Finite Group Databasebeta

← #30637338

Subgroups — #30637338

Order 16256.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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#22549617C212yesmaximal, complemented
4#22549615C212yesmaximal, complemented
5#432×C127C211yesmaximal, characteristic, complemented
6C2×#14523232—21no—
7#16861203—21no—
8#16861201C2211yescharacteristic
9#16861197C2211yescharacteristic
10#16861192—21no—
11#16861188C411yescharacteristic
12#16861183C411yescharacteristic
13C4×Dic254C411yescharacteristic
14C4×D508C411yescharacteristic
15C4×C22⋊D127—21no—
16#16861166—21no—
17#16861120C2212yes—
18C4×C127⋊C8—21no—
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—
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—
45C42×C127C2×C411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50Dic254—41no—
51D508—41no—
52C22⋊D127—42no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C4×D127—42no—
57C127⋊C8—24no—
58C23×C127C22⋊C411yescharacteristic
59C2×C4×C127—22no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22×C411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C8×C127—42no—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—21no—
72C22×C127C2×C22⋊C411yescharacteristic
73C2×D127—41no—
74C2×D127—41no—
75C2×D127—41no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127#10412yes—
82Dic127—41no—
83Dic127—41no—
84Dic127—41no—
85C2×C127—41no—
86C2×C127#41311yescharacteristic
87C2×C127C2×#10411yescharacteristic
88C2×C127#42111yescharacteristic
89D127—81no—
90#1529—1271nomaximal, Sylow
91C127#152911yescharacteristic, complemented, Sylow
92#529—1271no—
93#432D12711yescharacteristic, complemented
94C4×M16—1271no—
95#330—1272no—
96#329—1272no—
97C2×M16—2541no—
98#126—2541no—
99#125—1271no—
100#124—1271no—
101#123—2541no—
102C22⋊Q8—1271no—
103#121—1271no—
104C4×Q8—1271no—
105C4×D4—1271no—
106C4×D4—2541no—
107#117—2541no—
108C2×C42C2×D12711yescharacteristic
109C4⋊C8—21no—
110C22⋊C8—21no—
111#97—1272no—
112#97C2×D12712yes—
113C4×C8—2541no—
114C2×Q8—1271no—
115C2×D4—1271no—
116C2×D4—2541no—
117C22×C4—2541no—
118C22×C4C4×D12711yescharacteristic
119C22×C4C22⋊D12711yescharacteristic
120C22×C4D50811yescharacteristic
121M16—2544no—
122C2×C8—22no—
123C2×C8—2542no—
124C4⋊C4—1271no—
125C4⋊C4—1271no—
126C4⋊C4—2541no—
127C4⋊C4—2541no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C22⋊C4—2541no—
131C22⋊C4—2541no—
132C22⋊C4—2541no—
133C22⋊C4—2541no—
134C22⋊C4—2541no—
135C42—2541no—
136C42C4×D12711yescharacteristic
137C42C22⋊D12711yescharacteristic
138C42D50811yescharacteristic
139C42C22×D12711yescharacteristic
140C23—2541no—
141C23(C2×D127)⋊C411yescharacteristic
142Q8—5081no—
143D4—5081no—
144D4—5082no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—5082no—
149C2×C4—2541no—
150C2×C4—2541no—
151C2×C4—2541no—
152C2×C4(C2×D127)⋊C411yescharacteristic
153C2×C4(C2×D127)⋊C411yescharacteristic
154C2×C4(C2×D127)⋊C411yescharacteristic
155C2×C4C2×C4×D12711yescharacteristic
156C2×C4C2×C22⋊D12711yescharacteristic
157C2×C4C2×D50811yescharacteristic
158C8—42no—
159C8—2544no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—5081no—
164C22—5081no—
165C22—5081no—
166C22C2×(C2×D127)⋊C411yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—5081no—
173C4—5081no—
174C4—5081no—
175C4#1686113512yes—
176C2—41no—
177C2—10161no—
178C2#2254985811yescharacteristic
179C2C2×#1686113511yescharacteristic
180C2#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#1529
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. C4×#14523232C2semidirect
3. #22549617C2semidirect
4. #22549615C2semidirect
5. #432×C127C2semidirect
91. C127#1529semidirect
93. #432D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
C4×#145232322
#225496172
#225496152
#432×C1272
#1529127