Finite Group Databasebeta

← #30648197

Subgroups — #30648197

Order 16256.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16861238C212yesmaximal, complemented
2#22550424C211yesmaximal, characteristic
3C4×Q8×D127C211yesmaximal, characteristic, complemented
4#22550097C212yesmaximal, complemented
5#22550096C212yesmaximal, complemented
6#22550087C211yesmaximal, characteristic, complemented
7#22550086C211yesmaximal, characteristic, complemented
8#22550073C211yesmaximal, characteristic, complemented
9#22549898C211yesmaximal, characteristic, complemented
10#22549874C212yesmaximal, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C2×Q8×D127C2211yescharacteristic
14#16861238—28no—
15C2×Dic508C2212yes—
16C2×#14523238C2212yes—
17C2×#14523236C2212yes—
18C2×#14523231C2212yes—
19#16861209C2211yescharacteristic
20#16861196C2212yes—
21#16861180—21no—
22#16861180—21no—
23Q8×Dic127C2211yescharacteristic
24C4×Dic254—21no—
25C4×Dic254C2211yescharacteristic
26#16861174—21no—
27#16861174C2211yescharacteristic
28D4×Dic127—21no—
29D4×Dic127—21no—
30C4⋊C4×D127—21no—
31C4⋊C4×D127C2211yescharacteristic
32C42×D127—21no—
33C42×D127C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35Dic127⋊C8C2212yes—
36#16861134C2212yes—
37#16861132C2212yes—
38#16861130C2212yes—
39C2×SD16×C127C2212yes—
40#127×C127C2211yescharacteristic
41C4×Q8×C127C2211yescharacteristic
42C4⋊C8×C127C2211yescharacteristic
43#102×C127C2212yes—
44#14523247—24no—
45#14523247—24no—
46C2×Dic254—21no—
47C2×Dic254C2312yes—
48C2×Dic254C2311yescharacteristic
49Q8×D127—22no—
50Q8×D127D414yescomplemented
51C2×C22⋊D127—22no—
52C22×Dic127—22no—
53C2×C4×D127—21no—
54C2×C4×D127—21no—
55C2×C4×D127D411yescharacteristic
56C2×C4×D127D411yescharacteristic
57C2×C4×D127C2311yescharacteristic
58Dic508—24no—
59#14523238—24no—
60#14523236—24no—
61#14523231—24no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127—21no—
64C4⋊Dic127—21no—
65C4⋊Dic127—21no—
66C4⋊Dic127C2311yescharacteristic
67Dic127⋊C4—21no—
68Dic127⋊C4—21no—
69Dic127⋊C4—21no—
70C22⋊Dic127—22no—
71C22⋊Dic127—22no—
72C4×Dic127—21no—
73C4×Dic127—21no—
74C4×Dic127—21no—
75C4×Dic127D411yescharacteristic
76C4×Dic127D411yescharacteristic
77C4×Dic127C2311yescharacteristic
78C2×Q8×C127C2311yescharacteristic
79C2×D4×C127—21no—
80C2×D4×C127—21no—
81C2×D4×C127C2312yes—
82SD16×C127—24no—
83C2×C8×C127C2312yes—
84C4⋊C4×C127—21no—
85C4⋊C4×C127C2311yescharacteristic
86C42×C127—21no—
87C42×C127C2311yescharacteristic
88C22×D127C2×D411yescharacteristic
89Dic254—21no—
90Dic254—24no—
91Dic254—22no—
92Dic254—22no—
93Dic254C2×D412yes—
94C22⋊D127—44no—
95C2×Dic127—21no—
96C2×Dic127—21no—
97C2×Dic127—22no—
98C2×Dic127—44no—
99C2×Dic127C2×D411yescharacteristic
100C2×Dic127C2×D411yescharacteristic
101C2×Dic127C2×D411yescharacteristic
102C4×D127—24no—
103C4×D127—22no—
104C4×D127—41no—
105C4×D127—42no—
106C4×D127C2×D412yes—
107C4×D127#4112yes—
108C127⋊C8—22no—
109C23×C127—22no—
110Q8×C127—21no—
111Q8×C127C2×D412yes—
112D4×C127—22no—
113D4×C127—22no—
114D4×C127—42no—
115D4×C127—42no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127C22×D411yescharacteristic
126C2×D127—21no—
127C2×D127—21no—
128C2×D127#12112yes—
129C4×C127—22no—
130C4×C127—21no—
131C4×C127—41no—
132C4×C127#13712yescomplemented
133C4×C127C22×D411yescharacteristic
134C4×C127C2×#4111yescharacteristic
135Dic127—22no—
136Dic127—21no—
137Dic127—21no—
138Dic127—42no—
139Dic127—41no—
140Dic127#12112yes—
141C2×C127—82no—
142C2×C127C2×#12111yescharacteristic
143C2×C127C2×#13711yescharacteristic
144C2×C127#57511yescharacteristic
145D127—22no—
146#3037—1271nomaximal, Sylow
147C127#303711yescharacteristic, complemented, Sylow
148C2×#137—1272no—
149#532—1271no—
150C2×C4×Q8—1271no—
151#464—1272no—
152#464—1272no—
153#462—1271no—
154#462—1271no—
155#460—1271no—
156#460D12711yescharacteristic, complemented
157#423—1271no—
158#417—1272no—
159C2×#41—1272no—
160C22×Q8—1271no—
161#137—2548no—
162C2×Q16—1272no—
163C2×Q16—1272no—
164C2×SD16—1272no—
165C2×SD16C2×D12712yes—
166C2×M16—1272no—
167C4⋊Q8—1271no—
168#127C2×D12711yescharacteristic
169#124—1272no—
170#121—2541no—
171#121—2541no—
172C4×Q8—1271no—
173C4×Q8—1271no—
174C4×Q8—1271no—
175C4×Q8—2541no—
176C4×Q8—2541no—
177C4×Q8C2×D12711yescharacteristic
178C4×D4—2541no—
179C4×D4—2541no—
180C2×C4⋊C4—1271no—
181C2×C4⋊C4—2541no—
182C2×C42—1271no—
183C2×C42—2541no—
184C4⋊C8—1272no—
185C4⋊C8—1271no—
186C4⋊C8C2×D12711yescharacteristic
187#103—1272no—
188#103—1272no—
189#102—1272no—
190#102C2×D12712yes—
191#41—2544no—
192#41—2544no—
193C2×Q8—1274no—
194C2×Q8—1271no—
195C2×Q8—1272no—
196C2×Q8—2541no—
197C2×Q8—2542no—
198C2×Q8C22×D12711yescharacteristic
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2542no—
202C2×D4C22×D12712yes—
203C22×C4—1271no—
204C22×C4—1271no—
205C22×C4—1271no—
206C22×C4—2541no—
207C22×C4—2541no—
208C22×C4—2542no—
209Q16—2544no—
210Q16—2544no—
211SD16—24no—
212SD16—2544no—
213M16—2544no—
214C2×C8—1272no—
215C2×C8C22×D12712yes—
216C4⋊C4—21no—
217C4⋊C4—1271no—
218C4⋊C4—2541no—
219C4⋊C4—2541no—
220C4⋊C4—2541no—
221C4⋊C4—2541no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C4⋊C4C22×D12711yescharacteristic
225C22⋊C4—2542no—
226C22⋊C4—2542no—
227C42—21no—
228C42—1271no—
229C42—1271no—
230C42—1271no—
231C42—2541no—
232C42—2541no—
233C42—2541no—
234C42C22×D12711yescharacteristic
235C23—22no—
236C23—1271no—
237Q8—21no—
238Q8—1272no—
239Q8—2544no—
240Q8—2542no—
241Q8—2542no—
242Q8—2541no—
243Q8D4×D12712yes—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—42no—
248D4—5084no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—1271no—
252C2×C4—1272no—
253C2×C4—1272no—
254C2×C4—1271no—
255C2×C4—1271no—
256C2×C4—5081no—
257C2×C4—5082no—
258C2×C4—5084no—
259C2×C4—2544no—
260C2×C4—2541no—
261C2×C4—2542no—
262C2×C4—2541no—
263C2×C4—2542no—
264C2×C4D4×D12711yescharacteristic
265C2×C4D4×D12711yescharacteristic
266C2×C4C23×D12711yescharacteristic
267C8—22no—
268C8—2542no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—1272no—
273C22—2541no—
274C22—2541no—
275C22C2×D4×D12711yescharacteristic
276C4—22no—
277C4—21no—
278C4—41no—
279C4—1272no—
280C4—5082no—
281C4—5081no—
282C4—2542no—
283C4—2541no—
284C4—2541no—
285C4#1686123812yescomplemented
286C4C2×D4×D12711yescharacteristic
287C4#41×D12711yescharacteristic
288C2—82no—
289C2—2542no—
290C2#121×D12711yescharacteristic
291C2C2×#1686123811yescharacteristic
292C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550424C2
hypercenter#460D127
hypocenterC127#3037
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
3. C4×Q8×D127C2semidirect
4. #22550097C2semidirect
5. #22550096C2semidirect
6. #22550087C2semidirect
7. #22550086C2semidirect
8. #22550073C2semidirect
9. #22549898C2semidirect
10. #22549874C2semidirect
11. #460×C127C2semidirect
50. Q8×D127D4semidirect
132. C4×C127#137semidirect
147. C127#3037semidirect
156. #460D127semidirect
285. C4#16861238semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
#225504242
C4×Q8×D1272
#225500972
#225500962
#225500872
#225500862
#225500732
#225498982
#225498742
#460×C1272
#3037127