Finite Group Databasebeta

← #30648097

Subgroups — #30648097

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#22550676C212yesmaximal, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550061C214yesmaximal, complemented
5#22550051C214yesmaximal, complemented
6#22549892C212yesmaximal, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8#16861257—24no—
9#16861257—22no—
10#16861257C2214yes—
11C2×#14523249—22no—
12C2×#14523248—22no—
13C2×#14523248C2212yes—
14C2×#14523248C2211yescharacteristic
15C2×Q8×D127—22no—
16#16861242—22no—
17#16861242C2214yescomplemented
18#16861237—22no—
19#16861237C2214yescomplemented
20#16861231—24no—
21#16861207C2212yescomplemented
22#16861198C2212yescomplemented
23#16861192—22no—
24#16861185—22no—
25#16861168C2211yescharacteristic
26#16861135C2218yescomplemented
27C22×Q8×C127C2211yescharacteristic
28C2×M16×C127C2212yes—
29#101×C127C2214yescomplemented
30#14523249—28no—
31#14523249—28no—
32#14523248—28no—
33#14523248—28no—
34#14523248—22no—
35#14523248—22no—
36#14523248—22no—
37#14523248—22no—
38#14523248D414yes—
39#14523248C2314yes—
40C2×Dic254—22no—
41C2×Dic254D412yes—
42C2×Dic254C2311yescharacteristic
43Q8×D127—28no—
44Q8×D127—28no—
45C2×D508—22no—
46C2×D508D412yes—
47C2×D508C2311yescharacteristic
48C2×C22⋊D127—22no—
49C2×C22⋊D127—21no—
50C2×C4×D127—24no—
51C2×C4×D127—22no—
52C2×C4×D127—21no—
53Dic508—24no—
54#14523237—24no—
55#14523237—24no—
56D1016—24no—
57C4⋊Dic127—21no—
58Dic127⋊C4—24no—
59C22⋊Dic127—21no—
60(C2×D127)⋊C4—24no—
61C4×Dic127C2312yes—
62C2×Q8×C127—22no—
63C2×Q8×C127—22no—
64C2×Q8×C127D412yes—
65C2×Q8×C127C2312yes—
66C22×C4×C127D412yes—
67C22×C4×C127C2311yescharacteristic
68M16×C127—22no—
69M16×C127C2314yes—
70C2×C8×C127—22no—
71C22×D127—22no—
72C22×D127—21no—
73Dic254—28no—
74Dic254—22no—
75Dic254—22no—
76Dic254C2×D414yes—
77D508—28no—
78D508—22no—
79D508—22no—
80D508C2×D414yes—
81C22⋊D127—24no—
82C22⋊D127—24no—
83C22⋊D127—41no—
84C22⋊D127—41no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—21no—
88C4×D127—216no—
89C4×D127—24no—
90C4×D127—24no—
91C4×D127—42no—
92C23×C127C2×D411yescharacteristic
93Q8×C127—24no—
94Q8×C127—24no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127C2×D412yes—
98C2×C4×C127C2×D412yes—
99C2×C4×C127C2×D414yes—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2411yescharacteristic
102C8×C127—24no—
103C22×C127—41no—
104C22×C127#12011yescharacteristic
105C22×C127#12011yescharacteristic
106C22×C127C22×D411yescharacteristic
107C2×D127—24no—
108C2×D127—44no—
109C2×D127—42no—
110C2×D127—41no—
111C4×C127—24no—
112C4×C127#12012yes—
113C4×C127C22×D412yes—
114Dic127—24no—
115Dic127—42no—
116Dic127—41no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127C2×#12011yescharacteristic
121D127—44no—
122D127—81no—
123#3020—1271nomaximal, Sylow
124C127#302011yescharacteristic, complemented, Sylow
125C2×2−1+4—1271no—
126#575—1272no—
127#536—1271no—
128#456—1274no—
129#454—1274no—
130#421—1272no—
131C2×#101D12711yescharacteristic, complemented
1322−1+4—1274no—
1332−1+4—2544no—
1342−1+4—2542no—
135C2×#41—1272no—
136C2×#41—1271no—
137C2×#41—2542no—
138C2×#41—2542no—
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×M16C2×D12712yes—
147C4⋊Q8—1272no—
148#124—1272no—
149#123—2542no—
150C22⋊Q8—2542no—
151#117—1271no—
152#104—1278no—
153#101C2×D12714yescomplemented
154#41—1274no—
155#41—1274no—
156#41—2548no—
157#41—2542no—
158#41—2542no—
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—1271no—
168C2×Q8—2542no—
169C2×Q8—2548no—
170C2×Q8—2548no—
171C2×Q8D50812yes—
172C2×Q8C22×D12712yes—
173C2×D4—1272no—
174C2×D4—1271no—
175C2×D4—2542no—
176C2×D4—2541no—
177C2×D4—2542no—
178C22×C4—2544no—
179C22×C4—2541no—
180C22×C4—2542no—
181C22×C4D50812yes—
182C22×C4C22×D12711yescharacteristic
183Q16—2544no—
184SD16—2544no—
185SD16—2544no—
186D8—2544no—
187M16—22no—
188M16C22×D12714yes—
189C2×C8—22no—
190C4⋊C4—2544no—
191C4⋊C4—2541no—
192C22⋊C4—2541no—
193C22⋊C4—2544no—
194C42—1272no—
195C23—2541no—
196C23—2542no—
197C23C2×D50811yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—1274no—
201Q8—2548no—
202Q8—2542no—
203Q8—2542no—
204D4—1274no—
205D4—5081no—
206D4—5081no—
207D4—2548no—
208D4—2544no—
209D4—2544no—
210D4—2542no—
211D4—2542no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—5082no—
215C2×C4—25416no—
216C2×C4—2542no—
217C2×C4—2541no—
218C2×C4—2542no—
219C2×C4—2544no—
220C2×C4—2544no—
221C2×C4D4×D12712yes—
222C2×C4D4×D12712yes—
223C2×C4C2×D50814yes—
224C2×C4C2×D50811yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—24no—
227C22—41no—
228C22—5082no—
229C22—5084no—
230C22—5081no—
231C22—2544no—
232C22#1686117911yescharacteristic
233C22#1686117911yescharacteristic
234C22C22×D50811yescharacteristic
235C4—24no—
236C4—5082no—
237C4—5081no—
238C4—2544no—
239C4#1686117912yes—
240C4C2×D4×D12712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—10161no—
245C2—5084no—
246C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#101×C127C2
socleC2×C127C2×#120
hypercenterC2×#101D127
hypocenterC127#3020
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. #22550676C2semidirect
3. #22550450C2semidirect
4. #22550061C2semidirect
5. #22550051C2semidirect
6. #22549892C2semidirect
7. C2×#101×C127C2semidirect
17. #16861242C22semidirect
19. #16861237C22semidirect
21. #16861207C22semidirect
22. #16861198C22semidirect
26. #16861135C22semidirect
29. #101×C127C22semidirect
124. C127#3020semidirect
131. C2×#101D127semidirect
153. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
#225506762
#225504502
#225500612
#225500512
#225498922
C2×#101×C1272
#3020127