Finite Group Databasebeta

← C2×#22550201 #30648615

Subgroups — C2×#22550201 #30648615

Order 16256.

468 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861183C211yesmaximal, characteristic
2C2×#16861168C211yesmaximal, characteristic, complemented
3#22550201C218yesmaximal, direct product factor
4C2×#16861143C211yesmaximal, characteristic, complemented
5C2×#16861141C211yesmaximal, characteristic, complemented
6C2×#16861131C211yesmaximal, characteristic, complemented
7C2×#16861131C211yesmaximal, characteristic, complemented
8C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11#16861183—22no—
12#16861183C2214yes—
13#16861168—22no—
14#16861168C2214yescomplemented
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C22⋊Dic127—21no—
19C2×(C2×D127)⋊C4—21no—
20C2×C4×Dic127—21no—
21#16861143C2214yescomplemented
22#16861141C2214yescomplemented
23#16861131C2214yescomplemented
24#16861131C2214yescomplemented
25C23×C4×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C22×C8×C127C2211yescharacteristic
28C22⋊C8×C127C2214yescomplemented
29C23×D127—21no—
30C2×D508—21no—
31C2×D508—22no—
32C2×D508—22no—
33C2×D508C2312yes—
34C2×C22⋊D127—24no—
35C2×C22⋊D127—44no—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C22×Dic127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127C2312yes—
43C4⋊Dic127C2312yes—
44C4⋊Dic127C2312yes—
45C22⋊Dic127—24no—
46(C2×D127)⋊C4—24no—
47C4×Dic127—24no—
48C24×C127D411yescharacteristic
49C22×C4×C127—22no—
50C22×C4×C127—22no—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D412yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—22no—
56C2×C8×C127—22no—
57C2×C8×C127C2312yes—
58C2×C8×C127C2312yes—
59C22×D127—22no—
60C22×D127—42no—
61C22×D127—42no—
62C22×D127—42no—
63D508—22no—
64D508—22no—
65D508—42no—
66C22⋊D127—48no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—42no—
71C2×Dic127—42no—
72C2×Dic127—42no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127C2×D412yes—
77C23×C127C2×D411yescharacteristic
78C2×C4×C127—22no—
79C2×C4×C127—24no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127C2×D412yes—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4112yes—
85C2×C4×C127#4112yes—
86C2×C4×C127C2411yescharacteristic
87C8×C127—22no—
88C8×C127—22no—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—22no—
92C22×C127—44no—
93C22×C127#12312yes—
94C22×C127#13512yes—
95C22×C127AGL(1,ℤ/8)12yes—
96C22×C127C22×D411yescharacteristic
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—42no—
100C2×D127—84no—
101C4×C127—22no—
102C4×C127#12312yes—
103C4×C127C2×#4111yescharacteristic
104C4×C127C2×#4111yescharacteristic
105Dic127—42no—
106Dic127—42no—
107Dic127—42no—
108C2×C127—42no—
109C2×C127#48214yescomplemented
110C2×C127C2×#12311yescharacteristic
111C2×C127C2×#13511yescharacteristic
112C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
113D127—82no—
114C2×#482—1271nomaximal, Sylow
115C127C2×#48211yescharacteristic, complemented, Sylow
116C2×#121—1271no—
117C2×#117—1271no—
118#482—1278no—
119C2×#107—1271no—
120C2×#106—1271no—
121C2×#102—1271no—
122C2×#102—1271no—
123C2×C22⋊C8D12711yescharacteristic, complemented
124C22×D4—1271no—
125C22×D4—2541no—
126C23×C4C2×D12711yescharacteristic
127C22×C8C2×D12711yescharacteristic
128C22×C8C2×D12711yescharacteristic
129#121—1274no—
130#121—2542no—
131#117—1274no—
132#117—2542no—
133C2×C4⋊C4—1271no—
134C2×C4⋊C4—1271no—
135C2×C4⋊C4—1271no—
136C2×C22⋊C4—2541no—
137C2×C22⋊C4—2541no—
138C2×C42—2541no—
139#107—1274no—
140#106—1274no—
141#102—1274no—
142#102—1274no—
143C22⋊C8C2×D12714yescomplemented
144C24—2541no—
145C24D50811yescharacteristic
146C2×D4—1272no—
147C2×D4—5084no—
148C2×D4—2544no—
149C2×D4—2541no—
150C2×D4—2542no—
151C2×D4—2542no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—2541no—
155C22×C4—2541no—
156C22×C4—2541no—
157C22×C4D50812yes—
158C22×C4D50811yescharacteristic
159C22×C4C22×D12712yes—
160C22×C4C22×D12711yescharacteristic
161C2×C8—22no—
162C2×C8—22no—
163C2×C8C22×D12712yes—
164C2×C8C22×D12712yes—
165C4⋊C4—1272no—
166C4⋊C4—1272no—
167C4⋊C4—1272no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C4⋊C4—2541no—
171C22⋊C4—2544no—
172C22⋊C4—2544no—
173C42—2544no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—5082no—
178C23—5082no—
179C23—5082no—
180C23—2542no—
181C23C2×D50812yes—
182C23C2×D50811yescharacteristic
183D4—5088no—
184D4—5082no—
185D4—2542no—
186D4—2542no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—5082no—
192C2×C4—5082no—
193C2×C4—5082no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4—2542no—
197C2×C4C2×D50812yes—
198C2×C4C2×D50811yescharacteristic
199C2×C4#1452324712yes—
200C2×C4#1452324712yes—
201C2×C4C23×D12711yescharacteristic
202C8—22no—
203C8—22no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—10164no—
209C22—5082no—
210C22—5082no—
211C22—5082no—
212C22#1686119312yes—
213C22#1686123112yes—
214C22#1686123712yes—
215C22C22×D50811yescharacteristic
216C4—22no—
217C4—5082no—
218C4—5082no—
219C4—5082no—
220C4#1686119312yes—
221C4C2×#1452324711yescharacteristic
222C4C2×#1452324711yescharacteristic
223C2—42no—
224C2—10162no—
225C2#2255020114yesdirect product factor
226C2C2×#1686119311yescharacteristic
227C2C2×#1686123111yescharacteristic
228C2C2×#1686123711yescharacteristic
subgroupG/H
centerC23C2×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊C8×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
hypercenterC2×C22⋊C8D127
hypocenterC127C2×#482
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16861168C2semidirect
3. #22550201C2direct (N × H)
4. C2×#16861143C2semidirect
5. C2×#16861141C2semidirect
6. C2×#16861131C2semidirect
7. C2×#16861131C2semidirect
8. C2×C22⋊C8×C127C2semidirect
14. #16861168C22semidirect
21. #16861143C22semidirect
22. #16861141C22semidirect
23. #16861131C22semidirect
24. #16861131C22semidirect
28. C22⋊C8×C127C22semidirect
109. C2×C127#482semidirect
115. C127C2×#482semidirect
123. C2×C22⋊C8D127semidirect
143. C22⋊C8C2×D127semidirect
225. C2#22550201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×#168611682
#225502012
C2×#168611432
C2×#168611412
C2×#168611312
C2×C22⋊C8×C1272
C2×#482127