Finite Group Databasebeta

← #30647192

Subgroups — #30647192

Order 16256.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16861256C211yesmaximal, characteristic
2#22550353C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549849C212yesmaximal, complemented
5#22549848C218yesmaximal, complemented
6#410×C127C211yesmaximal, characteristic, complemented
7#16861256—24no—
8#16861256—24no—
9C2×#14523248C412yescomplemented
10C2×#14523248C2211yescharacteristic
11C2×#14523247C412yescomplemented
12C2×#14523247C2214yes—
13C22×Dic254C412yescomplemented
14C22×Dic254C2211yescharacteristic
15C2×Q8×D127C412yescomplemented
16C4×Dic254—24no—
17#16861174—24no—
18#16861168C2212yes—
19#16861165C2218yes—
20#16861124C2214yes—
21#16861123C2218yes—
22C2×#41×C127C2211yescharacteristic
23#117×C127C2212yes—
24#99×C127C2214yes—
25#14523248—24no—
26#14523248—24no—
27#14523248—22no—
28#14523248—22no—
29#14523247—28no—
30#14523247—216no—
31C2×Dic254—24no—
32C2×Dic254—24no—
33C2×Dic254C2×C414yes—
34C2×Dic254C2×C411yescharacteristic
35C2×Dic254C2×C412yes—
36C2×Dic254C2×C412yes—
37C2×Dic254D414yes—
38C2×Dic254D414yes—
39Q8×D127—28no—
40C2×D508C2×C411yescharacteristic
41C2×C22⋊D127C2×C414yes—
42C2×C22⋊D127C2312yes—
43C22×Dic127C2×C414yes—
44C22×Dic127C2312yes—
45C2×C4×D127C2×C412yes—
46C2×C4×D127C2×C414yes—
47C4⋊Dic127—22no—
48Dic127⋊C4—28no—
49C22⋊Dic127C2314yes—
50C4×Dic127—28no—
51C4×Dic127—22no—
52#41×C127—22no—
53#41×C127—22no—
54C2×Q8×C127C2×C411yescharacteristic
55C2×D4×C127C2×C411yescharacteristic
56C2×D4×C127C2312yes—
57C22×C4×C127C2×C412yes—
58C22×C4×C127C2311yescharacteristic
59C4⋊C4×C127—22no—
60C22⋊C4×C127C2314yes—
61C42×C127—22no—
62C22×D127C22×C412yes—
63Dic254—21no—
64Dic254—21no—
65Dic254—28no—
66Dic254C22⋊C418yes—
67Dic254C22⋊C414yes—
68Dic254C22⋊C414yes—
69D508—21no—
70D508—21no—
71C22⋊D127—24no—
72C22⋊D127—28no—
73C2×Dic127—28no—
74C2×Dic127—24no—
75C2×Dic127C22×C412yes—
76C2×Dic127C22×C414yes—
77C2×Dic127C2×D418yes—
78C4×D127—24no—
79C4×D127—28no—
80C23×C127C22×C412yes—
81C23×C127C2411yescharacteristic
82Q8×C127—22no—
83D4×C127—24no—
84D4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—24no—
88C2×C4×C127C22×C412yes—
89C2×C4×C127C22×C411yescharacteristic
90C2×C4×C127C22×C411yescharacteristic
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C22×C127—22no—
94C22×C127—41no—
95C22×C127—42no—
96C22×C127C23×C411yescharacteristic
97C22×C127C22×D412yes—
98C2×D127—22no—
99C2×D127—42no—
100C4×C127—22no—
101C4×C127—44no—
102C4×C127C2×C22⋊C411yescharacteristic
103C4×C127C2×C22⋊C411yescharacteristic
104C4×C127C2×C22⋊C412yes—
105Dic127—22no—
106Dic127—44no—
107Dic127C2×C22⋊C418yes—
108C2×C127—22no—
109C2×C127—21no—
110C2×C127—42no—
111C2×C127C22×C22⋊C411yescharacteristic
112D127—42no—
113#2884—1271nomaximal, Sylow
114C127#288411yescharacteristic, complemented, Sylow
115C2×2−1+4—1271no—
116#519—1272no—
117#410—1271no—
118#410—1272no—
119#410—1278no—
120#410D12711yescharacteristic, complemented
1212−1+4—2544no—
1222−1+4—2544no—
123C2×#41—1272no—
124C2×#41—1274no—
125C2×#41—1272no—
126C2×#41—1271no—
127C2×#41C2×D12711yescharacteristic
128C22×Q8—1272no—
129C22×Q8—1272no—
130C22×Q8—1271no—
131C4×Q8—2544no—
132C4×Q8—2544no—
133#117—1272no—
134#117—1278no—
135#117C2×D12712yes—
136#99—1274no—
137#99—1278no—
138#99C2×D12714yes—
139#41—22no—
140#41—22no—
141#41—2544no—
142#41—2544no—
143#41—25416no—
144#41—2548no—
145#41—2542no—
146#41—2542no—
147C2×Q8—1272no—
148C2×Q8—1272no—
149C2×Q8—1274no—
150C2×Q8—1271no—
151C2×Q8—1274no—
152C2×Q8—1274no—
153C2×Q8—2544no—
154C2×Q8—2544no—
155C2×Q8—2548no—
156C2×Q8C4×D12711yescharacteristic
157C2×D4—1274no—
158C2×D4—1272no—
159C2×D4—1271no—
160C2×D4C4×D12711yescharacteristic
161C2×D4C22×D12712yes—
162C22×C4—1272no—
163C22×C4—1274no—
164C22×C4—1274no—
165C22×C4—1272no—
166C22×C4C4×D12712yes—
167C22×C4C22×D12711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—2548no—
170C4⋊C4—2542no—
171C22⋊C4—1274no—
172C22⋊C4C22×D12714yes—
173C42—22no—
174C42—2548no—
175C42—2542no—
176C23—1272no—
177C23C2×C4×D12712yes—
178C23C23×D12711yescharacteristic
179Q8—22no—
180Q8—1274no—
181Q8—1274no—
182Q8—1278no—
183Q8—2548no—
184Q8—2541no—
185Q8—2541no—
186D4—24no—
187D4—22no—
188D4—2548no—
189D4—2544no—
190D4—2541no—
191D4—2541no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—1274no—
196C2×C4—1278no—
197C2×C4—1272no—
198C2×C4—2548no—
199C2×C4—2548no—
200C2×C4—2544no—
201C2×C4—2544no—
202C2×C4C2×C4×D12712yes—
203C2×C4C2×C4×D12711yescharacteristic
204C2×C4C2×C4×D12711yescharacteristic
205C2×C4D4×D12712yes—
206C2×C4D4×D12712yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—5082no—
211C22—2542no—
212C22C22×C4×D12711yescharacteristic
213C22C2×D4×D12712yes—
214C4—22no—
215C4—44no—
216C4—1278no—
217C4—5084no—
218C4—2542no—
219C4C22⋊C4×D12711yescharacteristic
220C4C22⋊C4×D12711yescharacteristic
221C4C22⋊C4×D12712yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—5082no—
226C2C2×C22⋊C4×D12711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D127
commutatorC22×C127C23×C4
FrattiniC23C23×D127
Fitting#410×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861256C2
cosocleC23×C127C24
hypercenter#410D127
hypocenterC127#2884
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22550353C2semidirect
3. #22549850C2semidirect
4. #22549849C2semidirect
5. #22549848C2semidirect
6. #410×C127C2semidirect
9. C2×#14523248C4semidirect
11. C2×#14523247C4semidirect
13. C22×Dic254C4semidirect
15. C2×Q8×D127C4semidirect
114. C127#2884semidirect
120. #410D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
#225503532
#225498502
#225498492
#225498482
#410×C1272
#2884127