Finite Group Databasebeta

← #30648135

Subgroups — #30648135

Order 16256.

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

normal subgroups only

This group has 716 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
2#22550450C212yesmaximal, complemented
3#22550065C218yesmaximal, complemented
4#22549850C211yesmaximal, characteristic, complemented
5#22549849C212yesmaximal, complemented
6#536×C127C211yesmaximal, characteristic, complemented
7#16861257—21no—
8#16861256—22no—
9#16861255—22no—
10#16861254—21no—
11#16861254C2214yes—
12C2×#14523248—22no—
13C2×#14523248C2212yes—
14#41×D127—24no—
15#16861207—22no—
16#16861198—22no—
17#16861192C2218yescomplemented
18#16861185—24no—
19#16861168C2212yescomplemented
20#16861124C2214yescomplemented
21#16861123C2218yescomplemented
22C2×#41×C127C2211yescharacteristic
23C22×Q8×C127C2211yescharacteristic, complemented
24C4⋊Q8×C127—21no—
25#124×C127—21no—
26#123×C127C2214yescomplemented
27C22⋊Q8×C127—22no—
28#117×C127C2211yescharacteristic, complemented
29#14523249—24no—
30#14523248—24no—
31#14523248—24no—
32#14523248—22no—
33#14523248—22no—
34#14523248—24no—
35#14523248—22no—
36#14523248—22no—
37#14523247—24no—
38#14523247—28no—
39C2×Dic254—22no—
40C2×Dic254D412yes—
41Q8×D127—24no—
42C2×D508—22no—
43C2×D508D412yes—
44C2×C22⋊D127—24no—
45C2×C22⋊D127C2314yes—
46D4×D127—24no—
47D4×D127—28no—
48C2×C4×D127—24no—
49C2×C4×D127D414yes—
50C4⋊Dic127—22no—
51Dic127⋊C4—28no—
52C22⋊Dic127C2314yes—
53(C2×D127)⋊C4—28no—
54C4×Dic127—22no—
55#41×C127—22no—
56#41×C127—22no—
57C2×Q8×C127—22no—
58C2×Q8×C127—22no—
59C2×Q8×C127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127D411yescharacteristic
62C2×D4×C127D411yescharacteristic
63C2×D4×C127C2312yes—
64C22×C4×C127D412yes—
65C22×C4×C127C2312yes—
66C22×C4×C127C2311yescharacteristic
67C4⋊C4×C127—21no—
68C4⋊C4×C127—24no—
69C22⋊C4×C127—24no—
70C22⋊C4×C127C2312yes—
71C42×C127—21no—
72C22×D127—24no—
73C22×D127C2×D414yes—
74Dic254—24no—
75Dic254—22no—
76Dic254—22no—
77D508—24no—
78D508—22no—
79D508—22no—
80C22⋊D127—28no—
81C22⋊D127—24no—
82C22⋊D127—24no—
83C2×Dic127—24no—
84C2×Dic127—24no—
85C2×Dic127C2×D414yes—
86C4×D127—28no—
87C4×D127—24no—
88C4×D127—24no—
89C23×C127C2×D412yes—
90C23×C127C2411yescharacteristic
91Q8×C127—22no—
92Q8×C127—41no—
93Q8×C127—41no—
94Q8×C127—42no—
95D4×C127—24no—
96D4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—24no—
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C22×C127—22no—
108C22×C127—41no—
109C22×C127—42no—
110C22×C127C22×D412yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—24no—
113C2×D127—44no—
114C2×D127—84no—
115C4×C127—22no—
116C4×C127—42no—
117C4×C127—42no—
118C4×C127#12011yescharacteristic
119C4×C127#12011yescharacteristic
120C4×C127#12012yes—
121Dic127—24no—
122Dic127—44no—
123C2×C127—22no—
124C2×C127—21no—
125C2×C127—42no—
126C2×C127C2×#12011yescharacteristic
127D127—44no—
128#3028—1271nomaximal, Sylow
129C127#302811yescharacteristic, complemented, Sylow
130#585—1271no—
131#536—1272no—
132#536D12711yescharacteristic, complemented
133#458—1278no—
134#410—1271no—
135#410—1272no—
1362−1+4—2542no—
1372−1+4—2541no—
138H(5,𝔽2)—1274no—
139H(5,𝔽2)—2542no—
140H(5,𝔽2)—2541no—
141C2×#41—1272no—
142C2×#41—2544no—
143C2×#41—2542no—
144C2×#41C2×D12711yescharacteristic
145C22×Q8C2×D12711yescharacteristic, complemented
146C4⋊Q8—21no—
147C4⋊Q8—2542no—
148#124—21no—
149#124—2542no—
150#123—1278no—
151#123C2×D12714yescomplemented
152C22⋊Q8—22no—
153C22⋊Q8—2544no—
154#117—1272no—
155#117C2×D12711yescharacteristic, complemented
156#99—1274no—
157#99—1278no—
158#41—22no—
159#41—22no—
160#41—2544no—
161#41—2544no—
162#41—2548no—
163#41—2544no—
164#41—2544no—
165#41—2542no—
166#41—2542no—
167#41—2542no—
168#41—2542no—
169#41—2544no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—1272no—
175C2×Q8—2542no—
176C2×Q8—2544no—
177C2×Q8C22⋊D12711yescharacteristic
178C2×D4—1274no—
179C2×D4—1272no—
180C2×D4—2542no—
181C2×D4—2548no—
182C2×D4—2544no—
183C2×D4—2544no—
184C2×D4C22⋊D12711yescharacteristic
185C2×D4C22×D12712yes—
186C22×C4—1274no—
187C22×C4—2544no—
188C22×C4C22⋊D12712yes—
189C22×C4C22×D12712yes—
190C22×C4C22×D12711yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—2548no—
194C4⋊C4—2542no—
195C22⋊C4—24no—
196C22⋊C4—1274no—
197C22⋊C4—2548no—
198C22⋊C4C22×D12712yes—
199C42—21no—
200C42—2542no—
201C23—1274no—
202C23—2544no—
203C23C2×C22⋊D12712yes—
204C23C23×D12711yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—2544no—
210Q8—2542no—
211Q8—2542no—
212D4—24no—
213D4—22no—
214D4—2544no—
215D4—2548no—
216D4—2544no—
217D4—2544no—
218D4—2542no—
219D4—2542no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—1274no—
226C2×C4—2548no—
227C2×C4—2544no—
228C2×C4—2544no—
229C2×C4—2544no—
230C2×C4—2544no—
231C2×C4D4×D12712yes—
232C2×C4D4×D12712yes—
233C2×C4C2×C22⋊D12712yes—
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—10164no—
240C22—5084no—
241C22—2544no—
242C22C2×D4×D12712yes—
243C22C22×C22⋊D12711yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—5084no—
248C4—2544no—
249C4#1686117711yescharacteristic
250C4#1686117711yescharacteristic
251C4#1686117712yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—5084no—
256C2C2×#1686117711yescharacteristic
subgroupG/H
centerC2C2×#16861177
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#536×C127C2
socleC2×C127C2×#120
prime radical#22550712C2
hypercenter#536D127
hypocenterC127#3028
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #22550450C2semidirect
3. #22550065C2semidirect
4. #22549850C2semidirect
5. #22549849C2semidirect
6. #536×C127C2semidirect
17. #16861192C22semidirect
19. #16861168C22semidirect
20. #16861124C22semidirect
21. #16861123C22semidirect
23. C22×Q8×C127C22semidirect
26. #123×C127C22semidirect
28. #117×C127C22semidirect
129. C127#3028semidirect
132. #536D127semidirect
145. C22×Q8C2×D127semidirect
151. #123C2×D127semidirect
155. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225504502
#225500652
#225498502
#225498492
#536×C1272
#3028127