Finite Group Databasebeta

← #30651306

Subgroups — #30651306

Order 16256.

336 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550567C212yesmaximal
2#22550278C211yesmaximal, characteristic, complemented
3#22550216C212yesmaximal, complemented
4#22550210C212yesmaximal, complemented
5#22550183C212yesmaximal, complemented
6#22550131C212yesmaximal, complemented
7#22549955C211yesmaximal, characteristic
8#22549917C212yesmaximal, complemented
9C4×Q16×C127C211yesmaximal, characteristic, complemented
10C2×#14523231C2212yes—
11#16861203—22no—
12#16861201C2212yes—
13#16861199—22no—
14#16861194—22no—
15#16861185C2212yes—
16#16861185C2212yes—
17C4×D508C2211yescharacteristic
18#16861167C2212yes—
19#16861166—22no—
20C4⋊C4×D127C2212yes—
21#16861143C2211yescharacteristic
22#16861143C2211yescharacteristic
23#16861142C2212yes—
24#16861141C2212yes—
25#16861140C2212yes—
26C4⋊(C127⋊C8)C2211yescharacteristic
27#16861133C2212yes—
28#16861133C2212yes—
29(C2×D127)⋊C8C2212yes—
30C2×Q16×C127C2211yescharacteristic
31C4×Q8×C127C2212yes—
32#107×C127C2211yescharacteristic
33#103×C127C2212yes—
34C4×C8×C127C2211yescharacteristic
35C2×D508D411yescharacteristic
36C2×C4×D127—22no—
37C2×C4×D127C2312yes—
38#14523231—24no—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127D411yescharacteristic
41C4⋊Dic127C2312yes—
42C4⋊Dic127C2312yes—
43Dic127⋊C4—22no—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48(C2×D127)⋊C4—22no—
49(C2×D127)⋊C4—22no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4D412yes—
52C4×Dic127—22no—
53C2×Q8×C127C2312yes—
54Q16×C127—22no—
55C2×C8×C127C2311yescharacteristic
56C2×C8×C127C2311yescharacteristic
57C4⋊C4×C127—22no—
58C4⋊C4×C127C2312yes—
59C42×C127—22no—
60C42×C127C2311yescharacteristic
61C22×D127C2×D412yes—
62D508—41no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127C2×D412yes—
66C4×D127—22no—
67C4×D127—22no—
68C4×D127—44no—
69C127⋊C8—22no—
70Q8×C127—22no—
71Q8×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2411yescharacteristic
77C8×C127—21no—
78C8×C127#4112yes—
79C22×C127C22×D411yescharacteristic
80C2×D127—22no—
81C2×D127—22no—
82C2×D127—22no—
83C4×C127—21no—
84C4×C127—21no—
85C4×C127—42no—
86C4×C127—42no—
87C4×C127C2×#4111yescharacteristic
88C4×C1272−1+411yescharacteristic
89Dic127—22no—
90Dic127—42no—
91Dic127—42no—
92C2×C127#54711yescharacteristic
93C2×C127#57511yescharacteristic
94C2×C127#57811yescharacteristic
95D127—42no—
96#3352—1271nomaximal, Sylow
97C127#335211yescharacteristic, complemented, Sylow
98#552—1272no—
99#499—1271no—
100#485—1272no—
101#484—1272no—
102#479—1272no—
103#470—1272no—
104C4×Q16D12711yescharacteristic, complemented
105#436—1271no—
106#428—1272no—
107C2×Q16C2×D12711yescharacteristic
108C2×M16—1272no—
109#126—2542no—
110#125—1272no—
111#125—2542no—
112#123—2542no—
113C22⋊Q8—1272no—
114C22⋊Q8—1272no—
115C4×Q8C2×D12712yes—
116C4×D4—1271no—
117#117—1272no—
118#117—2542no—
119C2×C4⋊C4—1272no—
120#107—1271no—
121#107—1271no—
122#107—1272no—
123#107C2×D12711yescharacteristic
124#106—1272no—
125#106—1272no—
126C4⋊C8—1271no—
127#103—1272no—
128#103—1272no—
129#103C2×D12712yes—
130C22⋊C8—1272no—
131C4×C8C2×D12711yescharacteristic
132C2×Q8C22×D12712yes—
133C2×D4—1271no—
134C22×C4—1272no—
135C22×C4—2542no—
136Q16—22no—
137M16—2544no—
138C2×C8—1272no—
139C2×C8C22×D12711yescharacteristic
140C2×C8C22×D12711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—1271no—
143C4⋊C4—1272no—
144C4⋊C4—1272no—
145C4⋊C4—2542no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4—2542no—
149C4⋊C4—2542no—
150C4⋊C4C22×D12712yes—
151C22⋊C4—1272no—
152C22⋊C4—2542no—
153C22⋊C4—2542no—
154C22⋊C4—2542no—
155C42—22no—
156C42—2542no—
157C42C22×D12711yescharacteristic
158C23—1272no—
159Q8—22no—
160Q8—22no—
161D4—5081no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—1272no—
165C2×C4—5084no—
166C2×C4—2542no—
167C2×C4—2542no—
168C2×C4—2542no—
169C2×C4—2542no—
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C2×C4C23×D12711yescharacteristic
173C8—21no—
174C8—2542no—
175C8#1452324812yes—
176C22—2542no—
177C22—2542no—
178C22—2542no—
179C22C2×D4×D12711yescharacteristic
180C4—21no—
181C4—21no—
182C4—42no—
183C4—42no—
184C4—5082no—
185C4—5082no—
186C4—2542no—
187C4C2×#1452324811yescharacteristic
188C4#1686125611yescharacteristic
189C2—5082no—
190C2#2255053111yescharacteristic
191C2#2255067311yescharacteristic
192C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×Q16×C127C2
socleC22×C127C22×D4
prime radicalC2×D508D4
hypercenterC4×Q16D127
hypocenterC127#3352
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #22550278C2semidirect
3. #22550216C2semidirect
4. #22550210C2semidirect
5. #22550183C2semidirect
6. #22550131C2semidirect
8. #22549917C2semidirect
9. C4×Q16×C127C2semidirect
97. C127#3352semidirect
104. C4×Q16D127semidirect

Maximal subgroups

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

subgroupindex
#225505672
#225502782
#225502162
#225502102
#225501832
#225501312
#225499552
#225499172
C4×Q16×C1272
#3352127