Finite Group Databasebeta

← #10595989⋊C3 #13152512

Subgroups — #10595989⋊C3 #13152512

Order 1728.

748 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3#64098⋊C3C214yesmaximal, complemented
4#64088⋊C3C214yesmaximal
5#64079⋊C3C211yesmaximal, characteristic, complemented
6#64078⋊C3C211yesmaximal, characteristic, complemented
7C4×#634⋊C3C212yesmaximal, complemented
8#10595989C311yesmaximal, characteristic, complemented
9C3×#5518—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C4⋊C4×H(3,𝔽3)C2212yes—
13C22⋊C4×H(3,𝔽3)C2212yes—
14C42×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3C2214yes—
17C4⋊((C3⋊Dic3)⋊C3)C2212yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yes—
19#3750⋊C3C2214yes—
20#3750⋊C3C2214yes—
21#3750⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2214yes—
23C32×#117—21no—
24C32×#117D311yescharacteristic
25C32×#117C2×C311yescharacteristic
26#64133C2×C312yes—
27#64098C2×C314yescomplemented
28#64088C2×C314yes—
29#64079C2×C311yescharacteristic, complemented
30#64078C2×C311yescharacteristic, complemented
31C4×#634C2×C312yescomplemented
32C3×#924—32no—
33C3×#889—34no—
34C3×#879—34no—
35C3×#870—31no—
36C3×#869—31no—
37C3×C4×Dic6—32no—
38C23×H(3,𝔽3)C2311yescharacteristic
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2312yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46#634⋊C3—24no—
47C3×#541—91no—
48#5518—31no—
49#5518—91no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C2×#634C22×C312yes—
54C2×C3×Dic6—32no—
55C32×C4⋊C4—22no—
56C32×C4⋊C4C2×D312yes—
57C32×C4⋊C4C22×C312yes—
58C32×C22⋊C4—22no—
59C32×C22⋊C4C2×D312yes—
60C32×C22⋊C4C22×C312yes—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42C2×D311yescharacteristic
64C32×C42C2×D311yescharacteristic
65C32×C42C22×C311yescharacteristic
66C32×C42C22×C311yescharacteristic
67#3757C22×C314yes—
68C4⋊(C3⋊Dic3)C22×C312yes—
69C4⋊(C3⋊Dic3)C22×C314yes—
70#3750C22×C314yes—
71#3750C22×C314yes—
72#3750C22×C314yes—
73C4×C3⋊Dic3C22×C314yes—
74C3×#217—34no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3—32no—
77C3×Dic3⋊C4—34no—
78C3×Dic3⋊C4—34no—
79C3×Dic3⋊C4—34no—
80C3×C4×Dic3—34no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)C2411yescharacteristic
84C4×H(3,𝔽3)—21no—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)—22no—
87C4×H(3,𝔽3)#4112yes—
88(C3⋊Dic3)⋊C3—24no—
89(C3⋊Dic3)⋊C3—24no—
90C3×C4⋊Q8—91no—
91C3×#126—94no—
92C3×#125—91no—
93C3×#125—94no—
94C3×C22⋊Q8—92no—
95C3×C4×Q8—92no—
96C3×#117—31no—
97C3×#117—31no—
98C3×#117—61no—
99C3×#117C3×D311yescharacteristic
100#924—32no—
101#924—92no—
102#889—34no—
103#889—94no—
104#879—34no—
105#879—94no—
106#870—31no—
107#870—91no—
108#869—31no—
109#869—91no—
110C4×Dic6—32no—
111C4×Dic6—92no—
112C23×C32—21no—
113C23×C32C22×D311yescharacteristic
114C23×C32C23×C311yescharacteristic
115C2×C32×C4—24no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D312yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C312yes—
127C2×C32×C4C23×C311yescharacteristic
128C2×C32×C4C23×C312yes—
129C2×C32×C4C23×C311yescharacteristic
130C2×C3⋊Dic3C23×C314yes—
131C2×C3⋊Dic3C23×C314yes—
132#634—24no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C3×Dic6—64no—
136#541—91noSylow
137C2×H(3,𝔽3)—41no—
138C2×H(3,𝔽3)C2×#4111yescharacteristic
139C2×H(3,𝔽3)2−1+412yes—
140C2×C3×Q8—92no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—61no—
144C22×C3×C4C2×C3×D311yescharacteristic
145C2×Dic6—32no—
146C2×Dic6—92no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—62no—
150C3×C4⋊C4—94no—
151C3×C4⋊C4—94no—
152C3×C4⋊C4—94no—
153C3×C4⋊C4—92no—
154C3×C4⋊C4—94no—
155C3×C4⋊C4C2×C3×D312yes—
156C3×C22⋊C4—32no—
157C3×C22⋊C4—32no—
158C3×C22⋊C4—62no—
159C3×C22⋊C4—94no—
160C3×C22⋊C4C2×C3×D312yes—
161C3×C42—31no—
162C3×C42—31no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42—61no—
167C3×C42—94no—
168C3×C42C2×C3×D311yescharacteristic
169C3×C42C2×C3×D311yescharacteristic
170#217—34no—
171#217—94no—
172C4⋊Dic3—32no—
173C4⋊Dic3—34no—
174C4⋊Dic3—92no—
175C4⋊Dic3—94no—
176Dic3⋊C4—34no—
177Dic3⋊C4—34no—
178Dic3⋊C4—34no—
179Dic3⋊C4—94no—
180Dic3⋊C4—94no—
181Dic3⋊C4—94no—
182C4×Dic3—34no—
183C4×Dic3—94no—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—42no—
191C22×C32C23×D311yescharacteristic
192C22×C32C24×C311yescharacteristic
193C32×C4—22no—
194C32×C4—21no—
195C32×C4—22no—
196C32×C4—22no—
197C32×C4—21no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—42no—
202C32×C4—42no—
203C32×C4#23512yes—
204C32×C4C3×#4112yes—
205C3⋊Dic3—24no—
206C3⋊Dic3—24no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C4⋊Q8—91no—
210#126—94no—
211#125—94no—
212#125—91no—
213C22⋊Q8—92no—
214C4×Q8—92no—
215#117(C3⋊D3)⋊C311yescharacteristic
216H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—61no—
220C23×C3C22×C3×D311yescharacteristic
221C3×Q8—184no—
222C2×C3×C4—22no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—32no—
226C2×C3×C4—32no—
227C2×C3×C4—31no—
228C2×C3×C4—32no—
229C2×C3×C4—31no—
230C2×C3×C4—32no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—64no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—94no—
239C2×C3×C4—94no—
240C2×C3×C4C22×C3×D312yes—
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×C3×C4C22×C3×D312yes—
243C2×C3×C4C22×C3×D311yescharacteristic
244C2×Dic3—34no—
245C2×Dic3—34no—
246C2×Dic3—94no—
247C2×Dic3—94no—
248Dic6—64no—
249Dic6—184no—
250C2×C32—21no—
251C2×C32—22no—
252C2×C32—42no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32C2×#23511yescharacteristic
256C2×C32#101012yes—
257C2×C32C2×C3×#4111yescharacteristic
258C2×C32C3×2−1+412yes—
259C2×Q8—92no—
260C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
261C4⋊C4—92no—
262C4⋊C4—94no—
263C4⋊C4—94no—
264C4⋊C4—94no—
265C4⋊C4—94no—
266C4⋊C4C2×(C3⋊D3)⋊C312yes—
267C22⋊C4—94no—
268C22⋊C4C2×(C3⋊D3)⋊C312yes—
269C42—94no—
270C42C2×(C3⋊D3)⋊C311yescharacteristic
271C42C2×(C3⋊D3)⋊C311yescharacteristic
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—31no—
275C22×C3—31no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—62no—
281C22×C3—61no—
282C22×C3—122no—
283C22×C3C23×C3×D311yescharacteristic
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—32no—
288C3×C4—32no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—62no—
292C3×C4—62no—
293C3×C4—62no—
294C3×C4—62no—
295C3×C4—62no—
296C3×C4—62no—
297C3×C4—122no—
298C3×C4—122no—
299C3×C4—184no—
300C3×C4—184no—
301C3×C4C3×#23512yes—
302Dic3—64no—
303Dic3—64no—
304Dic3—184no—
305Dic3—184no—
306C32—21no—
307C32#551811yescharacteristic, complemented
308C32C3×#54111yescharacteristic, complemented
309C23C22×(C3⋊D3)⋊C311yescharacteristic
310Q8—184no—
311C2×C4—22no—
312C2×C4—94no—
313C2×C4—94no—
314C2×C4C22×(C3⋊D3)⋊C312yes—
315C2×C4C22×(C3⋊D3)⋊C312yes—
316C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
317C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—41no—
323C2×C3—61no—
324C2×C3—62no—
325C2×C3—121no—
326C2×C3—121no—
327C2×C3—122no—
328C2×C3C2×C3×#23511yescharacteristic
329C2×C3C3×#101012yes—
330C22—21no—
331C22—22no—
332C22C23×(C3⋊D3)⋊C311yescharacteristic
333C4—21no—
334C4—22no—
335C4—22no—
336C4—184no—
337C4—184no—
338C4#3828⋊C312yes—
339C3—31no—
340C3—31no—
341C3—61no—
342C3C3×#551811yescharacteristic
343C2—41no—
344C2C2×#3828⋊C311yescharacteristic
345C2#1172263912yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radicalC23×H(3,𝔽3)C23
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #64098⋊C3C2semidirect
5. #64079⋊C3C2semidirect
6. #64078⋊C3C2semidirect
7. C4×#634⋊C3C2semidirect
8. #10595989C3semidirect
27. #64098C2×C3semidirect
29. #64079C2×C3semidirect
30. #64078C2×C3semidirect
31. C4×#634C2×C3semidirect
216. H(3,𝔽3)#541semidirect
307. C32#5518semidirect
308. 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)×#1172
#64133⋊C32
#64098⋊C32
#64088⋊C32
#64079⋊C32
#64078⋊C32
C4×#634⋊C32
#105959893
C3×#55183