Finite Group Databasebeta

← #30639802

Subgroups — #30639802

Order 16256.

510 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 93 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×D1016C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3C2×#16861205C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic, complemented
5#22549777C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—22no—
11C2×D1016—22no—
12C2×D1016—22no—
13C2×D1016—22no—
14C2×#14523233—22no—
15C2×#14523233—22no—
16C2×#14523233—22no—
17C22×C127⋊C8C2211yescharacteristic
18#16861205—22no—
19#16861205—22no—
20C2×(C2×D127)⋊C4—22no—
21C2×C4×Dic127C2211yescharacteristic
22#16861129—22no—
23C22×D4×C127C2211yescharacteristic
24C22×C8×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26#102×C127—22no—
27C23×D127—21no—
28C23×D127—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508D412yes—
40C2×D508D412yescomplemented
41C2×C22⋊D127—28no—
42C2×C22⋊D127—44no—
43C2×C22⋊D127—44no—
44C22×Dic127D412yes—
45D1016—28no—
46#14523233—28no—
47C2×C127⋊C8—21no—
48C2×C127⋊C8—21no—
49C2×C127⋊C8D412yescomplemented
50(C2×D127)⋊C4—44no—
51C4×Dic127—21no—
52C4×Dic127—21no—
53C24×C127—21no—
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
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127D412yescomplemented
65C4⋊C4×C127—21no—
66C4⋊C4×C127—21no—
67C22×D127—22no—
68C22×D127—22no—
69C22×D127—41no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127—41no—
73C22×D127—42no—
74C22×D127—42no—
75C22×D127—42no—
76C22×D127—42no—
77D508—22no—
78D508—22no—
79D508—22no—
80D508—22no—
81D508—41no—
82D508—41no—
83D508—41no—
84D508—41no—
85C22⋊D127—416no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127D814yescomplemented
89C127⋊C8—22no—
90C23×C127—22no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127—42no—
94C23×C127—42no—
95C23×C127C2×D411yescharacteristic
96D4×C127—22no—
97D4×C127—22no—
98D4×C127—41no—
99D4×C127—41no—
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×C127#4111yescharacteristic
108C8×C127—22no—
109C22×C127—42no—
110C22×C127—44no—
111C22×C127—82no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127#12011yescharacteristic
115C22×C127#12111yescharacteristic
116C22×C127#12711yescharacteristic
117C22×C127C2×D812yes—
118C22×C127AGL(1,ℤ/8)12yes—
119C2×D127—42no—
120C2×D127—42no—
121C2×D127—44no—
122C2×D127—44no—
123C2×D127—82no—
124C2×D127—82no—
125C2×D127—81no—
126C2×D127—81no—
127C2×D127—81no—
128C2×D127—81no—
129C4×C127—81no—
130C4×C127#12011yescharacteristic
131C4×C127#12011yescharacteristic
132C4×C127#12111yescharacteristic
133C4×C127#12111yescharacteristic
134Dic127—24no—
135C2×C127—82no—
136C2×C127#39211yescharacteristic
137C2×C127#44712yes—
138C2×C127#45912yes—
139C2×C127#49311yescharacteristic
140C2×C127#49611yescharacteristic
141D127—82no—
142D127—82no—
143#2011—1271nomaximal, Sylow
144C127#201111yescharacteristic, complemented, Sylow
145C22×D8—1271no—
146C22×D8—1271no—
147C2×#127—1271no—
148C2×#102—1271no—
149C2×#102D12711yescharacteristic, complemented
150#390—1271no—
151#336—1271no—
152C22×D4—1271no—
153C22×D4—1271no—
154C22×D4—2542no—
155C22×D4C2×D12711yescharacteristic
156C2×D8—2542no—
157C2×D8—2542no—
158C2×D8—2542no—
159C2×D8—2542no—
160C2×D8—2542no—
161C2×D8—2542no—
162C22×C8—1271no—
163C22×C8C2×D12711yescharacteristic
164#127—2542no—
165#127—2542no—
166C2×C4⋊C4C2×D12711yescharacteristic
167C2×C22⋊C4—2542no—
168C2×C42—1271no—
169#102—22no—
170#102—2542no—
171C24—21no—
172C24—2541no—
173C24—2541no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—1272no—
180C2×D4—1272no—
181C2×D4—5084no—
182C2×D4—5084no—
183C2×D4—2548no—
184C2×D4—2541no—
185C2×D4—2541no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C2×D4—2541no—
193C2×D4—2541no—
194C2×D4D50812yes—
195C22×C4—21no—
196C22×C4—1272no—
197C22×C4C22×D12711yescharacteristic
198D8—2548no—
199D8—2548no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—1272no—
203C2×C8—2541no—
204C2×C8—2541no—
205C2×C8C22⋊D12712yescomplemented
206C4⋊C4—21no—
207C4⋊C4—21no—
208C22⋊C4—5084no—
209C42—2541no—
210C42—2541no—
211C23—22no—
212C23—41no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—5081no—
217C23—5081no—
218C23—5082no—
219C23—5082no—
220C23—5082no—
221C23—5082no—
222C23—5081no—
223C23—5081no—
224C23—2542no—
225C23—2542no—
226C23D4×D12711yescharacteristic
227D4—22no—
228D4—22no—
229D4—41no—
230D4—41no—
231D4—50816no—
232D4—5081no—
233D4—5081no—
234D4—5081no—
235D4—5081no—
236D4—2542no—
237D4—2542no—
238D4—2542no—
239D4—2542no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—1274no—
243C2×C4—2542no—
244C2×C4—2542no—
245C2×C4D4×D12711yescharacteristic
246C2×C4D4×D12711yescharacteristic
247C2×C4D4×D12711yescharacteristic
248C2×C4C2×C22⋊D12711yescharacteristic
249C2×C4C2×D50811yescharacteristic
250C2×C4#1452324811yescharacteristic
251C8—22no—
252C8—2542no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—81no—
257C22—81no—
258C22—10162no—
259C22—10161no—
260C22—10161no—
261C22—10161no—
262C22—10161no—
263C22—10162no—
264C22—5082no—
265C22—5082no—
266C22—5084no—
267C22—5084no—
268C22#1686117911yescharacteristic
269C22#1686118211yescharacteristic
270C22#1686120511yescharacteristic
271C22D8×D12712yes—
272C22#1686123612yes—
273C4—81no—
274C4—2544no—
275C4#1686117711yescharacteristic
276C4#1686117911yescharacteristic
277C4#1686118211yescharacteristic
278C4#1686118311yescharacteristic
279C2—82no—
280C2—10162no—
281C2—10162no—
282C2#2254978511yescharacteristic
283C2#2255001112yes—
284C2#2255007012yes—
285C2#2255025111yescharacteristic
286C2#2255026211yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
hypercenterC2×#102D127
hypocenterC127#2011
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×D1016C2semidirect
2. C22×#14523233C2semidirect
3. C2×#16861205C2semidirect
4. C2×#16861129C2semidirect
5. #22549777C2semidirect
6. #22549630C2semidirect
7. C2×#102×C127C2semidirect
40. C2×D508D4semidirect
49. C2×C127⋊C8D4semidirect
64. C2×C8×C127D4semidirect
88. C2×Dic127D8semidirect
144. C127#2011semidirect
149. C2×#102D127semidirect
205. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×D10162
C22×#145232332
C2×#168612052
C2×#168611292
#225497772
#225496302
C2×#102×C1272
#2011127