Finite Group Databasebeta

← #30639836

Subgroups — #30639836

Order 16256.

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

normal subgroups only

This group has 382 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×#16861234C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3C2×#16861126C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic, complemented
5#22549843C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234—22no—
11C2×#14523235—21no—
12C2×#14523233—22no—
13C2×#14523233—22no—
14C2×#14523233—22no—
15C2×#14523233—21no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861126—22no—
20#16861125—22no—
21C22⋊(C127⋊C8)—22no—
22C2×#41×C127C2211yescharacteristic
23C22×D4×C127C2211yescharacteristic
24AGL(1,ℤ/8)×C127—22no—
25AGL(1,ℤ/8)×C127—22no—
26C2×SD16×C127—21no—
27C2×D8×C127—21no—
28C2×M16×C127C2211yescharacteristic
29C23×D127—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508D412yes—
36#14523235—24no—
37#14523233—28no—
38#14523233—24no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232D412yes—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8D412yes—
48C24×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53C2×Q8×C127D411yescharacteristic
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×C127D411yescharacteristic
61C2×D4×C127D412yes—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64SD16×C127—24no—
65D8×C127—24no—
66M16×C127—21no—
67M16×C127D412yes—
68C2×C8×C127—21no—
69C22×D127—22no—
70C22×D127—41no—
71C22×D127—41no—
72C22×D127—82no—
73D508—22no—
74D508—22no—
75D508—41no—
76D508—41no—
77C127⋊C8—22no—
78C127⋊C8—22no—
79C127⋊C8—41no—
80C23×C127—21no—
81C23×C127—22no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—82no—
85C23×C127#4111yescharacteristic
86Q8×C127—21no—
87Q8×C127—21no—
88D4×C127—21no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—21no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—41no—
96C2×C4×C127—21no—
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×C127C2×D411yescharacteristic
105C8×C127—22no—
106C22×C127—22no—
107C22×C127—41no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—82no—
111C22×C127—82no—
112C22×C127—82no—
113C22×C127#12111yescharacteristic
114C22×C127#12111yescharacteristic
115C22×C127#12111yescharacteristic
116C2×D127—42no—
117C2×D127—82no—
118C2×D127—82no—
119C2×D127—82no—
120C4×C127—41no—
121C4×C127#12011yescharacteristic
122C4×C127#12011yescharacteristic
123C4×C127#12011yescharacteristic
124C4×C127#12711yescharacteristic
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—81no—
128C2×C127—82no—
129C2×C127#39211yescharacteristic
130C2×C127#47212yes—
131D127—82no—
132#2018—1271nomaximal, Sylow
133C127#201811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1271no—
135C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
136C22×D8—1271no—
137C2×#100—1271no—
138C2×#100—1271no—
139#408—1271no—
140#341—1271no—
141C2×#41C2×D12711yescharacteristic
142C22×D4—1271no—
143C22×D4C2×D12711yescharacteristic
144AGL(1,ℤ/8)—22no—
145AGL(1,ℤ/8)—22no—
146AGL(1,ℤ/8)—2542no—
147AGL(1,ℤ/8)—2542no—
148C2×SD16—21no—
149C2×SD16—2541no—
150C2×D8—21no—
151C2×D8—2542no—
152C2×D8—2542no—
153C2×D8—2542no—
154C2×D8—2541no—
155C2×M16—1271no—
156C2×M16—1271no—
157C2×M16C2×D12711yescharacteristic
158C22×C8—1271no—
159#100—2542no—
160#100—2542no—
161C22⋊C8—2542no—
162C24—21no—
163C24—2541no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168C2×Q8C22⋊D12711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—1272no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C2×D4—2541no—
181C2×D4C22⋊D12711yescharacteristic
182C2×D4C22⋊D12712yes—
183C22×C4—21no—
184C22×C4C22×D12711yescharacteristic
185SD16—24no—
186SD16—2544no—
187D8—24no—
188D8—2548no—
189D8—2544no—
190M16—21no—
191M16—1272no—
192M16—2541no—
193M16—2541no—
194M16—2541no—
195M16C22⋊D12712yes—
196C2×C8—21no—
197C2×C8—1272no—
198C2×C8—2541no—
199C2×C8—2541no—
200C2×C8—2541no—
201C2×C8—2541no—
202C23—21no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—82no—
207C23—10162no—
208C23—5081no—
209C23—5081no—
210C23—2542no—
211C23#1452324711yescharacteristic
212Q8—21no—
213Q8—21no—
214D4—22no—
215D4—21no—
216D4—21no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—5081no—
223D4—5081no—
224D4—2542no—
225D4—2542no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C8—22no—
236C8—5081no—
237C8—2542no—
238C8—2542no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22—10162no—
247C22—10162no—
248C22—10162no—
249C22—5082no—
250C22#1686118011yescharacteristic
251C22#1686118111yescharacteristic
252C22#1686118111yescharacteristic
253C4—41no—
254C4#1686117711yescharacteristic
255C4#1686117711yescharacteristic
256C4#1686117811yescharacteristic
257C4#1686120511yescharacteristic
258C2—21no—
259C2—21no—
260C2—81no—
261C2—82no—
262C2—10162no—
263C2#2254978411yescharacteristic
264C2#2255014412yes—
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#121
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2018
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×#14523233C2semidirect
3. C2×#16861126C2semidirect
4. C2×#16861125C2semidirect
5. #22549843C2semidirect
6. #22549640C2semidirect
7. C2×AGL(1,ℤ/8)×C127C2semidirect
133. C127#2018semidirect
135. C2×AGL(1,ℤ/8)D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232332
C2×#168611262
C2×#168611252
#225498432
#225496402
C2×AGL(1,ℤ/8)×C1272
#2018127