Finite Group Databasebeta

← #72229068

Subgroups — #72229068

Order 2496.

396 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 66 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#88251⋊C3C211yesmaximal, characteristic, complemented
2#120×C13⋊C3C211yesmaximal, characteristic, complemented
3#11928318C212yesmaximal, complemented
4#88074⋊C3C211yesmaximal, characteristic, complemented
5#88046⋊C3C212yesmaximal, complemented
6#11718693C311yesmaximal, characteristic, complemented, Hall
7C24×C13⋊C3C2211yescharacteristic, complemented
8C2×D4×C13⋊C3—21no—
9C2×D4×C13⋊C3C2211yescharacteristic
10C2×(C22⋊D13)⋊C3—21no—
11C2×(C22⋊D13)⋊C3—22no—
12C2×(C22⋊D13)⋊C3C2212yes—
13#6129⋊C3—22no—
14D4×D13⋊C3—22no—
15#6127⋊C3—21no—
16C22⋊C4×C13⋊C3—21no—
17C22⋊C4×C13⋊C3C2211yescharacteristic, complemented
18(C22⋊Dic13)⋊C3—22no—
19(C22⋊Dic13)⋊C3C2212yescomplemented
20#88251C2×C311yescharacteristic, complemented
21C13×#120C2×C311yescharacteristic, complemented
22#88207C2×C312yescomplemented
23#88074C2×C311yescharacteristic, complemented
24#88046C2×C312yescomplemented
25C23×C13⋊C3—22no—
26C23×C13⋊C3—21no—
27C23×C13⋊C3—42no—
28C23×C13⋊C3D411yescharacteristic, complemented
29C23×C13⋊C3C2311yescharacteristic
30C22×D13⋊C3—21no—
31C22×D13⋊C3D412yescomplemented
32D4×C13⋊C3—22no—
33D4×C13⋊C3—42no—
34C2×C4×C13⋊C3—21no—
35C2×C4×C13⋊C3D411yescharacteristic, complemented
36(C22⋊D13)⋊C3—22no—
37(C22⋊D13)⋊C3—22no—
38(C22⋊D13)⋊C3—22no—
39(C22⋊D13)⋊C3—44no—
40C2×Dic13⋊C3—21no—
41C2×Dic13⋊C3—22no—
42C2×Dic13⋊C3D412yescomplemented
43C4⋊(D13⋊C3)—21no—
44C4×D13⋊C3—22no—
45(C13⋊C3)⋊Q8—21no—
46C24×C13C22×C311yescharacteristic, complemented
47C2×D4×C13—21no—
48C2×D4×C13C22×C311yescharacteristic
49C2×C22⋊D13—21no—
50C2×C22⋊D13—22no—
51C2×C22⋊D13C22×C312yes—
52#6129—22no—
53D4×D13—22no—
54#6127—21no—
55C13×C22⋊C4—21no—
56C13×C22⋊C4C22×C311yescharacteristic, complemented
57C22⋊Dic13—22no—
58C22⋊Dic13C22×C312yescomplemented
59C3×B(4,𝔽2)—131nomaximal, Hall
60C22×C13⋊C3—24no—
61C22×C13⋊C3—22no—
62C22×C13⋊C3—21no—
63C22×C13⋊C3—41no—
64C22×C13⋊C3—41no—
65C22×C13⋊C3—42no—
66C22×C13⋊C3—41no—
67C22×C13⋊C3—41no—
68C22×C13⋊C3C2×D412yes—
69C22×C13⋊C3C2×D411yescharacteristic
70C2×D13⋊C3—22no—
71C2×D13⋊C3—42no—
72C2×D13⋊C3—81no—
73C4×C13⋊C3—21no—
74C4×C13⋊C3—41no—
75Dic13⋊C3—22no—
76Dic13⋊C3—42no—
77C23×C13—22no—
78C23×C13—21no—
79C23×C13—42no—
80C23×C13C3×D411yescharacteristic, complemented
81C23×C13C23×C311yescharacteristic
82C22×D13—21no—
83C22×D13C3×D412yescomplemented
84D4×C13—22no—
85D4×C13—42no—
86C2×C4×C13—21no—
87C2×C4×C13C3×D411yescharacteristic, complemented
88C22⋊D13—22no—
89C22⋊D13—22no—
90C22⋊D13—22no—
91C22⋊D13—44no—
92C2×Dic13—21no—
93C2×Dic13—22no—
94C2×Dic13C3×D412yescomplemented
95D52—21no—
96C4×D13—22no—
97Dic26—21no—
98C3×H(5,𝔽2)—131no—
99C3×#120—132no—
100C3×#120—131no—
101C3×#99—132no—
102C3×#99—131no—
103C2×C13⋊C3—22no—
104C2×C13⋊C3—21no—
105C2×C13⋊C3—41no—
106C2×C13⋊C3—42no—
107C2×C13⋊C3#12011yescharacteristic
108D13⋊C3—42no—
109B(4,𝔽2)—131noSylow
110C22×C13—24no—
111C22×C13—22no—
112C22×C13—21no—
113C22×C13—41no—
114C22×C13—41no—
115C22×C13—42no—
116C22×C13—41no—
117C22×C13—41no—
118C22×C13C2×C3×D412yes—
119C22×C13C2×C3×D411yescharacteristic
120C2×D13—22no—
121C2×D13—42no—
122C2×D13—81no—
123C4×C13—21no—
124C4×C13—41no—
125Dic13—22no—
126Dic13—42no—
127C24×C3—131no—
128C3×#41—261no—
129C3×#41—262no—
130C2×C3×D4—132no—
131C2×C3×D4—131no—
132C2×C3×D4—261no—
133C2×C3×D4—262no—
134C2×C3×D4—261no—
135C2×C3×D4—262no—
136C3×C22⋊C4—132no—
137C3×C22⋊C4—131no—
138C3×C22⋊C4—261no—
139C3×C22⋊C4—262no—
140C13⋊C3B(4,𝔽2)11yescharacteristic, complemented, Hall
141H(5,𝔽2)—131no—
142#120—132no—
143#120D13⋊C311yescharacteristic, complemented
144#99—131no—
145#99—132no—
146C2×C13—22no—
147C2×C13—21no—
148C2×C13—41no—
149C2×C13—42no—
150C2×C13C3×#12011yescharacteristic
151D13—42no—
152C23×C3—132no—
153C23×C3—131no—
154C23×C3—131no—
155C23×C3—262no—
156C23×C3—261no—
157C23×C3—261no—
158C23×C3—522no—
159C3×Q8—261no—
160C3×D4—262no—
161C3×D4—262no—
162C3×D4—262no—
163C3×D4—261no—
164C3×D4—262no—
165C3×D4—522no—
166C3×D4—524no—
167C2×C3×C4—132no—
168C2×C3×C4—131no—
169C2×C3×C4—261no—
170C2×C3×C4—262no—
171C2×C3×C4—261no—
172C2×C3×C4—262no—
173C24C2×D13⋊C311yescharacteristic, complemented
174#41—262no—
175#41—261no—
176C2×D4—21no—
177C2×D4—132no—
178C2×D4—262no—
179C2×D4—261no—
180C2×D4—262no—
181C2×D4C2×D13⋊C311yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—132no—
184C22⋊C4—262no—
185C22⋊C4C2×D13⋊C311yescharacteristic, complemented
186C13C3×B(4,𝔽2)11yescharacteristic, complemented, Sylow
187C22×C3—132no—
188C22×C3—131no—
189C22×C3—261no—
190C22×C3—262no—
191C22×C3—264no—
192C22×C3—262no—
193C22×C3—521no—
194C22×C3—521no—
195C22×C3—521no—
196C22×C3—521no—
197C22×C3—522no—
198C22×C3—522no—
199C22×C3—1041no—
200C3×C4—261no—
201C3×C4—262no—
202C3×C4—521no—
203C3×C4—522no—
204C23—22no—
205C23—21no—
206C23—42no—
207C23—132no—
208C23—261no—
209C23(C22⋊D13)⋊C311yescharacteristic, complemented
210C23C22×D13⋊C311yescharacteristic
211Q8—261no—
212D4—22no—
213D4—42no—
214D4—261no—
215D4—262no—
216D4—262no—
217D4—262no—
218D4—524no—
219C2×C4—21no—
220C2×C4—132no—
221C2×C4—262no—
222C2×C4—261no—
223C2×C4—262no—
224C2×C4(C22⋊D13)⋊C311yescharacteristic, complemented
225C2×C3—131no—
226C2×C3—261no—
227C2×C3—262no—
228C2×C3—522no—
229C2×C3—521no—
230C2×C3—522no—
231C22—22no—
232C22—24no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—262no—
240C22—522no—
241C22—1041no—
242C22D4×D13⋊C312yes—
243C22C2×(C22⋊D13)⋊C311yescharacteristic
244C4—21no—
245C4—41no—
246C4—262no—
247C4—522no—
248C3—131noSylow
249C2—21no—
250C2—22no—
251C2—42no—
252C2—41no—
253C2—522no—
254C2#88191⋊C311yescharacteristic
subgroupG/H
centerC2#88191⋊C3
commutator, cosocleC23×C13C23×C3
FrattiniC23C22×D13⋊C3
FittingC13×#120C2×C3
socleC2×C13C3×#120
hypercenter#120D13⋊C3
hypocenterC13C3×B(4,𝔽2)
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #88251⋊C3C2semidirect
2. #120×C13⋊C3C2semidirect
3. #11928318C2semidirect
4. #88074⋊C3C2semidirect
5. #88046⋊C3C2semidirect
6. #11718693C3semidirect
7. C24×C13⋊C3C22semidirect
17. C22⋊C4×C13⋊C3C22semidirect
19. (C22⋊Dic13)⋊C3C22semidirect
20. #88251C2×C3semidirect
21. C13×#120C2×C3semidirect
22. #88207C2×C3semidirect
23. #88074C2×C3semidirect
24. #88046C2×C3semidirect
28. C23×C13⋊C3D4semidirect
31. C22×D13⋊C3D4semidirect
35. C2×C4×C13⋊C3D4semidirect
42. C2×Dic13⋊C3D4semidirect
46. C24×C13C22×C3semidirect
56. C13×C22⋊C4C22×C3semidirect
58. C22⋊Dic13C22×C3semidirect
80. C23×C13C3×D4semidirect
83. C22×D13C3×D4semidirect
87. C2×C4×C13C3×D4semidirect
94. C2×Dic13C3×D4semidirect
140. C13⋊C3B(4,𝔽2)semidirect
143. #120D13⋊C3semidirect
173. C24C2×D13⋊C3semidirect
185. C22⋊C4C2×D13⋊C3semidirect
186. C13C3×B(4,𝔽2)semidirect
209. C23(C22⋊D13)⋊C3semidirect
224. C2×C4(C22⋊D13)⋊C3semidirect

Maximal subgroups

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

subgroupindex
#88251⋊C32
#120×C13⋊C32
#119283182
#88074⋊C32
#88046⋊C32
#117186933
C3×B(4,𝔽2)13