Finite Group Databasebeta

← #30641440

Subgroups — #30641440

Order 16256.

612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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×#16861174C211yesmaximal, characteristic, complemented
3C2×D4×Dic127C211yesmaximal, characteristic
4#22549782C211yesmaximal, characteristic, complemented
5#22549770C214yesmaximal, complemented
6#22549748C214yesmaximal, complemented
7#22549745C211yesmaximal, characteristic, complemented
8#22549722C211yesmaximal, characteristic, complemented
9#390×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11C23×Dic127C2212yes—
12#16861196C418yescomplemented
13#16861174—24no—
14D4×Dic127—24no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4C2212yes—
17C2×Dic127⋊C4C2214yes—
18C2×C22⋊Dic127C2212yes—
19C2×C22⋊Dic127C2214yes—
20C2×C4×Dic127C2212yes—
21C2×C4×Dic127C2211yescharacteristic
22C2×C4×Dic127C2211yescharacteristic
23C127⋊#95C2212yes—
24C127⋊#95C2214yes—
25C127⋊#95C2212yes—
26C22×D4×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2214yes—
29C2×C42×C127C2211yescharacteristic
30C2×Dic254—22no—
31C2×Dic254—22no—
32C2×Dic254C2×C414yescomplemented
33C22×Dic127—24no—
34C22×Dic127—44no—
35C22×Dic127C2312yes—
36C22×Dic127C2312yes—
37C22×Dic127C2312yes—
38C22×Dic127C2311yescharacteristic
39C22×Dic127C2311yescharacteristic
40C4⋊Dic127—22no—
41Dic127⋊C4—24no—
42Dic127⋊C4—44no—
43C22⋊Dic127—24no—
44C22⋊Dic127C2×C4116yescomplemented
45C4×Dic127—24no—
46C4×Dic127C2×C414yescomplemented
47C4×Dic127D414yes—
48C24×C127C2312yes—
49C2×D4×C127—22no—
50C2×D4×C127—22no—
51C2×D4×C127C2×C414yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C22⋊C4×C127—44no—
57C42×C127—22no—
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—48no—
68C2×Dic127C22×C414yes—
69C2×Dic127C22×C414yes—
70C2×Dic127C2×D412yes—
71C2×Dic127C2×D412yes—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—24no—
75C23×C127—24no—
76C23×C127C22×C414yes—
77C23×C127C2411yescharacteristic
78D4×C127—28no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—24no—
83C2×C4×C127—22no—
84C2×C4×C127C22×C412yes—
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—42no—
98C4×C127—42no—
99C4×C127C4×D414yes—
100Dic127—24no—
101Dic127—24no—
102Dic127—22no—
103Dic127—22no—
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×D4—1271no—
116#391—1271no—
117#390D12711yescharacteristic
118#388—1274no—
119#385—1274no—
120#384—1271no—
121#376—1271no—
122C22×Q8—1271no—
123C22×D4C2×D12711yescharacteristic
124C23×C4—1272no—
125#124—1278no—
126C4×Q8—2544no—
127C4×D4—2544no—
128C2×C4⋊C4—1274no—
129C2×C4⋊C4—1271no—
130C2×C4⋊C4—1272no—
131C2×C4⋊C4C2×D12711yescharacteristic
132C2×C22⋊C4—1274no—
133C2×C22⋊C4—1272no—
134C2×C22⋊C4C2×D12714yes—
135C2×C42—1271no—
136C2×C42—1271no—
137C2×C42—1272no—
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—1271no—
150C22×C4—1272no—
151C22×C4—1272no—
152C22×C4—1271no—
153C22×C4—1272no—
154C22×C4—5084no—
155C22×C4—2544no—
156C22×C4C22×D12712yes—
157C22×C4C22×D12712yes—
158C22×C4C22×D12711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—5084no—
161C4⋊C4—2544no—
162C4⋊C4—2542no—
163C22⋊C4—44no—
164C22⋊C4—12716no—
165C22⋊C4—2544no—
166C42—22no—
167C42—1274no—
168C42—1274no—
169C42—2544no—
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—22no—
183C2×C4—1274no—
184C2×C4—1274no—
185C2×C4—1272no—
186C2×C4—1272no—
187C2×C4—5088no—
188C2×C4—2542no—
189C2×C4—2544no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4—2542no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4C2×C4×D12712yes—
197C2×C4D4×D12712yes—
198C2×C4#1452324712yes—
199C22—24no—
200C22—28no—
201C22—42no—
202C22—42no—
203C22—44no—
204C22C22×C4×D12711yescharacteristic
205C22C2×D4×D12711yescharacteristic
206C22C2×#1452324711yescharacteristic
207C22#1686125412yes—
208C22#1686125612yes—
209C4—42no—
210C4—42no—
211C4—5082no—
212C4—2544no—
213C4—2544no—
214C4—2542no—
215C4—2542no—
216C4(C22⋊D127)⋊C414yes—
217C2—44no—
218C2C2×(C22⋊D127)⋊C411yescharacteristic
219C2#2255035512yes—
220C2#2255043911yescharacteristic
221C2#2255044611yescharacteristic
222C2#2255049312yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#390×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D4×C127C22
hypercenter#390D127
hypocenterC127#2326
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16861174C2semidirect
4. #22549782C2semidirect
5. #22549770C2semidirect
6. #22549748C2semidirect
7. #22549745C2semidirect
8. #22549722C2semidirect
12. #16861196C4semidirect
32. C2×Dic254C2×C4semidirect
44. C22⋊Dic127C2×C4semidirect
46. 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×#168611742
C2×D4×Dic1272
#225497822
#225497702
#225497482
#225497452
#225497222
#390×C1272
#2326127