Finite Group Databasebeta

← #30648111

Subgroups — #30648111

Order 16256.

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

normal subgroups only

This group has 704 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
1#22550712C211yesmaximal, characteristic
2C2×#16861242C212yesmaximal, complemented
3#22550447C211yesmaximal, characteristic, complemented
4#22550061C214yesmaximal, complemented
5#22550056C214yesmaximal, complemented
6#22549892C212yesmaximal, complemented
7#413×C127C211yesmaximal, characteristic, complemented
8#16861257C2212yes—
9#16861256—22no—
10#16861255—22no—
11#16861254—22no—
12#16861254C2212yes—
13C2×#14523248—22no—
14C2×#14523248C2212yes—
15#41×D127—24no—
16C22×Dic254C2211yescharacteristic, complemented
17#16861242—24no—
18#16861242C2218yescomplemented
19C2×Dic508—22no—
20C2×#14523237—22no—
21#16861207C2212yescomplemented
22#16861196C2212yescomplemented
23#16861190—22no—
24#16861188—22no—
25#16861168C2211yescharacteristic, complemented
26#16861135C2218yescomplemented
27C2×#41×C127C2211yescharacteristic
28C2×M16×C127C2212yescomplemented
29#101×C127C2212yescomplemented
30#100×C127C2212yescomplemented
31#14523249—24no—
32#14523248—24no—
33#14523248—24no—
34#14523248—22no—
35#14523248—24no—
36#14523248—22no—
37#14523248D414yes—
38#14523248C2314yes—
39#14523247—28no—
40#14523247—24no—
41C2×Dic254—22no—
42C2×Dic254—21no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254D412yes—
46C2×Dic254C2312yes—
47Q8×D127—24no—
48C2×D508—22no—
49C2×D508D412yes—
50C2×C22⋊D127—24no—
51C2×C22⋊D127—22no—
52D4×D127—28no—
53D4×D127—24no—
54C22×Dic127—21no—
55C2×C4×D127—24no—
56C2×C4×D127—22no—
57Dic508—28no—
58#14523237—28no—
59C4⋊Dic127—21no—
60Dic127⋊C4—24no—
61C22⋊Dic127—24no—
62C22⋊Dic127—21no—
63C4×Dic127C2312yes—
64#41×C127—22no—
65#41×C127—22no—
66C2×Q8×C127C2311yescharacteristic
67C2×D4×C127D412yes—
68C2×D4×C127C2311yescharacteristic
69C22×C4×C127D412yes—
70C22×C4×C127C2311yescharacteristic
71M16×C127—22no—
72M16×C127C2314yes—
73C2×C8×C127—22no—
74C22×D127—24no—
75C22×D127—22no—
76Dic254—24no—
77Dic254—22no—
78Dic254—24no—
79Dic254—41no—
80Dic254—41no—
81Dic254C2×D414yes—
82D508—24no—
83D508—22no—
84D508C2×D414yes—
85C22⋊D127—28no—
86C22⋊D127—24no—
87C22⋊D127—24no—
88C2×Dic127—24no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—22no—
92C2×Dic127—42no—
93C4×D127—28no—
94C4×D127—24no—
95C4×D127—24no—
96C23×C127C2×D412yes—
97C23×C127C2×D411yescharacteristic
98Q8×C127—22no—
99D4×C127—24no—
100D4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2411yescharacteristic
108C8×C127—24no—
109C22×C127—22no—
110C22×C127—41no—
111C22×C127—42no—
112C22×C127#12012yes—
113C22×C127C22×D411yescharacteristic
114C2×D127—24no—
115C2×D127—82no—
116C2×D127—84no—
117C4×C127—22no—
118C4×C127#12011yescharacteristic
119C4×C127#12011yescharacteristic
120C4×C127C22×D412yes—
121Dic127—24no—
122Dic127—42no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—22no—
126C2×C127—42no—
127C2×C127C2×#12011yescharacteristic
128D127—44no—
129#3022—1271nomaximal, Sylow
130C127#302211yescharacteristic, complemented, Sylow
131#585—1271no—
132C2×#137—1272no—
133#536—1271no—
134#456—1274no—
135#455—1274no—
136#421—1272no—
137#413D12711yescharacteristic, complemented
1382−1+4—1272no—
1392−1+4—2542no—
140H(5,𝔽2)—1272no—
141H(5,𝔽2)—2542no—
142H(5,𝔽2)—2542no—
143C2×#41—1272no—
144C2×#41—2544no—
145C2×#41—2542no—
146C2×#41C2×D12711yescharacteristic
147C22×Q8—1271no—
148#137—1278no—
149#137—2544no—
150C2×Q16—2542no—
151C2×SD16—2542no—
152C2×M16C2×D12712yescomplemented
153C4⋊Q8—1272no—
154#124—1272no—
155#123—2542no—
156C22⋊Q8—2542no—
157#117—1271no—
158#104—1278no—
159#101C2×D12712yescomplemented
160#100C2×D12712yescomplemented
161#41—22no—
162#41—22no—
163#41—1274no—
164#41—1274no—
165#41—2544no—
166#41—2544no—
167#41—2544no—
168#41—2548no—
169#41—2544no—
170#41—2542no—
171#41—2542no—
172#41—2544no—
173C2×Q8—1272no—
174C2×Q8—1272no—
175C2×Q8—2542no—
176C2×Q8—2542no—
177C2×Q8—2542no—
178C2×Q8—2544no—
179C2×Q8—2541no—
180C2×Q8C22×D12711yescharacteristic
181C2×D4—1272no—
182C2×D4—2542no—
183C2×D4—2544no—
184C2×D4—2548no—
185C2×D4—2542no—
186C2×D4—2544no—
187C2×D4D50812yes—
188C2×D4C22×D12711yescharacteristic
189C22×C4—2544no—
190C22×C4—2541no—
191C22×C4—2542no—
192C22×C4D50812yes—
193C22×C4C22×D12711yescharacteristic
194Q16—2548no—
195SD16—2548no—
196M16—22no—
197M16C22×D12714yes—
198C2×C8—22no—
199C4⋊C4—2544no—
200C4⋊C4—2541no—
201C22⋊C4—2541no—
202C22⋊C4—2544no—
203C42—1272no—
204C23—2544no—
205C23—2542no—
206C23C2×D50812yes—
207C23C2×D50811yescharacteristic
208Q8—22no—
209Q8—1274no—
210Q8—5081no—
211Q8—5081no—
212Q8—2544no—
213Q8—2544no—
214Q8—2542no—
215D4—24no—
216D4—22no—
217D4—1274no—
218D4—2544no—
219D4—2548no—
220D4—2544no—
221D4—2544no—
222D4—2542no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—5082no—
226C2×C4—2548no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4—2544no—
230C2×C4—2542no—
231C2×C4—2544no—
232C2×C4—2544no—
233C2×C4D4×D12712yes—
234C2×C4D4×D12712yes—
235C2×C4C2×D50812yes—
236C2×C4C2×D50811yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—24no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—10162no—
243C22—10164no—
244C22—2544no—
245C22#1686117912yes—
246C22C22×D50811yescharacteristic
247C4—22no—
248C4—5082no—
249C4—5082no—
250C4—2544no—
251C4#1686117911yescharacteristic
252C4#1686117911yescharacteristic
253C4C2×D4×D12712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—5084no—
258C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#413×C127C2
socleC2×C127C2×#120
prime radical#22550712C2
hypercenter#413D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. #22550447C2semidirect
4. #22550061C2semidirect
5. #22550056C2semidirect
6. #22549892C2semidirect
7. #413×C127C2semidirect
16. C22×Dic254C22semidirect
18. #16861242C22semidirect
21. #16861207C22semidirect
22. #16861196C22semidirect
25. #16861168C22semidirect
26. #16861135C22semidirect
28. C2×M16×C127C22semidirect
29. #101×C127C22semidirect
30. #100×C127C22semidirect
130. C127#3022semidirect
137. #413D127semidirect
152. C2×M16C2×D127semidirect
159. #101C2×D127semidirect
160. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612422
#225504472
#225500612
#225500562
#225498922
#413×C1272
#3022127