Finite Group Databasebeta

← #30639745

Subgroups — #30639745

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×#16861235C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#127×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861205—21no—
16#16861196—21no—
17#16861180—22no—
18#16861178—22no—
19#16861168C2211yescharacteristic
20#16861137—22no—
21#16861130—22no—
22#16861126—22no—
23C22×D4×C127—22no—
24C22×D4×C127C2211yescharacteristic
25C22×D4×C127C2211yescharacteristic
26#127×C127—22no—
27#127×C127—22no—
28C2×C42×C127C2211yescharacteristic
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254D411yescharacteristic, complemented
34C2×D508D411yescharacteristic, complemented
35C2×C22⋊D127—24no—
36C2×C22⋊D127—21no—
37C2×C4×D127—21no—
38#14523236—24no—
39#14523233—24no—
40#14523232—21no—
41#14523232—21no—
42#14523232—21no—
43#14523232D412yescomplemented
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C4⋊Dic127D412yescomplemented
47C22⋊Dic127—24no—
48C22⋊Dic127—21no—
49C4×Dic127—21no—
50C24×C127—21no—
51C24×C127D412yes—
52C2×D4×C127—28no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—48no—
65C2×D4×C127D412yescomplemented
66C22×C4×C127—21no—
67C22×C4×C127C2311yescharacteristic
68C42×C127—21no—
69C42×C127D412yescomplemented
70C22×D127—21no—
71Dic254—21no—
72Dic254—21no—
73D508—21no—
74D508—21no—
75C22⋊D127—41no—
76C22⋊D127—41no—
77C22⋊D127—84no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C4×D127—41no—
81C4×D127—41no—
82C127⋊C8—22no—
83C127⋊C8—41no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—22no—
88C23×C127—44no—
89C23×C127—44no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127C2×D411yescharacteristic
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—416no—
96D4×C127—41no—
97D4×C127—41no—
98D4×C127—41no—
99D4×C127—41no—
100D4×C127—41no—
101D4×C127—41no—
102C2×C4×C127—24no—
103C2×C4×C127—22no—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127#4111yescharacteristic
110C22×C127—22no—
111C22×C127—44no—
112C22×C127—44no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—44no—
116C22×C127—82no—
117C22×C127—82no—
118C22×C127—82no—
119C22×C127#12011yescharacteristic
120C22×C127#12011yescharacteristic
121C22×C127#12111yescharacteristic
122C2×D127—41no—
123C2×D127—42no—
124C4×C127—24no—
125C4×C127#12011yescharacteristic
126C4×C127#12111yescharacteristic
127C4×C127#12111yescharacteristic
128C4×C127#12711yescharacteristic
129Dic127—41no—
130Dic127—42no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—82no—
134C2×C127—82no—
135C2×C127#39211yescharacteristic
136C2×C127#45312yes—
137D127—81no—
138#2002—1271nomaximal, Sylow
139C127#200211yescharacteristic, complemented, Sylow
140C2×AGL(1,ℤ/8)—1271no—
141#535—1271no—
142C2×#127D12711yescharacteristic, complemented
143C2×#104—1271no—
144#418—1271no—
145C2×#100—1271no—
146#339—1271no—
147C2×#41—1271no—
148C22×D4—22no—
149C22×D4C2×D12711yescharacteristic
150C22×D4C2×D12711yescharacteristic
151AGL(1,ℤ/8)—2542no—
152AGL(1,ℤ/8)—2542no—
153C2×SD16—2541no—
154C2×D8—2541no—
155C2×M16—1271no—
156C2×M16—1271no—
157#127—22no—
158#127—22no—
159#127—2541no—
160#124—2541no—
161#121—2542no—
162#120—2542no—
163#117—1271no—
164C2×C42C2×D12711yescharacteristic
165#104—2542no—
166#102—2542no—
167#100—2542no—
168C24—21no—
169C24C22⋊D12712yes—
170#41—2541no—
171#41—2541no—
172#41—2541no—
173#41—2541no—
174C2×Q8—1271no—
175C2×D4—28no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—48no—
188C2×D4—1271no—
189C2×D4—2544no—
190C2×D4—2541no—
191C2×D4C22⋊D12712yescomplemented
192C22×C4—21no—
193C22×C4—2541no—
194C22×C4C22×D12711yescharacteristic
195SD16—2544no—
196D8—2544no—
197M16—1272no—
198M16—2541no—
199M16—2541no—
200M16—2541no—
201C2×C8—2541no—
202C2×C8—2541no—
203C4⋊C4—1272no—
204C22⋊C4—2544no—
205C22⋊C4—2541no—
206C42—21no—
207C42—2541no—
208C42C22⋊D12712yescomplemented
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—44no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23—2541no—
218C23C2×C22⋊D12711yescharacteristic
219Q8—2541no—
220Q8—2541no—
221D4—22no—
222D4—22no—
223D4—416no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—10164no—
231D4—5081no—
232D4—5081no—
233D4—2541no—
234D4—2541no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—5081no—
238C2×C4—5081no—
239C2×C4—2542no—
240C2×C4—2541no—
241C2×C4D4×D12711yescharacteristic
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4C2×C22⋊D12711yescharacteristic
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4#1452324711yescharacteristic
247C8—5081no—
248C8—2542no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—5081no—
259C22—5082no—
260C22#1686117711yescharacteristic
261C22#1686117811yescharacteristic
262C22#1686118111yescharacteristic
263C4—24no—
264C4—5081no—
265C4—5082no—
266C4#1686117711yescharacteristic
267C4#1686118011yescharacteristic
268C4#1686118111yescharacteristic
269C4#1686120511yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—82no—
274C2—10161no—
275C2#2254978411yescharacteristic
276C2#2255004312yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#127×C127C2
socleC22×C127#120
hypercenterC2×#127D127
hypocenterC127#2002
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. #22550443C2semidirect
3. C2×#16861137C2semidirect
4. #22549881C2semidirect
5. C2×#16861126C2semidirect
6. #22549636C2semidirect
7. C2×#127×C127C2semidirect
33. C2×Dic254D4semidirect
34. C2×D508D4semidirect
43. #14523232D4semidirect
46. C4⋊Dic127D4semidirect
65. C2×D4×C127D4semidirect
69. C42×C127D4semidirect
139. C127#2002semidirect
142. C2×#127D127semidirect
191. C2×D4C22⋊D127semidirect
208. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504432
C2×#168611372
#225498812
C2×#168611262
#225496362
C2×#127×C1272
#2002127