Finite Group Databasebeta

← C24×C3×#1094 #72244761

Subgroups — C24×C3×#1094 #72244761

Order 5184.

30208 conjugacy classes of subgroups; only the 2814 normal subgroups are shown.

normal subgroups only

This group has 30208 conjugacy classes of subgroups — too many to draw. Only its 2814 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
1C23×C3×#1094C2115yesmaximal, direct product factor
2C26×C33C311yesmaximal, characteristic
3C24×#1094C319yesmaximal, direct product factor
4C24×C3×C22⋊C9C313yesmaximal, complemented
5C24×C3×3−1+2—41nomaximal
6C22×C3×#1094C22135yesdirect product factor
7C25×C33—31no—
8C25×C33—315no—
9C25×C33C2×C3115yes—
10C23×#1094C2×C31135yesdirect product factor
11C23×C3×C22⋊C9C2×C3145yescomplemented
12C23×C3×3−1+2—415no—
13C2×C3×#1094C23115yesdirect product factor
14C26×C32—33no—
15C26×C32C3213yes—
16C26×C32C3211yescharacteristic
17C24×C22⋊C9C3219yescomplemented
18C24×C33—3105no—
19C24×C33—370no—
20C24×C33—315no—
21C24×C33—315no—
22C24×C33A411yescharacteristic
23C24×C33C22×C3135yes—
24C24×3−1+2—49no—
25C24×C3×C9—43no—
26C22×#1094C22×C31315yesdirect product factor
27C22×C3×C22⋊C9C22×C31105yescomplemented
28C22×C3×3−1+2—435no—
29C3×#1094C2411yescharacteristic, direct product factor
30C25×C32—315no—
31C25×C32—345no—
32C25×C32—3135no—
33C25×C32—39no—
34C25×C32—33no—
35C25×C32—31no—
36C25×C32—345no—
37C25×C32C2×C32145yes—
38C25×C32C2×C32115yes—
39C23×C22⋊C9C2×C321135yescomplemented
40C23×C33—3210no—
41C23×C33—3105no—
42C23×C33—335no—
43C23×C33—3105no—
44C23×C33C2×A4115yes—
45C23×C33C23×C3115yes—
46C23×3−1+2—4135no—
47C23×C3×C9—445no—
48C2×#1094C23×C31135yesdirect product factor
49C2×C3×C22⋊C9C23×C3145yescomplemented
50C26×C3—33no—
51C26×C33−1+213yescomplemented
52C26×C3C3311yescharacteristic
53C2×C3×3−1+2—415no—
54C24×C32—3945no—
55C24×C32—3630no—
56C24×C32—3135no—
57C24×C32—3135no—
58C24×C32—3210no—
59C24×C32—370no—
60C24×C32—33no—
61C24×C32—3105no—
62C24×C32—345no—
63C24×C32—315no—
64C24×C32—345no—
65C24×C32—315no—
66C24×C32—3105no—
67C24×C32—3315no—
68C24×C32C3×A413yes—
69C24×C32C3×A411yescharacteristic
70C24×C32C22×C321105yes—
71C24×C32C22×C32135yes—
72C24×C9—49no—
73C22×C22⋊C9C22×C321315yescomplemented
74C22×C33—370no—
75C22×C33—3105no—
76C22×C33—315no—
77C22×C33—315no—
78C22×C33C22×A4135yes—
79C22×C33C24×C311yescharacteristic
80C22×3−1+2—4315no—
81C22×C3×C9—4105no—
82#1094C24×C319yesdirect product factor
83C3×C22⋊C9C24×C313yescomplemented
84C25×C3—345no—
85C25×C3—315no—
86C25×C3—3135no—
87C25×C3—39no—
88C25×C3—33no—
89C25×C3—31no—
90C25×C3—345no—
91C25×C3C2×3−1+2145yescomplemented
92C25×C3C2×C33115yes—
93C3×3−1+2—41noSylow
94C23×C32—3105no—
95C23×C32—3315no—
96C23×C32—3945no—
97C23×C32—31890no—
98C23×C32—3630no—
99C23×C32—3210no—
100C23×C32—3945no—
101C23×C32—3315no—
102C23×C32—3105no—
103C23×C32—335no—
104C23×C32—3315no—
105C23×C32—3105no—
106C23×C32—345no—
107C23×C32—345no—
108C23×C32C2×C3×A4145yes—
109C23×C32C2×C3×A4115yes—
110C23×C32C23×C32145yes—
111C23×C32C23×C32115yes—
112C23×C9—4135no—
113C2×C22⋊C9C23×C321135yescomplemented
114C26C3×3−1+211yescharacteristic, complemented, Sylow
115C2×C33—315no—
116C2×C33—31no—
117C2×C33C23×A4115yes—
118C2×3−1+2—4135no—
119C2×C3×C9—445no—
120C24×C3—3315no—
121C24×C3—3105no—
122C24×C3—3945no—
123C24×C3—3630no—
124C24×C3—370no—
125C24×C3—3210no—
126C24×C3—3135no—
127C24×C3—3135no—
128C24×C3—345no—
129C24×C3—345no—
130C24×C3—315no—
131C24×C3—315no—
132C24×C3—33no—
133C24×C3—3105no—
134C24×C3#109413yesdirect product factor
135C24×C3C22×3−1+21105yescomplemented
136C24×C3C32×A411yescharacteristic
137C24×C3C22×C33135yes—
138C22×C32—315no—
139C22×C32—345no—
140C22×C32—3135no—
141C22×C32—3630no—
142C22×C32—3210no—
143C22×C32—370no—
144C22×C32—3135no—
145C22×C32—3945no—
146C22×C32—3315no—
147C22×C32—3105no—
148C22×C32—345no—
149C22×C32—315no—
150C22×C32—3105no—
151C22×C32—33no—
152C22×C32C22×C3×A41105yes—
153C22×C32C22×C3×A4135yes—
154C22×C32C24×C3213yes—
155C22×C32C24×C3211yescharacteristic
156C22×C9—4315no—
157C22⋊C9C24×C3219yescomplemented
158C25—315no—
159C25—31no—
160C25C2×C3×3−1+2115yescomplemented
161C33C24×A411yescharacteristic
1623−1+2—49no—
163C3×C9—43no—
164C23×C3—3315no—
165C23×C3—3105no—
166C23×C3—3945no—
167C23×C3—31890no—
168C23×C3—3630no—
169C23×C3—3210no—
170C23×C3—3315no—
171C23×C3—3945no—
172C23×C3—3105no—
173C23×C3—335no—
174C23×C3—3315no—
175C23×C3—3105no—
176C23×C3—345no—
177C23×C3—345no—
178C23×C3C2×#1094145yesdirect product factor
179C23×C3C23×3−1+2145yescomplemented
180C23×C3C2×C32×A4115yes—
181C23×C3C23×C33115yes—
182C2×C32—39no—
183C2×C32—3135no—
184C2×C32—345no—
185C2×C32—315no—
186C2×C32—33no—
187C2×C32—31no—
188C2×C32—345no—
189C2×C32C23×C3×A4145yes—
190C2×C32C23×C3×A4115yes—
191C2×C9—4135no—
192C24—3105no—
193C24—370no—
194C24—315no—
195C24—315no—
196C24C3×#109411yescharacteristic, direct product factor
197C24C22×C3×3−1+2135yescomplemented
198C22×C3—345no—
199C22×C3—315no—
200C22×C3—3135no—
201C22×C3—3630no—
202C22×C3—370no—
203C22×C3—3210no—
204C22×C3—3135no—
205C22×C3—3945no—
206C22×C3—345no—
207C22×C3—3315no—
208C22×C3—315no—
209C22×C3—3105no—
210C22×C3—33no—
211C22×C3—3105no—
212C22×C3C22×#10941105yesdirect product factor
213C22×C3C24×3−1+213yescomplemented
214C22×C3C22×C32×A4135yes—
215C22×C3C24×C3311yescharacteristic
216C32—33no—
217C32C24×C3×A413yes—
218C32C24×C3×A411yescharacteristic
219C9—49no—
220C23—3105no—
221C23—3210no—
222C23—335no—
223C23—3105no—
224C23C2×C3×#1094115yesdirect product factor
225C23C23×C3×3−1+2115yescomplemented
226C2×C3—39no—
227C2×C3—3135no—
228C2×C3—31no—
229C2×C3—315no—
230C2×C3—33no—
231C2×C3—345no—
232C2×C3—345no—
233C2×C3C23×#1094145yesdirect product factor
234C2×C3C23×C32×A4115yes—
235C22—315no—
236C22—370no—
237C22—315no—
238C22—3105no—
239C22C22×C3×#1094135yesdirect product factor
240C22C24×C3×3−1+211yescharacteristic, complemented
241C3—33no—
242C3C24×#109413yesdirect product factor
243C3C24×C32×A411yescharacteristic
244C2—31no—
245C2—315no—
246C2C23×C3×#1094115yesdirect product factor
subgroupG/H
centerC24×C32C3×A4
commutator, cosocleC22×C3C24×C33
FrattiniC3C24×C32×A4
Fitting, prime radicalC26×C33C3
socleC26×C32C32
hypercenterC24×C33A4
hypocenterC22C24×C3×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C24×C32 ◁ C24×C33

Complements

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

NH ≅ G/Ntype
1. C23×C3×#1094C2direct (N × H)
3. C24×#1094C3direct (N × H)
4. C24×C3×C22⋊C9C3semidirect
6. C22×C3×#1094C22direct (N × H)
10. C23×#1094C2×C3direct (N × H)
11. C23×C3×C22⋊C9C2×C3semidirect
13. C2×C3×#1094C23direct (N × H)
17. C24×C22⋊C9C32semidirect
26. C22×#1094C22×C3direct (N × H)
27. C22×C3×C22⋊C9C22×C3semidirect
29. C3×#1094C24direct (N × H)
39. C23×C22⋊C9C2×C32semidirect
48. C2×#1094C23×C3direct (N × H)
49. C2×C3×C22⋊C9C23×C3semidirect
51. C26×C33−1+2semidirect
73. C22×C22⋊C9C22×C32semidirect
82. #1094C24×C3direct (N × H)
83. C3×C22⋊C9C24×C3semidirect
91. C25×C3C2×3−1+2semidirect
113. C2×C22⋊C9C23×C32semidirect
114. C26C3×3−1+2semidirect
134. C24×C3#1094direct (N × H)
135. C24×C3C22×3−1+2semidirect
157. C22⋊C9C24×C32semidirect
160. C25C2×C3×3−1+2semidirect
178. C23×C3C2×#1094direct (N × H)
179. C23×C3C23×3−1+2semidirect
196. C24C3×#1094direct (N × H)
197. C24C22×C3×3−1+2semidirect
212. C22×C3C22×#1094direct (N × H)
213. C22×C3C24×3−1+2semidirect
224. C23C2×C3×#1094direct (N × H)
225. C23C23×C3×3−1+2semidirect
233. C2×C3C23×#1094direct (N × H)
239. C22C22×C3×#1094direct (N × H)
240. C22C24×C3×3−1+2semidirect
242. C3C24×#1094direct (N × H)
246. C2C23×C3×#1094direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#10942
C26×C333
C24×#10943
C24×C3×C22⋊C93
C24×C3×3−1+24