Finite Group Databasebeta

← #125×(C3⋊D3)⋊C3 #13153178

Subgroups — #125×(C3⋊D3)⋊C3 #13153178

Order 1728.

847 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2C4⋊C4×(C3⋊D3)⋊C3C214yesmaximal
3C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal
4#64099⋊C3C212yesmaximal, complemented
5#64098⋊C3C214yesmaximal, complemented
6C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
7#64079⋊C3C211yesmaximal, characteristic, complemented
8C3⋊D3×#125C311yesmaximal, characteristic, complemented
9C3×D3×#125—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2214yes—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2214yes—
14C4⋊C4×H(3,𝔽3)C2212yes—
15C42×H(3,𝔽3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17C4⋊((C3⋊Dic3)⋊C3)C2214yes—
18#3750⋊C3C2214yes—
19#3750⋊C3C2214yes—
20#3750⋊C3C2214yes—
21C4×(C3⋊Dic3)⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
23C32×#125—21no—
24C32×#125D311yescharacteristic, complemented
25C32×#125C2×C311yescharacteristic, complemented
26C4⋊C4×C3⋊D3C2×C314yes—
27C4⋊C4×C3⋊D3C2×C312yes—
28#64099C2×C312yescomplemented
29#64098C2×C314yescomplemented
30C42×C3⋊D3C2×C311yescharacteristic
31#64079C2×C311yescharacteristic, complemented
32C3×D3×C4⋊C4—32no—
33C3×D3×C4⋊C4—34no—
34C3×#890—32no—
35C3×#889—34no—
36C3×C42×D3—31no—
37C3×#870—31no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2314yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2312yes—
43C2×(C3⋊Dic3)⋊C3C2314yes—
44C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
45C4×(C3⋊D3)⋊C3—24no—
46C4×(C3⋊D3)⋊C3—28no—
47C4×(C3⋊D3)⋊C3Q814yes—
48C2×C3×#125—91no—
49D3×#125—31no—
50D3×#125—91no—
51C2×C4×C3⋊D3C22×C312yes—
52C2×C4×C3⋊D3C22×C314yes—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—32no—
57C32×C4⋊C4—22no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4C2×D314yes—
60C32×C4⋊C4C2×D312yes—
61C32×C4⋊C4C22×C314yes—
62C32×C4⋊C4C22×C312yes—
63C32×C42—21no—
64C32×C42C2×D311yescharacteristic
65C32×C42C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C312yes—
67C4⋊(C3⋊Dic3)C22×C314yes—
68#3750C22×C314yes—
69#3750C22×C314yes—
70#3750C22×C314yes—
71C4×C3⋊Dic3C22×C312yes—
72C4×C3⋊Dic3C22×C311yescharacteristic
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—32no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—34no—
77C3×Dic3⋊C4—34no—
78C3×C4×Dic3—32no—
79C3×C4×Dic3—31no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×(C3⋊D3)⋊C3C2×Q812yes—
82C2×(C3⋊D3)⋊C3#4114yes—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)—24no—
85C4×H(3,𝔽3)C2×Q812yes—
86(C3⋊Dic3)⋊C3—22no—
87(C3⋊Dic3)⋊C3—24no—
88(C3⋊Dic3)⋊C3C2×Q812yes—
89C3×#125—31no—
90C3×#125—31no—
91C3×#125—61no—
92C3×#125—91no—
93C3×#125—94no—
94C3×#125—92no—
95C3×#125C3×D311yescharacteristic
96C2×C3×C4⋊C4—92no—
97C2×C3×C4⋊C4—94no—
98C2×C3×C42—91no—
99D3×C4⋊C4—34no—
100D3×C4⋊C4—32no—
101D3×C4⋊C4—94no—
102D3×C4⋊C4—92no—
103#890—32no—
104#890—92no—
105#889—34no—
106#889—94no—
107C42×D3—31no—
108C42×D3—91no—
109#870—31no—
110#870—91no—
111C22×C3⋊D3C23×C311yescharacteristic
112C22×C3×D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—24no—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C23×C312yes—
120C2×C32×C4C23×C314yes—
121C2×C32×C4C23×C311yescharacteristic
122C2×C3⋊Dic3C23×C312yes—
123C2×C3⋊Dic3C23×C314yes—
124C2×C3⋊Dic3C23×C311yescharacteristic
125C4×C3⋊D3—24no—
126C4×C3⋊D3—28no—
127C4×C3⋊D3C3×Q814yes—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—34no—
131C3×C4×D3—34no—
132C3×C4×D3—68no—
133C3×C4×D3—64no—
134C2×#125—91noSylow
135C2×H(3,𝔽3)C22×Q811yescharacteristic
136C2×H(3,𝔽3)C2×#4112yes—
137(C3⋊D3)⋊C3#12514yesdirect product factor
138C22×C3×C4—91no—
139C22×C3×C4—92no—
140C22×C3×C4—94no—
141C2×C4×D3—31no—
142C2×C4×D3—32no—
143C2×C4×D3—34no—
144C2×C4×D3—91no—
145C2×C4×D3—92no—
146C2×C4×D3—94no—
147C3×C4⋊C4—34no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—32no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4—62no—
152C3×C4⋊C4—64no—
153C3×C4⋊C4—94no—
154C3×C4⋊C4—94no—
155C3×C4⋊C4—94no—
156C3×C4⋊C4—92no—
157C3×C4⋊C4—94no—
158C3×C4⋊C4C2×C3×D314yes—
159C3×C4⋊C4C2×C3×D312yes—
160C3×C42—31no—
161C3×C42—31no—
162C3×C42—61no—
163C3×C42—91no—
164C3×C42—92no—
165C3×C42C2×C3×D311yescharacteristic
166C4⋊Dic3—32no—
167C4⋊Dic3—34no—
168C4⋊Dic3—92no—
169C4⋊Dic3—94no—
170Dic3⋊C4—34no—
171Dic3⋊C4—34no—
172Dic3⋊C4—34no—
173Dic3⋊C4—94no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176C4×Dic3—32no—
177C4×Dic3—31no—
178C4×Dic3—92no—
179C4×Dic3—91no—
180C22×C32—21no—
181C22×C32C23×D311yescharacteristic
182C22×C32C24×C311yescharacteristic
183C2×C3⋊D3C2×C3×Q812yes—
184C2×C3⋊D3C3×#4114yes—
185C2×C3×D3—32no—
186C2×C3×D3—34no—
187C32×C4—22no—
188C32×C4—24no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—44no—
194C32×C4D3×Q812yes—
195C32×C4C2×C3×Q812yes—
196C3⋊Dic3—22no—
197C3⋊Dic3—24no—
198C3⋊Dic3C2×C3×Q812yes—
199C3×Dic3—32no—
200C3×Dic3—62no—
201C3×Dic3—64no—
202#125—94no—
203#125—92no—
204#125—91no—
205#125(C3⋊D3)⋊C311yescharacteristic, direct product factor
206C2×C4⋊C4—92no—
207C2×C4⋊C4—94no—
208C2×C42—91no—
209H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C22×D3—31no—
212C22×D3—91no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—32no—
216C2×C3×C4—34no—
217C2×C3×C4—32no—
218C2×C3×C4—34no—
219C2×C3×C4—61no—
220C2×C3×C4—62no—
221C2×C3×C4—64no—
222C2×C3×C4—94no—
223C2×C3×C4—91no—
224C2×C3×C4—92no—
225C2×C3×C4—94no—
226C2×C3×C4—184no—
227C2×C3×C4—188no—
228C2×C3×C4C22×C3×D312yes—
229C2×C3×C4C22×C3×D314yes—
230C2×C3×C4C22×C3×D311yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—32no—
233C2×Dic3—34no—
234C2×Dic3—91no—
235C2×Dic3—92no—
236C2×Dic3—94no—
237C4×D3—34no—
238C4×D3—64no—
239C4×D3—68no—
240C4×D3—94no—
241C4×D3—184no—
242C4×D3—188no—
243C2×C32—21no—
244C2×C32—22no—
245C2×C32C2×D3×Q811yescharacteristic
246C2×C32D3×#4112yes—
247C2×C32C22×C3×Q811yescharacteristic
248C2×C32C2×C3×#4112yes—
249C3⋊D3C3×#12514yescomplemented
250C3×D3—34no—
251C22×C4—91no—
252C22×C4—92no—
253C22×C4—94no—
254C4⋊C4—94no—
255C4⋊C4—94no—
256C4⋊C4—94no—
257C4⋊C4—94no—
258C4⋊C4—92no—
259C4⋊C4C2×(C3⋊D3)⋊C314yes—
260C4⋊C4C2×(C3⋊D3)⋊C312yes—
261C42—92no—
262C42—91no—
263C42C2×(C3⋊D3)⋊C311yescharacteristic
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3—92no—
268C22×C3—94no—
269C22×C3C23×C3×D311yescharacteristic
270C2×D3—32no—
271C2×D3—34no—
272C2×D3—92no—
273C2×D3—94no—
274C3×C4—22no—
275C3×C4—24no—
276C3×C4—32no—
277C3×C4—32no—
278C3×C4—62no—
279C3×C4—64no—
280C3×C4—62no—
281C3×C4—64no—
282C3×C4—62no—
283C3×C4—92no—
284C3×C4—122no—
285C3×C4—124no—
286C3×C4—182no—
287C3×C4—184no—
288C3×C4C3×D3×Q812yes—
289Dic3—32no—
290Dic3—62no—
291Dic3—64no—
292Dic3—92no—
293Dic3—182no—
294Dic3—184no—
295C32—21no—
296C32D3×#12511yescharacteristic, complemented
297C32C2×C3×#12511yescharacteristic, complemented
298C23—91no—
299C2×C4—91no—
300C2×C4—94no—
301C2×C4—92no—
302C2×C4—94no—
303C2×C4—184no—
304C2×C4—188no—
305C2×C4C22×(C3⋊D3)⋊C312yes—
306C2×C4C22×(C3⋊D3)⋊C314yes—
307C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—61no—
313C2×C3—62no—
314C2×C3—94no—
315C2×C3C2×C3×D3×Q811yescharacteristic
316C2×C3C3×D3×#4112yes—
317D3—34no—
318D3—94no—
319C22—92no—
320C22—94no—
321C22C23×(C3⋊D3)⋊C311yescharacteristic
322C4—22no—
323C4—24no—
324C4—92no—
325C4—182no—
326C4—184no—
327C4Q8×(C3⋊D3)⋊C312yes—
328C3—31no—
329C3—31no—
330C3—61no—
331C3C3×D3×#12511yescharacteristic
332C2—94no—
333C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
334C2#41×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C2×C3×#125
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
4. #64099⋊C3C2semidirect
5. #64098⋊C3C2semidirect
7. #64079⋊C3C2semidirect
8. C3⋊D3×#125C3semidirect
24. C32×#125D3semidirect
25. C32×#125C2×C3semidirect
28. #64099C2×C3semidirect
29. #64098C2×C3semidirect
31. #64079C2×C3semidirect
137. (C3⋊D3)⋊C3#125direct (N × H)
205. #125(C3⋊D3)⋊C3direct (N × H)
209. H(3,𝔽3)C2×#125semidirect
249. C3⋊D3C3×#125semidirect
296. C32D3×#125semidirect
297. C32C2×C3×#125semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
C4⋊C4×(C3⋊D3)⋊C32
#64099⋊C32
#64098⋊C32
C42×(C3⋊D3)⋊C32
#64079⋊C32
C3⋊D3×#1253
C3×D3×#1253