Finite Group Databasebeta

← #30638834

Subgroups — #30638834

Order 16256.

262 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 89 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#22549860C211yesmaximal, characteristic
2#22549850C211yesmaximal, characteristic
3#22549636C212yesmaximal, complemented
4#22549632C212yesmaximal, complemented
5C4×#41×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7#16861168C2212yes—
8#16861128—21no—
9#16861126—21no—
10#16861124—22no—
11C2×#41×C127C2211yescharacteristic
12C4×Q8×C127—21no—
13C4×D4×C127—22no—
14C4×D4×C127—21no—
15#117×C127C412yescomplemented
16#117×C127C2211yescharacteristic
17C2×C42×C127C412yescomplemented
18C2×C42×C127C2211yescharacteristic
19#14523232—22no—
20#14523232—22no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—22no—
23C22⋊Dic127—22no—
24C4×Dic127—22no—
25#41×C127—22no—
26#41×C127—22no—
27C2×Q8×C127D411yescharacteristic
28C2×D4×C127D412yes—
29C2×D4×C127D411yescharacteristic
30C22×C4×C127—22no—
31C22×C4×C127—21no—
32C22×C4×C127C2×C412yes—
33C22×C4×C127C2311yescharacteristic
34C4⋊C4×C127—22no—
35C4⋊C4×C127—21no—
36C22⋊C4×C127—22no—
37C22⋊C4×C127—21no—
38C42×C127—22no—
39C42×C127—21no—
40C42×C127C2×C411yescharacteristic
41C42×C127C2×C412yes—
42C42×C127C2×C411yescharacteristic
43C2×Dic127—42no—
44C127⋊C8—42no—
45C23×C127C22⋊C412yes—
46C23×C127C2×D411yescharacteristic
47Q8×C127—22no—
48D4×C127—24no—
49D4×C127—22no—
50C2×C4×C127—21no—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—21no—
55C2×C4×C127—42no—
56C2×C4×C127—44no—
57C2×C4×C127C22⋊C412yes—
58C2×C4×C127C22×C411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4112yes—
61C2×C4×C127#4112yes—
62C22×C127—22no—
63C22×C127—41no—
64C22×C127—42no—
65C22×C127C2×C22⋊C411yescharacteristic
66C22×C127#12312yes—
67C4×C127—22no—
68C4×C127—22no—
69C4×C127—42no—
70C4×C127—41no—
71C4×C127#11711yescharacteristic
72C4×C127#11711yescharacteristic
73C4×C127#12312yes—
74Dic127—82no—
75C2×C127—22no—
76C2×C127—21no—
77C2×C127—42no—
78C2×C127#38111yescharacteristic
79#1827—1271nomaximal, Sylow
80C127#182711yescharacteristic, complemented, Sylow
81C4×#41D12711yescharacteristic
82#413—1271no—
83#410—1271no—
84#339—1272no—
85H(3,ℤ/4)—1272no—
86C2×#41C2×D12711yescharacteristic
87C2×M16—1272no—
88C4×Q8—21no—
89C4×D4—22no—
90C4×D4—21no—
91#117—1272no—
92#117Dic12712yescomplemented
93#117C2×D12711yescharacteristic
94C2×C42Dic12712yescomplemented
95C2×C42C2×D12711yescharacteristic
96#101—2541no—
97#100—2541no—
98#99—2542no—
99#41—22no—
100#41—22no—
101C2×Q8C22⋊D12711yescharacteristic
102C2×D4C22⋊D12712yes—
103C2×D4C22⋊D12711yescharacteristic
104C22×C4—22no—
105C22×C4—21no—
106C22×C4C2×Dic12712yes—
107C22×C4C22×D12711yescharacteristic
108M16—2542no—
109M16—2542no—
110C2×C8—2542no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—2542no—
114C22⋊C4—22no—
115C22⋊C4—21no—
116C22⋊C4—2542no—
117C42—22no—
118C42—21no—
119C42—2542no—
120C42C2×Dic12711yescharacteristic
121C42C2×Dic12712yes—
122C42C2×Dic12711yescharacteristic
123C23C22⋊Dic12712yes—
124C23C2×C22⋊D12711yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—5082no—
136C2×C4C22⋊Dic12712yes—
137C2×C4C22×Dic12711yescharacteristic
138C2×C4C2×C22⋊D12711yescharacteristic
139C2×C4#1452324812yes—
140C2×C4#1452324812yes—
141C8—5082no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×C22⋊Dic12711yescharacteristic
146C22#1686119212yes—
147C4—22no—
148C4—22no—
149C4—42no—
150C4—41no—
151C4—10162no—
152C4#1686116811yescharacteristic
153C4#1686116811yescharacteristic
154C4#1686119212yes—
155C2—21no—
156C2—22no—
157C2—42no—
158C2#2254973911yescharacteristic
subgroupG/H
centerC4#16861168
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC4×#41×C127C2
socleC2×C127#381
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC4×#41D127
hypocenterC127#1827
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ #117 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
3. #22549636C2semidirect
4. #22549632C2semidirect
15. #117×C127C4semidirect
17. C2×C42×C127C4semidirect
80. C127#1827semidirect
92. #117Dic127semidirect
94. C2×C42Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498602
#225498502
#225496362
#225496322
C4×#41×C1272
#1827127