Finite Group Databasebeta

← #10596190⋊C3 #13153277

Subgroups — #10596190⋊C3 #13153277

Order 1728.

948 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
2#64148⋊C3C214yesmaximal, complemented
3#64147⋊C3C212yesmaximal, complemented
4#64144⋊C3C214yesmaximal, complemented
5D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
6#64081⋊C3C211yesmaximal, characteristic, complemented
7#64079⋊C3C211yesmaximal, characteristic, complemented
8#10596190C311yesmaximal, characteristic, complemented
9C3×#5719—31nomaximal
10C2×D4×H(3,𝔽3)C2212yes—
11C2×D4×H(3,𝔽3)C2214yes—
12C2×#638⋊C3C2214yes—
13C22×(C3⋊Dic3)⋊C3C2214yes—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3C2214yes—
17#3757⋊C3C2214yes—
18#3752⋊C3C2212yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20#3750⋊C3C2214yes—
21#3750⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
23C32×#127—21no—
24C32×#127D311yescharacteristic, complemented
25C32×#127C2×C311yescharacteristic, complemented
26#64148C2×C314yescomplemented
27#64147C2×C312yescomplemented
28#64144C2×C314yescomplemented
29D4×C3⋊Dic3C2×C312yescomplemented
30#64081C2×C311yescharacteristic, complemented
31#64079C2×C311yescharacteristic, complemented
32C3×#939—34no—
33C3×#938—32no—
34C3×#935—34no—
35C3×D4×Dic3—32no—
36C3×#872—31no—
37C3×#870—31no—
38C23×H(3,𝔽3)C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40D4×H(3,𝔽3)—24no—
41D4×H(3,𝔽3)—44no—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—44no—
45C2×(C3⋊Dic3)⋊C3—44no—
46C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2314yes—
49C4×(C3⋊D3)⋊C3—22no—
50C3×#540—91no—
51#5719—31no—
52#5719—91no—
53C2×C32×D4—22no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D312yes—
56C2×C32×D4C2×D314yes—
57C2×C32×D4C22×C312yes—
58C2×C32×D4C22×C314yes—
59C2×#638C22×C314yes—
60C22×C3⋊Dic3C22×C314yes—
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C2×C3×#67—34no—
63C22×C3×Dic3—34no—
64C2×C3×C4×D3—31no—
65C32×C42—21no—
66C32×C42C2×D311yescharacteristic
67C32×C42C22×C311yescharacteristic
68#3757C22×C314yes—
69#3757C22×C314yes—
70#3752C22×C312yes—
71C4⋊(C3⋊Dic3)C22×C312yes—
72#3750C22×C314yes—
73#3750C22×C312yes—
74C4×C3⋊Dic3C22×C311yescharacteristic
75C3×#217—34no—
76C3×#217—34no—
77C3×(C2×D3)⋊C4—32no—
78C3×C4⋊Dic3—32no—
79C3×Dic3⋊C4—32no—
80C3×Dic3⋊C4—34no—
81C3×C4×Dic3—31no—
82C22×H(3,𝔽3)—24no—
83C22×H(3,𝔽3)—24no—
84C22×H(3,𝔽3)—24no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×(C3⋊D3)⋊C3—21no—
87C2×(C3⋊D3)⋊C3—22no—
88C4×H(3,𝔽3)—22no—
89C4×H(3,𝔽3)#4112yes—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—22no—
92(C3⋊Dic3)⋊C3—24no—
93C3×#127—31no—
94C3×#127—31no—
95C3×#127—61no—
96C3×#127C3×D311yescharacteristic
97C3×#125—91no—
98C3×#123—94no—
99C3×#121—92no—
100C3×#121—94no—
101C3×C4×D4—92no—
102C3×#117—91no—
103#939—34no—
104#939—94no—
105#938—32no—
106#938—92no—
107#935—34no—
108#935—94no—
109D4×Dic3—32no—
110D4×Dic3—92no—
111#872—31no—
112#872—91no—
113#870—31no—
114#870—91no—
115C23×C32—24no—
116C23×C32C22×D314yes—
117C23×C32C23×C314yes—
118C22×C3⋊D3C23×C311yescharacteristic
119C22×C3×D3—31no—
120C32×D4—24no—
121C32×D4—24no—
122C32×D4—44no—
123C32×D4—44no—
124C32×D4—44no—
125C32×D4—48no—
126C2×C32×C4—22no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D312yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C312yes—
131C2×C32×C4C23×C311yescharacteristic
132#638—44no—
133C2×C3⋊Dic3—44no—
134C2×C3⋊Dic3C23×C312yes—
135C2×C3⋊Dic3C23×C314yes—
136C2×C3⋊Dic3C23×C311yescharacteristic
137C4×C3⋊D3—22no—
138C3×#67—124no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—124no—
143C3×C4×D3—62no—
144#540—91noSylow
145C2×H(3,𝔽3)—44no—
146C2×H(3,𝔽3)C2×#4111yescharacteristic
147C2×H(3,𝔽3)H(5,𝔽2)12yes—
148(C3⋊D3)⋊C3—41no—
149C2×C3×D4—32no—
150C2×C3×D4—32no—
151C2×C3×D4—34no—
152C2×C3×D4—34no—
153C2×C3×D4—62no—
154C2×C3×D4—64no—
155C2×C3×D4—94no—
156C2×C3×D4C2×C3×D312yes—
157C2×C3×D4C2×C3×D314yes—
158C22×C3×C4—91no—
159C22×C3×C4—94no—
160C2×#67—34no—
161C2×#67—94no—
162C22×Dic3—34no—
163C22×Dic3—94no—
164C2×C4×D3—31no—
165C2×C4×D3—91no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—94no—
168C3×C4⋊C4—92no—
169C3×C22⋊C4—92no—
170C3×C22⋊C4—94no—
171C3×C22⋊C4—94no—
172C3×C42—31no—
173C3×C42—31no—
174C3×C42—61no—
175C3×C42—91no—
176C3×C42C2×C3×D311yescharacteristic
177#217—34no—
178#217—34no—
179#217—94no—
180#217—94no—
181(C2×D3)⋊C4—32no—
182(C2×D3)⋊C4—92no—
183C4⋊Dic3—32no—
184C4⋊Dic3—92no—
185Dic3⋊C4—34no—
186Dic3⋊C4—32no—
187Dic3⋊C4—94no—
188Dic3⋊C4—92no—
189C4×Dic3—31no—
190C4×Dic3—91no—
191C22×C32—24no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32—21no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32C23×D311yescharacteristic
202C22×C32C24×C311yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—22no—
205C2×C3×D3—61no—
206C2×C3×D3—62no—
207C32×C4—24no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4#23712yes—
212C32×C4C3×#4112yes—
213C3⋊Dic3—21no—
214C3⋊Dic3—22no—
215C3⋊Dic3—24no—
216C3×Dic3—61no—
217C3×Dic3—62no—
218C3×Dic3—64no—
219#127(C3⋊D3)⋊C311yescharacteristic, complemented
220#125—91no—
221#123—94no—
222#121—92no—
223#121—94no—
224C4×D4—92no—
225#117—91no—
226H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
227C23×C3—34no—
228C23×C3—34no—
229C23×C3—64no—
230C23×C3—91no—
231C23×C3C22×C3×D314yes—
232C22×D3—31no—
233C22×D3—91no—
234C3×D4—24no—
235C3×D4—44no—
236C3×D4—64no—
237C3×D4—64no—
238C3×D4—124no—
239C3×D4—124no—
240C3×D4—124no—
241C3×D4—128no—
242C3×D4—364no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—62no—
248C2×C3×C4—61no—
249C2×C3×C4—92no—
250C2×C3×C4—91no—
251C2×C3×C4—94no—
252C2×C3×C4—182no—
253C2×C3×C4—364no—
254C2×C3×C4C22×C3×D312yes—
255C2×C3×C4C22×C3×D311yescharacteristic
256#67—124no—
257#67—364no—
258C2×Dic3—32no—
259C2×Dic3—34no—
260C2×Dic3—31no—
261C2×Dic3—92no—
262C2×Dic3—94no—
263C2×Dic3—91no—
264C2×Dic3—124no—
265C2×Dic3—364no—
266C4×D3—62no—
267C4×D3—182no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32—44no—
271C2×C32—44no—
272C2×C32—84no—
273C2×C32C2×#23711yescharacteristic
274C2×C32#100412yes—
275C2×C32C2×C3×#4111yescharacteristic
276C2×C32C3×H(5,𝔽2)12yes—
277C3⋊D3—41no—
278C3×D3—121no—
279C2×D4—94no—
280C2×D4C2×(C3⋊D3)⋊C312yes—
281C2×D4C2×(C3⋊D3)⋊C314yes—
282C22×C4—94no—
283C22×C4—91no—
284C4⋊C4—92no—
285C4⋊C4—92no—
286C4⋊C4—94no—
287C22⋊C4—94no—
288C22⋊C4—94no—
289C22⋊C4—92no—
290C42—91no—
291C42C2×(C3⋊D3)⋊C311yescharacteristic
292C22×C3—24no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—64no—
298C22×C3—64no—
299C22×C3—64no—
300C22×C3—64no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—61no—
304C22×C3—124no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3—181no—
308C22×C3—182no—
309C22×C3C23×C3×D311yescharacteristic
310C2×D3—61no—
311C2×D3—62no—
312C2×D3—181no—
313C2×D3—182no—
314C3×C4—22no—
315C3×C4—32no—
316C3×C4—32no—
317C3×C4—62no—
318C3×C4—62no—
319C3×C4—64no—
320C3×C4—62no—
321C3×C4—181no—
322C3×C4—182no—
323C3×C4—184no—
324C3×C4C3×#23712yes—
325Dic3—61no—
326Dic3—62no—
327Dic3—64no—
328Dic3—181no—
329Dic3—182no—
330Dic3—184no—
331C32—21no—
332C32#571911yescharacteristic, complemented
333C32C3×#54011yescharacteristic, complemented
334C23—91no—
335C23C22×(C3⋊D3)⋊C314yes—
336D4—24no—
337D4—44no—
338D4—364no—
339C2×C4—92no—
340C2×C4—91no—
341C2×C4—94no—
342C2×C4—182no—
343C2×C4—364no—
344C2×C4C22×(C3⋊D3)⋊C312yes—
345C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—32no—
349C2×C3—32no—
350C2×C3—44no—
351C2×C3—61no—
352C2×C3—62no—
353C2×C3—124no—
354C2×C3—124no—
355C2×C3—244no—
356C2×C3—361no—
357C2×C3C2×C3×#23711yescharacteristic
358C2×C3C3×#100412yes—
359D3—121no—
360D3—361no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—181no—
365C22—182no—
366C22C23×(C3⋊D3)⋊C311yescharacteristic
367C4—22no—
368C4—181no—
369C4—182no—
370C4—184no—
371C4#3830⋊C312yes—
372C3—31no—
373C3—31no—
374C3—61no—
375C3C3×#571911yescharacteristic
376C2—44no—
377C2—361no—
378C2C2×#3830⋊C311yescharacteristic
379C2#64361⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#127C2semidirect
2. #64148⋊C3C2semidirect
3. #64147⋊C3C2semidirect
4. #64144⋊C3C2semidirect
5. D4×(C3⋊Dic3)⋊C3C2semidirect
6. #64081⋊C3C2semidirect
7. #64079⋊C3C2semidirect
8. #10596190C3semidirect
24. C32×#127D3semidirect
25. C32×#127C2×C3semidirect
26. #64148C2×C3semidirect
27. #64147C2×C3semidirect
28. #64144C2×C3semidirect
29. D4×C3⋊Dic3C2×C3semidirect
30. #64081C2×C3semidirect
31. #64079C2×C3semidirect
219. #127(C3⋊D3)⋊C3semidirect
226. H(3,𝔽3)#540semidirect
332. C32#5719semidirect
333. C32C3×#540semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
#64148⋊C32
#64147⋊C32
#64144⋊C32
D4×(C3⋊Dic3)⋊C32
#64081⋊C32
#64079⋊C32
#105961903
C3×#57193