Finite Group Databasebeta

← #10596169⋊C3 #13153188

Subgroups — #10596169⋊C3 #13153188

Order 1728.

728 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64107⋊C3C214yesmaximal
3#64106⋊C3C212yesmaximal
4#64099⋊C3C214yesmaximal, complemented
5#64096⋊C3C212yesmaximal, complemented
6#64081⋊C3C211yesmaximal, characteristic
7#64078⋊C3C211yesmaximal, characteristic, complemented
8#10596169C311yesmaximal, characteristic, complemented
9C3×#5698—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C4⋊C4×H(3,𝔽3)C2214yes—
13C4⋊C4×H(3,𝔽3)C2212yes—
14C42×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3C2212yes—
16#3752⋊C3C2214yes—
17C4⋊((C3⋊Dic3)⋊C3)C2214yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yes—
19#3750⋊C3C2214yes—
20#3750⋊C3C2212yes—
21C4×(C3⋊Dic3)⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
23C32×#125—21no—
24C32×#125D311yescharacteristic, complemented
25C32×#125C2×C311yescharacteristic, complemented
26#64107C2×C314yes—
27#64106C2×C312yes—
28#64099C2×C314yescomplemented
29#64096C2×C312yescomplemented
30#64081C2×C311yescharacteristic
31#64078C2×C311yescharacteristic, complemented
32C3×#898—34no—
33C3×#897—32no—
34C3×#890—34no—
35C3×#887—32no—
36C3×#872—31no—
37C3×#869—31no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2314yes—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2314yes—
43C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C4×(C3⋊D3)⋊C3—22no—
46#634⋊C3—24no—
47C3×#541—91no—
48#5698—31no—
49#5698—91no—
50C2×C4×C3⋊D3C22×C311yescharacteristic
51C2×#634C22×C312yes—
52C2×C3×C4×D3—31no—
53C2×C3×Dic6—32no—
54C32×C4⋊C4—22no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4C2×D312yes—
57C32×C4⋊C4C2×D314yes—
58C32×C4⋊C4C22×C312yes—
59C32×C4⋊C4C22×C314yes—
60C32×C42—21no—
61C32×C42C2×D311yescharacteristic
62C32×C42C22×C311yescharacteristic
63#3752C22×C312yes—
64#3752C22×C314yes—
65C4⋊(C3⋊Dic3)C22×C314yes—
66C4⋊(C3⋊Dic3)C22×C314yes—
67#3750C22×C312yes—
68#3750C22×C314yes—
69C4×C3⋊Dic3C22×C311yescharacteristic
70C4×C3⋊Dic3C22×C314yes—
71C3×(C2×D3)⋊C4—34no—
72C3×(C2×D3)⋊C4—32no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—34no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—32no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—34no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—22no—
82C4×H(3,𝔽3)—24no—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)#4112yes—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—22no—
88C3×C4⋊Q8—91no—
89C3×#126—94no—
90C3×#125—31no—
91C3×#125—31no—
92C3×#125—61no—
93C3×#125—94no—
94C3×#125C3×D311yescharacteristic
95C3×C22⋊Q8—92no—
96C3×C4×Q8—92no—
97C3×#117—91no—
98#898—34no—
99#898—94no—
100#897—32no—
101#897—92no—
102#890—34no—
103#890—94no—
104#887—32no—
105#887—92no—
106#872—31no—
107#872—91no—
108#869—31no—
109#869—91no—
110C22×C3⋊D3C23×C311yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—24no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C23×C314yes—
119C2×C32×C4C23×C312yes—
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3C23×C312yes—
122C2×C3⋊Dic3C23×C314yes—
123C2×C3⋊Dic3C23×C311yescharacteristic
124C4×C3⋊D3—22no—
125#634—24no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—31no—
129C3×C4×D3—62no—
130C3×Dic6—64no—
131#541—91noSylow
132C2×H(3,𝔽3)C2×#4111yescharacteristic
133C2×H(3,𝔽3)2−1+412yes—
134(C3⋊D3)⋊C3—41no—
135C2×C3×Q8—92no—
136C22×C3×C4—91no—
137C2×C4×D3—31no—
138C2×C4×D3—91no—
139C2×Dic6—32no—
140C2×Dic6—92no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—34no—
143C3×C4⋊C4—32no—
144C3×C4⋊C4—34no—
145C3×C4⋊C4—62no—
146C3×C4⋊C4—64no—
147C3×C4⋊C4—92no—
148C3×C4⋊C4—94no—
149C3×C4⋊C4—94no—
150C3×C4⋊C4—94no—
151C3×C4⋊C4C2×C3×D312yes—
152C3×C4⋊C4C2×C3×D314yes—
153C3×C22⋊C4—92no—
154C3×C22⋊C4—94no—
155C3×C42—31no—
156C3×C42—31no—
157C3×C42—61no—
158C3×C42—91no—
159C3×C42—94no—
160C3×C42C2×C3×D311yescharacteristic
161(C2×D3)⋊C4—32no—
162(C2×D3)⋊C4—34no—
163(C2×D3)⋊C4—92no—
164(C2×D3)⋊C4—94no—
165C4⋊Dic3—34no—
166C4⋊Dic3—34no—
167C4⋊Dic3—94no—
168C4⋊Dic3—94no—
169Dic3⋊C4—32no—
170Dic3⋊C4—34no—
171Dic3⋊C4—92no—
172Dic3⋊C4—94no—
173C4×Dic3—31no—
174C4×Dic3—34no—
175C4×Dic3—91no—
176C4×Dic3—94no—
177C22×C32—21no—
178C22×C32C23×D311yescharacteristic
179C22×C32C24×C311yescharacteristic
180C2×C3⋊D3—21no—
181C2×C3⋊D3—22no—
182C2×C3×D3—61no—
183C2×C3×D3—62no—
184C32×C4—24no—
185C32×C4—22no—
186C32×C4—24no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—42no—
190C32×C4—44no—
191C32×C4#23912yes—
192C32×C4C3×#4112yes—
193C3⋊Dic3—24no—
194C3⋊Dic3—21no—
195C3⋊Dic3—22no—
196C3×Dic3—64no—
197C3×Dic3—61no—
198C3×Dic3—62no—
199C4⋊Q8—91no—
200#126—94no—
201#125—94no—
202#125(C3⋊D3)⋊C311yescharacteristic, complemented
203C22⋊Q8—92no—
204C4×Q8—92no—
205#117—91no—
206H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
207C23×C3—91no—
208C22×D3—31no—
209C22×D3—91no—
210C3×Q8—184no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—32no—
217C2×C3×C4—61no—
218C2×C3×C4—62no—
219C2×C3×C4—64no—
220C2×C3×C4—92no—
221C2×C3×C4—94no—
222C2×C3×C4—91no—
223C2×C3×C4—182no—
224C2×C3×C4C22×C3×D314yes—
225C2×C3×C4C22×C3×D312yes—
226C2×C3×C4C22×C3×D311yescharacteristic
227C2×Dic3—32no—
228C2×Dic3—34no—
229C2×Dic3—31no—
230C2×Dic3—92no—
231C2×Dic3—94no—
232C2×Dic3—91no—
233C4×D3—62no—
234C4×D3—182no—
235Dic6—64no—
236Dic6—184no—
237C2×C32—21no—
238C2×C32—22no—
239C2×C32C2×#23911yescharacteristic
240C2×C32#101012yes—
241C2×C32C2×C3×#4111yescharacteristic
242C2×C32C3×2−1+412yes—
243C3⋊D3—41no—
244C3×D3—121no—
245C2×Q8—92no—
246C22×C4—91no—
247C4⋊C4—92no—
248C4⋊C4—94no—
249C4⋊C4—94no—
250C4⋊C4—94no—
251C4⋊C4C2×(C3⋊D3)⋊C312yes—
252C4⋊C4C2×(C3⋊D3)⋊C314yes—
253C22⋊C4—92no—
254C22⋊C4—94no—
255C42—91no—
256C42—94no—
257C42C2×(C3⋊D3)⋊C311yescharacteristic
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—61no—
261C22×C3—181no—
262C22×C3—182no—
263C22×C3C23×C3×D311yescharacteristic
264C2×D3—61no—
265C2×D3—62no—
266C2×D3—181no—
267C2×D3—182no—
268C3×C4—24no—
269C3×C4—22no—
270C3×C4—32no—
271C3×C4—32no—
272C3×C4—64no—
273C3×C4—62no—
274C3×C4—62no—
275C3×C4—64no—
276C3×C4—62no—
277C3×C4—122no—
278C3×C4—124no—
279C3×C4—181no—
280C3×C4—184no—
281C3×C4—182no—
282C3×C4C3×#23912yes—
283Dic3—64no—
284Dic3—61no—
285Dic3—62no—
286Dic3—184no—
287Dic3—181no—
288Dic3—182no—
289C32—21no—
290C32#569811yescharacteristic, complemented
291C32C3×#54111yescharacteristic, complemented
292C23—91no—
293Q8—184no—
294C2×C4—92no—
295C2×C4—94no—
296C2×C4—91no—
297C2×C4—182no—
298C2×C4C22×(C3⋊D3)⋊C314yes—
299C2×C4C22×(C3⋊D3)⋊C312yes—
300C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—32no—
304C2×C3—32no—
305C2×C3—61no—
306C2×C3—62no—
307C2×C3—361no—
308C2×C3C2×C3×#23911yescharacteristic
309C2×C3C3×#101012yes—
310D3—121no—
311D3—361no—
312C22—181no—
313C22—182no—
314C22C23×(C3⋊D3)⋊C311yescharacteristic
315C4—24no—
316C4—22no—
317C4—181no—
318C4—184no—
319C4—182no—
320C4#8872212yes—
321C3—31no—
322C3—31no—
323C3—61no—
324C3C3×#569811yescharacteristic
325C2—361no—
326C2C2×#8872211yescharacteristic
327C2#1172263912yes—
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
hypocenterC32C3×#541
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. #64096⋊C3C2semidirect
7. #64078⋊C3C2semidirect
8. #10596169C3semidirect
24. C32×#125D3semidirect
25. C32×#125C2×C3semidirect
28. #64099C2×C3semidirect
29. #64096C2×C3semidirect
31. #64078C2×C3semidirect
202. #125(C3⋊D3)⋊C3semidirect
206. H(3,𝔽3)#541semidirect
290. C32#5698semidirect
291. C32C3×#541semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64107⋊C32
#64106⋊C32
#64099⋊C32
#64096⋊C32
#64081⋊C32
#64078⋊C32
#105961693
C3×#56983