Finite Group Databasebeta

← #30648093

Subgroups — #30648093

Order 16256.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16861257C211yesmaximal, characteristic, complemented
2#22550674C212yesmaximal, complemented
3#22550058C214yesmaximal, complemented
4#22550048C214yesmaximal, complemented
5#22549894C212yesmaximal, complemented
6C2×#16861128C211yesmaximal, characteristic, complemented
7#536×C127C211yesmaximal, characteristic, complemented
8#16861257—24no—
9#16861257—22no—
10#16861257C2214yes—
11C2×#14523249—22no—
12C2×#14523248—22no—
13C2×#14523248C2212yes—
14C2×Q8×D127—22no—
15#16861240C2214yescomplemented
16#16861239—22no—
17#16861235C2214yescomplemented
18#16861234—22no—
19#16861229—24no—
20C2×#14523232C2212yes—
21#16861137C2218yescomplemented
22#16861128C2214yescomplemented
23C2×#41×C127C2211yescharacteristic
24C22×Q8×C127C2211yescharacteristic
25C4⋊Q8×C127C2212yescomplemented
26#124×C127C2212yescomplemented
27#123×C127—22no—
28C22⋊Q8×C127—22no—
29#117×C127C2211yescharacteristic
30#14523249—28no—
31#14523249—28no—
32#14523248—28no—
33#14523248—28no—
34#14523248—22no—
35#14523248—22no—
36#14523248D414yes—
37#14523248C2314yes—
38C2×Dic254—22no—
39C2×Dic254D412yes—
40Q8×D127—28no—
41Q8×D127—28no—
42C2×D508—22no—
43C2×D508D412yes—
44C2×C22⋊D127—22no—
45C2×C4×D127—24no—
46C2×C4×D127—22no—
47#14523238—24no—
48#14523236—24no—
49#14523235—24no—
50#14523233—24no—
51#14523232—22no—
52#14523232C2314yes—
53C2×C127⋊C8—22no—
54#41×C127—22no—
55#41×C127—22no—
56C2×Q8×C127—22no—
57C2×Q8×C127—22no—
58C2×Q8×C127D412yes—
59C2×Q8×C127C2311yescharacteristic
60C2×Q8×C127C2312yes—
61C2×D4×C127—21no—
62C2×D4×C127C2311yescharacteristic
63C22×C4×C127—21no—
64C22×C4×C127D412yes—
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—21no—
67C4⋊C4×C127—24no—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—24no—
70C42×C127C2312yes—
71C22×D127—22no—
72Dic254—28no—
73Dic254—22no—
74Dic254C2×D414yes—
75D508—28no—
76D508—22no—
77D508C2×D414yes—
78C22⋊D127—24no—
79C22⋊D127—24no—
80C2×Dic127—22no—
81C4×D127—216no—
82C4×D127—24no—
83C4×D127—24no—
84C127⋊C8—24no—
85C23×C127—21no—
86C23×C127C2×D411yescharacteristic
87Q8×C127—24no—
88Q8×C127—24no—
89Q8×C127—22no—
90D4×C127—22no—
91D4×C127—41no—
92D4×C127—41no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D414yes—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2411yescharacteristic
103C22×C127—41no—
104C22×C127—42no—
105C22×C127—41no—
106C22×C127#12011yescharacteristic
107C22×C127#12011yescharacteristic
108C22×C127C22×D411yescharacteristic
109C2×D127—24no—
110C2×D127—44no—
111C4×C127—24no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127#12012yes—
115C4×C127C22×D412yes—
116Dic127—24no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—81no—
121C2×C127C2×#12011yescharacteristic
122D127—44no—
123#3020—1271nomaximal, Sylow
124C127#302011yescharacteristic, complemented, Sylow
125C2×2−1+4—1271no—
126#575—1272no—
127#536D12711yescharacteristic, complemented
128#456—1274no—
129#454—1274no—
130#421—1272no—
131C2×#101—1271no—
1322−1+4—1274no—
1332−1+4—2544no—
1342−1+4—2542no—
135C2×#41—1272no—
136C2×#41—2542no—
137C2×#41—2542no—
138C2×#41C2×D12711yescharacteristic
139C22×Q8—2542no—
140C22×Q8C2×D12711yescharacteristic
141#137—1274no—
142#137—2542no—
143AGL(1,ℤ/8)—1274no—
144AGL(1,ℤ/8)—2542no—
145#135—2544no—
146C2×M16—1272no—
147C4⋊Q8C2×D12712yescomplemented
148#124C2×D12712yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D12711yescharacteristic
152#104—1278no—
153#101—1274no—
154#41—22no—
155#41—22no—
156#41—1274no—
157#41—1274no—
158#41—2548no—
159#41—2548no—
160#41—2548no—
161#41—2542no—
162#41—2542no—
163#41—2548no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—1272no—
167C2×Q8—2542no—
168C2×Q8—2548no—
169C2×Q8—2548no—
170C2×Q8C22⋊D12712yes—
171C2×Q8C22×D12711yescharacteristic
172C2×Q8C22×D12712yes—
173C2×D4—21no—
174C2×D4—1272no—
175C2×D4—2542no—
176C2×D4—2542no—
177C2×D4C22×D12711yescharacteristic
178C22×C4—21no—
179C22×C4—2544no—
180C22×C4—2542no—
181C22×C4C22⋊D12712yes—
182C22×C4C22×D12711yescharacteristic
183Q16—2544no—
184SD16—2544no—
185SD16—2544no—
186D8—2544no—
187M16—1274no—
188M16—2542no—
189C2×C8—2542no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D12712yes—
195C23—21no—
196C23—2542no—
197C23C2×C22⋊D12711yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—1274no—
202Q8—2548no—
203Q8—2542no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—1274no—
208D4—2548no—
209D4—2544no—
210D4—2544no—
211D4—2542no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—25416no—
218C2×C4—2542no—
219C2×C4—2544no—
220C2×C4—2544no—
221C2×C4D4×D12712yes—
222C2×C4D4×D12712yes—
223C2×C4C2×C22⋊D12714yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—2544no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—5084no—
231C22—2544no—
232C22#1686117711yescharacteristic
233C22#1686117711yescharacteristic
234C22C22×C22⋊D12711yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—2544no—
239C4#1686117712yes—
240C4C2×D4×D12712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—5084no—
246C2C2×#1686117711yescharacteristic
subgroupG/H
centerC2C2×#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC2×C127C2×#120
hypercenter#536D127
hypocenterC127#3020
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. #22550674C2semidirect
3. #22550058C2semidirect
4. #22550048C2semidirect
5. #22549894C2semidirect
6. C2×#16861128C2semidirect
7. #536×C127C2semidirect
15. #16861240C22semidirect
17. #16861235C22semidirect
21. #16861137C22semidirect
22. #16861128C22semidirect
25. C4⋊Q8×C127C22semidirect
26. #124×C127C22semidirect
124. C127#3020semidirect
127. #536D127semidirect
147. C4⋊Q8C2×D127semidirect
148. #124C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
#225506742
#225500582
#225500482
#225498942
C2×#168611282
#536×C1272
#3020127