Finite Group Databasebeta

← #30639748

Subgroups — #30639748

Order 16256.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861237C211yesmaximal, characteristic, complemented
2#22550445C211yesmaximal, characteristic, complemented
3C2×#16861206C211yesmaximal, characteristic, complemented
4C2×#16861137C211yesmaximal, characteristic, complemented
5#22549880C211yesmaximal, characteristic, complemented
6C2×#16861125C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508—22no—
10C22×D508C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861237—22no—
13#16861237—22no—
14C2×#14523237—21no—
15C2×D1016—21no—
16C2×#14523232C2211yescharacteristic
17#16861206—22no—
18#16861206—21no—
19#16861206—22no—
20#16861197—21no—
21#16861184—22no—
22#16861179—22no—
23#16861137—22no—
24#16861129—22no—
25#16861125—22no—
26C2×M16×C127C2211yescharacteristic
27#117×C127C2211yescharacteristic
28C2×C42×C127C2211yescharacteristic
29C23×D127—21no—
30C23×D127D412yes—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254D411yescharacteristic, complemented
36C2×D508—28no—
37C2×D508—24no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508—21no—
43C2×D508—21no—
44C2×D508—21no—
45C2×D508—21no—
46C2×D508—21no—
47C2×D508—21no—
48C2×D508—21no—
49C2×D508—48no—
50C2×D508D411yescharacteristic, complemented
51C2×D508D412yescomplemented
52C2×C22⋊D127—21no—
53C2×C4×D127—21no—
54#14523237—24no—
55D1016—24no—
56#14523232—21no—
57#14523232—21no—
58C2×C127⋊C8—21no—
59(C2×D127)⋊C4—24no—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62M16×C127—21no—
63M16×C127D412yescomplemented
64C2×C8×C127—21no—
65C4⋊C4×C127D412yescomplemented
66C22⋊C4×C127—21no—
67C42×C127—21no—
68C42×C127—21no—
69C42×C127D412yescomplemented
70C22×D127—22no—
71C22×D127—21no—
72C22×D127—22no—
73C22×D127—22no—
74C22×D127—22no—
75C22×D127—44no—
76C22×D127—44no—
77C22×D127—41no—
78C22×D127—41no—
79Dic254—21no—
80Dic254—21no—
81D508—21no—
82D508—22no—
83D508—22no—
84D508—21no—
85D508—416no—
86D508—41no—
87D508—41no—
88D508—41no—
89D508—41no—
90D508—41no—
91D508—41no—
92D508—84no—
93C22⋊D127—41no—
94C22⋊D127—41no—
95C2×Dic127—21no—
96C4×D127—41no—
97C4×D127—41no—
98C127⋊C8—41no—
99C23×C127C2×D411yescharacteristic
100C2×C4×C127—24no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127#4111yescharacteristic
109C8×C127—22no—
110C22×C127—22no—
111C22×C127#12011yescharacteristic
112C22×C127#12011yescharacteristic
113C22×C127#12111yescharacteristic
114C2×D127—44no—
115C2×D127—44no—
116C2×D127—41no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—42no—
120C2×D127—44no—
121C2×D127—82no—
122C2×D127—82no—
123C2×D127—82no—
124C4×C127—24no—
125C4×C127—42no—
126C4×C127#12011yescharacteristic
127C4×C127#12111yescharacteristic
128C4×C127#12111yescharacteristic
129C4×C127#12711yescharacteristic
130Dic127—41no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127#39211yescharacteristic
134C2×C127#45312yes—
135D127—81no—
136D127—82no—
137D127—82no—
138#2002—1271nomaximal, Sylow
139C127#200211yescharacteristic, complemented, Sylow
140C2×AGL(1,ℤ/8)—1271no—
141#535—1271no—
142C2×#127—1271no—
143C2×#104—1271no—
144#418—1271no—
145C2×#100—1271no—
146#339D12711yescharacteristic, complemented
147C2×#41—1271no—
148C22×D4—1271no—
149C22×D4—1271no—
150C22×D4—2542no—
151AGL(1,ℤ/8)—2542no—
152AGL(1,ℤ/8)—2542no—
153C2×SD16—2541no—
154C2×D8—2541no—
155C2×M16—1271no—
156C2×M16C2×D12711yescharacteristic
157#127—2542no—
158#127—2541no—
159#127—2542no—
160#124—2541no—
161#121—2542no—
162#120—2542no—
163#117C2×D12711yescharacteristic
164C2×C42C2×D12711yescharacteristic
165#104—2542no—
166#102—2542no—
167#100—2542no—
168C24—1272no—
169C24—2541no—
170#41—2541no—
171#41—2541no—
172#41—2541no—
173#41—2541no—
174C2×Q8—1271no—
175C2×D4—1271no—
176C2×D4—1272no—
177C2×D4—5088no—
178C2×D4—2544no—
179C2×D4—2548no—
180C2×D4—2541no—
181C2×D4—2541no—
182C2×D4—2541no—
183C2×D4—2541no—
184C2×D4—2541no—
185C2×D4—2541no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C22×C4—21no—
193C22×C4—2541no—
194C22×C4C22×D12711yescharacteristic
195SD16—2544no—
196D8—2544no—
197M16—21no—
198M16—2541no—
199M16—2541no—
200M16D50812yescomplemented
201C2×C8—21no—
202C2×C8—2541no—
203C4⋊C4D50812yescomplemented
204C22⋊C4—21no—
205C22⋊C4—2544no—
206C42—21no—
207C42—21no—
208C42D50812yescomplemented
209C23—5084no—
210C23—5084no—
211C23—5081no—
212C23—5081no—
213C23—2542no—
214C23—2542no—
215C23—2542no—
216C23—2542no—
217C23—2541no—
218C23D4×D12711yescharacteristic
219Q8—2541no—
220Q8—2541no—
221D4—10164no—
222D4—50816no—
223D4—5081no—
224D4—5081no—
225D4—5081no—
226D4—5081no—
227D4—5081no—
228D4—5081no—
229D4—5081no—
230D4—5081no—
231D4—2542no—
232D4—2541no—
233D4—2541no—
234D4—2542no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—5081no—
239C2×C4—5081no—
240C2×C4—2541no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4C2×D50811yescharacteristic
244C2×C4C2×D50811yescharacteristic
245C2×C4C2×D50811yescharacteristic
246C2×C4#1452324911yescharacteristic
247C8—22no—
248C8—5081no—
249C22—22no—
250C22—10162no—
251C22—10162no—
252C22—10162no—
253C22—5084no—
254C22—5084no—
255C22—5081no—
256C22—5082no—
257C22—5082no—
258C22—5082no—
259C22—5084no—
260C22#1686117911yescharacteristic
261C22#1686117911yescharacteristic
262C22#1686118411yescharacteristic
263C4—24no—
264C4—42no—
265C4—5081no—
266C4#1686117911yescharacteristic
267C4#1686118411yescharacteristic
268C4#1686118411yescharacteristic
269C4#1686120611yescharacteristic
270C2—21no—
271C2—21no—
272C2—10162no—
273C2—10162no—
274C2—10161no—
275C2#2254978611yescharacteristic
276C2#2255004512yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127#120
hypercenter#339D127
hypocenterC127#2002
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. #22550445C2semidirect
3. C2×#16861206C2semidirect
4. C2×#16861137C2semidirect
5. #22549880C2semidirect
6. C2×#16861125C2semidirect
7. #339×C127C2semidirect
35. C2×Dic254D4semidirect
50. C2×D508D4semidirect
51. C2×D508D4semidirect
63. M16×C127D4semidirect
65. C4⋊C4×C127D4semidirect
69. C42×C127D4semidirect
139. C127#2002semidirect
146. #339D127semidirect
200. M16D508semidirect
203. C4⋊C4D508semidirect
208. C42D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225504452
C2×#168612062
C2×#168611372
#225498802
C2×#168611252
#339×C1272
#2002127