Finite Group Databasebeta

← #10595954⋊C3 #13152359

Subgroups — #10595954⋊C3 #13152359

Order 1728.

979 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 979 conjugacy classes of subgroups — too many to draw. Only its 252 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
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3757⋊C3C212yesmaximal
3#64133⋊C3C214yesmaximal, complemented
4#64087⋊C3C218yesmaximal, complemented
5#10595954C311yesmaximal, characteristic, complemented
6C3×#5483—31nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2212yes—
9C22×(C3⋊Dic3)⋊C3C2214yes—
10C2×#634⋊C3C2214yescomplemented
11C22⋊C4×H(3,𝔽3)C2214yes—
12#3757⋊C3C2214yes—
13#3757⋊C3C2214yes—
14C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
15#3750⋊C3C2218yescomplemented
16C2×C32×C22⋊C4—21no—
17C2×C32×C22⋊C4D311yescharacteristic
18C2×C32×C22⋊C4C2×C311yescharacteristic
19C2×#3757C2×C312yes—
20#64133C2×C314yescomplemented
21#64087C2×C318yescomplemented
22C2×C3×#217—32no—
23C3×#924—34no—
24C3×#878—38no—
25C23×H(3,𝔽3)—42no—
26C23×H(3,𝔽3)Q814yescomplemented
27C23×H(3,𝔽3)C2312yes—
28C23×H(3,𝔽3)C2311yescharacteristic
29C2×C4×H(3,𝔽3)—42no—
30C2×C4×H(3,𝔽3)C2314yes—
31C2×(C3⋊Dic3)⋊C3—44no—
32C2×(C3⋊Dic3)⋊C3C2318yes—
33#634⋊C3—44no—
34C3×#543—91no—
35#5483—31no—
36#5483—91no—
37C24×C32—21no—
38C24×C32C2×D311yescharacteristic
39C24×C32C22×C311yescharacteristic
40C22×C32×C4—22no—
41C22×C32×C4C2×D312yes—
42C22×C32×C4C22×C312yes—
43C22×C3⋊Dic3C22×C314yes—
44C2×#634C22×C314yescomplemented
45C22×C3×Dic3—34no—
46C2×C3×Dic6—34no—
47C32×C22⋊C4—24no—
48C32×C22⋊C4C2×D314yes—
49C32×C22⋊C4C22×C314yes—
50#3757C22×C314yes—
51#3757C22×C314yes—
52C4⋊(C3⋊Dic3)C22×C314yescomplemented
53#3750C22×C318yescomplemented
54C3×#217—34no—
55C3×#217—34no—
56C3×C4⋊Dic3—34no—
57C3×Dic3⋊C4—38no—
58C22×H(3,𝔽3)—22no—
59C22×H(3,𝔽3)—24no—
60C22×H(3,𝔽3)—44no—
61C22×H(3,𝔽3)C2×Q812yes—
62C22×H(3,𝔽3)C2×Q814yes—
63C22×H(3,𝔽3)C2411yescharacteristic
64C4×H(3,𝔽3)—24no—
65(C3⋊Dic3)⋊C3—28no—
66C3×C22⋊Q8—98no—
67C3×C22⋊Q8—94no—
68C2×C3×C22⋊C4—31no—
69C2×C3×C22⋊C4—31no—
70C2×C3×C22⋊C4—61no—
71C2×C3×C22⋊C4—92no—
72C2×C3×C22⋊C4C3×D311yescharacteristic
73C2×#217—32no—
74C2×#217—92no—
75#924—34no—
76#924—94no—
77#878—38no—
78#878—98no—
79C23×C32—24no—
80C23×C32—21no—
81C23×C32—22no—
82C23×C32—42no—
83C23×C32—42no—
84C23×C32—44no—
85C23×C32Dic614yescomplemented
86C23×C32C3×Q814yescomplemented
87C23×C32C22×D311yescharacteristic
88C23×C32C22×D312yes—
89C23×C32C23×C311yescharacteristic
90C23×C32C23×C312yes—
91C2×C32×C4—24no—
92C2×C32×C4—42no—
93C2×C32×C4—42no—
94C2×C32×C4—44no—
95C2×C32×C4C22×D314yes—
96C2×C32×C4C23×C314yes—
97C2×C3⋊Dic3—44no—
98C2×C3⋊Dic3C23×C318yes—
99#634—44no—
100C2×C3×Dic3—38no—
101C2×C3×Dic3—124no—
102C3×Dic6—124no—
103#543—91noSylow
104C2×H(3,𝔽3)—22no—
105C2×H(3,𝔽3)—24no—
106C2×H(3,𝔽3)C22×Q811yescharacteristic
107C2×H(3,𝔽3)H(5,𝔽2)12yes—
108C24×C3—31no—
109C24×C3—31no—
110C24×C3—61no—
111C24×C3C2×C3×D311yescharacteristic
112C2×C3×Q8—94no—
113C22×C3×C4—32no—
114C22×C3×C4—32no—
115C22×C3×C4—62no—
116C22×C3×C4—94no—
117C22×C3×C4C2×C3×D312yes—
118C22×Dic3—34no—
119C22×Dic3—94no—
120C2×Dic6—34no—
121C2×Dic6—94no—
122C3×C4⋊C4—98no—
123C3×C4⋊C4—94no—
124C3×C22⋊C4—34no—
125C3×C22⋊C4—34no—
126C3×C22⋊C4—64no—
127C3×C22⋊C4—94no—
128C3×C22⋊C4—94no—
129C3×C22⋊C4C2×C3×D314yes—
130#217—34no—
131#217—34no—
132#217—94no—
133#217—94no—
134C4⋊Dic3—34no—
135C4⋊Dic3—94no—
136Dic3⋊C4—38no—
137Dic3⋊C4—98no—
138C22×C32—24no—
139C22×C32—24no—
140C22×C32—22no—
141C22×C32—24no—
142C22×C32—22no—
143C22×C32—21no—
144C22×C32—22no—
145C22×C32—24no—
146C22×C32—44no—
147C22×C32—44no—
148C22×C32—44no—
149C22×C32—48no—
150C22×C32C2×Dic612yes—
151C22×C32C2×Dic614yes—
152C22×C32C2×C3×Q812yes—
153C22×C32C2×C3×Q814yes—
154C22×C32C23×D311yescharacteristic
155C22×C32C24×C311yescharacteristic
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—44no—
159C3⋊Dic3—28no—
160C3×Dic3—68no—
161C22⋊Q8—94no—
162C22⋊Q8—98no—
163C2×C22⋊C4—92no—
164C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
165H(3,𝔽3)#54311yescharacteristic, complemented, Sylow
166C23×C3—34no—
167C23×C3—34no—
168C23×C3—32no—
169C23×C3—31no—
170C23×C3—31no—
171C23×C3—32no—
172C23×C3—42no—
173C23×C3—64no—
174C23×C3—61no—
175C23×C3—62no—
176C23×C3—122no—
177C23×C3—122no—
178C23×C3—124no—
179C23×C3C3×Dic614yes—
180C23×C3C22×C3×D311yescharacteristic
181C23×C3C22×C3×D312yes—
182C3×Q8—364no—
183C2×C3×C4—34no—
184C2×C3×C4—34no—
185C2×C3×C4—42no—
186C2×C3×C4—64no—
187C2×C3×C4—98no—
188C2×C3×C4—122no—
189C2×C3×C4—122no—
190C2×C3×C4—124no—
191C2×C3×C4—364no—
192C2×C3×C4C22×C3×D314yes—
193C2×Dic3—38no—
194C2×Dic3—98no—
195C2×Dic3—124no—
196C2×Dic3—364no—
197Dic6—124no—
198Dic6—364no—
199C2×C32—24no—
200C2×C32—22no—
201C2×C32—24no—
202C2×C32—22no—
203C2×C32—24no—
204C2×C32—21no—
205C2×C32—22no—
206C2×C32—44no—
207C2×C32C22×Dic611yescharacteristic
208C2×C32#100412yes—
209C2×C32C22×C3×Q811yescharacteristic
210C2×C32C3×H(5,𝔽2)12yes—
211C24C2×(C3⋊D3)⋊C311yescharacteristic
212C2×Q8—94no—
213C22×C4—94no—
214C22×C4C2×(C3⋊D3)⋊C312yes—
215C4⋊C4—98no—
216C4⋊C4—94no—
217C22⋊C4—94no—
218C22⋊C4—94no—
219C22⋊C4C2×(C3⋊D3)⋊C314yes—
220C22×C3—24no—
221C22×C3—22no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—34no—
225C22×C3—32no—
226C22×C3—32no—
227C22×C3—34no—
228C22×C3—44no—
229C22×C3—64no—
230C22×C3—62no—
231C22×C3—62no—
232C22×C3—64no—
233C22×C3—64no—
234C22×C3—61no—
235C22×C3—62no—
236C22×C3—64no—
237C22×C3—124no—
238C22×C3—124no—
239C22×C3—124no—
240C22×C3—128no—
241C22×C3C2×C3×Dic612yes—
242C22×C3C2×C3×Dic614yes—
243C22×C3C23×C3×D311yescharacteristic
244C3×C4—24no—
245C3×C4—64no—
246C3×C4—64no—
247C3×C4—124no—
248C3×C4—188no—
249Dic3—68no—
250Dic3—188no—
251C32—21no—
252C32#548311yescharacteristic, complemented
253C32C3×#54311yescharacteristic, complemented
254C23—42no—
255C23#634⋊C314yescomplemented
256C23C22×(C3⋊D3)⋊C311yescharacteristic
257C23C22×(C3⋊D3)⋊C312yes—
258Q8—364no—
259C2×C4—42no—
260C2×C4—98no—
261C2×C4—364no—
262C2×C4C22×(C3⋊D3)⋊C314yes—
263C2×C3—22no—
264C2×C3—24no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—32no—
268C2×C3—32no—
269C2×C3—62no—
270C2×C3—62no—
271C2×C3—64no—
272C2×C3—64no—
273C2×C3—64no—
274C2×C3—61no—
275C2×C3—62no—
276C2×C3—124no—
277C2×C3C22×C3×Dic611yescharacteristic
278C2×C3C3×#100412yes—
279C22—22no—
280C22—24no—
281C22—44no—
282C22C2×#634⋊C312yes—
283C22C2×#634⋊C314yes—
284C22C23×(C3⋊D3)⋊C311yescharacteristic
285C4—24no—
286C4—188no—
287C3—31no—
288C3—31no—
289C3—61no—
290C3C3×#548311yescharacteristic
291C2—22no—
292C2—24no—
293C2C22×#634⋊C311yescharacteristic
294C2#64361⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #64133⋊C3C2semidirect
4. #64087⋊C3C2semidirect
5. #10595954C3semidirect
10. C2×#634⋊C3C22semidirect
14. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
15. #3750⋊C3C22semidirect
20. #64133C2×C3semidirect
21. #64087C2×C3semidirect
26. C23×H(3,𝔽3)Q8semidirect
44. C2×#634C22×C3semidirect
52. C4⋊(C3⋊Dic3)C22×C3semidirect
53. #3750C22×C3semidirect
85. C23×C32Dic6semidirect
86. C23×C32C3×Q8semidirect
165. H(3,𝔽3)#543semidirect
252. C32#5483semidirect
253. C32C3×#543semidirect
255. C23#634⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
#64133⋊C32
#64087⋊C32
#105959543
C3×#54833