Finite Group Databasebeta

← #30637249

Subgroups — #30637249

Order 16256.

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

normal subgroups only

This group has 294 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
1C2×#16861188C211yesmaximal, characteristic
2#22549840C211yesmaximal, characteristic
3#22549612C214yesmaximal, complemented
4C4×C22⋊C4×C127C211yesmaximal, characteristic
5C22×Dic254C411yescharacteristic, complemented
6C2×#14523232—21no—
7#16861188—21no—
8#16861188—21no—
9#16861188C412yescomplemented
10#16861188C2212yes—
11C2×C4⋊Dic127C411yescharacteristic, complemented
12C2×Dic127⋊C4—21no—
13C2×C22⋊Dic127—21no—
14C22⋊(C127⋊C8)—21no—
15C22⋊(C127⋊C8)C2212yes—
16C23×C4×C127C2211yescharacteristic
17C2×C22⋊C4×C127—21no—
18C2×C42×C127—21no—
19#95×C127C2212yes—
20C2×Dic254—21no—
21C2×Dic254—42no—
22C2×Dic254C2×C412yes—
23C22×Dic127—21no—
24C22×Dic127—21no—
25#14523232—42no—
26C2×C127⋊C8—22no—
27C4⋊Dic127—21no—
28C4⋊Dic127C2×C412yes—
29Dic127⋊C4—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33C24×C127D411yescharacteristic
34C22×C4×C127—22no—
35C22×C4×C127—42no—
36C22×C4×C127C2×C411yescharacteristic
37C22×C4×C127C2×C411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2311yescharacteristic
42C22⋊C4×C127—24no—
43C42×C127—42no—
44Dic254—42no—
45Dic254—42no—
46C2×Dic127—22no—
47C2×Dic127—22no—
48C2×Dic127—42no—
49C2×Dic127—42no—
50C127⋊C8—42no—
51C23×C127—22no—
52C23×C127—22no—
53C23×C127—41no—
54C23×C127C22⋊C411yescharacteristic
55C23×C127C22⋊C411yescharacteristic
56C23×C127C2×D411yescharacteristic
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—24no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—42no—
65C2×C4×C127—42no—
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C22×C127—21no—
71C22×C127—21no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—44no—
78C22×C127#10412yes—
79C22×C127C2×C22⋊C411yescharacteristic
80C22×C127#13712yes—
81C4×C127—21no—
82C4×C127—21no—
83C4×C127—21no—
84C4×C127—44no—
85C4×C127#9912yes—
86Dic127—42no—
87Dic127—42no—
88C2×C127—22no—
89C2×C127—22no—
90C2×C127—41no—
91C2×C127C2×#9911yescharacteristic
92C2×C127#41911yescharacteristic
93C2×C127C2×#10411yescharacteristic
94#1508—1271nomaximal, Sylow
95C127#150811yescharacteristic, complemented, Sylow
96C2×C22⋊Q8—1271no—
97#407—1271no—
98C4×C22⋊C4D12711yescharacteristic
99#328—1274no—
100C22×Q8—1271no—
101C23×C4C2×D12711yescharacteristic
102C2×M16—2541no—
103C22⋊Q8—1272no—
104C22⋊Q8—1272no—
105C22⋊Q8—2541no—
106C22⋊Q8—2541no—
107C2×C4⋊C4—1271no—
108C2×C4⋊C4—2541no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—2541no—
111C2×C42—21no—
112C22⋊C8—1272no—
113C22⋊C8—2541no—
114#95C2×D12712yes—
115C24C22⋊D12711yescharacteristic
116C2×Q8—1272no—
117C2×Q8—5082no—
118C2×Q8—2541no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—2541no—
122C22×C4—2541no—
123C22×C4C4×D12711yescharacteristic
124C22×C4C4×D12711yescharacteristic
125C22×C4C22⋊D12711yescharacteristic
126C22×C4D50811yescharacteristic
127C22×C4D50811yescharacteristic
128C22×C4C22×D12711yescharacteristic
129M16—5082no—
130C2×C8—2542no—
131C4⋊C4—1272no—
132C4⋊C4—2542no—
133C4⋊C4—2541no—
134C4⋊C4—2542no—
135C22⋊C4—24no—
136C22⋊C4—2542no—
137C22⋊C4—2542no—
138C42—42no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23(C2×D127)⋊C411yescharacteristic
143C23(C2×D127)⋊C411yescharacteristic
144C23C2×C22⋊D12711yescharacteristic
145Q8—5082no—
146Q8—5082no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—5082no—
157C2×C4—5082no—
158C2×C4—2542no—
159C2×C4—2542no—
160C2×C4(C2×D127)⋊C411yescharacteristic
161C2×C4(C2×D127)⋊C411yescharacteristic
162C2×C4C2×C4×D12711yescharacteristic
163C2×C4C2×D50811yescharacteristic
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22#1686113712yes—
174C22C2×(C2×D127)⋊C411yescharacteristic
175C22#1686123912yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—44no—
180C4—5082no—
181C4—5082no—
182C4#1686112312yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#1686112311yescharacteristic
187C2#2254988411yescharacteristic
188C2C2×#1686113711yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC4×C22⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC4×C22⋊C4D127
hypocenterC127#1508
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #22549612C2semidirect
5. C22×Dic254C4semidirect
9. #16861188C4semidirect
11. C2×C4⋊Dic127C4semidirect
95. C127#1508semidirect

Maximal subgroups

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

subgroupindex
C2×#168611882
#225498402
#225496122
C4×C22⋊C4×C1272
#1508127