Finite Group Databasebeta

← #30651311

Subgroups — #30651311

Order 16256.

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

normal subgroups only

This group has 374 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#22550600C211yesmaximal, characteristic, complemented
2#22550531C211yesmaximal, characteristic
3#22550279C211yesmaximal, characteristic, complemented
4#22550213C212yesmaximal, complemented
5#22550199C212yesmaximal, complemented
6#22550162C211yesmaximal, characteristic, complemented
7#22550121C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9#22549953C211yesmaximal, characteristic, complemented
10#22549917C212yesmaximal, complemented
11#474×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—21no—
13C2×#14523236C2211yescharacteristic
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic
16#16861204—22no—
17#16861198—21no—
18#16861191—22no—
19#16861189C2212yes—
20#16861188—21no—
21#16861180C2212yes—
22Q8×Dic127—21no—
23C4×Dic254C2211yescharacteristic
24C4×D508C2211yescharacteristic
25#16861167C2212yes—
26C4⋊C4×D127C2212yes—
27#16861142C2212yes—
28#16861142C2212yes—
29#16861141C2212yes—
30#16861140C2211yescharacteristic
31#16861134C2212yes—
32#16861132C2211yescharacteristic
33#16861130C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C4⋊Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#107×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523248—44no—
42C2×Dic254—21no—
43C2×Dic254C2311yescharacteristic
44C2×D508D411yescharacteristic
45C2×C22⋊D127—22no—
46C2×C4×D127—22no—
47C2×C4×D127C2312yes—
48#14523236—22no—
49#14523231—24no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—22no—
53C4⋊Dic127—21no—
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127C2312yes—
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58Dic127⋊C4—21no—
59C22⋊Dic127—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4D412yes—
63C4×Dic127—22no—
64C4×Dic127—21no—
65C2×Q8×C127—21no—
66C2×D4×C127C2311yescharacteristic
67C22×C4×C127—21no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127C2×D412yes—
75Dic254—21no—
76Dic254—21no—
77Dic254—42no—
78D508—22no—
79C22⋊D127—44no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C2×Dic127C2×D412yes—
84C4×D127—22no—
85C4×D127—22no—
86C4×D127—44no—
87C127⋊C8—21no—
88C127⋊C8#4112yescomplemented
89C23×C127—21no—
90Q8×C127—42no—
91D4×C127—21no—
92D4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—42no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2411yescharacteristic
99C8×C127—22no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—22no—
106C2×D127—22no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—42no—
110C4×C127AGL(1,ℤ/8)12yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—22no—
114Dic127—41no—
115Dic127—42no—
116Dic127—41no—
117C2×C127—81no—
118C2×C127#54711yescharacteristic
119C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C127#57811yescharacteristic
121D127—42no—
122#3353—1271nomaximal, Sylow
123C127#335311yescharacteristic, complemented, Sylow
124#556—1271no—
125#547—1271no—
126#500—1271no—
127#484—1272no—
128#482—1272no—
129#475—1271no—
130#474D12711yescharacteristic, complemented
131#468—1272no—
132C4×SD16—1271no—
133#436—1271no—
134#428—1272no—
135C2×#41—2541no—
136C2×SD16—1271no—
137C2×M16—1272no—
138C4⋊Q8—1271no—
139C4⋊Q8C2×D12711yescharacteristic
140#126—2542no—
141#124—2541no—
142#123—2542no—
143C22⋊Q8—1272no—
144C22⋊Q8—2541no—
145#121—1272no—
146C4×Q8—1271no—
147C4×Q8—2541no—
148C4×D4—1271no—
149C4×D4C2×D12711yescharacteristic
150#117—1272no—
151C2×C4⋊C4—1272no—
152#107—1272no—
153#107—1272no—
154#107C2×D12712yes—
155#106—1272no—
156#106—1271no—
157C4⋊C8C2×D12711yescharacteristic
158#103—1272no—
159#103—1271no—
160#102—1271no—
161#102C2×D12712yes—
162C22⋊C8—1272no—
163C4×C8—1271no—
164#41—5084no—
165C2×Q8—21no—
166C2×Q8—1271no—
167C2×Q8—2541no—
168C2×D4—1271no—
169C2×D4—2542no—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—1272no—
173C22×C4—2542no—
174SD16—2542no—
175M16—2544no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8C22×D12712yes—
179C4⋊C4—21no—
180C4⋊C4—1271no—
181C4⋊C4—1271no—
182C4⋊C4—1272no—
183C4⋊C4—2542no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4C22×D12712yes—
188C4⋊C4C22×D12711yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—1272no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C22⋊C4—2542no—
194C42—2542no—
195C42—2541no—
196C42C22×D12711yescharacteristic
197C23—21no—
198C23—1272no—
199Q8—42no—
200Q8—5082no—
201Q8—2541no—
202Q8—2541no—
203D4—21no—
204D4—21no—
205D4—5084no—
206D4—2542no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—1272no—
211C2×C4—5084no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—1272no—
222C8—2541no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—2542no—
227C22—2542no—
228C22—2542no—
229C22C2×D4×D12711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—5081no—
234C4—5082no—
235C4—5081no—
236C4—2542no—
237C4#1686123312yes—
238C4#41×D12711yescharacteristic
239C4#1686125711yescharacteristic
240C2—81no—
241C2—5082no—
242C2#2255052811yescharacteristic
243C2C2×#1686123311yescharacteristic
244C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
prime radical#22550531C2
hypercenter#474D127
hypocenterC127#3353
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550600C2semidirect
3. #22550279C2semidirect
4. #22550213C2semidirect
5. #22550199C2semidirect
6. #22550162C2semidirect
7. #22550121C2semidirect
8. C4×#14523236C2semidirect
9. #22549953C2semidirect
10. #22549917C2semidirect
11. #474×C127C2semidirect
88. C127⋊C8#41semidirect
123. C127#3353semidirect
130. #474D127semidirect

Maximal subgroups

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

subgroupindex
#225506002
#225505312
#225502792
#225502132
#225501992
#225501622
#225501212
C4×#145232362
#225499532
#225499172
#474×C1272
#3353127