Finite Group Databasebeta

← #72243411

Subgroups — #72243411

Order 4374.

13240 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 13240 conjugacy classes of subgroups — too many to draw. Only its 269 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
1C32×#6891C211yesmaximal, characteristic, complemented, Sylow
2C32×C3⋊(C3⋊(C3⋊D3))C311yesmaximal, characteristic
33−1+2×C3⋊(C3⋊D3)—31nomaximal
4C3⋊#92259—312nomaximal
5C3⋊(C3⋊((C3⋊D3)⋊C9))C313yesmaximal, complemented
6C36C2×C311yescharacteristic
7C33×3−1+2—21no—
8C33×3−1+2D311yescharacteristic, complemented
9C3×#6891—236no—
10C3×#6891—212no—
11C3×#6891D3112yescomplemented
12C32×C32⋊C9—23no—
13C32×C32⋊C9C2×C313yescomplemented
14C3×C3⋊(C3⋊(C3⋊D3))—31no—
15C3×C3⋊(C3⋊(C3⋊D3))C3211yescharacteristic
16C32×C3⋊(C3⋊D3)—31no—
17C32×C3⋊(C3⋊D3)—312no—
18C32×C3⋊(C3⋊D3)—99no—
19C9×C3⋊(C3⋊D3)—33no—
20C3⋊D3×3−1+2—99no—
21C3⋊D3×3−1+2—94no—
22C3⋊((C3⋊D3)⋊C9)—336no—
23#92259—99no—
24C35—21no—
25C35—212no—
26C35—31no—
27C35—39no—
28C35—627no—
29C35—69no—
30C35—61no—
31C35—612no—
32C35C3×D311yescharacteristic
33C35C3×D3112yes—
34C35C2×C3211yescharacteristic
35C32×3−1+2—224no—
36C32×3−1+2—24no—
37C32×3−1+2—212no—
38C32×3−1+2—212no—
39C32×3−1+2—39no—
40C32×3−1+2—69no—
41C32×3−1+2—618no—
42C32×3−1+2—69no—
43C32×3−1+2C3⋊D314yescomplemented
44C33×C9—23no—
45C33×C9C3×D313yescomplemented
46#6891—2216no—
47#6891—236no—
48#6891—2108no—
49#6891C3⋊D319yescomplemented
50C3×C32⋊C9—272no—
51C3×C32⋊C9—236no—
52C3×C32⋊C9—236no—
53C3×C32⋊C9C3×D3136yescomplemented
54C3⋊(C3⋊(C3⋊D3))3−1+211yescharacteristic, complemented
55C3×C3⋊(C3⋊D3)—31no—
56C3×C3⋊(C3⋊D3)—312no—
57C3×C3⋊(C3⋊D3)—99no—
58C3×C3⋊(C3⋊D3)—91no—
59C3×C3⋊(C3⋊D3)—912no—
60C3×C3⋊(C3⋊D3)—927no—
61C32×C3⋊D3—99no—
62C32×C3⋊D3—94no—
63C32×C3⋊D3—99no—
64C32×C3⋊D3—2736no—
65C9×C3⋊D3—927no—
66C9×C3⋊D3—912no—
67D3×3−1+2—2712no—
68D3×3−1+2—271no—
69(C3⋊D3)⋊C9—927no—
70C34—227no—
71C34—29no—
72C34—21no—
73C34—212no—
74C34—24no—
75C34—212no—
76C34—236no—
77C34—327no—
78C34—39no—
79C34—31no—
80C34—312no—
81C34—336no—
82C34—69no—
83C34—64no—
84C34—6108no—
85C34—636no—
86C34—612no—
87C34—636no—
88C34—69no—
89C34—6108no—
90C34—69no—
91C34—68no—
92C34—6216no—
93C34—624no—
94C34—672no—
95C34—627no—
96C34—6324no—
97C34—627no—
98C34—627no—
99C34—6648no—
100C34—627no—
101C34—69no—
102C34—61no—
103C34—612no—
104C34(C3⋊D3)⋊C319yes—
105C34C2×3−1+211yescharacteristic, complemented
106C34C32×D311yescharacteristic
107C34C32×D3112yes—
108C34C3×C3⋊D314yes—
109C34C3×C3⋊D319yes—
110C3×3−1+2—224no—
111C3×3−1+2—272no—
112C3×3−1+2—29no—
113C3×3−1+2—24no—
114C3×3−1+2—212no—
115C3×3−1+2—312no—
116C3×3−1+2—612no—
117C3×3−1+2—6108no—
118C3×3−1+2—624no—
119C3×3−1+2—6216no—
120C3×3−1+2—672no—
121C3×3−1+2—612no—
122C3×3−1+2—636no—
123C3×3−1+2C3⋊(C3⋊D3)11yescharacteristic, complemented
124C32×C9—236no—
125C32×C9—212no—
126C32×C9—327no—
127C32×C9—627no—
128C32×C9C3×C3⋊D3112yescomplemented
129C32⋊C9—2216no—
130C32⋊C9—2108no—
131C32⋊C9—227no—
132C32⋊C9C3×C3⋊D3127yescomplemented
133C3⋊(C3⋊D3)—31no—
134C3⋊(C3⋊D3)—312no—
135C3⋊(C3⋊D3)—99no—
136C3×C3⋊D3—99no—
137C3×C3⋊D3—94no—
138C3×C3⋊D3—99no—
139C3×C3⋊D3—2736no—
140C3×C3⋊D3—279no—
141C3×C3⋊D3—274no—
142C3×C3⋊D3—279no—
143C3×C3⋊D3—27108no—
144C32×D3—2712no—
145C32×D3—271no—
146C32×D3—819no—
147C2×3−1+2—811no—
148D3×C9—2736no—
149D3×C9—273no—
150C33—29no—
151C33—24no—
152C33—2108no—
153C33—236no—
154C33—212no—
155C33—212no—
156C33—236no—
157C33—29no—
158C33—24no—
159C33—39no—
160C33—3108no—
161C33—39no—
162C33—336no—
163C33—34no—
164C33—39no—
165C33—39no—
166C33—636no—
167C33—636no—
168C33—612no—
169C33—64no—
170C33—627no—
171C33—69no—
172C33—69no—
173C33—6108no—
174C33—6108no—
175C33—624no—
176C33—636no—
177C33—6216no—
178C33—672no—
179C33—68no—
180C33—624no—
181C33—6216no—
182C33—672no—
183C33—6324no—
184C33—6108no—
185C33—6324no—
186C33—6108no—
187C33—6108no—
188C33—6648no—
189C33—61944no—
190C33—6108no—
191C33—6648no—
192C33—627no—
193C33—69no—
194C33—69no—
195C33—636no—
196C33—64no—
197C33—612no—
198C33—636no—
199C33C3×(C3⋊D3)⋊C319yes—
200C33D3×3−1+211yescharacteristic
201C33D3×3−1+2112yescomplemented
202C33C3⋊((C3⋊D3)⋊C3)112yes—
203C33C32×C3⋊D314yes—
204C33C32×C3⋊D319yes—
205C33C3×C3⋊(C3⋊D3)11yescharacteristic
2063−1+2—31no—
2073−1+2—61no—
2083−1+2—636no—
2093−1+2—672no—
2103−1+2—68no—
2113−1+2—6216no—
2123−1+2—618no—
2133−1+2—64no—
2143−1+2—69no—
215C3×C9—212no—
216C3×C9—227no—
217C3×C9—336no—
218C3×C9—636no—
219C3×C9—6108no—
220C3×C9C3×C3⋊(C3⋊D3)13yescomplemented
221C2×C32—811no—
222C3⋊D3—99no—
223C3⋊D3—94no—
224C3⋊D3—99no—
225C3⋊D3—2736no—
226C3×D3—2712no—
227C3×D3—271no—
228C3×D3—8112no—
229C3×D3—811no—
230C3×D3—819no—
231C3×D3—8127no—
232C2×C9—813no—
233C32—212no—
234C32—236no—
235C32—24no—
236C32—227no—
237C32—212no—
238C32—21no—
239C32—29no—
240C32—336no—
241C32—327no—
242C32—312no—
243C32—39no—
244C32—31no—
245C32—612no—
246C32—69no—
247C32—61no—
248C32—6108no—
249C32—69no—
250C32—68no—
251C32—6216no—
252C32—672no—
253C32—624no—
254C32—6324no—
255C32—627no—
256C32—627no—
257C32—627no—
258C32—627no—
259C32—6648no—
260C32—6108no—
261C32—636no—
262C32—612no—
263C32—636no—
264C32—69no—
265C32—69no—
266C32—64no—
267C32#9225919yescomplemented
268C32C3×C3⋊((C3⋊D3)⋊C3)112yes—
269C32C3⋊D3×3−1+214yes—
270C32C3⋊D3×3−1+219yescomplemented
271C32C3⋊(C3⋊((C3⋊D3)⋊C3))11yescharacteristic
272C32C32×C3⋊(C3⋊D3)11yescharacteristic
273C9—33no—
274C9—612no—
275C9—627no—
276C2×C3—811no—
277C2×C3—2431no—
278D3—2712no—
279D3—271no—
280D3—819no—
281C3—21no—
282C3—212no—
283C3—39no—
284C3—31no—
285C3—627no—
286C3—612no—
287C3—61no—
288C3—69no—
289C3C3⋊#92259112yescomplemented
290C3C3×C3⋊(C3⋊((C3⋊D3)⋊C3))11yescharacteristic
291C33−1+2×C3⋊(C3⋊D3)11yescharacteristic
292C2—811noSylow
subgroupG/H
centerC3C3×C3⋊(C3⋊((C3⋊D3)⋊C3))
commutator, cosocleC35C2×C32
FrattiniC32C32×C3⋊(C3⋊D3)
FittingC32×#6891C2
socleC34C32×D3
prime radicalC32×C3⋊(C3⋊(C3⋊D3))C3
hypercenterC32C3⋊(C3⋊((C3⋊D3)⋊C3))
hypocenterC34C2×3−1+2
LCSG ▷ C35 ▷ C34
UCSC1 ◁ C3 ◁ C32

Complements

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

NH ≅ G/Ntype
1. C32×#6891C2semidirect
5. C3⋊(C3⋊((C3⋊D3)⋊C9))C3semidirect
8. C33×3−1+2D3semidirect
11. C3×#6891D3semidirect
13. C32×C32⋊C9C2×C3semidirect
43. C32×3−1+2C3⋊D3semidirect
45. C33×C9C3×D3semidirect
49. #6891C3⋊D3semidirect
53. C3×C32⋊C9C3×D3semidirect
54. C3⋊(C3⋊(C3⋊D3))3−1+2semidirect
105. C34C2×3−1+2semidirect
123. C3×3−1+2C3⋊(C3⋊D3)semidirect
128. C32×C9C3×C3⋊D3semidirect
132. C32⋊C9C3×C3⋊D3semidirect
201. C33D3×3−1+2semidirect
220. C3×C9C3×C3⋊(C3⋊D3)semidirect
267. C32#92259semidirect
270. C32C3⋊D3×3−1+2semidirect
289. C3C3⋊#92259semidirect

Maximal subgroups

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

subgroupindex
C32×#68912
C32×C3⋊(C3⋊(C3⋊D3))3
3−1+2×C3⋊(C3⋊D3)3
C3⋊#922593
C3⋊(C3⋊((C3⋊D3)⋊C9))3