Finite Group Databasebeta

← #30650595

Subgroups — #30650595

Order 16256.

420 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 141 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#22550530C211yesmaximal, characteristic
2#22550226C211yesmaximal, characteristic, complemented
3#22550204C212yesmaximal, complemented
4#22550161C211yesmaximal, characteristic, complemented
5#22550107C212yesmaximal, complemented
6#22550078C211yesmaximal, characteristic, complemented
7#22550015C212yesmaximal, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10C2×#16861133C212yesmaximal, complemented
11D4×Q8×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×#14523235C2211yescharacteristic
14C2×#14523235C2212yes—
15C22×C127⋊C8C2212yes—
16#16861209C2211yescharacteristic
17#16861193—22no—
18#16861189—21no—
19#16861183C2212yes—
20C4×D508C2211yescharacteristic
21C2×C4⋊Dic127C2212yes—
22#16861140C2212yes—
23#16861140C2211yescharacteristic
24C4⋊(C127⋊C8)C2211yescharacteristic
25#16861133C2212yes—
26#16861133C2212yes—
27#16861133C2211yescharacteristic
28#16861133C2212yes—
29#16861133C2212yes—
30#16861129C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C22×Q8×C127C2212yes—
34C4⋊Q8×C127—21no—
35C4⋊Q8×C127C2211yescharacteristic
36C22⋊Q8×C127—22no—
37C22⋊Q8×C127C2212yes—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127—21no—
40C4×D4×C127C2211yescharacteristic
41#14523247—44no—
42C2×Dic254—21no—
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—22no—
45C22×Dic127—22no—
46C2×C4×D127—21no—
47#14523235—22no—
48#14523235—42no—
49C2×C127⋊C8—42no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2312yes—
52C2×C127⋊C8C2311yescharacteristic
53C4⋊Dic127—22no—
54C4⋊Dic127—21no—
55C4⋊Dic127C2312yes—
56C4⋊Dic127C2311yescharacteristic
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—21no—
59C2×Q8×C127—22no—
60C2×Q8×C127—24no—
61C2×Q8×C127—42no—
62C2×Q8×C127C2312yes—
63C2×Q8×C127C2311yescharacteristic
64C2×D4×C127D411yescharacteristic
65C22×C4×C127—22no—
66C22×C4×C127C2312yes—
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127—22no—
71C4⋊C4×C127D411yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—22no—
74C22⋊C4×C127D412yes—
75C42×C127—21no—
76C42×C127C2311yescharacteristic
77C22×D127—21no—
78Dic254—42no—
79D508—21no—
80D508—21no—
81C22⋊D127—44no—
82C2×Dic127—21no—
83C2×Dic127—22no—
84C2×Dic127—44no—
85C4×D127—42no—
86C127⋊C8—21no—
87C127⋊C8—21no—
88C127⋊C8—22no—
89C23×C127C2×D412yes—
90Q8×C127—21no—
91Q8×C127—22no—
92Q8×C127—22no—
93Q8×C127—44no—
94Q8×C127#4112yescomplemented
95D4×C127SD1614yescomplemented
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—44no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2411yescharacteristic
106C22×C127—22no—
107C22×C127—22no—
108C22×C127C2×SD1614yes—
109C22×C127C22×D411yescharacteristic
110C2×D127—41no—
111C2×D127—41no—
112C2×D127—41no—
113C4×C127—22no—
114C4×C127—22no—
115C4×C127—21no—
116C4×C127—42no—
117C4×C127—41no—
118C4×C127C2×SD1612yes—
119C4×C127C2×#4111yescharacteristic
120C4×C127H(5,𝔽2)11yescharacteristic
121Dic127—41no—
122Dic127—42no—
123C2×C127—24no—
124C2×C127#54611yescharacteristic
125C2×C127C22×SD1611yescharacteristic
126C2×C127#57811yescharacteristic
127D127—81no—
128#3298—1271nomaximal, Sylow
129C127#329811yescharacteristic, complemented, Sylow
130D4×Q8D12711yescharacteristic, complemented
131#547—1271no—
132#487—1271no—
133#483—1272no—
134#475—1271no—
135#465—1272no—
136#461—1271no—
137#448—1272no—
138C4×SD16—1271no—
139C8×D4—1271no—
140C2×#103—1272no—
141C2×#41—2541no—
142C22×Q8C2×D12712yes—
143C2×SD16—1272no—
144C2×SD16—1271no—
145C22×C8—1272no—
146C4⋊Q8—21no—
147C4⋊Q8—1271no—
148C4⋊Q8C2×D12711yescharacteristic
149#123—2542no—
150C22⋊Q8—22no—
151C22⋊Q8—2541no—
152C22⋊Q8C2×D12712yes—
153#121—1272no—
154C4×Q8C2×D12711yescharacteristic
155C4×D4—21no—
156C4×D4—1271no—
157C4×D4C2×D12711yescharacteristic
158C2×C4⋊C4—1272no—
159#106—1271no—
160#106—1272no—
161C4⋊C8—1271no—
162#103—1271no—
163#103—1272no—
164#103—1272no—
165#103—1272no—
166#103—1272no—
167#102—1271no—
168C22⋊C8—1272no—
169C4×C8—1271no—
170#41—5084no—
171C2×Q8—22no—
172C2×Q8—24no—
173C2×Q8—42no—
174C2×Q8—2541no—
175C2×Q8C22×D12712yes—
176C2×Q8C22×D12711yescharacteristic
177C2×D4—1271no—
178C2×D4—2542no—
179C2×D4C22⋊D12711yescharacteristic
180C22×C4—22no—
181C22×C4—2542no—
182C22×C4—2541no—
183C22×C4C22×D12712yes—
184SD16—5082no—
185SD16—2542no—
186C2×C8—1271no—
187C2×C8—1271no—
188C2×C8—1272no—
189C2×C8—5082no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—1271no—
195C4⋊C4—1272no—
196C4⋊C4—2542no—
197C4⋊C4—2541no—
198C4⋊C4C22⋊D12711yescharacteristic
199C4⋊C4C22×D12711yescharacteristic
200C22⋊C4—22no—
201C22⋊C4—2541no—
202C22⋊C4—2542no—
203C22⋊C4C22⋊D12712yes—
204C42—21no—
205C42C22×D12711yescharacteristic
206C23—2541no—
207C23C2×C22⋊D12712yes—
208Q8—21no—
209Q8—22no—
210Q8—22no—
211Q8—44no—
212Q8—5082no—
213Q8#1452324712yescomplemented
214D4—5084no—
215D4—2541no—
216D4—2541no—
217D4#1452323514yescomplemented
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—44no—
224C2×C4—5084no—
225C2×C4—5082no—
226C2×C4—2541no—
227C2×C4—2542no—
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4C2×C22⋊D12712yes—
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4C23×D12711yescharacteristic
232C8—2541no—
233C8—2541no—
234C8—2542no—
235C22—22no—
236C22—22no—
237C22—5081no—
238C22—5081no—
239C22—5081no—
240C22C2×#1452323514yes—
241C22C22×C22⋊D12711yescharacteristic
242C4—21no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—5081no—
248C4—5082no—
249C4C2×#1452323512yes—
250C4C2×#1452324711yescharacteristic
251C4#1686125411yescharacteristic
252C2—24no—
253C2—10161no—
254C2#2255052011yescharacteristic
255C2C22×#1452323511yescharacteristic
256C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD4×Q8×C127C2
socleC22×C127C22×D4
prime radical#22550530C2
hypercenterD4×Q8D127
hypocenterC127#3298
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #22550226C2semidirect
3. #22550204C2semidirect
4. #22550161C2semidirect
5. #22550107C2semidirect
6. #22550078C2semidirect
7. #22550015C2semidirect
8. C4×#14523235C2semidirect
9. D4×C127⋊C8C2semidirect
10. C2×#16861133C2semidirect
11. D4×Q8×C127C2semidirect
94. Q8×C127#41semidirect
95. D4×C127SD16semidirect
129. C127#3298semidirect
130. D4×Q8D127semidirect
213. Q8#14523247semidirect
217. D4#14523235semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502262
#225502042
#225501612
#225501072
#225500782
#225500152
C4×#145232352
D4×C127⋊C82
C2×#168611332
D4×Q8×C1272
#3298127