Finite Group Databasebeta

← #10596202⋊C3 #13153327

Subgroups — #10596202⋊C3 #13153327

Order 1728.

740 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64155⋊C3C212yesmaximal
3Q8×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
4#64107⋊C3C214yesmaximal
5#64098⋊C3C214yesmaximal, complemented
6#64081⋊C3C211yesmaximal, characteristic
7#64079⋊C3C211yesmaximal, characteristic, complemented
8#10596202C311yesmaximal, characteristic, complemented
9C3×#5731—31nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2214yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3C2212yes—
15#3752⋊C3C2214yes—
16C4⋊((C3⋊Dic3)⋊C3)C2214yes—
17C4⋊((C3⋊Dic3)⋊C3)C2212yes—
18#3750⋊C3C2214yes—
19#3750⋊C3C2212yes—
20#3750⋊C3C2214yes—
21C4×(C3⋊Dic3)⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
23C32×C4⋊Q8—21no—
24C32×C4⋊Q8D311yescharacteristic, complemented
25C32×C4⋊Q8C2×C311yescharacteristic, complemented
26#64155C2×C312yes—
27Q8×C3⋊Dic3C2×C312yescomplemented
28#64107C2×C314yes—
29#64098C2×C314yescomplemented
30#64081C2×C311yescharacteristic
31#64079C2×C311yescharacteristic, complemented
32C3×#946—32no—
33C3×Q8×Dic3—32no—
34C3×#898—34no—
35C3×#889—34no—
36C3×#872—31no—
37C3×#870—31no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39Q8×H(3,𝔽3)—24no—
40C2×C4×H(3,𝔽3)C2314yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C4×(C3⋊D3)⋊C3—22no—
47C3×#541—91no—
48#5731—31no—
49#5731—91no—
50C2×C32×Q8—22no—
51C2×C32×Q8C2×D312yes—
52C2×C32×Q8C22×C312yes—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×C3×C4×D3—31no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C22×C314yes—
58C32×C42—21no—
59C32×C42C2×D311yescharacteristic
60C32×C42C22×C311yescharacteristic
61#3752C22×C312yes—
62#3752C22×C314yes—
63C4⋊(C3⋊Dic3)C22×C312yes—
64C4⋊(C3⋊Dic3)C22×C314yes—
65#3750C22×C314yes—
66#3750C22×C312yes—
67#3750C22×C314yes—
68C4×C3⋊Dic3C22×C311yescharacteristic
69C4×C3⋊Dic3C22×C314yes—
70C3×(C2×D3)⋊C4—34no—
71C3×(C2×D3)⋊C4—32no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—32no—
74C3×Dic3⋊C4—34no—
75C3×Dic3⋊C4—32no—
76C3×Dic3⋊C4—34no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—31no—
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—21no—
86(C3⋊Dic3)⋊C3—22no—
87(C3⋊Dic3)⋊C3—24no—
88C3×C4⋊Q8—31no—
89C3×C4⋊Q8—31no—
90C3×C4⋊Q8—61no—
91C3×C4⋊Q8C3×D311yescharacteristic
92C3×#126—94no—
93C3×#125—91no—
94C3×#125—94no—
95C3×C22⋊Q8—92no—
96C3×C4×Q8—92no—
97C3×#117—91no—
98#946—32no—
99#946—92no—
100Q8×Dic3—32no—
101Q8×Dic3—92no—
102#898—34no—
103#898—94no—
104#889—34no—
105#889—94no—
106#872—31no—
107#872—91no—
108#870—31no—
109#870—91no—
110C22×C3⋊D3C23×C311yescharacteristic
111C22×C3×D3—31no—
112C32×Q8—24no—
113C32×Q8—24no—
114C32×Q8—44no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D312yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C23×C314yes—
122C2×C32×C4C23×C312yes—
123C2×C32×C4C23×C311yescharacteristic
124C2×C3⋊Dic3C23×C312yes—
125C2×C3⋊Dic3C23×C314yes—
126C2×C3⋊Dic3C23×C311yescharacteristic
127C4×C3⋊D3—22no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—31no—
131C3×C4×D3—62no—
132#541—91noSylow
133C2×H(3,𝔽3)C2×#4111yescharacteristic
134C2×H(3,𝔽3)2−1+412yes—
135(C3⋊D3)⋊C3—41no—
136C2×C3×Q8—32no—
137C2×C3×Q8—32no—
138C2×C3×Q8—62no—
139C2×C3×Q8C2×C3×D312yes—
140C22×C3×C4—91no—
141C2×C4×D3—31no—
142C2×C4×D3—91no—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—34no—
145C3×C4⋊C4—64no—
146C3×C4⋊C4—92no—
147C3×C4⋊C4—94no—
148C3×C4⋊C4—94no—
149C3×C4⋊C4—92no—
150C3×C4⋊C4—94no—
151C3×C4⋊C4C2×C3×D314yes—
152C3×C22⋊C4—92no—
153C3×C22⋊C4—94no—
154C3×C42—31no—
155C3×C42—31no—
156C3×C42—61no—
157C3×C42—91no—
158C3×C42—94no—
159C3×C42C2×C3×D311yescharacteristic
160(C2×D3)⋊C4—32no—
161(C2×D3)⋊C4—34no—
162(C2×D3)⋊C4—92no—
163(C2×D3)⋊C4—94no—
164C4⋊Dic3—32no—
165C4⋊Dic3—34no—
166C4⋊Dic3—92no—
167C4⋊Dic3—94no—
168Dic3⋊C4—34no—
169Dic3⋊C4—32no—
170Dic3⋊C4—34no—
171Dic3⋊C4—94no—
172Dic3⋊C4—92no—
173Dic3⋊C4—94no—
174C4×Dic3—31no—
175C4×Dic3—34no—
176C4×Dic3—91no—
177C4×Dic3—94no—
178C22×C32—21no—
179C22×C32C23×D311yescharacteristic
180C22×C32C24×C311yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—22no—
183C2×C3×D3—61no—
184C2×C3×D3—62no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—22no—
188C32×C4—24no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—44no—
192C32×C4#23712yes—
193C32×C4C3×#4112yes—
194C3⋊Dic3—21no—
195C3⋊Dic3—22no—
196C3⋊Dic3—24no—
197C3×Dic3—61no—
198C3×Dic3—62no—
199C3×Dic3—64no—
200C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
201#126—94no—
202#125—94no—
203#125—91no—
204C22⋊Q8—92no—
205C4×Q8—92no—
206#117—91no—
207H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—31no—
210C22×D3—91no—
211C3×Q8—24no—
212C3×Q8—64no—
213C3×Q8—64no—
214C3×Q8—124no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—32no—
218C2×C3×C4—34no—
219C2×C3×C4—34no—
220C2×C3×C4—32no—
221C2×C3×C4—62no—
222C2×C3×C4—61no—
223C2×C3×C4—64no—
224C2×C3×C4—92no—
225C2×C3×C4—94no—
226C2×C3×C4—91no—
227C2×C3×C4—182no—
228C2×C3×C4C22×C3×D314yes—
229C2×C3×C4C22×C3×D312yes—
230C2×C3×C4C22×C3×D311yescharacteristic
231C2×Dic3—32no—
232C2×Dic3—34no—
233C2×Dic3—31no—
234C2×Dic3—92no—
235C2×Dic3—94no—
236C2×Dic3—91no—
237C4×D3—62no—
238C4×D3—182no—
239C2×C32—21no—
240C2×C32—22no—
241C2×C32C2×#23711yescharacteristic
242C2×C32#100712yes—
243C2×C32C2×C3×#4111yescharacteristic
244C2×C32C3×2−1+412yes—
245C3⋊D3—41no—
246C3×D3—121no—
247C2×Q8C2×(C3⋊D3)⋊C312yes—
248C22×C4—91no—
249C4⋊C4—92no—
250C4⋊C4—92no—
251C4⋊C4—94no—
252C4⋊C4—94no—
253C4⋊C4—94no—
254C4⋊C4C2×(C3⋊D3)⋊C314yes—
255C22⋊C4—92no—
256C22⋊C4—94no—
257C42—91no—
258C42—94no—
259C42C2×(C3⋊D3)⋊C311yescharacteristic
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—61no—
263C22×C3—181no—
264C22×C3—182no—
265C22×C3C23×C3×D311yescharacteristic
266C2×D3—61no—
267C2×D3—62no—
268C2×D3—181no—
269C2×D3—182no—
270C3×C4—24no—
271C3×C4—22no—
272C3×C4—32no—
273C3×C4—32no—
274C3×C4—64no—
275C3×C4—64no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—64no—
279C3×C4—62no—
280C3×C4—124no—
281C3×C4—181no—
282C3×C4—182no—
283C3×C4—184no—
284C3×C4C3×#23712yes—
285Dic3—61no—
286Dic3—62no—
287Dic3—64no—
288Dic3—181no—
289Dic3—182no—
290Dic3—184no—
291C32—21no—
292C32#573111yescharacteristic, complemented
293C32C3×#54111yescharacteristic, complemented
294C23—91no—
295Q8—24no—
296C2×C4—92no—
297C2×C4—94no—
298C2×C4—91no—
299C2×C4—182no—
300C2×C4C22×(C3⋊D3)⋊C314yes—
301C2×C4C22×(C3⋊D3)⋊C312yes—
302C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—32no—
306C2×C3—32no—
307C2×C3—61no—
308C2×C3—62no—
309C2×C3—361no—
310C2×C3C2×C3×#23711yescharacteristic
311C2×C3C3×#100712yes—
312D3—121no—
313D3—361no—
314C22—181no—
315C22—182no—
316C22C23×(C3⋊D3)⋊C311yescharacteristic
317C4—24no—
318C4—22no—
319C4—181no—
320C4—182no—
321C4—184no—
322C4#3830⋊C312yes—
323C3—31no—
324C3—31no—
325C3—61no—
326C3C3×#573111yescharacteristic
327C2—361no—
328C2C2×#3830⋊C311yescharacteristic
329C2#1172262712yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. #64098⋊C3C2semidirect
7. #64079⋊C3C2semidirect
8. #10596202C3semidirect
24. C32×C4⋊Q8D3semidirect
25. C32×C4⋊Q8C2×C3semidirect
27. Q8×C3⋊Dic3C2×C3semidirect
29. #64098C2×C3semidirect
31. #64079C2×C3semidirect
200. C4⋊Q8(C3⋊D3)⋊C3semidirect
207. H(3,𝔽3)#541semidirect
292. C32#5731semidirect
293. 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)×C4⋊Q82
#64155⋊C32
Q8×(C3⋊Dic3)⋊C32
#64107⋊C32
#64098⋊C32
#64081⋊C32
#64079⋊C32
#105962023
C3×#57313