Finite Group Databasebeta

← #30639816

Subgroups — #30639816

Order 16256.

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

normal subgroups only

This group has 474 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
1C22×#14523235C211yesmaximal, characteristic, complemented
2C2×#16861205C211yesmaximal, characteristic
3C2×#16861131C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic, complemented
5#22549783C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127—22no—
10C2×#14523235—22no—
11C2×#14523235—22no—
12C2×#14523235—22no—
13C22×C127⋊C8C2211yescharacteristic
14#16861205—22no—
15#16861205—22no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4×Dic127C2211yescharacteristic
18#16861131—22no—
19#16861130—22no—
20C127⋊#95—22no—
21C22×Q8×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23C2×SD16×C127—22no—
24C2×SD16×C127—22no—
25C2×SD16×C127—22no—
26C22×C8×C127C2211yescharacteristic
27C23×D127—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508D412yes—
34C2×C22⋊D127—28no—
35C2×C22⋊D127—44no—
36C2×C22⋊D127—44no—
37C22×Dic127—21no—
38C22×Dic127D412yes—
39#14523235—28no—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8D412yescomplemented
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C24×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127D412yescomplemented
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127D412yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62SD16×C127—28no—
63C2×C8×C127—21no—
64C2×C8×C127—21no—
65C2×C8×C127D412yescomplemented
66C22×D127—22no—
67C22×D127—41no—
68C22×D127—41no—
69C22×D127—42no—
70C22×D127—42no—
71D508—22no—
72D508—22no—
73D508—41no—
74D508—41no—
75C22⋊D127—416no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—41no—
79C2×Dic127—42no—
80C2×Dic127SD1614yescomplemented
81C127⋊C8—22no—
82C23×C127—22no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—42no—
86C23×C127—42no—
87C23×C127C2×D411yescharacteristic
88Q8×C127—22no—
89Q8×C127—22no—
90Q8×C127—41no—
91Q8×C127—41no—
92D4×C127—22no—
93D4×C127—22no—
94D4×C127—41no—
95D4×C127—41no—
96C2×C4×C127—22no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4111yescharacteristic
105C8×C127—22no—
106C22×C127—42no—
107C22×C127—44no—
108C22×C127—82no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127#12011yescharacteristic
112C22×C127#12111yescharacteristic
113C22×C127#12711yescharacteristic
114C22×C127C2×SD1612yes—
115C22×C127AGL(1,ℤ/8)12yes—
116C2×D127—42no—
117C2×D127—44no—
118C2×D127—82no—
119C2×D127—81no—
120C2×D127—81no—
121C4×C127—42no—
122C4×C127#12011yescharacteristic
123C4×C127#12011yescharacteristic
124C4×C127#12111yescharacteristic
125C4×C127#12111yescharacteristic
126Dic127—24no—
127Dic127—81no—
128C2×C127—82no—
129C2×C127#39211yescharacteristic
130C2×C127#45012yes—
131C2×C127#46012yes—
132C2×C127#49211yescharacteristic
133C2×C127#49611yescharacteristic
134D127—82no—
135#2014—1271nomaximal, Sylow
136C127#201411yescharacteristic, complemented, Sylow
137C22×SD16—1271no—
138C22×SD16D12711yescharacteristic, complemented
139C2×#127—1271no—
140C2×#102—1271no—
141C2×#102—1271no—
142#391—1271no—
143#336—1271no—
144C22×Q8C2×D12711yescharacteristic
145C22×D4—1271no—
146C22×D4—2542no—
147C22×D4C2×D12711yescharacteristic
148C2×SD16—22no—
149C2×SD16—22no—
150C2×SD16—22no—
151C2×SD16—2542no—
152C2×SD16—2542no—
153C2×SD16—2542no—
154C22×C8—1271no—
155C22×C8C2×D12711yescharacteristic
156#127—2542no—
157#127—2542no—
158C2×C4⋊C4—1271no—
159C2×C42—1271no—
160#102—2542no—
161#102—2542no—
162#95—2542no—
163C24—21no—
164C24—2541no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8C22⋊D12712yescomplemented
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—1272no—
177C2×D4—5084no—
178C2×D4—5084no—
179C2×D4—2548no—
180C2×D4—2541no—
181C2×D4—2541no—
182C2×D4—2541no—
183C2×D4—2541no—
184C2×D4—2541no—
185C2×D4C22⋊D12712yes—
186C22×C4—21no—
187C22×C4—1272no—
188C22×C4—2541no—
189C22×C4C22×D12711yescharacteristic
190SD16—28no—
191SD16—2548no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—1272no—
195C2×C8—2541no—
196C2×C8—2541no—
197C2×C8C22⋊D12712yescomplemented
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C42—2541no—
201C42—2541no—
202C23—22no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—5081no—
208C23—5082no—
209C23—5082no—
210C23—5081no—
211C23—2542no—
212C23D4×D12711yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—41no—
216Q8—41no—
217D4—22no—
218D4—22no—
219D4—41no—
220D4—41no—
221D4—50816no—
222D4—5081no—
223D4—5081no—
224D4—2542no—
225D4—2542no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—1274no—
230C2×C4—5081no—
231C2×C4—5082no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4D4×D12711yescharacteristic
235C2×C4D4×D12711yescharacteristic
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4C2×C22⋊D12711yescharacteristic
238C2×C4C2×C22⋊D12711yescharacteristic
239C2×C4#1452324711yescharacteristic
240C8—22no—
241C8—2542no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22—10162no—
248C22—10161no—
249C22—10161no—
250C22—5084no—
251C22—5082no—
252C22#1686117711yescharacteristic
253C22#1686118111yescharacteristic
254C22#1686120511yescharacteristic
255C22SD16×D12712yes—
256C22#1686123612yes—
257C4—42no—
258C4—10161no—
259C4—2544no—
260C4#1686117711yescharacteristic
261C4#1686117811yescharacteristic
262C4#1686118011yescharacteristic
263C4#1686118111yescharacteristic
264C2—82no—
265C2—10162no—
266C2#2254978411yescharacteristic
267C2#2255002512yes—
268C2#2255007412yes—
269C2#2255024711yescharacteristic
270C2#2255026211yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861205C2
hypercenterC22×SD16D127
hypocenterC127#2014
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
3. C2×#16861131C2semidirect
4. C2×#16861130C2semidirect
5. #22549783C2semidirect
6. #22549630C2semidirect
7. C22×SD16×C127C2semidirect
42. C2×C127⋊C8D4semidirect
53. C2×Q8×C127D4semidirect
65. C2×C8×C127D4semidirect
80. C2×Dic127SD16semidirect
136. C127#2014semidirect
138. C22×SD16D127semidirect
170. C2×Q8C22⋊D127semidirect
197. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168612052
C2×#168611312
C2×#168611302
#225497832
#225496302
C22×SD16×C1272
#2014127