Finite Group Databasebeta

← #30637363

Subgroups — #30637363

Order 16256.

238 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 59 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#22550425C211yesmaximal, characteristic
2#22549931C212yesmaximal, complemented
3#22549616C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic, complemented
6C2×#14523249C411yescharacteristic
7C2×#14523247C411yescharacteristic
8#16861207C411yescharacteristic
9#16861205C411yescharacteristic
10#16861197C2211yescharacteristic
11#16861182—22no—
12(C22⋊D127)⋊C4—22no—
13C42×D127C2211yescharacteristic
14Dic127⋊C8—22no—
15(C2×D127)⋊C8—22no—
16#16861120C2212yes—
17#124×C127C2211yescharacteristic
18#97×C127C2212yes—
19#14523249—42no—
20#14523247—42no—
21C2×Dic254C2×C411yescharacteristic
22C2×D508—21no—
23C2×D508C2×C411yescharacteristic
24C2×C22⋊D127—21no—
25C2×C22⋊D127—22no—
26C22×Dic127—21no—
27C2×C4×D127—21no—
28C2×C4×D127C2×C411yescharacteristic
29C2×C4×D127D412yes—
30C2×C127⋊C8—22no—
31Dic127⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C4×Dic127C2×C411yescharacteristic
34C4×Dic127D412yes—
35C2×Q8×C127C2×C411yescharacteristic
36C2×D4×C127C2×C411yescharacteristic
37C2×C8×C127—22no—
38C22⋊C4×C127—22no—
39C42×C127C2311yescharacteristic
40C22×D127—21no—
41C22×D127C22⋊C411yescharacteristic
42Dic254—41no—
43D508—41no—
44D508—42no—
45C22⋊D127—44no—
46C22⋊D127—42no—
47C2×Dic127—21no—
48C2×Dic127—42no—
49C2×Dic127C22⋊C412yes—
50C2×Dic127C22⋊C411yescharacteristic
51C4×D127—22no—
52C4×D127—42no—
53C4×D127—42no—
54C127⋊C8—42no—
55C23×C127—21no—
56Q8×C127—41no—
57D4×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127C22×C411yescharacteristic
60C2×C4×C127C2×D412yes—
61C8×C127—42no—
62C22×C127—41no—
63C22×C127—42no—
64C22×C127C2×C22⋊C411yescharacteristic
65C2×D127—21no—
66C2×D127—22no—
67C2×D127—41no—
68C2×D127—42no—
69C4×C127—21no—
70C4×C127—22no—
71C4×C127—41no—
72Dic127—22no—
73Dic127—41no—
74Dic127#10012yes—
75C2×C127—81no—
76C2×C127C2×#10011yescharacteristic
77C2×C127#42112yes—
78D127—41no—
79D127—81no—
80#1536—1271nomaximal, Sylow
81C127#153611yescharacteristic, complemented, Sylow
82#532—1271no—
83#432—1272no—
84#329—1271no—
85#329—1272no—
86#329D12711yescharacteristic, complemented
87C2×#41—1271no—
88C2×#41—1271no—
89C4⋊Q8—1271no—
90#127—1271no—
91#124—1271no—
92#124C2×D12711yescharacteristic
93#121—2542no—
94C4×D4—2542no—
95C2×C42—1271no—
96C4⋊C8—2542no—
97C22⋊C8—2542no—
98#97—1272no—
99#97C2×D12712yes—
100#41—5082no—
101#41—5082no—
102C2×Q8—1271no—
103C2×Q8C4×D12711yescharacteristic
104C2×D4—1271no—
105C2×D4—2541no—
106C2×D4—2542no—
107C2×D4—2541no—
108C2×D4C4×D12711yescharacteristic
109C22×C4—1272no—
110C22×C4—1271no—
111C22×C4—2541no—
112C22×C4—2541no—
113C2×C8—22no—
114C2×C8—2542no—
115C4⋊C4—2542no—
116C22⋊C4—22no—
117C22⋊C4—2542no—
118C42—1271no—
119C42—1272no—
120C42C22×D12711yescharacteristic
121C23—21no—
122C23—1271no—
123C23—2541no—
124Q8—41no—
125Q8—5081no—
126D4—41no—
127D4—5081no—
128D4—5082no—
129D4—5082no—
130D4—5084no—
131C2×C4—21no—
132C2×C4—1271no—
133C2×C4—1272no—
134C2×C4—5082no—
135C2×C4—5082no—
136C2×C4—5082no—
137C2×C4—2542no—
138C2×C4—2541no—
139C2×C4C2×C4×D12711yescharacteristic
140C2×C4D4×D12712yes—
141C8—42no—
142C8—5082no—
143C22—41no—
144C22—42no—
145C22—5081no—
146C22—5082no—
147C22—2541no—
148C22—2542no—
149C22C22⋊C4×D12711yescharacteristic
150C4—21no—
151C4—22no—
152C4—41no—
153C4—1272no—
154C4—5081no—
155C4—2542no—
156C2—81no—
157C2—10161no—
158C2—5081no—
159C2#100×D12711yescharacteristic
160C2#2254989112yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550425C2
cosocleC42×C127C23
hypercenter#329D127
hypocenterC127#1536
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22549931C2semidirect
3. #22549616C2semidirect
4. #22549615C2semidirect
5. #329×C127C2semidirect
81. C127#1536semidirect
86. #329D127semidirect

Maximal subgroups

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

subgroupindex
#225504252
#225499312
#225496162
#225496152
#329×C1272
#1536127