Finite Group Databasebeta

← #10596171⋊C3 #13153190

Subgroups — #10596171⋊C3 #13153190

Order 1728.

826 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 202 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64105⋊C3C214yesmaximal, complemented
3#64104⋊C3C214yesmaximal, complemented
4#64103⋊C3C214yesmaximal, complemented
5#64084⋊C3C212yesmaximal, complemented
6#10596171C311yesmaximal, characteristic, complemented
7C3×#5700—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
10C2×C4×(C3⋊D3)⋊C3C2214yes—
11C2×#634⋊C3C2212yes—
12C4⋊C4×H(3,𝔽3)C2212yes—
13C4⋊C4×H(3,𝔽3)C2214yes—
14C42×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3C2214yes—
16#3752⋊C3C2214yes—
17#3752⋊C3C2214yes—
18#3750⋊C3C2214yes—
19C32×#125—21no—
20C32×#125D311yescharacteristic, complemented
21C32×#125C2×C311yescharacteristic, complemented
22#64105C2×C314yescomplemented
23#64104C2×C314yescomplemented
24#64103C2×C314yescomplemented
25#64084C2×C312yescomplemented
26C3×#896—34no—
27C3×#895—34no—
28C3×#894—34no—
29C3×#875—32no—
30C22×(C3⋊D3)⋊C3C2314yes—
31C2×C4×H(3,𝔽3)C2312yes—
32C2×C4×H(3,𝔽3)C2314yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3C2314yes—
35C4⋊((C3⋊D3)⋊C3)—42no—
36C4⋊((C3⋊D3)⋊C3)—44no—
37C4×(C3⋊D3)⋊C3—44no—
38#634⋊C3—42no—
39C3×#562—91no—
40#5700—31no—
41#5700—91no—
42C2×C4⋊(C3⋊D3)C22×C312yes—
43C2×C4⋊(C3⋊D3)C22×C314yes—
44C2×C4×C3⋊D3C22×C314yes—
45C2×#634C22×C312yes—
46C2×C3×D12—34no—
47C2×C3×D12—32no—
48C2×C3×C4×D3—34no—
49C2×C3×Dic6—32no—
50C32×C4⋊C4—22no—
51C32×C4⋊C4—24no—
52C32×C4⋊C4C2×D312yes—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C22×C312yes—
55C32×C4⋊C4C22×C314yes—
56C32×C42—21no—
57C32×C42C2×D311yescharacteristic
58C32×C42C22×C311yescharacteristic
59#3752C22×C314yes—
60#3752C22×C314yes—
61#3752C22×C314yes—
62#3750C22×C314yes—
63C3×(C2×D3)⋊C4—34no—
64C3×(C2×D3)⋊C4—34no—
65C3×(C2×D3)⋊C4—34no—
66C3×Dic3⋊C4—34no—
67C22×H(3,𝔽3)C2411yescharacteristic
68C2×(C3⋊D3)⋊C3—24no—
69C2×(C3⋊D3)⋊C3—24no—
70C2×(C3⋊D3)⋊C3—24no—
71C4×H(3,𝔽3)—22no—
72C4×H(3,𝔽3)—24no—
73C4×H(3,𝔽3)—21no—
74(C3⋊Dic3)⋊C3—24no—
75C3×#125—31no—
76C3×#125—31no—
77C3×#125—61no—
78C3×#125C3×D311yescharacteristic
79C3×#124—92no—
80C3×#123—94no—
81C3×C22⋊Q8—94no—
82C3×#121—94no—
83#896—34no—
84#896—94no—
85#895—34no—
86#895—94no—
87#894—34no—
88#894—94no—
89#875—32no—
90#875—92no—
91C22×C3⋊D3C23×C314yes—
92C22×C3×D3—34no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—24no—
96C2×C32×C4C22×D314yes—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D312yes—
99C2×C32×C4C23×C314yes—
100C2×C32×C4C23×C311yescharacteristic
101C2×C32×C4C23×C312yes—
102C2×C3⋊Dic3C23×C314yes—
103C4⋊(C3⋊D3)—42no—
104C4⋊(C3⋊D3)—44no—
105C4×C3⋊D3—44no—
106#634—42no—
107C2×C3×Dic3—34no—
108C3×D12—122no—
109C3×D12—124no—
110C3×C4×D3—124no—
111C3×Dic6—122no—
112#562—91noSylow
113C2×H(3,𝔽3)H(5,𝔽2)12yes—
114C2×H(3,𝔽3)2−1+411yescharacteristic
115(C3⋊D3)⋊C3—44no—
116C2×C3×Q8—92no—
117C2×C3×D4—94no—
118C2×C3×D4—92no—
119C22×C3×C4—94no—
120C2×D12—32no—
121C2×D12—34no—
122C2×D12—92no—
123C2×D12—94no—
124C2×C4×D3—34no—
125C2×C4×D3—94no—
126C2×Dic6—32no—
127C2×Dic6—92no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—34no—
130C3×C4⋊C4—34no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—64no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—94no—
135C3×C4⋊C4C2×C3×D312yes—
136C3×C4⋊C4C2×C3×D314yes—
137C3×C22⋊C4—94no—
138C3×C22⋊C4—94no—
139C3×C22⋊C4—94no—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—61no—
143C3×C42C2×C3×D311yescharacteristic
144(C2×D3)⋊C4—34no—
145(C2×D3)⋊C4—34no—
146(C2×D3)⋊C4—34no—
147(C2×D3)⋊C4—94no—
148(C2×D3)⋊C4—94no—
149(C2×D3)⋊C4—94no—
150Dic3⋊C4—34no—
151Dic3⋊C4—94no—
152C22×C32—21no—
153C22×C32C23×D311yescharacteristic
154C22×C32C24×C311yescharacteristic
155C2×C3⋊D3—24no—
156C2×C3⋊D3—24no—
157C2×C3⋊D3—24no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—64no—
161C32×C4—22no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—24no—
165C32×C4—22no—
166C32×C4—24no—
167C32×C4—21no—
168C32×C4—42no—
169C32×C4—44no—
170C3⋊Dic3—24no—
171C3×Dic3—64no—
172#125(C3⋊D3)⋊C311yescharacteristic, complemented
173#124—92no—
174#123—94no—
175C22⋊Q8—94no—
176#121—94no—
177H(3,𝔽3)#56211yescharacteristic, complemented, Sylow
178C23×C3—94no—
179C22×D3—34no—
180C22×D3—94no—
181C3×Q8—362no—
182C3×D4—364no—
183C3×D4—362no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—34no—
187C2×C3×C4—34no—
188C2×C3×C4—31no—
189C2×C3×C4—32no—
190C2×C3×C4—61no—
191C2×C3×C4—62no—
192C2×C3×C4—64no—
193C2×C3×C4—94no—
194C2×C3×C4—364no—
195C2×C3×C4C22×C3×D314yes—
196C2×C3×C4C22×C3×D311yescharacteristic
197C2×C3×C4C22×C3×D312yes—
198C2×Dic3—34no—
199C2×Dic3—94no—
200D12—122no—
201D12—124no—
202D12—362no—
203D12—364no—
204C4×D3—124no—
205C4×D3—364no—
206Dic6—122no—
207Dic6—362no—
208C2×C32—21no—
209C2×C32—22no—
210C2×C32#100711yescharacteristic
211C2×C32#100912yes—
212C2×C32C3×H(5,𝔽2)12yes—
213C2×C32C3×2−1+411yescharacteristic
214C3⋊D3—44no—
215C3×D3—124no—
216C2×Q8—92no—
217C2×D4—94no—
218C2×D4—92no—
219C22×C4—94no—
220C4⋊C4—94no—
221C4⋊C4C2×(C3⋊D3)⋊C314yes—
222C4⋊C4C2×(C3⋊D3)⋊C312yes—
223C22⋊C4—94no—
224C22⋊C4—94no—
225C22⋊C4—94no—
226C42C2×(C3⋊D3)⋊C311yescharacteristic
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—61no—
230C22×C3—184no—
231C22×C3—184no—
232C22×C3—184no—
233C22×C3C23×C3×D311yescharacteristic
234C2×D3—64no—
235C2×D3—64no—
236C2×D3—64no—
237C2×D3—184no—
238C2×D3—184no—
239C2×D3—184no—
240C3×C4—21no—
241C3×C4—22no—
242C3×C4—24no—
243C3×C4—62no—
244C3×C4—64no—
245C3×C4—64no—
246C3×C4—62no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4—124no—
251C3×C4—122no—
252C3×C4—184no—
253Dic3—64no—
254Dic3—184no—
255C32—21no—
256C32#570011yescharacteristic, complemented
257C32C3×#56211yescharacteristic, complemented
258C23—94no—
259Q8—362no—
260D4—364no—
261D4—362no—
262C2×C4—94no—
263C2×C4—364no—
264C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
265C2×C4C22×(C3⋊D3)⋊C312yes—
266C2×C4C22×(C3⋊D3)⋊C314yes—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—32no—
270C2×C3—32no—
271C2×C3—61no—
272C2×C3—62no—
273C2×C3—364no—
274C2×C3C3×#100711yescharacteristic
275C2×C3C3×#100912yes—
276D3—124no—
277D3—364no—
278C22—184no—
279C22—184no—
280C22—184no—
281C22C23×(C3⋊D3)⋊C311yescharacteristic
282C4—21no—
283C4—24no—
284C4—22no—
285C4—184no—
286C3—31no—
287C3—31no—
288C3—61no—
289C3C3×#570011yescharacteristic
290C2—364no—
291C2#1172262711yescharacteristic
292C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #64105⋊C3C2semidirect
3. #64104⋊C3C2semidirect
4. #64103⋊C3C2semidirect
5. #64084⋊C3C2semidirect
6. #10596171C3semidirect
20. C32×#125D3semidirect
21. C32×#125C2×C3semidirect
22. #64105C2×C3semidirect
23. #64104C2×C3semidirect
24. #64103C2×C3semidirect
25. #64084C2×C3semidirect
172. #125(C3⋊D3)⋊C3semidirect
177. H(3,𝔽3)#562semidirect
256. C32#5700semidirect
257. C32C3×#562semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64105⋊C32
#64104⋊C32
#64103⋊C32
#64084⋊C32
#105961713
C3×#57003