Finite Group Databasebeta

← #30648901

Subgroups — #30648901

Order 16256.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#22550410C211yesmaximal, characteristic
2#22550211C211yesmaximal, characteristic, complemented
3#22550194C211yesmaximal, characteristic
4#22550183C212yesmaximal, complemented
5#22550098C212yesmaximal, complemented
6#22550032C211yesmaximal, characteristic, complemented
7#22550015C211yesmaximal, characteristic
8#22549981C212yesmaximal, complemented
9#22549974C212yesmaximal, complemented
10#22549933C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C127C211yesmaximal, characteristic, complemented
12C2×#14523238C2212yes—
13C2×#14523235C2212yes—
14#16861203—21no—
15#16861201C2211yescharacteristic
16#16861197C2211yescharacteristic
17#16861192—21no—
18#16861188C2211yescharacteristic
19#16861183C2211yescharacteristic
20C4×Dic254C2211yescharacteristic
21C4×D508C2211yescharacteristic
22C4×C22⋊D127—21no—
23#16861166—21no—
24#16861142C2212yes—
25#16861140C2212yes—
26C4⋊(C127⋊C8)C2212yes—
27#16861133C2212yes—
28#16861133C2212yes—
29#16861132C2212yes—
30#16861129C2212yes—
31C22⋊(C127⋊C8)C2211yescharacteristic
32C22⋊(C127⋊C8)C2211yescharacteristic
33#16861120C2212yes—
34C22×Q8×C127C2211yescharacteristic
35C4×Q8×C127—21no—
36C4×Q8×C127—21no—
37C4×Q8×C127C2212yes—
38C4×Q8×C127C2212yes—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127C2211yescharacteristic
41C2×C42×C127—21no—
42C2×C42×C127C2211yescharacteristic
43C2×Dic254C2311yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—21no—
46C2×C4×D127—21no—
47#14523238—24no—
48#14523235—24no—
49C2×C127⋊C8C2312yes—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—21no—
54Dic127⋊C4—21no—
55Dic127⋊C4—21no—
56Dic127⋊C4—21no—
57C22⋊Dic127—21no—
58(C2×D127)⋊C4—21no—
59(C2×D127)⋊C4—21no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—21no—
63C2×Q8×C127—24no—
64C2×Q8×C127—21no—
65C2×Q8×C127C2312yes—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127D411yescharacteristic
69C22×C4×C127D411yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127—22no—
72C4⋊C4×C127—22no—
73C4⋊C4×C127—21no—
74C4⋊C4×C127C2312yes—
75C42×C127—22no—
76C42×C127—22no—
77C42×C127D411yescharacteristic
78C42×C127D411yescharacteristic
79C42×C127C2311yescharacteristic
80C42×C127C2311yescharacteristic
81C22×D127—21no—
82Dic254—22no—
83D508—22no—
84C22⋊D127—42no—
85C2×Dic127—21no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C4×D127—42no—
89C127⋊C8—22no—
90C127⋊C8—22no—
91C23×C127C2×D411yescharacteristic
92Q8×C127—24no—
93Q8×C127—22no—
94Q8×C127#4114yes—
95C2×C4×C127—24no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—41no—
101C2×C4×C127—42no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C22×C127—21no—
109C22×C127—21no—
110C22×C127#13712yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—41no—
113C2×D127—41no—
114C2×D127—41no—
115C4×C127—24no—
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127—21no—
121C4×C127—42no—
122C4×C127C2×#4112yes—
123Dic127—41no—
124Dic127—41no—
125Dic127—41no—
126C2×C127—22no—
127C2×C127#52511yescharacteristic
128C2×C127C2×#13711yescharacteristic
129C2×C127#57511yescharacteristic
130D127—81no—
131#3117—1271nomaximal, Sylow
132C127#311711yescharacteristic, complemented, Sylow
133#529—1271no—
134C2×C4×Q8D12711yescharacteristic, complemented
135#484—1271no—
136#481—1271no—
137#479—1272no—
138#464—1272no—
139#451—1271no—
140#448—1271no—
141#441—1272no—
142#440—1272no—
143#432—1271no—
144C22×Q8C2×D12711yescharacteristic
145C2×Q16—1272no—
146C2×SD16—1272no—
147#126—2541no—
148#125—1271no—
149#124—1271no—
150#123—2541no—
151C22⋊Q8—1271no—
152#121—1271no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—1271no—
156C4×Q8C2×D12712yes—
157C4×Q8C2×D12712yes—
158C4×D4—1271no—
159C4×D4—2541no—
160#117—2541no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D12711yescharacteristic
163C2×C42—21no—
164C2×C42C2×D12711yescharacteristic
165#107—1272no—
166#106—1272no—
167C4⋊C8—1272no—
168#103—1272no—
169#103—1272no—
170#103—1272no—
171#102—1272no—
172C22⋊C8—1271no—
173C22⋊C8—1271no—
174#97—1272no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1271no—
178C2×Q8C22×D12712yes—
179C2×D4—1271no—
180C2×D4—2541no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—2541no—
184C22×C4C22⋊D12711yescharacteristic
185C22×C4C22⋊D12711yescharacteristic
186C22×C4C22×D12711yescharacteristic
187Q16—2544no—
188SD16—2544no—
189C2×C8—1272no—
190C2×C8—1272no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—1271no—
195C4⋊C4—1271no—
196C4⋊C4—2541no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C4⋊C4C22×D12712yes—
201C22⋊C4—2541no—
202C22⋊C4—2541no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C42—22no—
207C42—22no—
208C42—2541no—
209C42C22⋊D12711yescharacteristic
210C42C22⋊D12711yescharacteristic
211C42C22×D12711yescharacteristic
212C42C22×D12711yescharacteristic
213C23—2541no—
214C23C2×C22⋊D12711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—2542no—
218Q8#1452324814yes—
219D4—5082no—
220D4—2542no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—5082no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—2542no—
239C8—2542no—
240C22—21no—
241C22—21no—
242C22—5081no—
243C22—5081no—
244C22—5081no—
245C22#1686124012yes—
246C22C22×C22⋊D12711yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—5081no—
255C4—5081no—
256C4—5081no—
257C4C2×#1452324812yes—
258C2—22no—
259C2—10161no—
260C2#2255038711yescharacteristic
261C2C2×#1686124011yescharacteristic
262C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×Q8×C127C2
socleC22×C127C22×D4
prime radical#16861183C22
hypercenterC2×C4×Q8D127
hypocenterC127#3117
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22550211C2semidirect
4. #22550183C2semidirect
5. #22550098C2semidirect
6. #22550032C2semidirect
8. #22549981C2semidirect
9. #22549974C2semidirect
10. #22549933C2semidirect
11. C2×C4×Q8×C127C2semidirect
132. C127#3117semidirect
134. C2×C4×Q8D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225502112
#225501942
#225501832
#225500982
#225500322
#225500152
#225499812
#225499742
#225499332
C2×C4×Q8×C1272
#3117127