C127—21no—
29C2×D4×C127C2×C411yescharacteristic
30C22×C4×C127—21no—
31C22×C4×C127C2×C411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127C2×C411yescharacteristic
39C4⋊C4×C127C2×C411yescharacteristic
40C22⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—21no—
43C22⋊C4×C127—21no—
44C22⋊C4×C127—21no—
45C42×C127—21no—
46C42×C127C2×C411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C42×C127C2311yescharacteristic
50C127⋊C8—24no—
51C127⋊C8—24no—
52C23×C127—21no—
53C23×C127C22⋊C411yescharacteristic
54Q8×C127—41no—
55D4×C127—41no—
56D4×C127—42no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—41no—
63C2×C4×C127—42no—
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22×C411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C22×C127—21no—
71C22×C127—21no—
72C22×C127—41no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127#10412yes—
76C22×C127C2×C22⋊C411yescharacteristic
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127—41no—
84C4×C127—41no—
85C4×C127#10412yes—
86C2×C127—22no—
87C2×C127—81no—
88C2×C127#41311yescharacteristic
89C2×C127C2×#10411yescharacteristic
90C2×C127C2×#10411yescharacteristic
91#1524—1271nomaximal, Sylow
92C127#152411yescharacteristic, complemented, Sylow
93#529D12711yescharacteristic
94C4×M16—1271no—
95C2×#97—1271no—
96#330—1272no—
97#329—1272no—
98C2×M16—2541no—
99C22×C8—2541no—
100#126—21no—
101#125C2×D12711yescharacteristic
102#124C2×D12711yescharacteristic
103#123—21no—
104C22⋊Q8Dic12711yescharacteristic
105#121Dic12711yescharacteristic
106C4×Q8Dic12711yescharacteristic
107C4×D4—21no—
108C4×D4Dic12711yescharacteristic
109#117—21no—
110C2×C42C2×D12711yescharacteristic
111#97—1272no—
112#97—1272no—
113#97—2541no—
114C4×C8—2541no—
115C2×Q8C2×Dic12711yescharacteristic
116C2×D4—21no—
117C2×D4C2×Dic12711yescharacteristic
118C22×C4—21no—
119C22×C4C2×Dic12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22⋊D12711yescharacteristic
122M16—2544no—
123C2×C8—2544no—
124C2×C8—2542no—
125C2×C8—2542no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4C2×Dic12711yescharacteristic
131C4⋊C4C2×Dic12711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42—21no—
138C42C2×Dic12711yescharacteristic
139C42C22⋊D12711yescharacteristic
140C42C22⋊D12711yescharacteristic
141C42C22×D12711yescharacteristic
142C23—21no—
143C23C22⋊Dic12711yescharacteristic
144Q8—41no—
145D4—41no—
146D4—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4C22⋊Dic12711yescharacteristic
155C2×C4C22⋊Dic12711yescharacteristic
156C2×C4C22⋊Dic12711yescharacteristic
157C2×C4C22×Dic12711yescharacteristic
158C2×C4C2×C22⋊D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—2544no—
161C8—2544no—
162C22—21no—
163C22—21no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22#1686113612yes—
168C22C2×C22⋊Dic12711yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—41no—
176C4—41no—
177C4#1686113612yes—
178C2—22no—
179C2—81no—
180C2#2254986011yescharacteristic
181C2C2×#1686113611yescharacteristic
182C2C2×#1686113611yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#529×C127C2
socleC22×C127C2×C22⋊C4
prime radical#121×C127C4
cosocleC42×C127C23
hypercenter#529D127
hypocenterC127#1524
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C4×#14523232C2semidirect
2. C2×#16861120C2semidirect
3. #22549618C2semidirect
4. #22549616C2semidirect
92. C127#1524semidirect

Maximal subgroups

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

subgroupindex
C4×#145232322
C2×#168611202
#225496182
#225496162
#529×C1272
#1524127