Finite Group Databasebeta

← #30649107

Subgroups — #30649107

Order 16256.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550431C211yesmaximal, characteristic
2C4⋊Q8×D127C211yesmaximal, characteristic
3#22550243C214yesmaximal, complemented
4#22550228C211yesmaximal, characteristic, complemented
5#22550227C211yesmaximal, characteristic, complemented
6#22550226C211yesmaximal, characteristic, complemented
7#22549883C214yesmaximal
8C4×#14523231C211yesmaximal, characteristic
9#487×C127C211yesmaximal, characteristic, complemented
10C2×Q8×D127C2212yes—
11C2×#14523231C2212yes—
12#16861209C2211yescharacteristic
13#16861209C2211yescharacteristic
14#16861208—21no—
15#16861208—21no—
16#16861207C2212yes—
17#16861200C2212yes—
18#16861189—21no—
19#16861185—21no—
20Q8×Dic127—21no—
21#16861174—21no—
22#16861167C2212yes—
23C4⋊C4×D127—21no—
24C4⋊C4×D127—21no—
25C42×D127C2211yescharacteristic
26#16861134C2214yes—
27#16861133C2214yes—
28#16861132C2214yes—
29#16861119C2212yes—
30C4×C127⋊C8C2211yescharacteristic
31C4⋊Q8×C127C2211yescharacteristic
32C4⋊Q8×C127C2211yescharacteristic
33#103×C127C2214yes—
34C4×C8×C127C2211yescharacteristic
35C2×Dic254—22no—
36C2×Dic254—21no—
37C2×Dic254C2312yes—
38Q8×D127—28no—
39C2×C4×D127—22no—
40C2×C4×D127D411yescharacteristic
41C2×C4×D127D411yescharacteristic
42C2×C4×D127C2311yescharacteristic
43#14523231—24no—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—22no—
46C4⋊Dic127—21no—
47C4⋊Dic127—21no—
48C4⋊Dic127—21no—
49C4⋊Dic127C2312yes—
50Dic127⋊C4—22no—
51Dic127⋊C4—22no—
52Dic127⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—22no—
55C4×Dic127D411yescharacteristic
56C4×Dic127D411yescharacteristic
57C4×Dic127C2311yescharacteristic
58C2×Q8×C127—21no—
59C2×Q8×C127C2312yes—
60C2×C8×C127C2312yes—
61C4⋊C4×C127—21no—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127C2312yes—
65C42×C127C2311yescharacteristic
66C22×D127C2×D411yescharacteristic
67Dic254—24no—
68Dic254—42no—
69Dic254—44no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127C2×D411yescharacteristic
73C2×Dic127C2×D411yescharacteristic
74C2×Dic127C2×D411yescharacteristic
75C4×D127—22no—
76C4×D127—22no—
77C4×D127—44no—
78C4×D127#4114yes—
79C127⋊C8—22no—
80Q8×C127—24no—
81Q8×C127—42no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2411yescharacteristic
87C8×C127—22no—
88C22×C127C22×D411yescharacteristic
89C2×D127—21no—
90C2×D127—21no—
91C2×D127#12412yes—
92C4×C127—42no—
93C4×C127—42no—
94C4×C127#13712yes—
95C4×C127#13712yes—
96C4×C127C2×#4112yes—
97Dic127—21no—
98Dic127—21no—
99Dic127—42no—
100Dic127—42no—
101Dic127#12412yes—
102C2×C127C2×#12411yescharacteristic
103C2×C127C2×#13711yescharacteristic
104C2×C127C2×#13711yescharacteristic
105D127—22no—
106#3139—1271nomaximal, Sylow
107C127#313911yescharacteristic, complemented, Sylow
108#533—1271no—
109C2×C4⋊Q8—1271no—
110#491—1274no—
111#487—1271no—
112#487—1271no—
113#487—1271no—
114#487D12711yescharacteristic, complemented
115#419—1274no—
116C4×M16—1271no—
117C22×Q8—1272no—
118C2×M16—1272no—
119C4⋊Q8—1272no—
120C4⋊Q8—1271no—
121C4⋊Q8—1271no—
122C4⋊Q8—2541no—
123C4⋊Q8—2541no—
124C4⋊Q8C2×D12711yescharacteristic
125C4⋊Q8C2×D12711yescharacteristic
126#125—1272no—
127C22⋊Q8—2541no—
128C22⋊Q8—2541no—
129C4×Q8—2541no—
130C4×Q8—2541no—
131#117—1272no—
132C2×C4⋊C4—2541no—
133C2×C4⋊C4—2541no—
134C2×C42—1271no—
135#103—1274no—
136#103—1274no—
137#103—1274no—
138#103C2×D12714yes—
139#97—1272no—
140C4×C8—1271no—
141C4×C8C2×D12711yescharacteristic
142C2×Q8—21no—
143C2×Q8—1272no—
144C2×Q8—2542no—
145C2×Q8—2548no—
146C2×Q8—2541no—
147C2×Q8C22×D12712yes—
148C22×C4—1271no—
149C22×C4—1271no—
150C22×C4—1271no—
151C22×C4—2542no—
152M16—2544no—
153C2×C8—1272no—
154C2×C8C22×D12712yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—1272no—
159C4⋊C4—2542no—
160C4⋊C4—2542no—
161C4⋊C4—2542no—
162C4⋊C4—2542no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4C22×D12712yes—
167C22⋊C4—2542no—
168C42—1271no—
169C42—1271no—
170C42—1271no—
171C42—2542no—
172C42C22×D12711yescharacteristic
173C23—1271no—
174Q8—24no—
175Q8—42no—
176Q8—5084no—
177Q8—5082no—
178Q8—2544no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—1271no—
182C2×C4—1274no—
183C2×C4—1271no—
184C2×C4—1271no—
185C2×C4—5084no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4D4×D12711yescharacteristic
191C2×C4D4×D12711yescharacteristic
192C2×C4C23×D12711yescharacteristic
193C8—22no—
194C8—2542no—
195C22—1272no—
196C22—2541no—
197C22—2541no—
198C22C2×D4×D12711yescharacteristic
199C4—42no—
200C4—42no—
201C4—1272no—
202C4—5082no—
203C4—5082no—
204C4—2541no—
205C4—2541no—
206C4#1686123812yes—
207C4#1686124112yes—
208C4#41×D12712yes—
209C2—2542no—
210C2#124×D12711yescharacteristic
211C2C2×#1686123811yescharacteristic
212C2C2×#1686124111yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#487×C127C2
socleC22×C127C22×D4
prime radicalC22×D127C2×D4
hypercenter#487D127
hypocenterC127#3139
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #22550243C2semidirect
4. #22550228C2semidirect
5. #22550227C2semidirect
6. #22550226C2semidirect
9. #487×C127C2semidirect
107. C127#3139semidirect
114. #487D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
C4⋊Q8×D1272
#225502432
#225502282
#225502272
#225502262
#225498832
C4×#145232312
#487×C1272
#3139127