Finite Group Databasebeta

← #30637362

Subgroups — #30637362

Order 16256.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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#22549933C211yesmaximal, characteristic, complemented
3#22549617C212yesmaximal, complemented
4#22549615C212yesmaximal, complemented
5#432×C127C211yesmaximal, characteristic, complemented
6#16861203—21no—
7#16861201C2211yescharacteristic
8#16861197C2211yescharacteristic
9#16861192—21no—
10#16861188C411yescharacteristic
11#16861183C411yescharacteristic
12C4×Dic254C411yescharacteristic
13C4×D508C411yescharacteristic
14C4×C22⋊D127—21no—
15#16861166—21no—
16C4⋊(C127⋊C8)—21no—
17C22⋊(C127⋊C8)—21no—
18#16861120C2212yes—
19C2×C42×C127C2211yescharacteristic
20C4⋊C8×C127—21no—
21C22⋊C8×C127—21no—
22#97×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
43C2×C8×C127—22no—
44C42×C127C2×C411yescharacteristic
45C42×C127D411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—41no—
50D508—41no—
51C22⋊D127—42no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C4×D127—42no—
56C127⋊C8—42no—
57C23×C127C22⋊C411yescharacteristic
58C2×C4×C127—41no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127C22⋊C411yescharacteristic
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22×C411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C8×C127—42no—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127C2×C22⋊C411yescharacteristic
72C2×D127—41no—
73C2×D127—41no—
74C2×D127—41no—
75C4×C127—21no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81Dic127—41no—
82Dic127—41no—
83Dic127—41no—
84C2×C127—41no—
85C2×C127#41311yescharacteristic
86C2×C127#42111yescharacteristic
87C2×C127#42111yescharacteristic
88D127—81no—
89#1535—1271nomaximal, Sylow
90C127#153511yescharacteristic, complemented, Sylow
91#529—1271no—
92#432—1271no—
93#432D12711yescharacteristic, complemented
94#330—1272no—
95#329—1272no—
96#126—2541no—
97#125—1271no—
98#124—1271no—
99#123—2541no—
100C22⋊Q8—1271no—
101#121—1271no—
102C4×Q8—1271no—
103C4×D4—1271no—
104C4×D4—2541no—
105#117—2541no—
106C2×C42C2×D12711yescharacteristic
107C4⋊C8—21no—
108C4⋊C8—2541no—
109C22⋊C8—21no—
110C22⋊C8—2541no—
111#97—1272no—
112#97C2×D12712yes—
113C2×Q8—1271no—
114C2×D4—1271no—
115C2×D4—2541no—
116C22×C4—2541no—
117C22×C4C4×D12711yescharacteristic
118C22×C4C22⋊D12711yescharacteristic
119C22×C4D50811yescharacteristic
120C2×C8—22no—
121C2×C8—2542no—
122C4⋊C4—1271no—
123C4⋊C4—1271no—
124C4⋊C4—2541no—
125C4⋊C4—2541no—
126C4⋊C4—2541no—
127C4⋊C4—2541no—
128C22⋊C4—2541no—
129C22⋊C4—2541no—
130C22⋊C4—2541no—
131C22⋊C4—2541no—
132C22⋊C4—2541no—
133C42—2541no—
134C42C4×D12711yescharacteristic
135C42C22⋊D12711yescharacteristic
136C42D50811yescharacteristic
137C42C22×D12711yescharacteristic
138C23—2541no—
139C23(C2×D127)⋊C411yescharacteristic
140Q8—5081no—
141D4—5081no—
142D4—5082no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—5082no—
147C2×C4—2541no—
148C2×C4—2541no—
149C2×C4—2541no—
150C2×C4(C2×D127)⋊C411yescharacteristic
151C2×C4(C2×D127)⋊C411yescharacteristic
152C2×C4(C2×D127)⋊C411yescharacteristic
153C2×C4C2×C4×D12711yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×D50811yescharacteristic
156C8—42no—
157C8—5082no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—5081no—
162C22—5081no—
163C22—5081no—
164C22C2×(C2×D127)⋊C411yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—5081no—
172C4—5081no—
173C4—5081no—
174C2—41no—
175C2—10161no—
176C2#2254985811yescharacteristic
177C2#2254989211yescharacteristic
178C2#2254989411yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#432×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C4
cosocleC42×C127C23
hypercenter#432D127
hypocenterC127#1535
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #22549933C2semidirect
3. #22549617C2semidirect
4. #22549615C2semidirect
5. #432×C127C2semidirect
90. C127#1535semidirect
93. #432D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225499332
#225496172
#225496152
#432×C1272
#1535127