Finite Group Databasebeta

← #30648679

Subgroups — #30648679

Order 16256.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861188C211yesmaximal, characteristic
2C2×#16861168C211yesmaximal, characteristic
3#22550218C214yesmaximal, complemented
4#22550212C212yesmaximal, complemented
5#22550207C212yesmaximal, complemented
6#22549918C212yesmaximal, complemented
7#22549886C212yesmaximal, complemented
8#407×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10#16861188—21no—
11#16861188—21no—
12#16861188C2212yes—
13#16861188C2212yes—
14#16861168—21no—
15#16861168—21no—
16#16861168C2212yes—
17#16861168C2212yes—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4—21no—
22C2×C22⋊Dic127—21no—
23C2×C22⋊Dic127—21no—
24C2×C4×Dic127—21no—
25#16861143C2214yescomplemented
26#16861141C2214yescomplemented
27#16861134C2214yescomplemented
28#16861134C2214yescomplemented
29C23×C4×C127C2211yescharacteristic
30C2×M16×C127C2212yes—
31C22⋊C8×C127C2212yes—
32C22⋊C8×C127C2212yes—
33C2×Dic254—21no—
34C2×Dic254—24no—
35C2×Dic254C2312yes—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43C4⋊Dic127C2312yes—
44C4⋊Dic127C2312yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48C22⋊Dic127—22no—
49C22⋊Dic127—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127—22no—
52C4×Dic127—22no—
53C4×Dic127—22no—
54C24×C127D411yescharacteristic
55C22×C4×C127—24no—
56C22×C4×C127D411yescharacteristic
57C22×C4×C127D411yescharacteristic
58C22×C4×C127D411yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C22×C4×C127C2311yescharacteristic
61C22×C4×C127C2311yescharacteristic
62M16×C127—24no—
63C2×C8×C127C2314yes—
64Dic254—24no—
65Dic254—42no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—42no—
71C2×Dic127—42no—
72C2×Dic127—42no—
73C2×Dic127—42no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—41no—
77C23×C127C2×D411yescharacteristic
78C23×C127C2×D411yescharacteristic
79C23×C127C2×D411yescharacteristic
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4114yes—
89C2×C4×C127C2411yescharacteristic
90C8×C127—24no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—44no—
99C22×C127#12312yes—
100C22×C127#13712yes—
101C22×C127C22×D411yescharacteristic
102C4×C127—21no—
103C4×C127—21no—
104C4×C127#12312yes—
105C4×C127C2×#4112yes—
106Dic127—42no—
107Dic127—42no—
108Dic127—42no—
109Dic127—42no—
110C2×C127—22no—
111C2×C127—22no—
112C2×C127—41no—
113C2×C127C2×#12311yescharacteristic
114C2×C127C2×#13711yescharacteristic
115C2×C127#57511yescharacteristic
116#3095—1271nomaximal, Sylow
117C127#309511yescharacteristic, complemented, Sylow
118C2×C22⋊Q8—1271no—
119C2×#117—1271no—
120#485—1274no—
121#484—1272no—
122#483—1272no—
123#428—1272no—
124#419—1272no—
125#407D12711yescharacteristic
126C22×Q8—1271no—
127C23×C4C2×D12711yescharacteristic
128C2×M16C2×D12712yes—
129C22⋊Q8—1272no—
130C22⋊Q8—1272no—
131C22⋊Q8—2541no—
132C22⋊Q8—2541no—
133#117—1272no—
134#117—1272no—
135#117—2541no—
136#117—2541no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4—1271no—
139C2×C4⋊C4—1271no—
140C2×C4⋊C4—2541no—
141C2×C22⋊C4—2541no—
142C2×C22⋊C4—2541no—
143C2×C42—2541no—
144#107—1274no—
145#106—1274no—
146#103—1274no—
147#103—1274no—
148C22⋊C8C2×D12712yes—
149C22⋊C8C2×D12712yes—
150C24D50811yescharacteristic
151C2×Q8—1272no—
152C2×Q8—2544no—
153C2×Q8—2541no—
154C22×C4—24no—
155C22×C4—2541no—
156C22×C4—2541no—
157C22×C4—2541no—
158C22×C4—2541no—
159C22×C4D50811yescharacteristic
160C22×C4D50811yescharacteristic
161C22×C4D50811yescharacteristic
162C22×C4C22×D12711yescharacteristic
163C22×C4C22×D12711yescharacteristic
164C22×C4C22×D12711yescharacteristic
165M16—24no—
166C2×C8C22×D12714yes—
167C4⋊C4—1272no—
168C4⋊C4—1272no—
169C4⋊C4—1272no—
170C4⋊C4—2542no—
171C4⋊C4—2542no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C22⋊C4—2542no—
176C22⋊C4—2542no—
177C22⋊C4—2542no—
178C22⋊C4—2542no—
179C42—2542no—
180C42—2542no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×D50811yescharacteristic
185C23C2×D50811yescharacteristic
186C23C2×D50811yescharacteristic
187Q8—5082no—
188Q8—2544no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—5082no—
195C2×C4—5082no—
196C2×C4—5082no—
197C2×C4—5082no—
198C2×C4—2542no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4C2×D50811yescharacteristic
203C2×C4C2×D50811yescharacteristic
204C2×C4C2×D50811yescharacteristic
205C2×C4#1452324714yes—
206C2×C4C23×D12711yescharacteristic
207C8—24no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1686119312yes—
217C22#1686124212yes—
218C22C22×D50811yescharacteristic
219C4—21no—
220C4—21no—
221C4—5082no—
222C4—5082no—
223C4—5082no—
224C4—5082no—
225C4#1686119312yes—
226C4C2×#1452324712yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1686119311yescharacteristic
231C2C2×#1686124211yescharacteristic
232C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#407×C127C2
socleC22×C127C22×D4
prime radicalC24×C127D4
hypercenter#407D127
hypocenterC127#3095
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #22550218C2semidirect
4. #22550212C2semidirect
5. #22550207C2semidirect
6. #22549918C2semidirect
7. #22549886C2semidirect
25. #16861143C22semidirect
26. #16861141C22semidirect
27. #16861134C22semidirect
28. #16861134C22semidirect
117. C127#3095semidirect

Maximal subgroups

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

subgroupindex
C2×#168611882
C2×#168611682
#225502182
#225502122
#225502072
#225499182
#225498862
#407×C1272
#3095127