Finite Group Databasebeta

← #30651314

Subgroups — #30651314

Order 16256.

374 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 123 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#22550596C211yesmaximal, characteristic, complemented
2#22550279C211yesmaximal, characteristic, complemented
3#22550211C212yesmaximal, complemented
4#22550197C212yesmaximal, complemented
5#22550158C211yesmaximal, characteristic, complemented
6#22550156C211yesmaximal, characteristic, complemented
7#22550120C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9#22549954C211yesmaximal, characteristic, complemented
10#22549916C212yesmaximal, complemented
11#547×C127C211yesmaximal, characteristic
12C2×#14523236C2211yescharacteristic
13C2×#14523232C2212yes—
14#16861209C2211yescharacteristic
15#16861202—22no—
16#16861196—21no—
17#16861188C2212yes—
18C4×Dic254C2211yescharacteristic
19#16861174—21no—
20#16861168C2212yes—
21#16861142C2212yes—
22#16861142C2212yes—
23#16861142C2212yes—
24#16861140C2212yes—
25#16861140C2211yescharacteristic
26C4⋊(C127⋊C8)C2211yescharacteristic
27#16861132C2212yes—
28#16861132C2211yescharacteristic
29#16861130C2212yes—
30#16861130C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C2×#41×C127—21no—
34C4⋊Q8×C127C2211yescharacteristic
35#123×C127—22no—
36C22⋊Q8×C127—21no—
37#121×C127C2212yes—
38C4×D4×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2212yes—
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43#14523236—22no—
44#14523232—24no—
45C2×C127⋊C8C2312yes—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2311yescharacteristic
48C4⋊Dic127—21no—
49C4⋊Dic127C2312yes—
50C4⋊Dic127C2311yescharacteristic
51Dic127⋊C4—22no—
52Dic127⋊C4—21no—
53C22⋊Dic127—22no—
54C22⋊Dic127—22no—
55C4×Dic127—21no—
56C4×Dic127—22no—
57#41×C127—44no—
58C2×Q8×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127D411yescharacteristic
61C2×D4×C127C2311yescharacteristic
62C22×C4×C127—22no—
63C22×C4×C127—21no—
64C22×C4×C127C2312yes—
65C4⋊C4×C127—22no—
66C4⋊C4×C127—21no—
67C4⋊C4×C127D411yescharacteristic
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2312yes—
70C22⋊C4×C127—21no—
71C22⋊C4×C127—22no—
72C22⋊C4×C127D412yes—
73C42×C127C2311yescharacteristic
74Dic254—21no—
75Dic254—21no—
76Dic254—42no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C127⋊C8—22no—
81C127⋊C8—21no—
82C127⋊C8#4112yescomplemented
83C23×C127—21no—
84C23×C127C2×D412yes—
85Q8×C127—42no—
86D4×C127—22no—
87D4×C127—21no—
88D4×C127—21no—
89D4×C127—44no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—44no—
95C2×C4×C127—42no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—22no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127C22×D411yescharacteristic
107C4×C127—22no—
108C4×C127—21no—
109C4×C127—41no—
110C4×C127—42no—
111C4×C127AGL(1,ℤ/8)12yes—
112C4×C127C2×#4111yescharacteristic
113C4×C1272−1+411yescharacteristic
114Dic127—41no—
115Dic127—42no—
116Dic127—41no—
117C2×C127—42no—
118C2×C127—81no—
119C2×C127#54711yescharacteristic
120C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C127#57811yescharacteristic
122#3353—1271nomaximal, Sylow
123C127#335311yescharacteristic, complemented, Sylow
124#556—1271no—
125#547D12711yescharacteristic
126#500—1271no—
127#484—1272no—
128#482—1272no—
129#475—1271no—
130#474—1271no—
131#468—1272no—
132C4×SD16—1271no—
133#436—1271no—
134#428—1272no—
135C2×#41—21no—
136C2×SD16—1271no—
137C2×M16—1272no—
138C4⋊Q8—1271no—
139C4⋊Q8C2×D12711yescharacteristic
140#126—2542no—
141#124—2541no—
142#123—22no—
143C22⋊Q8—21no—
144C22⋊Q8—1272no—
145#121C2×D12712yes—
146C4×Q8—1271no—
147C4×Q8—2541no—
148C4×D4C2×D12711yescharacteristic
149C4×D4C2×D12711yescharacteristic
150#117—1272no—
151C2×C4⋊C4C2×D12712yes—
152#107—1272no—
153#107—1272no—
154#107—1272no—
155#106—1271no—
156#106—1272no—
157C4⋊C8—1271no—
158#103—1271no—
159#103—1272no—
160#102—1271no—
161#102—1272no—
162C22⋊C8—1272no—
163C4×C8—1271no—
164#41—44no—
165C2×Q8—21no—
166C2×Q8—1271no—
167C2×Q8—2541no—
168C2×D4—22no—
169C2×D4C22⋊D12711yescharacteristic
170C2×D4C22×D12711yescharacteristic
171C22×C4—22no—
172C22×C4—21no—
173C22×C4C22×D12712yes—
174SD16—2542no—
175M16—2544no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8—1272no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—1271no—
182C4⋊C4—1272no—
183C4⋊C4—2541no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4C22⋊D12711yescharacteristic
187C4⋊C4C22×D12711yescharacteristic
188C4⋊C4C22×D12712yes—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C22⋊C4C22⋊D12712yes—
194C42—2542no—
195C42—2541no—
196C42C22×D12711yescharacteristic
197C23—21no—
198C23C2×C22⋊D12712yes—
199Q8—42no—
200Q8—5082no—
201Q8—2541no—
202Q8—2541no—
203D4—22no—
204D4—21no—
205D4—21no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—2541no—
214C2×C4—2542no—
215C2×C4—2541no—
216C2×C4C2×C22⋊D12712yes—
217C2×C4C2×C22⋊D12711yescharacteristic
218C2×C4C2×C22⋊D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—1272no—
221C8—2542no—
222C8—2541no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22C22×C22⋊D12711yescharacteristic
230C4—21no—
231C4—22no—
232C4—41no—
233C4—42no—
234C4—5082no—
235C4—5081no—
236C4—5081no—
237C4#1686123512yes—
238C4#41×D12711yescharacteristic
239C4#1686125611yescharacteristic
240C2—42no—
241C2—81no—
242C2#2255052511yescharacteristic
243C2C2×#1686123511yescharacteristic
244C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3353
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22550596C2semidirect
2. #22550279C2semidirect
3. #22550211C2semidirect
4. #22550197C2semidirect
5. #22550158C2semidirect
6. #22550156C2semidirect
7. #22550120C2semidirect
8. C4×#14523236C2semidirect
9. #22549954C2semidirect
10. #22549916C2semidirect
82. C127⋊C8#41semidirect
123. C127#3353semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225502792
#225502112
#225501972
#225501582
#225501562
#225501202
C4×#145232362
#225499542
#225499162
#547×C1272
#3353127