Finite Group Databasebeta

← #10596006⋊C3 #13152526

Subgroups — #10596006⋊C3 #13152526

Order 1728.

916 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2#64139⋊C3C212yesmaximal, complemented
3#64103⋊C3C214yesmaximal, complemented
4#64093⋊C3C214yesmaximal, complemented
5#11721385C211yesmaximal, characteristic, complemented
6C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
7#64079⋊C3C211yesmaximal, characteristic, complemented
8#10596006C311yesmaximal, characteristic, complemented
9C3×#5535—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C2214yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
14C2×C4×(C3⋊D3)⋊C3C2214yes—
15C4⋊C4×H(3,𝔽3)C2212yes—
16C22⋊C4×H(3,𝔽3)C2212yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18C42×H(3,𝔽3)C2211yescharacteristic
19#3752⋊C3C2214yes—
20#3752⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22#3750⋊C3C2214yes—
23C32×#117—21no—
24C32×#117D311yescharacteristic, complemented
25C32×#117C2×C311yescharacteristic, complemented
26#64139C2×C312yescomplemented
27#64103C2×C314yescomplemented
28#64093C2×C314yescomplemented
29#64083C2×C311yescharacteristic, complemented
30C4×C4⋊(C3⋊D3)C2×C312yescomplemented
31#64079C2×C311yescharacteristic, complemented
32C3×#930—32no—
33C3×#894—34no—
34C3×#884—34no—
35C3×#874—31no—
36C3×C4×D12—32no—
37C3×#870—31no—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3C2314yes—
40C2×C4×H(3,𝔽3)—22no—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—44no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C4⋊((C3⋊D3)⋊C3)—24no—
48C4⋊((C3⋊D3)⋊C3)—44no—
49C4×(C3⋊D3)⋊C3—44no—
50C3×#540—91no—
51#5535—31no—
52#5535—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638C22×C314yes—
57C2×C4⋊(C3⋊D3)C22×C312yes—
58C2×C4⋊(C3⋊D3)C22×C314yes—
59C2×C4×C3⋊D3C22×C314yes—
60C2×C3×#67—34no—
61C2×C3×D12—32no—
62C2×C3×D12—34no—
63C2×C3×C4×D3—34no—
64C32×C4⋊C4—22no—
65C32×C4⋊C4C2×D312yes—
66C32×C4⋊C4C22×C312yes—
67C32×C22⋊C4—22no—
68C32×C22⋊C4C2×D312yes—
69C32×C22⋊C4C22×C312yes—
70C32×C42—21no—
71C32×C42—21no—
72C32×C42C2×D311yescharacteristic
73C32×C42C2×D311yescharacteristic
74C32×C42C22×C311yescharacteristic
75C32×C42C22×C311yescharacteristic
76#3752C22×C314yes—
77#3752C22×C314yes—
78C4⋊(C3⋊Dic3)C22×C312yes—
79#3750C22×C314yes—
80C3×(C2×D3)⋊C4—34no—
81C3×(C2×D3)⋊C4—34no—
82C3×C4⋊Dic3—32no—
83C3×Dic3⋊C4—34no—
84C22×H(3,𝔽3)—21no—
85C22×H(3,𝔽3)—22no—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×(C3⋊D3)⋊C3—24no—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C4×H(3,𝔽3)—21no—
91C4×H(3,𝔽3)—22no—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)#4112yes—
94(C3⋊Dic3)⋊C3—24no—
95C3×#127—91no—
96C3×#125—91no—
97C3×#123—94no—
98C3×#121—94no—
99C3×#121—92no—
100C3×C4×D4—92no—
101C3×#117—31no—
102C3×#117—31no—
103C3×#117—61no—
104C3×#117C3×D311yescharacteristic
105#930—32no—
106#930—92no—
107#894—34no—
108#894—94no—
109#884—34no—
110#884—94no—
111#874—31no—
112#874—91no—
113C4×D12—32no—
114C4×D12—92no—
115#870—31no—
116#870—91no—
117C23×C32—21no—
118C23×C32C22×D311yescharacteristic
119C23×C32C23×C311yescharacteristic
120C22×C3⋊D3C23×C314yes—
121C22×C3×D3—34no—
122C2×C32×C4—24no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4C22×D312yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D312yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C23×C312yes—
134C2×C32×C4C23×C311yescharacteristic
135C2×C32×C4C23×C312yes—
136C2×C32×C4C23×C311yescharacteristic
137#638—44no—
138C2×C3⋊Dic3C23×C314yes—
139C4⋊(C3⋊D3)—24no—
140C4⋊(C3⋊D3)—44no—
141C4×C3⋊D3—44no—
142C3×#67—124no—
143C2×C3×Dic3—34no—
144C3×D12—64no—
145C3×D12—124no—
146C3×C4×D3—124no—
147#540—91noSylow
148C2×H(3,𝔽3)—41no—
149C2×H(3,𝔽3)C2×#4111yescharacteristic
150C2×H(3,𝔽3)H(5,𝔽2)12yes—
151(C3⋊D3)⋊C3—44no—
152C2×C3×D4—92no—
153C2×C3×D4—94no—
154C2×C3×D4—94no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—61no—
158C22×C3×C4—94no—
159C22×C3×C4C2×C3×D311yescharacteristic
160C2×#67—34no—
161C2×#67—94no—
162C2×D12—32no—
163C2×D12—34no—
164C2×D12—92no—
165C2×D12—94no—
166C2×C4×D3—34no—
167C2×C4×D3—94no—
168C3×C4⋊C4—32no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—62no—
171C3×C4⋊C4—94no—
172C3×C4⋊C4—92no—
173C3×C4⋊C4C2×C3×D312yes—
174C3×C22⋊C4—32no—
175C3×C22⋊C4—32no—
176C3×C22⋊C4—62no—
177C3×C22⋊C4—94no—
178C3×C22⋊C4—94no—
179C3×C22⋊C4C2×C3×D312yes—
180C3×C42—31no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—31no—
184C3×C42—61no—
185C3×C42—61no—
186C3×C42C2×C3×D311yescharacteristic
187C3×C42C2×C3×D311yescharacteristic
188(C2×D3)⋊C4—34no—
189(C2×D3)⋊C4—34no—
190(C2×D3)⋊C4—94no—
191(C2×D3)⋊C4—94no—
192C4⋊Dic3—32no—
193C4⋊Dic3—92no—
194Dic3⋊C4—34no—
195Dic3⋊C4—94no—
196C22×C32—22no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—42no—
203C22×C32C23×D311yescharacteristic
204C22×C32C24×C311yescharacteristic
205C2×C3⋊D3—24no—
206C2×C3⋊D3—24no—
207C2×C3⋊D3—24no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C32×C4—22no—
212C32×C4—21no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—21no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4#23512yes—
222C32×C4C3×#4112yes—
223C3⋊Dic3—24no—
224C3×Dic3—64no—
225#127—91no—
226#125—91no—
227#123—94no—
228#121—92no—
229#121—94no—
230C4×D4—92no—
231#117(C3⋊D3)⋊C311yescharacteristic, complemented
232H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—61no—
236C23×C3—94no—
237C23×C3C22×C3×D311yescharacteristic
238C22×D3—34no—
239C22×D3—94no—
240C3×D4—184no—
241C3×D4—364no—
242C3×D4—364no—
243C2×C3×C4—22no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—31no—
249C2×C3×C4—32no—
250C2×C3×C4—31no—
251C2×C3×C4—32no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—64no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—94no—
260C2×C3×C4—364no—
261C2×C3×C4C22×C3×D312yes—
262C2×C3×C4C22×C3×D311yescharacteristic
263C2×C3×C4C22×C3×D312yes—
264C2×C3×C4C22×C3×D311yescharacteristic
265#67—124no—
266#67—364no—
267C2×Dic3—34no—
268C2×Dic3—94no—
269D12—64no—
270D12—124no—
271D12—184no—
272D12—364no—
273C4×D3—124no—
274C4×D3—364no—
275C2×C32—21no—
276C2×C32—22no—
277C2×C32—42no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32C2×#23511yescharacteristic
281C2×C32#100912yes—
282C2×C32C2×C3×#4111yescharacteristic
283C2×C32C3×H(5,𝔽2)12yes—
284C3⋊D3—44no—
285C3×D3—124no—
286C2×D4—94no—
287C2×D4—92no—
288C2×D4—94no—
289C22×C4—94no—
290C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
291C4⋊C4—92no—
292C4⋊C4—94no—
293C4⋊C4C2×(C3⋊D3)⋊C312yes—
294C22⋊C4—94no—
295C22⋊C4—94no—
296C22⋊C4C2×(C3⋊D3)⋊C312yes—
297C42C2×(C3⋊D3)⋊C311yescharacteristic
298C42C2×(C3⋊D3)⋊C311yescharacteristic
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—62no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—122no—
310C22×C3—184no—
311C22×C3—184no—
312C22×C3—184no—
313C22×C3C23×C3×D311yescharacteristic
314C2×D3—64no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—184no—
318C2×D3—184no—
319C2×D3—184no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—32no—
324C3×C4—32no—
325C3×C4—62no—
326C3×C4—62no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—62no—
330C3×C4—62no—
331C3×C4—62no—
332C3×C4—62no—
333C3×C4—122no—
334C3×C4—122no—
335C3×C4—184no—
336C3×C4C3×#23512yes—
337Dic3—64no—
338Dic3—184no—
339C32—21no—
340C32#553511yescharacteristic, complemented
341C32C3×#54011yescharacteristic, complemented
342C23—94no—
343C23C22×(C3⋊D3)⋊C311yescharacteristic
344D4—184no—
345D4—364no—
346D4—364no—
347C2×C4—22no—
348C2×C4—94no—
349C2×C4—364no—
350C2×C4C22×(C3⋊D3)⋊C312yes—
351C2×C4C22×(C3⋊D3)⋊C312yes—
352C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
353C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—32no—
357C2×C3—32no—
358C2×C3—41no—
359C2×C3—61no—
360C2×C3—62no—
361C2×C3—121no—
362C2×C3—121no—
363C2×C3—122no—
364C2×C3—364no—
365C2×C3C2×C3×#23511yescharacteristic
366C2×C3C3×#100912yes—
367D3—124no—
368D3—364no—
369C22—21no—
370C22—22no—
371C22—184no—
372C22—184no—
373C22—184no—
374C22C23×(C3⋊D3)⋊C311yescharacteristic
375C4—21no—
376C4—22no—
377C4—22no—
378C4—184no—
379C4#3828⋊C312yes—
380C3—31no—
381C3—31no—
382C3—61no—
383C3C3×#553511yescharacteristic
384C2—41no—
385C2—364no—
386C2C2×#3828⋊C311yescharacteristic
387C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#117C2semidirect
2. #64139⋊C3C2semidirect
3. #64103⋊C3C2semidirect
4. #64093⋊C3C2semidirect
5. #11721385C2semidirect
6. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
7. #64079⋊C3C2semidirect
8. #10596006C3semidirect
24. C32×#117D3semidirect
25. C32×#117C2×C3semidirect
26. #64139C2×C3semidirect
27. #64103C2×C3semidirect
28. #64093C2×C3semidirect
29. #64083C2×C3semidirect
30. C4×C4⋊(C3⋊D3)C2×C3semidirect
31. #64079C2×C3semidirect
231. #117(C3⋊D3)⋊C3semidirect
232. H(3,𝔽3)#540semidirect
340. C32#5535semidirect
341. 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)×#1172
#64139⋊C32
#64103⋊C32
#64093⋊C32
#117213852
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
#105960063
C3×#55353