Finite Group Databasebeta

← #30640414

Subgroups — #30640414

Order 16256.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550427C211yesmaximal, characteristic
2#22549859C212yesmaximal, complemented
3#22549675C211yesmaximal, characteristic, complemented
4#22549672C212yesmaximal, complemented
5#356×C127C211yesmaximal, characteristic, complemented
6C2×#14523249C411yescharacteristic
7C2×#14523249C2211yescharacteristic
8M16×D127—22no—
9#16861206C411yescharacteristic
10#16861198C411yescharacteristic
11#16861198C2211yescharacteristic
12#16861184—22no—
13#16861171—22no—
14C42×D127C411yescharacteristic
15#16861128C2212yes—
16#16861125—22no—
17C4⋊Q8×C127C2211yescharacteristic
18#101×C127C2212yes—
19#14523249—41no—
20#14523249—41no—
21#14523249—41no—
22#14523249—41no—
23C2×D508—21no—
24C2×D508—22no—
25C2×D508C2×C411yescharacteristic
26C2×D508C2×C411yescharacteristic
27C2×D508D412yes—
28C2×C4×D127—21no—
29C2×C4×D127—21no—
30C2×C4×D127C2×C411yescharacteristic
31C2×C4×D127D412yes—
32#14523232—22no—
33#14523231—42no—
34C8×D127—42no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37C4×Dic127—21no—
38C4×Dic127C2×C411yescharacteristic
39C2×Q8×C127C2×C411yescharacteristic
40C2×Q8×C127C2311yescharacteristic
41M16×C127—22no—
42C4⋊C4×C127—22no—
43C42×C127C2×C411yescharacteristic
44C22×D127—21no—
45C22×D127C22⋊C412yes—
46C22×D127C22⋊C411yescharacteristic
47D508—41no—
48D508—42no—
49D508—41no—
50D508—44no—
51D508—42no—
52C2×Dic127—21no—
53C2×Dic127C22⋊C411yescharacteristic
54C4×D127—22no—
55C4×D127—22no—
56C4×D127—21no—
57C4×D127—21no—
58C4×D127—42no—
59C4×D127—41no—
60C4×D127—41no—
61C127⋊C8—42no—
62Q8×C127—41no—
63Q8×C127—41no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127C22×C411yescharacteristic
67C2×C4×C127C2×D412yes—
68C8×C127—42no—
69C22×C127C2×C22⋊C411yescharacteristic
70C2×D127—21no—
71C2×D127—22no—
72C2×D127—41no—
73C2×D127—42no—
74C2×D127—42no—
75C2×D127—41no—
76C4×C127—22no—
77C4×C127—22no—
78C4×C127—21no—
79C4×C127—41no—
80Dic127—41no—
81Dic127#9911yescharacteristic
82Dic127#9911yescharacteristic
83C2×C127—21no—
84C2×C127C2×#9911yescharacteristic
85D127—41no—
86D127—42no—
87D127—81no—
88#2132—1271nomaximal, Sylow
89C127#213211yescharacteristic, complemented, Sylow
90#532—1271no—
91#413—1272no—
92#356—1271no—
93#356D12711yescharacteristic, complemented
94#355—1272no—
95C2×#41—1271no—
96C2×#41—1271no—
97C2×M16—2542no—
98C4⋊Q8C2×D12711yescharacteristic
99#127—1271no—
100#124—1271no—
101#124—1271no—
102#121—2542no—
103C4×D4—2542no—
104C2×C42—1271no—
105#101—1272no—
106#101C2×D12712yes—
107#100—2542no—
108#41—5081no—
109#41—5081no—
110#41—5081no—
111#41—5081no—
112C2×Q8C4×D12711yescharacteristic
113C2×Q8C22×D12711yescharacteristic
114C2×D4—1271no—
115C2×D4—1271no—
116C2×D4—1272no—
117C2×D4—2542no—
118C2×D4—2541no—
119C22×C4—1271no—
120C22×C4—1272no—
121C22×C4—2541no—
122C22×C4—2541no—
123M16—22no—
124M16—5082no—
125M16—2542no—
126C2×C8—5082no—
127C4⋊C4—22no—
128C22⋊C4—2542no—
129C22⋊C4—2542no—
130C42—1271no—
131C42—2541no—
132C42C4×D12711yescharacteristic
133C23—1272no—
134C23—1271no—
135C23—2541no—
136Q8—41no—
137Q8—41no—
138D4—5081no—
139D4—5084no—
140D4—5082no—
141D4—5081no—
142D4—5082no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—1271no—
146C2×C4—5082no—
147C2×C4—5081no—
148C2×C4—5081no—
149C2×C4—2542no—
150C2×C4—2542no—
151C2×C4—2541no—
152C2×C4—2541no—
153C2×C4—2541no—
154C2×C4C2×C4×D12711yescharacteristic
155C2×C4D4×D12712yes—
156C8—42no—
157C8—5082no—
158C22—5082no—
159C22—5081no—
160C22—5082no—
161C22—5081no—
162C22—2541no—
163C22—2542no—
164C22C22⋊C4×D12711yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—1271no—
170C4—1271no—
171C4—5081no—
172C2—21no—
173C2—10161no—
174C2—5081no—
175C2—5082no—
176C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
Fitting#356×C127C2
socleC2×C127C2×#99
prime radical#22550427C2
cosocleC2×Q8×C127C23
hypercenter#356D127
hypocenterC127#2132
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #22549859C2semidirect
3. #22549675C2semidirect
4. #22549672C2semidirect
5. #356×C127C2semidirect
89. C127#2132semidirect
93. #356D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
#225498592
#225496752
#225496722
#356×C1272
#2132127