Finite Group Databasebeta

← #30638809

Subgroups — #30638809

Order 16256.

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

normal subgroups only

This group has 316 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
1C2×C4⋊C4×D127C211yesmaximal, characteristic
2#22549839C212yesmaximal
3#22549645C211yesmaximal, characteristic, complemented
4#22549644C212yesmaximal, complemented
5#343×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C412yes—
7C22×C4×D127C2211yescharacteristic
8C2×#14523232C2212yes—
9C2×C4⋊Dic127C2211yescharacteristic
10C2×Dic127⋊C4C412yes—
11C4⋊C4×D127—22no—
12C4⋊C4×D127—22no—
13(C2×D127)⋊C8—22no—
14(C2×D127)⋊C8—22no—
15C2×M16×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C23×D127C2×C411yescharacteristic
18C22×Dic127C2×C411yescharacteristic
19C22×Dic127C2×C412yes—
20C2×C4×D127—22no—
21C2×C4×D127—22no—
22C2×C4×D127—21no—
23C2×C4×D127—22no—
24C2×C4×D127—21no—
25C2×C4×D127—44no—
26C2×C4×D127D412yes—
27C2×C4×D127D412yes—
28#14523232—22no—
29#14523232—22no—
30C2×C127⋊C8—22no—
31C4⋊Dic127—22no—
32Dic127⋊C4—24no—
33C22×C4×C127C2×C412yes—
34C22×C4×C127C2311yescharacteristic
35M16×C127—22no—
36M16×C127—22no—
37C2×C8×C127—22no—
38C4⋊C4×C127—22no—
39C22×D127—21no—
40C22×D127—21no—
41C22×D127—22no—
42C22×D127—22no—
43C22×D127C22⋊C411yescharacteristic
44C22×D127C22⋊C411yescharacteristic
45C2×Dic127—21no—
46C2×Dic127—21no—
47C2×Dic127—22no—
48C2×Dic127—42no—
49C2×Dic127C22⋊C412yes—
50C4×D127—22no—
51C4×D127—22no—
52C4×D127—22no—
53C4×D127—22no—
54C4×D127—44no—
55C4×D127—44no—
56C127⋊C8—42no—
57C23×C127C22×C411yescharacteristic
58C2×C4×C127—22no—
59C2×C4×C127—42no—
60C2×C4×C127C2×D412yes—
61C2×C4×C127#4111yescharacteristic
62C2×C4×C127#4112yes—
63C2×C4×C127#4111yescharacteristic
64C8×C127—42no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127C2×C22⋊C411yescharacteristic
68C22×C127#11711yescharacteristic
69C22×C127#11711yescharacteristic
70C2×D127—24no—
71C2×D127—24no—
72C2×D127—21no—
73C2×D127—21no—
74C2×D127—21no—
75C2×D127—21no—
76C2×D127#10012yes—
77C2×D127#10112yes—
78C4×C127—42no—
79C4×C127#12312yes—
80C4×C127#12312yes—
81Dic127—22no—
82Dic127—42no—
83C2×C127—21no—
84C2×C127—21no—
85C2×C127#38111yescharacteristic
86C2×C127C2×#10011yescharacteristic
87C2×C127C2×#10111yescharacteristic
88D127—22no—
89D127—22no—
90#1821—1271nomaximal, Sylow
91C127#182111yescharacteristic, complemented, Sylow
92C22×C4⋊C4—1271no—
93#407—1272no—
94#343—1272no—
95#343—1271no—
96#343D12711yescharacteristic, complemented
97C23×C4—1271no—
98C23×C4—1272no—
99C2×M16—1272no—
100C2×M16C2×D12712yes—
101C2×C4⋊C4—1272no—
102C2×C4⋊C4—1271no—
103C2×C4⋊C4—2542no—
104C2×C4⋊C4—2542no—
105C2×C4⋊C4C2×D12711yescharacteristic
106C22⋊C8—2542no—
107C22⋊C8—2542no—
108C24—1271no—
109C22×C4—1271no—
110C22×C4—1272no—
111C22×C4—1272no—
112C22×C4—1272no—
113C22×C4—5084no—
114C22×C4—2542no—
115C22×C4—2542no—
116C22×C4—2541no—
117C22×C4—2542no—
118C22×C4—2541no—
119C22×C4C4×D12712yes—
120C22×C4C22×D12711yescharacteristic
121M16—22no—
122M16—22no—
123M16—2542no—
124M16—2542no—
125C2×C8—22no—
126C2×C8—2542no—
127C4⋊C4—22no—
128C4⋊C4—2544no—
129C4⋊C4—2542no—
130C23—1271no—
131C23—1271no—
132C23—2542no—
133C23—2542no—
134C23—2541no—
135C23—2541no—
136C23C2×C4×D12711yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—1272no—
140C2×C4—5084no—
141C2×C4—5084no—
142C2×C4—5082no—
143C2×C4—2542no—
144C2×C4—2541no—
145C2×C4—2541no—
146C2×C4—2542no—
147C2×C4—2542no—
148C2×C4—2542no—
149C2×C4—2542no—
150C2×C4D4×D12712yes—
151C2×C4#1452324711yescharacteristic
152C2×C4#1452324812yes—
153C2×C4#1452324911yescharacteristic
154C8—42no—
155C8—5082no—
156C22—21no—
157C22—21no—
158C22—1272no—
159C22—1272no—
160C22—2544no—
161C22—2544no—
162C22—2541no—
163C22—2541no—
164C22—2541no—
165C22—2541no—
166C22C22⋊C4×D12711yescharacteristic
167C22#1686116611yescharacteristic
168C22#1686116711yescharacteristic
169C4—42no—
170C4—5082no—
171C4—2542no—
172C4#1686119112yes—
173C4#1686119412yes—
174C2—21no—
175C2—21no—
176C2—2542no—
177C2—2542no—
178C2#2254973811yescharacteristic
179C2#100×D12711yescharacteristic
180C2#101×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#343D127
hypocenterC127#1821
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #22549645C2semidirect
4. #22549644C2semidirect
5. #343×C127C2semidirect
91. C127#1821semidirect
96. #343D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1272
#225498392
#225496452
#225496442
#343×C1272
#1821127