Finite Group Databasebeta

← #30641442

Subgroups — #30641442

Order 16256.

612 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 205 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×#16861196C211yesmaximal, characteristic
2C2×C4×Dic254C211yesmaximal, characteristic, complemented
3#22549781C211yesmaximal, characteristic, complemented
4#22549779C211yesmaximal, characteristic
5#22549764C214yesmaximal, complemented
6#22549749C214yesmaximal, complemented
7#22549745C211yesmaximal, characteristic, complemented
8#22549722C211yesmaximal, characteristic, complemented
9C2×C4×D4×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861196C418yescomplemented
12C4×Dic254—24no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4C2212yes—
15C2×Dic127⋊C4C2214yes—
16C2×C22⋊Dic127C2214yes—
17C2×C22⋊Dic127C2214yes—
18C2×C4×Dic127C2212yes—
19C2×C4×Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21C127⋊#95C2212yes—
22C127⋊#95C2214yes—
23C127⋊#95C2212yes—
24C22×D4×C127C2211yescharacteristic
25C23×C4×C127C2212yes—
26C4×D4×C127—24no—
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2212yes—
29C2×C42×C127C2211yescharacteristic
30C2×Dic254—22no—
31C2×Dic254—22no—
32C2×Dic254C2×C414yescomplemented
33C22×Dic127C2312yes—
34C22×Dic127C2312yes—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C4⋊Dic127—22no—
38Dic127⋊C4—24no—
39Dic127⋊C4—44no—
40C22⋊Dic127—44no—
41C22⋊Dic127C2×C4116yescomplemented
42C4×Dic127—24no—
43C4×Dic127—22no—
44C4×Dic127C2×C414yescomplemented
45C24×C127C2312yes—
46C2×D4×C127—22no—
47C2×D4×C127—22no—
48C2×D4×C127C2×C414yes—
49C22×C4×C127—24no—
50C22×C4×C127—44no—
51C22×C4×C127C2312yes—
52C22×C4×C127C2311yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C22⋊C4×C127—24no—
57C42×C127D414yes—
58Dic254—28no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—24no—
67C2×Dic127—22no—
68C2×Dic127C22×C414yes—
69C2×Dic127C22×C414yes—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—24no—
73C23×C127—24no—
74C23×C127C22×C414yes—
75C23×C127C2411yescharacteristic
76D4×C127—28no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—24no—
81C2×C4×C127—48no—
82C2×C4×C127C22×C412yes—
83C2×C4×C127C2×D412yes—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D412yes—
86C2×C4×C127#4112yes—
87C22×C127—24no—
88C22×C127—28no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127—44no—
92C22×C127C23×C411yescharacteristic
93C22×C127C22×D411yescharacteristic
94C22×C127C2×#4111yescharacteristic
95C22×C127H(5,𝔽2)12yes—
96C22×C1272−1+412yes—
97C4×C127—22no—
98C4×C127—22no—
99C4×C127—42no—
100C4×C127C4×D414yes—
101Dic127—24no—
102Dic127—24no—
103Dic127—42no—
104Dic127—42no—
105C2×C127—44no—
106C2×C127C2×C4×D411yescharacteristic
107C2×C127#52012yes—
108C2×C127#53511yescharacteristic
109C2×C127#53611yescharacteristic
110C2×C127#54212yes—
111#2326—1271nomaximal, Sylow
112C127#232611yescharacteristic, complemented, Sylow
113C2×#124—1271no—
114C2×C4×Q8—1271no—
115C2×C4×D4D12711yescharacteristic
116#391—1271no—
117#390—1271no—
118#388—1274no—
119#385—1274no—
120#384—1271no—
121#376—1271no—
122C22×Q8—1271no—
123C22×D4C2×D12711yescharacteristic
124C23×C4C2×D12712yes—
125#124—1278no—
126C4×Q8—2544no—
127C4×D4—24no—
128C2×C4⋊C4—1274no—
129C2×C4⋊C4—1271no—
130C2×C4⋊C4—1272no—
131C2×C4⋊C4C2×D12711yescharacteristic
132C2×C22⋊C4—1274no—
133C2×C22⋊C4—1274no—
134C2×C22⋊C4C2×D12712yes—
135C2×C42—1271no—
136C2×C42—1272no—
137C2×C42—1271no—
138C2×C42C2×D12711yescharacteristic
139#95—1272no—
140#95—1272no—
141#95—1274no—
142C24C22×D12712yes—
143C2×Q8—1274no—
144C2×Q8—2542no—
145C2×Q8—2542no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4C4×D12714yes—
149C22×C4—24no—
150C22×C4—44no—
151C22×C4—1272no—
152C22×C4—1272no—
153C22×C4—1272no—
154C22×C4—1272no—
155C22×C4C22×D12712yes—
156C22×C4C22×D12711yescharacteristic
157C22×C4C22×D12711yescharacteristic
158C22×C4C22×D12711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—5084no—
161C4⋊C4—2544no—
162C4⋊C4—2542no—
163C22⋊C4—24no—
164C22⋊C4—12716no—
165C22⋊C4—5084no—
166C42—1274no—
167C42—2544no—
168C42—2542no—
169C42C22⋊D12714yes—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—24no—
174C23C2×C4×D12714yes—
175C23C23×D12711yescharacteristic
176Q8—2548no—
177D4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—48no—
183C2×C4—1274no—
184C2×C4—1274no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4—2544no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4C2×C4×D12712yes—
195C2×C4C2×C22⋊D12712yes—
196C2×C4C2×C22⋊D12712yes—
197C2×C4C2×C22⋊D12712yes—
198C2×C4#1452324812yes—
199C22—24no—
200C22—28no—
201C22—42no—
202C22—42no—
203C22—44no—
204C22C22×C4×D12711yescharacteristic
205C22C22×C22⋊D12711yescharacteristic
206C22C2×#1452324811yescharacteristic
207C22#1686125412yes—
208C22#1686125612yes—
209C4—22no—
210C4—22no—
211C4—42no—
212C4—5082no—
213C4—5082no—
214C4—2544no—
215C4—2544no—
216C4C4×C22⋊D12714yes—
217C2—44no—
218C2C2×C4×C22⋊D12711yescharacteristic
219C2#2255035512yes—
220C2#2255044311yescharacteristic
221C2#2255044711yescharacteristic
222C2#2255049112yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×C4×D4×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D4×C127C22
hypercenterC2×C4×D4D127
hypocenterC127#2326
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic254C2semidirect
3. #22549781C2semidirect
5. #22549764C2semidirect
6. #22549749C2semidirect
7. #22549745C2semidirect
8. #22549722C2semidirect
11. #16861196C4semidirect
32. C2×Dic254C2×C4semidirect
41. C22⋊Dic127C2×C4semidirect
44. C4×Dic127C2×C4semidirect
112. C127#2326semidirect

Maximal subgroups

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

subgroupindex
C2×#168611962
C2×C4×Dic2542
#225497812
#225497792
#225497642
#225497492
#225497452
#225497222
C2×C4×D4×C1272
#2326127