Finite Group Databasebeta

← #30638040

Subgroups — #30638040

Order 16256.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550630C211yesmaximal, characteristic
2#22550282C211yesmaximal, characteristic, complemented
3#22550268C211yesmaximal, characteristic
4#22549618C211yesmaximal, characteristic, complemented
5#22549617C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic
7#501×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9C2×#14523235—21no—
10#16861208—21no—
11#16861204—21no—
12#16861204—21no—
13#16861201C2211yescharacteristic
14#16861197C2211yescharacteristic
15#16861194—21no—
16#16861189—21no—
17#16861187—21no—
18#16861142—21no—
19#16861140—21no—
20#16861120C2211yescharacteristic
21#16861120C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#125×C127C2211yescharacteristic
24#107×C127—21no—
25#106×C127—21no—
26#97×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C4×D127—21no—
31#14523238—42no—
32#14523235—42no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127—21no—
47C2×Q8×C127—21no—
48C2×C8×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127D411yescharacteristic
54C4⋊C4×C127D411yescharacteristic
55C42×C127C2311yescharacteristic
56C22×D127—21no—
57Dic254—41no—
58Dic254—81no—
59D508—41no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C4×D127—81no—
64C127⋊C8—22no—
65C127⋊C8—22no—
66Q8×C127—41no—
67Q8×C127—41no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—22no—
75C22×C127#12011yescharacteristic
76C2×D127—41no—
77C2×D127—41no—
78C2×D127—41no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127—41no—
84C4×C127—81no—
85Dic127—41no—
86Dic127—41no—
87Dic127—41no—
88C2×C127#45311yescharacteristic
89C2×C127#45611yescharacteristic
90C2×C127#45611yescharacteristic
91D127—81no—
92#1660—1271nomaximal, Sylow
93C127#166011yescharacteristic, complemented, Sylow
94#563—1271no—
95#501—1271no—
96#501D12711yescharacteristic, complemented
97#497—1271no—
98#330—1271no—
99#330—1271no—
100#329—1271no—
101C2×Q16—2541no—
102C2×SD16—2541no—
103C4⋊Q8—2541no—
104C4⋊Q8C2×D12711yescharacteristic
105#126—2541no—
106#126—2541no—
107#125—1271no—
108#125C2×D12711yescharacteristic
109#124—1271no—
110#123—2541no—
111C22⋊Q8—2541no—
112C22⋊Q8—2541no—
113#107—21no—
114#107—2541no—
115#106—21no—
116#106—2541no—
117#97—1271no—
118#97—1271no—
119#97C2×D12711yescharacteristic
120C2×Q8—21no—
121C2×Q8—1271no—
122C2×Q8—2541no—
123C2×D4—1271no—
124C22×C4—2541no—
125Q16—5082no—
126SD16—5082no—
127C2×C8—21no—
128C2×C8—2541no—
129C2×C8—2541no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1271no—
135C4⋊C4—1271no—
136C4⋊C4—2541no—
137C4⋊C4—2541no—
138C4⋊C4—2541no—
139C4⋊C4—2541no—
140C4⋊C4C22⋊D12711yescharacteristic
141C4⋊C4C22⋊D12711yescharacteristic
142C22⋊C4—2541no—
143C22⋊C4—2541no—
144C22⋊C4—2541no—
145C22⋊C4—2541no—
146C22⋊C4—2541no—
147C42—2541no—
148C42C22×D12711yescharacteristic
149C23—2541no—
150Q8—41no—
151Q8—41no—
152Q8—10161no—
153Q8—5081no—
154D4—5081no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—10161no—
159C2×C4—2541no—
160C2×C4—2541no—
161C2×C4—2541no—
162C2×C4D4×D12711yescharacteristic
163C2×C4D4×D12711yescharacteristic
164C2×C4C2×C22⋊D12711yescharacteristic
165C8—22no—
166C8—2542no—
167C8—2542no—
168C22—5081no—
169C22—5081no—
170C22—5081no—
171C22#1686117711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—5081no—
179C4—5081no—
180C4—5081no—
181C2—10161no—
182C2#2255004211yescharacteristic
183C2#2255005711yescharacteristic
184C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#501×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#501D127
hypocenterC127#1660
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #22550282C2semidirect
4. #22549618C2semidirect
5. #22549617C2semidirect
7. #501×C127C2semidirect
93. C127#1660semidirect
96. #501D127semidirect

Maximal subgroups

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

subgroupindex
#225506302
#225502822
#225502682
#225496182
#225496172
#225496152
#501×C1272
#1660127