Finite Group Databasebeta

← #30637141

Subgroups — #30637141

Order 16256.

234 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C4×#14523232C211yesmaximal, characteristic, complemented
3#22549606C212yesmaximal, complemented
4#22549603C212yesmaximal, complemented
5C2×C4⋊C8×C127C211yesmaximal, characteristic, complemented
6C2×#14523232—21no—
7#16861203—21no—
8#16861201C411yescharacteristic
9#16861197C411yescharacteristic
10#16861192—21no—
11#16861188C411yescharacteristic
12#16861183C411yescharacteristic
13C4×Dic254C2211yescharacteristic
14C4×D508C2211yescharacteristic
15C4×C22⋊D127—21no—
16#16861166—21no—
17#16861120—21no—
18C4×C127⋊C8C2212yes—
19C22×C8×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—
27#14523232—24no—
28C2×C127⋊C8—22no—
29C4⋊Dic127C2×C411yescharacteristic
30C4⋊Dic127C2×C411yescharacteristic
31Dic127⋊C4—21no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35C22⋊Dic127—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C22×C4×C127C2×C411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44C2×C8×C127—22no—
45C2×C8×C127—42no—
46C42×C127C2×C411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—22no—
52D508—22no—
53C22⋊D127—42no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C4×D127—41no—
58C4×D127—41no—
59C127⋊C8—24no—
60C23×C127C22⋊C411yescharacteristic
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—42no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127#10412yes—
75C22×C127C2×C22⋊C411yescharacteristic
76C2×D127—41no—
77C2×D127—41no—
78C2×D127—41no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—21no—
83C4×C127#13112yes—
84C4×C127AGL(1,ℤ/8)11yescharacteristic
85C4×C127#13711yescharacteristic
86Dic127—41no—
87Dic127—41no—
88Dic127—41no—
89C2×C127—22no—
90C2×C127#40811yescharacteristic
91C2×C127#41711yescharacteristic
92C2×C127C2×#10411yescharacteristic
93D127—81no—
94#1483—1271nomaximal, Sylow
95C127#148311yescharacteristic, complemented, Sylow
96#529—1271no—
97C2×C4⋊C8D12711yescharacteristic, complemented
98C4×M16—1271no—
99Q8⋊C8—1272no—
100D4⋊C8—1272no—
101C2×M16—2541no—
102C22×C8—21no—
103#126—2541no—
104#125—1271no—
105#124—1271no—
106#123—2541no—
107C22⋊Q8—1271no—
108#121—1271no—
109C4×Q8—1271no—
110C4×D4—1271no—
111C4×D4—2541no—
112#117—2541no—
113C2×C42C2×D12711yescharacteristic
114C4⋊C8—21no—
115C4⋊C8C2×D12712yes—
116#97—2541no—
117C4×C8—1272no—
118C2×Q8—1271no—
119C2×D4—1271no—
120C2×D4—2541no—
121C22×C4—2541no—
122C22×C4C4×D12711yescharacteristic
123C22×C4C22⋊D12711yescharacteristic
124C22×C4D50811yescharacteristic
125M16—2544no—
126C2×C8—22no—
127C2×C8—42no—
128C2×C8—2542no—
129C4⋊C4—1271no—
130C4⋊C4—1271no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C4⋊C4—2541no—
134C4⋊C4—2541no—
135C22⋊C4—2541no—
136C22⋊C4—2541no—
137C22⋊C4—2541no—
138C22⋊C4—2541no—
139C22⋊C4—2541no—
140C42—2541no—
141C42C4×D12711yescharacteristic
142C42C22⋊D12711yescharacteristic
143C42D50811yescharacteristic
144C42C22×D12711yescharacteristic
145C23—2541no—
146C23(C2×D127)⋊C411yescharacteristic
147Q8—2542no—
148D4—5082no—
149D4—2542no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—5081no—
155C2×C4—5081no—
156C2×C4—2541no—
157C2×C4—2541no—
158C2×C4—2541no—
159C2×C4(C2×D127)⋊C411yescharacteristic
160C2×C4(C2×D127)⋊C411yescharacteristic
161C2×C4(C2×D127)⋊C411yescharacteristic
162C2×C4C2×C4×D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C2×C4C2×D50811yescharacteristic
165C8—42no—
166C8—2544no—
167C22—21no—
168C22—21no—
169C22—5081no—
170C22—5081no—
171C22—5081no—
172C22#1686113512yes—
173C22C2×(C2×D127)⋊C411yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—5081no—
179C4—5081no—
180C4—5081no—
181C4#1686121512yes—
182C4#1686123511yescharacteristic
183C4#1686124011yescharacteristic
184C2—22no—
185C2—10161no—
186C2#2254984211yescharacteristic
187C2#2254987511yescharacteristic
188C2C2×#1686113511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC2×C4⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C4
cosocleC42×C127C23
hypercenterC2×C4⋊C8D127
hypocenterC127#1483
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#14523232C2semidirect
3. #22549606C2semidirect
4. #22549603C2semidirect
5. C2×C4⋊C8×C127C2semidirect
95. C127#1483semidirect
97. C2×C4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
C4×#145232322
#225496062
#225496032
C2×C4⋊C8×C1272
#1483127