Finite Group Databasebeta

← #30638052

Subgroups — #30638052

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#22550620C211yesmaximal, characteristic
2#22550244C211yesmaximal, characteristic, complemented
3#22550238C211yesmaximal, characteristic, complemented
4#22549617C211yesmaximal, characteristic, complemented
5#22549616C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#489×C127C211yesmaximal, characteristic, complemented
8#16861201C2211yescharacteristic
9#16861197C2211yescharacteristic
10#16861193—21no—
11#16861191—21no—
12#16861186—21no—
13#16861185—21no—
14#16861182—21no—
15#16861181—21no—
16#16861133—21no—
17#16861133—21no—
18#16861132—21no—
19#16861129—21no—
20#16861120C2211yescharacteristic
21#16861120C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#124×C127C2211yescharacteristic
24#103×C127—21no—
25#102×C127—21no—
26#97×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C22×Dic127—21no—
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37C4⋊Dic127D411yescharacteristic
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127D411yescharacteristic
47C2×D4×C127D411yescharacteristic
48C2×C8×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C42×C127C2311yescharacteristic
54C22×D127—21no—
55Dic254—41no—
56D508—41no—
57C22⋊D127—81no—
58C22⋊D127—81no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—81no—
63C4×D127—81no—
64C127⋊C8—41no—
65C127⋊C8—41no—
66C23×C127—21no—
67Q8×C127—41no—
68Q8×C127—42no—
69D4×C127—41no—
70C2×C4×C127—21no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C8×C127—41no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127#12011yescharacteristic
80C2×D127—41no—
81C2×D127—41no—
82C2×D127—41no—
83C4×C127—21no—
84C4×C127—21no—
85C4×C127—21no—
86C4×C127—41no—
87C4×C127—81no—
88Dic127—41no—
89Dic127—41no—
90Dic127—41no—
91C2×C127—81no—
92C2×C127#45411yescharacteristic
93C2×C127#45511yescharacteristic
94C2×C127#45511yescharacteristic
95D127—81no—
96#1662—1271nomaximal, Sylow
97C127#166211yescharacteristic, complemented, Sylow
98#562—1271no—
99#491—1271no—
100#489—1271no—
101#489D12711yescharacteristic, complemented
102#330—1271no—
103#329—1271no—
104#329—1271no—
105C4⋊Q8C2×D12711yescharacteristic
106#125—1271no—
107#124—1271no—
108#124C2×D12711yescharacteristic
109#123—2541no—
110#123—2541no—
111C22⋊Q8—2541no—
112C22⋊Q8—2541no—
113#121—2541no—
114#121—2541no—
115#103—21no—
116#103—2541no—
117#103—2541no—
118#103—2541no—
119#102—21no—
120#102—2541no—
121#97—1271no—
122#97—1271no—
123#97C2×D12711yescharacteristic
124C2×Q8—21no—
125C2×Q8—1271no—
126C2×Q8C22⋊D12711yescharacteristic
127C2×D4—1271no—
128C2×D4—2541no—
129C2×D4—2541no—
130C2×D4C22⋊D12711yescharacteristic
131C22×C4—2541no—
132C22×C4—2541no—
133C2×C8—21no—
134C2×C8—2541no—
135C2×C8—2541no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1271no—
139C4⋊C4—1271no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C4⋊C4—2541no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—2541no—
147C22⋊C4—2541no—
148C22⋊C4—2541no—
149C22⋊C4—2541no—
150C42C22×D12711yescharacteristic
151C23—21no—
152C23—2541no—
153Q8—41no—
154Q8—42no—
155Q8—5081no—
156D4—41no—
157D4—10161no—
158D4—10161no—
159D4—5081no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—10161no—
163C2×C4—10161no—
164C2×C4—2541no—
165C2×C4—2541no—
166C2×C4—2541no—
167C2×C4D4×D12711yescharacteristic
168C2×C4D4×D12711yescharacteristic
169C2×C4C2×C22⋊D12711yescharacteristic
170C8—41no—
171C8—5081no—
172C8—5081no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—5081no—
177C22—5081no—
178C22—5081no—
179C22#1686117711yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—81no—
185C4—5081no—
186C4—5081no—
187C4—5081no—
188C2—81no—
189C2—10161no—
190C2#2255004711yescharacteristic
191C2#2255005211yescharacteristic
192C2#2255005311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#489×C127C2
socleC22×C127#120
prime radical#22550620C2
hypercenter#489D127
hypocenterC127#1662
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #22550244C2semidirect
3. #22550238C2semidirect
4. #22549617C2semidirect
5. #22549616C2semidirect
6. #22549615C2semidirect
7. #489×C127C2semidirect
97. C127#1662semidirect
101. #489D127semidirect

Maximal subgroups

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

subgroupindex
#225506202
#225502442
#225502382
#225496172
#225496162
#225496152
#489×C1272
#1662127