Finite Group Databasebeta

← #30637892

Subgroups — #30637892

Order 16256.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550541C211yesmaximal, characteristic
2#22550168C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic, complemented
5#22549608C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic
7#487×C127C211yesmaximal, characteristic, complemented
8C2×Q8×D127—21no—
9#16861208—21no—
10#16861187—22no—
11#16861185—21no—
12C4×Dic254C2211yescharacteristic
13C4×D508C2211yescharacteristic
14#16861171—21no—
15#16861142—21no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861132—21no—
20#16861129—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#103×C127—22no—
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—22no—
26C2×Dic254D411yescharacteristic
27Q8×D127—42no—
28Q8×D127—42no—
29C2×D508D411yescharacteristic
30C2×C4×D127—22no—
31C2×C4×D127—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—21no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D412yes—
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50Dic254—21no—
51Dic254—44no—
52Dic254SD1612yes—
53D508—21no—
54D508Q1612yes—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C4×D127—44no—
59C4×D127—41no—
60C4×D127—41no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63Q8×C127—42no—
64Q8×C127—41no—
65Q8×C127—41no—
66C2×C4×C127—22no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—41no—
72C22×C127#12011yescharacteristic
73C2×D127—22no—
74C2×D127—41no—
75C2×D127—41no—
76C4×C127—21no—
77C4×C127—42no—
78C4×C127—81no—
79C4×C127C2×SD1611yescharacteristic
80C4×C127C2×Q1611yescharacteristic
81C4×C127AGL(1,ℤ/8)11yescharacteristic
82C4×C127#13711yescharacteristic
83Dic127—22no—
84Dic127—41no—
85Dic127—41no—
86C2×C127#45011yescharacteristic
87C2×C127#45111yescharacteristic
88C2×C127#45411yescharacteristic
89D127—42no—
90#1637—1271nomaximal, Sylow
91C127#163711yescharacteristic, complemented, Sylow
92D4×Q8—1271no—
93#487D12711yescharacteristic, complemented
94#477—1271no—
95#476—1271no—
96#332—1271no—
97Q8⋊C8—1271no—
98D4⋊C8—1271no—
99C22×Q8—2541no—
100C4⋊Q8—2541no—
101C4⋊Q8C2×D12711yescharacteristic
102C4⋊Q8C2×D12711yescharacteristic
103C22⋊Q8—2541no—
104C22⋊Q8—2542no—
105C4×Q8—1271no—
106C4×D4—1271no—
107C4×D4—2541no—
108#107—2541no—
109#106—2541no—
110C4⋊C8—1271no—
111C4⋊C8—1271no—
112#103—22no—
113#103—2541no—
114#102—2541no—
115C4×C8C2×D12711yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—1271no—
119C2×Q8—5082no—
120C2×Q8—5082no—
121C2×Q8—2542no—
122C2×D4—1271no—
123C22×C4—2542no—
124C22×C4—2541no—
125C2×C8—21no—
126C2×C8—2541no—
127C2×C8—2541no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1271no—
132C4⋊C4—1271no—
133C4⋊C4—2542no—
134C4⋊C4—2541no—
135C4⋊C4C22⋊D12712yes—
136C22⋊C4—2542no—
137C22⋊C4—2541no—
138C42—2541no—
139C42C22×D12711yescharacteristic
140C23—2541no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—1272no—
145Q8—5084no—
146Q8—2541no—
147D4—1272no—
148D4—2541no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—5084no—
152C2×C4—5081no—
153C2×C4—5081no—
154C2×C4—2541no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—41no—
161C8—5081no—
162C8—5081no—
163C22—5081no—
164C22—5081no—
165C22—2542no—
166C22#1686117711yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—5081no—
171C4—5081no—
172C4—2542no—
173C4SD16×D12711yescharacteristic
174C4Q16×D12711yescharacteristic
175C4#1686123611yescharacteristic
176C4#1686124111yescharacteristic
177C2—5082no—
178C2#2255002511yescharacteristic
179C2#2255003011yescharacteristic
180C2#2255004811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#487×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#487D127
hypocenterC127#1637
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #22550168C2semidirect
4. #22549622C2semidirect
5. #22549608C2semidirect
7. #487×C127C2semidirect
91. C127#1637semidirect
93. #487D127semidirect

Maximal subgroups

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

subgroupindex
#225505412
#225501682
#225501632
#225496222
#225496082
#225496052
#487×C1272
#1637127