Finite Group Databasebeta

← #30639108

Subgroups — #30639108

Order 16256.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#22550359C211yesmaximal, characteristic, complemented
2C2×#16861136C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic, complemented
4#22549779C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127—21no—
10D4×Dic127—22no—
11D4×Dic127—21no—
12#16861168—21no—
13#16861168C2211yescharacteristic
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861136—22no—
19#16861130—21no—
20#16861130—21no—
21C2×#41×C127C2211yescharacteristic
22C22×D4×C127C2211yescharacteristic
23AGL(1,ℤ/8)×C127—22no—
24AGL(1,ℤ/8)×C127C414yescomplemented
25C2×SD16×C127—21no—
26C2×D8×C127—21no—
27C2×M16×C127C2211yescharacteristic
28C22×Dic127—22no—
29C22×Dic127—21no—
30C22×Dic127D412yes—
31#14523232—21no—
32#14523232—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—22no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39C22⋊Dic127—22no—
40C22⋊Dic127—21no—
41C22⋊Dic127—44no—
42C4×Dic127—22no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45C4×Dic127—21no—
46C24×C127—21no—
47#41×C127—21no—
48#41×C127—21no—
49#41×C127C2×C412yes—
50#41×C127D412yes—
51C2×Q8×C127D411yescharacteristic
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127C2×C412yescomplemented
59C2×D4×C127D411yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62SD16×C127—24no—
63D8×C127—24no—
64M16×C127—21no—
65M16×C127C2×C412yescomplemented
66C2×C8×C127—21no—
67C2×Dic127—22no—
68C2×Dic127—21no—
69C2×Dic127—22no—
70C2×Dic127—21no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127—42no—
74C2×Dic127—82no—
75C127⋊C8—41no—
76C23×C127—21no—
77C23×C127—22no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—21no—
84Q8×C127C22⋊C412yescomplemented
85D4×C127—22no—
86D4×C127—22no—
87D4×C127—22no—
88D4×C127—22no—
89D4×C127—21no—
90D4×C127—41no—
91D4×C127—41no—
92D4×C127C22⋊C412yescomplemented
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4111yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C8×C127—22no—
103C22×C127—21no—
104C22×C127—21no—
105C22×C127—22no—
106C22×C127—44no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—42no—
110C22×C127—82no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127C4×D411yescharacteristic
114C22×C127#12011yescharacteristic
115C22×C127#12111yescharacteristic
116C4×C127—22no—
117C4×C127C2×C22⋊C411yescharacteristic
118C4×C127C4×D411yescharacteristic
119C4×C127#12111yescharacteristic
120C4×C127#12311yescharacteristic
121Dic127—42no—
122Dic127—42no—
123Dic127—41no—
124Dic127—41no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—42no—
128C2×C127—82no—
129C2×C127#38611yescharacteristic
130C2×C127#45311yescharacteristic
131C2×C127#45411yescharacteristic
132#1883—1271nomaximal, Sylow
133C127#188311yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)D12711yescharacteristic
135#520—1271no—
136C2×#104—1271no—
137#418—1271no—
138#390—1271no—
139#343—1271no—
140#339—1271no—
141C2×#41C2×D12711yescharacteristic
142C22×D4C2×D12711yescharacteristic
143AGL(1,ℤ/8)—22no—
144AGL(1,ℤ/8)Dic12714yescomplemented
145C2×SD16—21no—
146C2×D8—21no—
147C2×M16—1271no—
148C2×M16C2×D12711yescharacteristic
149C4×Q8—2541no—
150C4×D4—2542no—
151C4×D4—2541no—
152#117—1271no—
153#117—2541no—
154C2×C4⋊C4—1271no—
155C2×C4⋊C4—2541no—
156C2×C22⋊C4—2542no—
157C2×C42—1271no—
158#104—2542no—
159#102—2541no—
160#102—2541no—
161C24—21no—
162#41—21no—
163#41—21no—
164#41C2×Dic12712yes—
165#41C22⋊D12712yes—
166C2×Q8C22⋊D12711yescharacteristic
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4C2×Dic12712yescomplemented
174C2×D4C22⋊D12711yescharacteristic
175C22×C4—21no—
176C22×C4—1272no—
177C22×C4—2542no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180SD16—24no—
181D8—24no—
182M16—21no—
183M16—2541no—
184M16—2541no—
185M16C2×Dic12712yescomplemented
186C2×C8—21no—
187C2×C8—2541no—
188C4⋊C4—1271no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2541no—
192C4⋊C4—2541no—
193C22⋊C4—5084no—
194C22⋊C4—2542no—
195C22⋊C4—2541no—
196C42—2542no—
197C42—2541no—
198C42—2541no—
199C42—2541no—
200C23—21no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23D4×D12711yescharacteristic
207Q8—21no—
208Q8C22⋊Dic12712yescomplemented
209D4—22no—
210D4—22no—
211D4—21no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4C22⋊Dic12712yescomplemented
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—10162no—
221C2×C4—5081no—
222C2×C4—5081no—
223C2×C4—5082no—
224C2×C4—2542no—
225C2×C4—2541no—
226C2×C4—2541no—
227C2×C4—2542no—
228C2×C4C22×Dic12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4C2×C22⋊D12711yescharacteristic
232C2×C4#1452324711yescharacteristic
233C2×C4#1452324711yescharacteristic
234C8—22no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—44no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—81no—
245C22—81no—
246C22D4×Dic12711yescharacteristic
247C22#1686117711yescharacteristic
248C22#1686118111yescharacteristic
249C4—22no—
250C4—5082no—
251C4—5081no—
252C4—5082no—
253C4—5081no—
254C4C2×C22⋊Dic12711yescharacteristic
255C4D4×Dic12711yescharacteristic
256C4#1686118111yescharacteristic
257C4#1686119011yescharacteristic
258C2—21no—
259C2—21no—
260C2—42no—
261C2—82no—
262C2#2254975611yescharacteristic
263C2#2255004211yescharacteristic
264C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#120
cosocleC22×C4×C127C23
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#1883
LCSG ▷ C2×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. #22550359C2semidirect
2. C2×#16861136C2semidirect
3. #22549881C2semidirect
4. #22549779C2semidirect
5. #22549644C2semidirect
6. #22549637C2semidirect
24. AGL(1,ℤ/8)×C127C4semidirect
58. C2×D4×C127C2×C4semidirect
65. M16×C127C2×C4semidirect
84. Q8×C127C22⋊C4semidirect
92. D4×C127C22⋊C4semidirect
133. C127#1883semidirect
144. AGL(1,ℤ/8)Dic127semidirect
173. C2×D4C2×Dic127semidirect
185. M16C2×Dic127semidirect
208. Q8C22⋊Dic127semidirect
216. D4C22⋊Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503592
C2×#168611362
#225498812
#225497792
#225496442
#225496372
C2×AGL(1,ℤ/8)×C1272
#1883127