Finite Group Databasebeta

← #30647085

Subgroups — #30647085

Order 16256.

752 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×#16861207C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic
3C2×#16861198C211yesmaximal, characteristic, complemented
4C2×#16861197C211yesmaximal, characteristic, complemented
5C2×#16861196C211yesmaximal, characteristic, complemented
6#22549776C218yesmaximal, complemented
7C42×Dic127C211yesmaximal, characteristic, complemented
8C2×#124×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×D508C2211yescharacteristic
11C22×C22⋊D127C2214yes—
12#16861207—24no—
13#16861205—24no—
14#16861198—24no—
15#16861197—24no—
16#16861196—24no—
17C2×Dic127⋊C4C2214yes—
18C2×C22⋊Dic127C2214yes—
19C2×(C2×D127)⋊C4C2214yes—
20C2×(C2×D127)⋊C4C2214yes—
21C2×C4×Dic127C2214yes—
22C2×C4×Dic127C2211yescharacteristic
23C2×C4×Dic127C2211yescharacteristic
24C22×Q8×C127C2211yescharacteristic
25C22×D4×C127C2211yescharacteristic
26#124×C127—24no—
27C2×C22⋊C4×C127C2214yes—
28C2×C42×C127C2211yescharacteristic
29C23×D127C2312yes—
30C2×Dic254—22no—
31C2×Dic254—22no—
32C2×Dic254—22no—
33C2×D508—22no—
34C2×D508—22no—
35C2×D508—22no—
36C2×C22⋊D127—28no—
37C2×C22⋊D127—216no—
38C22×Dic127C2312yes—
39C22×Dic127C2312yes—
40C22×Dic127C2312yes—
41Dic127⋊C4—28no—
42C22⋊Dic127—28no—
43(C2×D127)⋊C4—28no—
44(C2×D127)⋊C4—28no—
45C4×Dic127—28no—
46C4×Dic127D414yes—
47C4×Dic127D414yescomplemented
48C24×C127C2312yes—
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127—22no—
52C2×D4×C127—22no—
53C2×D4×C127—22no—
54C2×D4×C127—22no—
55C22×C4×C127C2312yes—
56C22×C4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58C22⋊C4×C127—28no—
59C42×C127D414yescomplemented
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—22no—
63C22×D127—28no—
64Dic254—44no—
65D508—44no—
66C22⋊D127—416no—
67C2×Dic127—24no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127C2×D414yes—
72C2×Dic127C2×D414yes—
73C2×Dic127#4118yes—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—28no—
78C23×C127C2411yescharacteristic
79Q8×C127—44no—
80D4×C127—44no—
81C2×C4×C127—24no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2×D412yes—
87C2×C4×C127C2×D414yes—
88C2×C4×C127C2×D412yes—
89C22×C127—44no—
90C22×C127—42no—
91C22×C127—44no—
92C22×C127—44no—
93C22×C127C22×D411yescharacteristic
94C22×C127C22×D411yescharacteristic
95C22×C127C22×D411yescharacteristic
96C22×C127C2×#4114yes—
97C2×D127—44no—
98C2×D127—42no—
99C2×D127—44no—
100C2×D127—44no—
101C4×C127—24no—
102C4×C127—42no—
103C4×C127#12714yes—
104Dic127—24no—
105Dic127—42no—
106Dic127#12418yescomplemented
107C2×C127—82no—
108C2×C127C2×#12412yes—
109C2×C127C2×#12711yescharacteristic
110C2×C127#53212yes—
111C2×C127#53212yes—
112D127—82no—
113#2858—1271nomaximal, Sylow
114C127#285811yescharacteristic, complemented, Sylow
115C2×C4⋊Q8—1271no—
116C2×#127—1271no—
117C2×#124—1271no—
118C2×#124—1271no—
119C2×#124—1271no—
120C2×#124D12711yescharacteristic, complemented
121#390—1278no—
122C43—1271no—
123C22×Q8—1271no—
124C22×Q8C2×D12711yescharacteristic
125C22×D4—1274no—
126C22×D4—1271no—
127C22×D4C2×D12711yescharacteristic
128C4⋊Q8—2544no—
129#127—2544no—
130#124—24no—
131#124—2544no—
132#124—2544no—
133#124—2544no—
134C2×C4⋊C4—1274no—
135C2×C22⋊C4—1274no—
136C2×C22⋊C4—1274no—
137C2×C22⋊C4—1274no—
138C2×C22⋊C4C2×D12714yes—
139C2×C42—1271no—
140C2×C42—1271no—
141C2×C42—1274no—
142C2×C42C2×D12711yescharacteristic
143C24—1272no—
144C24C22×D12712yes—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—2542no—
149C2×Q8—2542no—
150C2×Q8—2542no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—2548no—
155C2×D4—25416no—
156C2×D4—2542no—
157C2×D4—2542no—
158C2×D4—2542no—
159C22×C4—1272no—
160C22×C4—1272no—
161C22×C4—1272no—
162C22×C4C22×D12712yes—
163C22×C4C22×D12712yes—
164C22×C4C22×D12711yescharacteristic
165C4⋊C4—2548no—
166C22⋊C4—28no—
167C22⋊C4—2548no—
168C22⋊C4—2548no—
169C22⋊C4—2548no—
170C42—1274no—
171C42—1274no—
172C42—2548no—
173C42C22⋊D12714yescomplemented
174C23—22no—
175C23—22no—
176C23—22no—
177C23—28no—
178C23—2542no—
179C23—2542no—
180C23—2542no—
181C23—2548no—
182C23C23×D12711yescharacteristic
183Q8—44no—
184Q8—5084no—
185D4—44no—
186D4—50816no—
187D4—5084no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—1274no—
193C2×C4—1278no—
194C2×C4—1274no—
195C2×C4—2542no—
196C2×C4—2542no—
197C2×C4—2542no—
198C2×C4—2544no—
199C2×C4D4×D12712yes—
200C2×C4D4×D12712yes—
201C2×C4C2×C22⋊D12714yes—
202C2×C4C2×C22⋊D12712yes—
203C22—44no—
204C22—42no—
205C22—44no—
206C22—44no—
207C22—5084no—
208C22—5082no—
209C22—5084no—
210C22—5084no—
211C22C2×D4×D12711yescharacteristic
212C22C2×D4×D12711yescharacteristic
213C22C22×C22⋊D12711yescharacteristic
214C22#41×D12714yes—
215C4—24no—
216C4—42no—
217C4—1278no—
218C4—5082no—
219C4—2544no—
220C4#1686120514yes—
221C2—82no—
222C2—10162no—
223C2#124×D12712yes—
224C2C2×#1686120511yescharacteristic
225C2#2255042212yes—
226C2#2255042512yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#124×C127C2
prime radicalC2×#16861205C2
hypercenterC2×#124D127
hypocenterC127#2858
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16861207C2semidirect
3. C2×#16861198C2semidirect
4. C2×#16861197C2semidirect
5. C2×#16861196C2semidirect
6. #22549776C2semidirect
7. C42×Dic127C2semidirect
8. C2×#124×C127C2semidirect
47. C4×Dic127D4semidirect
59. C42×C127D4semidirect
106. Dic127#124semidirect
114. C127#2858semidirect
120. C2×#124D127semidirect
173. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×#168612052
C2×#168611982
C2×#168611972
C2×#168611962
#225497762
C42×Dic1272
C2×#124×C1272
#2858127