Finite Group Databasebeta

← #30638716

Subgroups — #30638716

Order 16256.

380 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 95 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×D4×Dic127C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic
3#22549734C212yesmaximal, complemented
4#22549642C212yesmaximal, complemented
5C2×#99×C127C211yesmaximal, characteristic
6C23×Dic127C411yescharacteristic, complemented
7C23×Dic127C2211yescharacteristic
8D4×Dic127—22no—
9D4×Dic127—22no—
10C2×C4⋊Dic127C411yescharacteristic, complemented
11C2×Dic127⋊C4—22no—
12C2×C22⋊Dic127C411yescharacteristic, complemented
13C2×C22⋊Dic127C2212yes—
14C2×C22⋊Dic127C2211yescharacteristic
15C2×C4×Dic127C411yescharacteristic, complemented
16#16861124—22no—
17C127⋊#95—22no—
18C22×D4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20#99×C127—22no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127—22no—
24C22×Dic127—42no—
25C22×Dic127—42no—
26C22×Dic127C2×C411yescharacteristic
27C22×Dic127C2×C411yescharacteristic
28C22×Dic127C2×C411yescharacteristic
29C22×Dic127C2×C411yescharacteristic
30C22×Dic127D412yes—
31C22×Dic127D412yes—
32C4⋊Dic127—22no—
33Dic127⋊C4—44no—
34C22⋊Dic127—22no—
35C22⋊Dic127—22no—
36C22⋊Dic127—22no—
37C22⋊Dic127—22no—
38C4×Dic127—21no—
39C4×Dic127—21no—
40C24×C127C2×C411yescharacteristic
41C24×C127C2311yescharacteristic
42C2×D4×C127—22no—
43C2×D4×C127—22no—
44C2×D4×C127D412yes—
45C2×D4×C127Q812yes—
46C22×C4×C127—22no—
47C22×C4×C127C2×C411yescharacteristic
48C22⋊C4×C127—22no—
49C22⋊C4×C127—22no—
50C2×Dic127—24no—
51C2×Dic127—22no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—41no—
55C2×Dic127—42no—
56C2×Dic127—42no—
57C2×Dic127—42no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C2×Dic127—42no—
61C2×Dic127—42no—
62C2×Dic127—42no—
63C2×Dic127C22⋊C411yescharacteristic
64C2×Dic127C22⋊C412yes—
65C2×Dic127C22⋊C411yescharacteristic
66C23×C127—21no—
67C23×C127—21no—
68C23×C127—41no—
69C23×C127—41no—
70C23×C127—41no—
71C23×C127—41no—
72C23×C127C4⋊C412yescomplemented
73C23×C127C22×C411yescharacteristic
74C23×C127C2×D412yes—
75C23×C127C2×D411yescharacteristic
76C23×C127C2×Q811yescharacteristic
77C23×C127#4112yes—
78D4×C127—24no—
79D4×C127—24no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127—42no—
85C2×C4×C127C4⋊C412yescomplemented
86C22×C127—22no—
87C22×C127—21no—
88C22×C127—22no—
89C22×C127—21no—
90C22×C127—22no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127C2×C22⋊C411yescharacteristic
99C22×C127C2×C4⋊C411yescharacteristic
100C22×C127#11711yescharacteristic
101C22×C127#12112yes—
102C22×C127C22⋊Q812yes—
103C4×C127—22no—
104C4×C127—82no—
105Dic127—21no—
106Dic127—22no—
107Dic127—41no—
108Dic127—41no—
109Dic127—82no—
110Dic127#9912yescomplemented
111C2×C127—22no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—22no—
115C2×C127—42no—
116C2×C127#38011yescharacteristic
117C2×C127C2×#9911yescharacteristic
118C2×C127#41011yescharacteristic
119#1799—1271nomaximal, Sylow
120C127#179911yescharacteristic, complemented, Sylow
121C2×C4×D4—1271no—
122C2×#99—1271no—
123C2×#99D12711yescharacteristic
124#380—1272no—
125#342—1272no—
126C22×D4C2×D12711yescharacteristic
127C23×C4—1271no—
128C23×C4—1271no—
129C4×D4—2542no—
130C4×D4—2542no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—2542no—
133C2×C22⋊C4—1271no—
134C2×C22⋊C4—1271no—
135C2×C22⋊C4—1272no—
136C2×C22⋊C4C2×D12712yes—
137C2×C42—1271no—
138#99—22no—
139#99—2542no—
140#95—2542no—
141C24C4×D12711yescharacteristic
142C24C22×D12711yescharacteristic
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22⋊D12712yes—
146C2×D4Dic25412yes—
147C22×C4—22no—
148C22×C4—1271no—
149C22×C4—1272no—
150C22×C4—1271no—
151C22×C4—1272no—
152C22×C4—1271no—
153C22×C4—1271no—
154C22×C4—5082no—
155C22×C4—5082no—
156C22×C4—2541no—
157C22×C4—2541no—
158C22×C4—2542no—
159C22×C4C4×D12711yescharacteristic
160C4⋊C4—5084no—
161C4⋊C4—2542no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—2542no—
165C22⋊C4—2542no—
166C22⋊C4—2542no—
167C22⋊C4—2542no—
168C42—2541no—
169C42—2541no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23Dic127⋊C412yescomplemented
177C23C2×C4×D12711yescharacteristic
178C23D4×D12712yes—
179C23C2×C22⋊D12711yescharacteristic
180C23C2×Dic25411yescharacteristic
181C23#1452324712yes—
182D4—24no—
183D4—24no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—1271no—
190C2×C4—1271no—
191C2×C4—1272no—
192C2×C4—5082no—
193C2×C4—5082no—
194C2×C4—5082no—
195C2×C4—5082no—
196C2×C4—5082no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—5082no—
200C2×C4—5081no—
201C2×C4—2544no—
202C2×C4—2542no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4Dic127⋊C412yescomplemented
206C22—22no—
207C22—21no—
208C22—22no—
209C22—21no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22C22⋊C4×D12711yescharacteristic
219C22C2×Dic127⋊C411yescharacteristic
220C22#1686116511yescharacteristic
221C22#1686118112yes—
222C22#1686118612yes—
223C4—22no—
224C4—82no—
225C4—1272no—
226C4—10162no—
227C4—5081no—
228C4—5081no—
229C4—2541no—
230C4—2542no—
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2#2254973411yescharacteristic
237C2#99×D12711yescharacteristic
238C2#2254984811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C22
cosocleC24×C127C23
hypercenterC2×#99D127
hypocenterC127#1799
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #22549734C2semidirect
4. #22549642C2semidirect
6. C23×Dic127C4semidirect
10. C2×C4⋊Dic127C4semidirect
12. C2×C22⋊Dic127C4semidirect
15. C2×C4×Dic127C4semidirect
72. C23×C127C4⋊C4semidirect
85. C2×C4×C127C4⋊C4semidirect
110. Dic127#99semidirect
120. C127#1799semidirect
176. C23Dic127⋊C4semidirect
205. C2×C4Dic127⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C2×#168611242
#225497342
#225496422
C2×#99×C1272
#1799127