Finite Group Databasebeta

← #30651396

Subgroups — #30651396

Order 16256.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic254C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic
3#22550222C211yesmaximal, characteristic, complemented
4#22550221C212yesmaximal, complemented
5#22550167C211yesmaximal, characteristic, complemented
6#22550164C212yesmaximal, complemented
7#22550074C212yesmaximal, complemented
8#22550000C211yesmaximal, characteristic, complemented
9SD16×Dic127C212yesmaximal, complemented
10C4×#14523235C211yesmaximal, characteristic, complemented
11#460×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—21no—
13C2×#14523235C2211yescharacteristic
14#16861209—21no—
15#16861209C2211yescharacteristic
16#16861208—22no—
17#16861208C2212yes—
18#16861205C2212yes—
19#16861181—22no—
20#16861180—21no—
21Q8×Dic127C2212yes—
22C4×Dic254—21no—
23C4×Dic254C2211yescharacteristic
24C4×D508C2211yescharacteristic
25D4×Dic127C2212yes—
26#16861141C2212yes—
27#16861140C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861133C2211yescharacteristic
30#16861131C2212yes—
31#16861130C2212yes—
32#16861130C2212yes—
33#16861129C2211yescharacteristic
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C2×SD16×C127C2212yes—
37#127×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41#14523247—44no—
42C2×Dic254—22no—
43C2×Dic254—21no—
44C2×Dic254D411yescharacteristic
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C22×Dic127—22no—
49C2×C4×D127—21no—
50#14523235—22no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C4⋊Dic127—22no—
54C4⋊Dic127—21no—
55C4⋊Dic127D411yescharacteristic
56C4⋊Dic127C2312yes—
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—22no—
59Dic127⋊C4D412yes—
60C22⋊Dic127—22no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—22no—
63C4×Dic127C2312yes—
64C2×Q8×C127C2311yescharacteristic
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127C2312yes—
68SD16×C127—24no—
69C2×C8×C127C2312yes—
70C4⋊C4×C127—21no—
71C4⋊C4×C127C2311yescharacteristic
72C42×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75Dic254—82no—
76Dic254—81no—
77Dic254SD1614yescomplemented
78D508—21no—
79D508—21no—
80C22⋊D127—44no—
81C22⋊D127—82no—
82C2×Dic127—22no—
83C2×Dic127—21no—
84C2×Dic127—44no—
85C2×Dic127C2×D412yes—
86C2×Dic127C2×D412yes—
87C4×D127—42no—
88C127⋊C8—21no—
89C127⋊C8—21no—
90C23×C127—22no—
91Q8×C127—21no—
92Q8×C127#4112yescomplemented
93D4×C127—22no—
94D4×C127—22no—
95D4×C127—42no—
96D4×C127—81no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2411yescharacteristic
102C8×C127—22no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127C22×D411yescharacteristic
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—41no—
110C4×C127—22no—
111C4×C127—21no—
112C4×C127—41no—
113C4×C127C2×SD1612yes—
114C4×C127C2×#4111yescharacteristic
115C4×C1272−1+411yescharacteristic
116Dic127—22no—
117Dic127—41no—
118Dic127—42no—
119Dic127C2×SD1614yes—
120C2×C127—82no—
121C2×C127#54811yescharacteristic
122C2×C127C22×SD1611yescharacteristic
123C2×C127#57611yescharacteristic
124D127—81no—
125#3362—1271nomaximal, Sylow
126C127#336211yescharacteristic, complemented, Sylow
127Q82—1271no—
128#550—1271no—
129#486—1271no—
130#486—1272no—
131#476—1271no—
132#476—1272no—
133#460—1272no—
134#460D12711yescharacteristic, complemented
135C8×Q8—1271no—
136C4×SD16—1272no—
137C4×SD16—1271no—
138C2×#41—2541no—
139C2×SD16—1271no—
140C2×SD16C2×D12712yes—
141C4⋊Q8—1272no—
142C4⋊Q8—1271no—
143C4⋊Q8—2541no—
144C4⋊Q8—2542no—
145#127—1272no—
146#127C2×D12711yescharacteristic
147#121—2541no—
148#121—2542no—
149C4×Q8—1271no—
150C4×Q8—1272no—
151C4×Q8—2541no—
152C4×Q8C2×D12711yescharacteristic
153C4×D4—1272no—
154C4×D4—1271no—
155#106—1272no—
156#106—1271no—
157C4⋊C8—1272no—
158C4⋊C8C2×D12711yescharacteristic
159#103—1271no—
160#102—1272no—
161#102—1272no—
162#102—1272no—
163#102—1271no—
164#102C2×D12712yes—
165C4×C8—1272no—
166C4×C8—1271no—
167#41—5084no—
168C2×Q8—1271no—
169C2×Q8—2542no—
170C2×Q8—2541no—
171C2×Q8C22×D12711yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1271no—
175C2×D4—2542no—
176C2×D4—2542no—
177C2×D4C22×D12712yes—
178C22×C4—2542no—
179C22×C4—2541no—
180SD16—24no—
181SD16—2542no—
182C2×C8—1271no—
183C2×C8—1271no—
184C2×C8C22×D12712yes—
185C4⋊C4—21no—
186C4⋊C4—1272no—
187C4⋊C4—1271no—
188C4⋊C4—1272no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2542no—
192C4⋊C4—2541no—
193C4⋊C4C22×D12711yescharacteristic
194C22⋊C4—2542no—
195C22⋊C4—2541no—
196C42—21no—
197C42—1272no—
198C42—2542no—
199C42C22×D12711yescharacteristic
200C23—22no—
201C23—2541no—
202Q8—21no—
203Q8—1274no—
204Q8—10161no—
205Q8—10162no—
206Q8#1452324712yescomplemented
207D4—22no—
208D4—22no—
209D4—42no—
210D4—81no—
211D4—10162no—
212D4—5084no—
213D4—2541no—
214D4—2541no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—1272no—
218C2×C4—1272no—
219C2×C4—5084no—
220C2×C4—5082no—
221C2×C4—2542no—
222C2×C4—2541no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—22no—
227C8—2541no—
228C8—2541no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—5081no—
233C22—5081no—
234C22—5081no—
235C22C2×D4×D12711yescharacteristic
236C4—22no—
237C4—21no—
238C4—41no—
239C4—1274no—
240C4—5081no—
241C4—5082no—
242C4—2542no—
243C4SD16×D12712yes—
244C4C2×#1452324711yescharacteristic
245C4#1686125611yescharacteristic
246C2—82no—
247C2—10161no—
248C2#2255053211yescharacteristic
249C2C2×SD16×D12711yescharacteristic
250C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#460×C127C2
socleC22×C127C22×D4
prime radical#22550546C2
hypercenter#460D127
hypocenterC127#3362
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. Q8×Dic254C2semidirect
3. #22550222C2semidirect
4. #22550221C2semidirect
5. #22550167C2semidirect
6. #22550164C2semidirect
7. #22550074C2semidirect
8. #22550000C2semidirect
9. SD16×Dic127C2semidirect
10. C4×#14523235C2semidirect
11. #460×C127C2semidirect
77. Dic254SD16semidirect
92. Q8×C127#41semidirect
126. C127#3362semidirect
134. #460D127semidirect
206. Q8#14523247semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225505462
#225502222
#225502212
#225501672
#225501642
#225500742
#225500002
SD16×Dic1272
C4×#145232352
#460×C1272
#3362127