Finite Group Databasebeta

← #30650661

Subgroups — #30650661

Order 16256.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550591C211yesmaximal, characteristic, complemented
2#22550529C211yesmaximal, characteristic, complemented
3#22550226C211yesmaximal, characteristic, complemented
4#22550215C212yesmaximal, complemented
5#22550167C211yesmaximal, characteristic, complemented
6#22550102C212yesmaximal, complemented
7#22550020C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9#22549941C211yesmaximal, characteristic, complemented
10#22549871C212yesmaximal, complemented
11#461×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×#14523233C2211yescharacteristic
14C2×C8×D127C2212yes—
15#16861209C2211yescharacteristic
16#16861195—22no—
17#16861194—22no—
18#16861185C2212yes—
19#16861184—21no—
20#16861183—21no—
21#16861180C2212yes—
22C4×D508C2211yescharacteristic
23C4×D508C2211yescharacteristic
24#16861168—21no—
25#16861167C2212yes—
26C4⋊C4×D127—21no—
27#16861142C2211yescharacteristic
28#16861141C2212yes—
29#16861133C2212yes—
30#16861133C2212yes—
31#16861131C2212yes—
32#16861130C2211yescharacteristic
33#16861129C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×SD16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41#14523249—24no—
42#14523249—42no—
43C2×Dic254—21no—
44C2×D508—22no—
45C2×D508D411yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—22no—
48C2×C4×D127—22no—
49C2×C4×D127—21no—
50C2×C4×D127C2312yes—
51#14523233—22no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127—42no—
55C4⋊Dic127—21no—
56C4⋊Dic127D411yescharacteristic
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2312yes—
59Dic127⋊C4—22no—
60C22⋊Dic127—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—21no—
64(C2×D127)⋊C4D412yes—
65C4×Dic127—22no—
66C2×Q8×C127C2312yes—
67C2×D4×C127C2311yescharacteristic
68C22×C4×C127—21no—
69SD16×C127—42no—
70C2×C8×C127C2312yes—
71C4⋊C4×C127—21no—
72C4⋊C4×C127—21no—
73C4⋊C4×C127C2311yescharacteristic
74C22⋊C4×C127—21no—
75C42×C127C2311yescharacteristic
76C22×D127—21no—
77C22×D127C2×D412yes—
78Dic254—81no—
79D508—22no—
80D508—21no—
81D508—44no—
82D508#4112yes—
83C22⋊D127—82no—
84C2×Dic127—22no—
85C2×Dic127—21no—
86C2×Dic127C2×D412yes—
87C4×D127—22no—
88C4×D127—22no—
89C4×D127—44no—
90C4×D127—42no—
91C127⋊C8—21no—
92C127⋊C8—21no—
93C23×C127—21no—
94Q8×C127—22no—
95Q8×C127—22no—
96D4×C127—21no—
97D4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—42no—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2411yescharacteristic
104C8×C127—22no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127C22×D411yescharacteristic
109C2×D127—22no—
110C2×D127—22no—
111C2×D127—22no—
112C2×D127—22no—
113C2×D127—41no—
114C2×D127—41no—
115C4×C127—21no—
116C4×C127—42no—
117C4×C127—41no—
118C4×C127#13512yes—
119C4×C127C2×#4111yescharacteristic
120C4×C127H(5,𝔽2)11yescharacteristic
121Dic127—22no—
122Dic127—41no—
123Dic127—42no—
124C2×C127—81no—
125C2×C127#54611yescharacteristic
126C2×C127C2×#13511yescharacteristic
127C2×C127#57711yescharacteristic
128D127—42no—
129D127—42no—
130#3303—1271nomaximal, Sylow
131C127#330311yescharacteristic, complemented, Sylow
132#555—1271no—
133#547—1271no—
134#487—1271no—
135#485—1272no—
136#476—1271no—
137#465—1272no—
138#461D12711yescharacteristic, complemented
139#449—1272no—
140C4×D8—1271no—
141C8×D4—1271no—
142#416—1272no—
143C2×#41—1272no—
144C2×SD16C2×D12712yes—
145C2×D8—1271no—
146C22×C8—1272no—
147C4⋊Q8—1271no—
148C4⋊Q8C2×D12711yescharacteristic
149#124—2542no—
150#123—2542no—
151C22⋊Q8—1272no—
152#121—1272no—
153#121—2541no—
154#121—2541no—
155C4×D4—1271no—
156C4×D4—1271no—
157C4×D4C2×D12711yescharacteristic
158#117—1272no—
159#117—2541no—
160C2×C4⋊C4—2541no—
161#107—1271no—
162#106—1272no—
163C4⋊C8C2×D12711yescharacteristic
164#103—1272no—
165#103—1272no—
166#103C2×D12712yes—
167#102—1271no—
168#102—1272no—
169#102—1271no—
170C22⋊C8—1272no—
171C4×C8—1271no—
172#41—5082no—
173#41—2544no—
174C2×Q8—2541no—
175C2×Q8C22×D12712yes—
176C2×D4—1271no—
177C2×D4—1271no—
178C2×D4—2542no—
179C2×D4—2542no—
180C2×D4C22×D12711yescharacteristic
181C22×C4—21no—
182C22×C4—1272no—
183C22×C4—2542no—
184C22×C4—2541no—
185SD16—42no—
186D8—2542no—
187C2×C8—1271no—
188C2×C8—1271no—
189C2×C8—5082no—
190C2×C8C22×D12712yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—1271no—
194C4⋊C4—1272no—
195C4⋊C4—1271no—
196C4⋊C4—2542no—
197C4⋊C4—2541no—
198C4⋊C4C22×D12711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—1272no—
201C22⋊C4—2542no—
202C22⋊C4—2542no—
203C22⋊C4—2542no—
204C22⋊C4—2541no—
205C42—2542no—
206C42C22×D12711yescharacteristic
207C23—21no—
208C23—1272no—
209C23—2541no—
210Q8—22no—
211Q8—22no—
212Q8—10161no—
213D4—21no—
214D4—21no—
215D4—1272no—
216D4—10162no—
217D4—5084no—
218D4—2542no—
219D4—2541no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—1272no—
224C2×C4—5084no—
225C2×C4—5082no—
226C2×C4—2542no—
227C2×C4—2541no—
228C2×C4—2542no—
229C2×C4—2542no—
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4C23×D12711yescharacteristic
233C8—22no—
234C8—2541no—
235C8—2541no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—5081no—
240C22—5081no—
241C22—2542no—
242C22—2542no—
243C22—2542no—
244C22—2542no—
245C22C2×D4×D12711yescharacteristic
246C4—21no—
247C4—41no—
248C4—42no—
249C4—5081no—
250C4—5082no—
251C4—2542no—
252C4#1686123012yes—
253C4#41×D12711yescharacteristic
254C4#1686125411yescharacteristic
255C2—81no—
256C2—5082no—
257C2—5082no—
258C2#2255052311yescharacteristic
259C2C2×#1686123011yescharacteristic
260C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
hypercenter#461D127
hypocenterC127#3303
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22550591C2semidirect
2. #22550529C2semidirect
3. #22550226C2semidirect
4. #22550215C2semidirect
5. #22550167C2semidirect
6. #22550102C2semidirect
7. #22550020C2semidirect
8. C4×#14523233C2semidirect
9. #22549941C2semidirect
10. #22549871C2semidirect
11. #461×C127C2semidirect
131. C127#3303semidirect
138. #461D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225505292
#225502262
#225502152
#225501672
#225501022
#225500202
C4×#145232332
#225499412
#225498712
#461×C1272
#3303127