Finite Group Databasebeta

← #30637193

Subgroups — #30637193

Order 16256.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550410C211yesmaximal, characteristic
2#22549929C211yesmaximal, characteristic, complemented
3#22549606C212yesmaximal, complemented
4#22549603C212yesmaximal, complemented
5#423×C127C211yesmaximal, characteristic, complemented
6#16861203—21no—
7#16861201C411yescharacteristic
8#16861197C411yescharacteristic
9#16861192—21no—
10#16861188C411yescharacteristic
11#16861183C411yescharacteristic
12C4×Dic254C2211yescharacteristic
13C4×D508C2211yescharacteristic
14C4×C22⋊D127—21no—
15#16861166—21no—
16C4⋊(C127⋊C8)—21no—
17C22⋊(C127⋊C8)—21no—
18C4×C127⋊C8C2212yes—
19C2×M16×C127—21no—
20C2×C42×C127C2211yescharacteristic
21C4⋊C8×C127—21no—
22C4⋊C8×C127C2212yes—
23C2×Dic254C2×C411yescharacteristic
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26C2×C4×D127—21no—
27C2×C127⋊C8—22no—
28C4⋊Dic127C2×C411yescharacteristic
29C4⋊Dic127C2×C411yescharacteristic
30Dic127⋊C4—21no—
31Dic127⋊C4—21no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34C22⋊Dic127—21no—
35(C2×D127)⋊C4—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39C4×Dic127—21no—
40C22×C4×C127C2×C411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43M16×C127—42no—
44C2×C8×C127—22no—
45C42×C127C2×C411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50Dic254—22no—
51D508—22no—
52C22⋊D127—42no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C4×D127—41no—
57C4×D127—41no—
58C127⋊C8—42no—
59C23×C127C22⋊C411yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22×C411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—42no—
71C22×C127—21no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127C2×C22⋊C411yescharacteristic
75C2×D127—41no—
76C2×D127—41no—
77C2×D127—41no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127#13112yes—
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127#13711yescharacteristic
85Dic127—41no—
86Dic127—41no—
87Dic127—41no—
88C2×C127—41no—
89C2×C127#40811yescharacteristic
90C2×C127#41711yescharacteristic
91C2×C127#42111yescharacteristic
92D127—81no—
93#1495—1271nomaximal, Sylow
94C127#149511yescharacteristic, complemented, Sylow
95#529—1271no—
96#431—1271no—
97#423D12711yescharacteristic, complemented
98Q8⋊C8—1272no—
99D4⋊C8—1272no—
100C2×M16—21no—
101#126—2541no—
102#125—1271no—
103#124—1271no—
104#123—2541no—
105C22⋊Q8—1271no—
106#121—1271no—
107C4×Q8—1271no—
108C4×D4—1271no—
109C4×D4—2541no—
110#117—2541no—
111C2×C42C2×D12711yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—2541no—
114C4⋊C8C2×D12712yes—
115C22⋊C8—2541no—
116C4×C8—1272no—
117C2×Q8—1271no—
118C2×D4—1271no—
119C2×D4—2541no—
120C22×C4—2541no—
121C22×C4C4×D12711yescharacteristic
122C22×C4C22⋊D12711yescharacteristic
123C22×C4D50811yescharacteristic
124M16—42no—
125C2×C8—22no—
126C2×C8—2542no—
127C4⋊C4—1271no—
128C4⋊C4—1271no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C22⋊C4—2541no—
134C22⋊C4—2541no—
135C22⋊C4—2541no—
136C22⋊C4—2541no—
137C22⋊C4—2541no—
138C42—2541no—
139C42C4×D12711yescharacteristic
140C42C22⋊D12711yescharacteristic
141C42D50811yescharacteristic
142C42C22×D12711yescharacteristic
143C23—2541no—
144C23(C2×D127)⋊C411yescharacteristic
145Q8—2542no—
146D4—5082no—
147D4—2542no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—5081no—
153C2×C4—5081no—
154C2×C4—2541no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4(C2×D127)⋊C411yescharacteristic
158C2×C4(C2×D127)⋊C411yescharacteristic
159C2×C4(C2×D127)⋊C411yescharacteristic
160C2×C4C2×C4×D12711yescharacteristic
161C2×C4C2×C22⋊D12711yescharacteristic
162C2×C4C2×D50811yescharacteristic
163C8—42no—
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—5081no—
169C22—5081no—
170C22—5081no—
171C22C2×(C2×D127)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—5081no—
177C4—5081no—
178C4—5081no—
179C4#1686121712yes—
180C4#1686123511yescharacteristic
181C4#1686124011yescharacteristic
182C2—41no—
183C2—10161no—
184C2#2254984411yescharacteristic
185C2#2254987511yescharacteristic
186C2#2254989211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#423×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C4
cosocleC42×C127C23
hypercenter#423D127
hypocenterC127#1495
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #22549929C2semidirect
3. #22549606C2semidirect
4. #22549603C2semidirect
5. #423×C127C2semidirect
94. C127#1495semidirect
97. #423D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225499292
#225496062
#225496032
#423×C1272
#1495127