Finite Group Databasebeta

← #30651452

Subgroups — #30651452

Order 16256.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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
2D4×Dic254C211yesmaximal, characteristic, complemented
3#22550237C212yesmaximal, complemented
4#22550226C211yesmaximal, characteristic, complemented
5#22550169C212yesmaximal, complemented
6#22550157C211yesmaximal, characteristic, complemented
7#22550093C212yesmaximal
8#22550003C211yesmaximal, characteristic, complemented
9#22549977C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic
11#461×C127C211yesmaximal, characteristic, complemented
12C22×Dic254—21no—
13C2×#14523233C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861208C2212yes—
16#16861204—22no—
17#16861198C2212yes—
18#16861189—21no—
19#16861188—21no—
20#16861186—22no—
21Q8×Dic127C2212yes—
22C4×Dic254C2211yescharacteristic
23C4×D508C2211yescharacteristic
24D4×Dic127C2212yes—
25#16861167—21no—
26#16861143C2212yes—
27#16861142C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861133C2212yes—
30#16861133C2212yes—
31#16861131C2212yes—
32#16861130C2211yescharacteristic
33#16861129C2211yescharacteristic
34#16861119C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×SD16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254—21no—
43C2×Dic254—44no—
44C2×Dic254D411yescharacteristic
45C2×D508C2311yescharacteristic
46C22×Dic127—22no—
47C2×C4×D127—21no—
48#14523233—22no—
49C2×C127⋊C8C2311yescharacteristic
50C2×C127⋊C8C2311yescharacteristic
51C4⋊Dic127—22no—
52C4⋊Dic127—21no—
53C4⋊Dic127D411yescharacteristic
54C4⋊Dic127C2312yes—
55C4⋊Dic127C2311yescharacteristic
56Dic127⋊C4—22no—
57Dic127⋊C4D412yes—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—22no—
63C4×Dic127C2312yes—
64C2×Q8×C127C2312yes—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67SD16×C127—24no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127—21no—
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75Dic254—22no—
76Dic254—44no—
77Dic254—42no—
78D508—21no—
79D508—21no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C2×Dic127—44no—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×D412yes—
85C4×D127—42no—
86C127⋊C8—21no—
87C127⋊C8—21no—
88C23×C127—21no—
89Q8×C127—22no—
90Q8×C127—22no—
91D4×C127—21no—
92D4×C127#4112yes—
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—22no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127C22×D411yescharacteristic
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—21no—
108C4×C127—42no—
109C4×C127—41no—
110C4×C127AGL(1,ℤ/8)12yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—22no—
114Dic127—22no—
115Dic127—41no—
116Dic127—42no—
117C2×C127—42no—
118C2×C127#54811yescharacteristic
119C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C127#57811yescharacteristic
121D127—81no—
122#3367—1271nomaximal, Sylow
123C127#336711yescharacteristic, complemented, Sylow
124#556—1271no—
125D4×Q8—1271no—
126#489—1272no—
127#487—1271no—
128#477—1272no—
129#474—1271no—
130#463—1272no—
131#461D12711yescharacteristic, complemented
132#446—1271no—
133#440—1272no—
134C4×D8—1271no—
135C22×Q8—2541no—
136C2×SD16C2×D12712yes—
137C2×D8—1271no—
138C4⋊Q8—1272no—
139C4⋊Q8—1271no—
140C4⋊Q8C2×D12711yescharacteristic
141#126—2542no—
142#124—1272no—
143C22⋊Q8—2542no—
144C22⋊Q8—2541no—
145C22⋊Q8—2541no—
146C4×Q8—1271no—
147C4×Q8—1272no—
148C4×D4—1272no—
149C4×D4—1271no—
150C4×D4C2×D12711yescharacteristic
151#117—2541no—
152#107—1272no—
153#107—1271no—
154C4⋊C8—1272no—
155C4⋊C8C2×D12711yescharacteristic
156#103—1272no—
157#103—1272no—
158#103C2×D12712yes—
159#102—1271no—
160#102—1272no—
161#102—1271no—
162#97—1272no—
163C4×C8—1271no—
164C2×Q8—1271no—
165C2×Q8—5084no—
166C2×Q8—2542no—
167C2×Q8—2541no—
168C2×Q8C22×D12712yes—
169C2×D4—1271no—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—2542no—
173C22×C4—2541no—
174SD16—24no—
175D8—2542no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8C22×D12712yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1272no—
182C4⋊C4—1271no—
183C4⋊C4—1272no—
184C4⋊C4—1271no—
185C4⋊C4—2542no—
186C4⋊C4—2542no—
187C4⋊C4—2541no—
188C4⋊C4C22×D12711yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—2542no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C22⋊C4—2541no—
194C42—1272no—
195C42—2542no—
196C42C22×D12711yescharacteristic
197C23—21no—
198C23—2541no—
199Q8—22no—
200Q8—22no—
201Q8—5084no—
202Q8—5082no—
203Q8—2542no—
204D4—21no—
205D4—2541no—
206D4—2541no—
207D4#1452324712yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—1272no—
212C2×C4—1272no—
213C2×C4—5084no—
214C2×C4—5082no—
215C2×C4—2541no—
216C2×C4—2542no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—2541no—
222C8—2541no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—5081no—
227C22—5081no—
228C22—5081no—
229C22C2×D4×D12711yescharacteristic
230C4—21no—
231C4—42no—
232C4—41no—
233C4—5082no—
234C4—5081no—
235C4—2542no—
236C4—2542no—
237C4#1686123612yes—
238C4C2×#1452324711yescharacteristic
239C4#1686125611yescharacteristic
240C2—42no—
241C2—10161no—
242C2#2255053211yescharacteristic
243C2C2×#1686123611yescharacteristic
244C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523233C22
hypercenter#461D127
hypocenterC127#3367
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. #22550600C2semidirect
2. D4×Dic254C2semidirect
3. #22550237C2semidirect
4. #22550226C2semidirect
5. #22550169C2semidirect
6. #22550157C2semidirect
8. #22550003C2semidirect
9. #22549977C2semidirect
11. #461×C127C2semidirect
123. C127#3367semidirect
131. #461D127semidirect

Maximal subgroups

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

subgroupindex
#225506002
D4×Dic2542
#225502372
#225502262
#225501692
#225501572
#225500932
#225500032
#225499772
C4×#145232332
#461×C1272
#3367127