Finite Group Databasebeta

← #10596154⋊C3 #13153124

Subgroups — #10596154⋊C3 #13153124

Order 1728.

986 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2#64156⋊C3C212yesmaximal, complemented
3#64146⋊C3C212yesmaximal, complemented
4#64094⋊C3C214yesmaximal, complemented
5#64091⋊C3C214yesmaximal, complemented
6#64084⋊C3C212yesmaximal, complemented
7#10596154C311yesmaximal, characteristic, complemented
8C3×#5683—31nomaximal
9C23×(C3⋊D3)⋊C3C2212yescomplemented
10C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3C2214yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15C2×#634⋊C3C2212yescomplemented
16C22⋊C4×H(3,𝔽3)C2214yescomplemented
17C42×H(3,𝔽3)C2211yescharacteristic, complemented
18#3757⋊C3C2212yescomplemented
19#3752⋊C3C2214yescomplemented
20#3752⋊C3C2214yescomplemented
21#3752⋊C3C2214yescomplemented
22C4×(C3⋊Dic3)⋊C3C2212yescomplemented
23C32×#124—21no—
24C32×#124D311yescharacteristic, complemented
25C32×#124C2×C311yescharacteristic, complemented
26#64156C2×C312yescomplemented
27#64146C2×C312yescomplemented
28#64094C2×C314yescomplemented
29#64091C2×C314yescomplemented
30#64084C2×C312yescomplemented
31C3×#947—32no—
32C3×#937—32no—
33C3×#885—34no—
34C3×#882—34no—
35C3×#875—32no—
36C23×H(3,𝔽3)C2312yes—
37C22×(C3⋊D3)⋊C3—42no—
38C22×(C3⋊D3)⋊C3—44no—
39C22×(C3⋊D3)⋊C3C2314yes—
40Q8×H(3,𝔽3)—41no—
41D4×H(3,𝔽3)—41no—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×C4×H(3,𝔽3)C2312yes—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—44no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C4⋊((C3⋊D3)⋊C3)—42no—
48C4⋊((C3⋊D3)⋊C3)—42no—
49#634⋊C3—42no—
50C3×H(3,𝔽4)—91no—
51#5683—31no—
52#5683—91no—
53C23×C3⋊D3C22×C312yescomplemented
54C23×C3×D3—32no—
55C2×C32×Q8—21no—
56C2×C32×Q8C2×D311yescharacteristic, complemented
57C2×C32×Q8C22×C311yescharacteristic, complemented
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C22×C311yescharacteristic
61C2×#638C22×C314yes—
62C2×C4⋊(C3⋊D3)C22×C312yes—
63C2×C4⋊(C3⋊D3)C22×C312yes—
64C2×#634C22×C312yescomplemented
65C2×C3×#67—34no—
66C2×C3×D12—32no—
67C2×C3×D12—32no—
68C2×C3×Dic6—32no—
69C32×C22⋊C4—24no—
70C32×C22⋊C4C2×D314yescomplemented
71C32×C22⋊C4C22×C314yescomplemented
72C32×C42—21no—
73C32×C42C2×D311yescharacteristic, complemented
74C32×C42C22×C311yescharacteristic, complemented
75#3757C22×C312yescomplemented
76#3752C22×C314yescomplemented
77#3752C22×C314yescomplemented
78#3752C22×C314yescomplemented
79C4×C3⋊Dic3C22×C312yescomplemented
80C3×#217—32no—
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—34no—
84C3×C4×Dic3—32no—
85C22×H(3,𝔽3)—24no—
86C22×H(3,𝔽3)—22no—
87C22×H(3,𝔽3)C2411yescharacteristic
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3—24no—
91C2×(C3⋊D3)⋊C3—48no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)—22no—
94C4×H(3,𝔽3)—22no—
95(C3⋊Dic3)⋊C3—24no—
96C3×#124—31no—
97C3×#124—31no—
98C3×#124—61no—
99C3×#124—92no—
100C3×#124—94no—
101C3×#124—92no—
102C3×#124C3×D311yescharacteristic
103C3×#120—94no—
104C3×#120—92no—
105#947—32no—
106#947—92no—
107#937—32no—
108#937—92no—
109#885—34no—
110#885—94no—
111#882—34no—
112#882—94no—
113#875—32no—
114#875—92no—
115C23×C32—22no—
116C23×C32C22×D312yes—
117C23×C32C23×C312yes—
118C22×C3⋊D3—42no—
119C22×C3⋊D3—44no—
120C22×C3⋊D3C23×C314yes—
121C22×C3×D3—34no—
122C22×C3×D3—122no—
123C22×C3×D3—124no—
124C32×Q8—41no—
125C32×Q8—41no—
126C32×Q8—42no—
127C32×D4—41no—
128C32×D4—41no—
129C32×D4—42no—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4C22×D312yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D312yes—
136C2×C32×C4C23×C312yes—
137C2×C32×C4C23×C311yescharacteristic
138C2×C32×C4C23×C312yes—
139#638—44no—
140C2×C3⋊Dic3C23×C314yes—
141C4⋊(C3⋊D3)—42no—
142C4⋊(C3⋊D3)—42no—
143#634—42no—
144C3×#67—124no—
145C2×C3×Dic3—34no—
146C3×D12—122no—
147C3×D12—122no—
148C3×Dic6—122no—
149H(3,𝔽4)—91noSylow
150C2×H(3,𝔽3)—42no—
151C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
152C2×H(3,𝔽3)H(5,𝔽2)12yes—
153(C3⋊D3)⋊C3—44no—
154C24×C3—92no—
155C23×D3—32no—
156C23×D3—92no—
157C2×C3×Q8—31no—
158C2×C3×Q8—31no—
159C2×C3×Q8—61no—
160C2×C3×Q8—92no—
161C2×C3×Q8C2×C3×D311yescharacteristic
162C2×C3×D4—31no—
163C2×C3×D4—31no—
164C2×C3×D4—61no—
165C2×C3×D4—92no—
166C2×C3×D4—92no—
167C2×C3×D4—94no—
168C2×C3×D4C2×C3×D311yescharacteristic
169C2×#67—34no—
170C2×#67—94no—
171C2×D12—32no—
172C2×D12—32no—
173C2×D12—92no—
174C2×D12—92no—
175C2×Dic6—32no—
176C2×Dic6—92no—
177C3×C22⋊C4—34no—
178C3×C22⋊C4—34no—
179C3×C22⋊C4—64no—
180C3×C22⋊C4—94no—
181C3×C22⋊C4—94no—
182C3×C22⋊C4—94no—
183C3×C22⋊C4—92no—
184C3×C22⋊C4C2×C3×D314yes—
185C3×C42—31no—
186C3×C42—31no—
187C3×C42—61no—
188C3×C42—92no—
189C3×C42C2×C3×D311yescharacteristic
190#217—32no—
191#217—92no—
192(C2×D3)⋊C4—34no—
193(C2×D3)⋊C4—34no—
194(C2×D3)⋊C4—34no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—94no—
197(C2×D3)⋊C4—94no—
198C4×Dic3—32no—
199C4×Dic3—92no—
200C22×C32—22no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—42no—
206C22×C32—44no—
207C22×C32C23×D311yescharacteristic
208C22×C32C24×C311yescharacteristic
209C2×C3⋊D3—24no—
210C2×C3⋊D3—24no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3—48no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—128no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—21no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—42no—
225C32×C4—42no—
226C3⋊Dic3—24no—
227C3×Dic3—64no—
228#124—92no—
229#124—94no—
230#124—92no—
231#124(C3⋊D3)⋊C311yescharacteristic, complemented
232#120—92no—
233#120—94no—
234H(3,𝔽3)H(3,𝔽4)11yescharacteristic, complemented, Sylow
235C23×C3—32no—
236C23×C3—32no—
237C23×C3—62no—
238C23×C3—94no—
239C23×C3—362no—
240C23×C3—364no—
241C23×C3C22×C3×D312yes—
242C22×D3—34no—
243C22×D3—94no—
244C22×D3—122no—
245C22×D3—124no—
246C22×D3—362no—
247C22×D3—364no—
248C3×Q8—41no—
249C3×Q8—121no—
250C3×Q8—121no—
251C3×Q8—122no—
252C3×Q8—362no—
253C3×D4—41no—
254C3×D4—121no—
255C3×D4—121no—
256C3×D4—122no—
257C3×D4—362no—
258C3×D4—362no—
259C3×D4—364no—
260C2×C3×C4—32no—
261C2×C3×C4—31no—
262C2×C3×C4—32no—
263C2×C3×C4—32no—
264C2×C3×C4—31no—
265C2×C3×C4—32no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—94no—
270C2×C3×C4C22×C3×D312yes—
271C2×C3×C4C22×C3×D311yescharacteristic
272C2×C3×C4C22×C3×D312yes—
273#67—124no—
274#67—364no—
275C2×Dic3—34no—
276C2×Dic3—94no—
277D12—122no—
278D12—122no—
279D12—362no—
280D12—362no—
281Dic6—122no—
282Dic6—362no—
283C2×C32—21no—
284C2×C32—22no—
285C2×C32—42no—
286C2×C32—42no—
287C2×C32—82no—
288C2×C32#100411yescharacteristic
289C2×C32#100912yes—
290C2×C32C3×H(5,𝔽2)11yescharacteristic
291C2×C32C3×H(5,𝔽2)12yes—
292C3⋊D3—44no—
293C3×D3—124no—
294C24—92no—
295C2×Q8—92no—
296C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
297C2×D4—94no—
298C2×D4—92no—
299C2×D4—92no—
300C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
301C22⋊C4—92no—
302C22⋊C4—94no—
303C22⋊C4—94no—
304C22⋊C4—94no—
305C22⋊C4C2×(C3⋊D3)⋊C314yescomplemented
306C42—92no—
307C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
308C22×C3—22no—
309C22×C3—24no—
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—64no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—64no—
316C22×C3—61no—
317C22×C3—122no—
318C22×C3—124no—
319C22×C3—184no—
320C22×C3—184no—
321C22×C3—184no—
322C22×C3—368no—
323C22×C3C23×C3×D311yescharacteristic
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—128no—
328C2×D3—184no—
329C2×D3—184no—
330C2×D3—184no—
331C2×D3—368no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—62no—
336C3×C4—62no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—62no—
340C3×C4—62no—
341C3×C4—62no—
342C3×C4—122no—
343C3×C4—122no—
344C3×C4—184no—
345Dic3—64no—
346Dic3—184no—
347C32—21no—
348C32#568311yescharacteristic, complemented
349C32C3×H(3,𝔽4)11yescharacteristic, complemented
350C23—94no—
351C23—362no—
352C23—364no—
353C23C22×(C3⋊D3)⋊C312yes—
354Q8—41no—
355Q8—362no—
356D4—41no—
357D4—362no—
358D4—362no—
359D4—364no—
360C2×C4—94no—
361C2×C4C22×(C3⋊D3)⋊C312yes—
362C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
363C2×C4C22×(C3⋊D3)⋊C312yes—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—32no—
367C2×C3—32no—
368C2×C3—42no—
369C2×C3—61no—
370C2×C3—62no—
371C2×C3—122no—
372C2×C3—122no—
373C2×C3—242no—
374C2×C3—364no—
375C2×C3C3×#100411yescharacteristic
376C2×C3C3×#100912yes—
377D3—124no—
378D3—364no—
379C22—22no—
380C22—24no—
381C22—184no—
382C22—184no—
383C22—184no—
384C22—368no—
385C22C23×(C3⋊D3)⋊C311yescharacteristic
386C4—21no—
387C4—22no—
388C4—22no—
389C4—184no—
390C3—31no—
391C3—31no—
392C3—61no—
393C3C3×#568311yescharacteristic
394C2—42no—
395C2—364no—
396C2#64361⋊C311yescharacteristic
397C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×H(3,𝔽4)
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. #64156⋊C3C2semidirect
3. #64146⋊C3C2semidirect
4. #64094⋊C3C2semidirect
5. #64091⋊C3C2semidirect
6. #64084⋊C3C2semidirect
7. #10596154C3semidirect
9. C23×(C3⋊D3)⋊C3C22semidirect
10. C2×Q8×H(3,𝔽3)C22semidirect
15. C2×#634⋊C3C22semidirect
16. C22⋊C4×H(3,𝔽3)C22semidirect
17. C42×H(3,𝔽3)C22semidirect
18. #3757⋊C3C22semidirect
19. #3752⋊C3C22semidirect
20. #3752⋊C3C22semidirect
21. #3752⋊C3C22semidirect
22. C4×(C3⋊Dic3)⋊C3C22semidirect
24. C32×#124D3semidirect
25. C32×#124C2×C3semidirect
26. #64156C2×C3semidirect
27. #64146C2×C3semidirect
28. #64094C2×C3semidirect
29. #64091C2×C3semidirect
30. #64084C2×C3semidirect
53. C23×C3⋊D3C22×C3semidirect
56. C2×C32×Q8C2×D3semidirect
57. C2×C32×Q8C22×C3semidirect
64. C2×#634C22×C3semidirect
70. C32×C22⋊C4C2×D3semidirect
71. C32×C22⋊C4C22×C3semidirect
73. C32×C42C2×D3semidirect
74. C32×C42C22×C3semidirect
75. #3757C22×C3semidirect
76. #3752C22×C3semidirect
77. #3752C22×C3semidirect
78. #3752C22×C3semidirect
79. C4×C3⋊Dic3C22×C3semidirect
231. #124(C3⋊D3)⋊C3semidirect
234. H(3,𝔽3)H(3,𝔽4)semidirect
296. C2×Q8C2×(C3⋊D3)⋊C3semidirect
305. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
307. C42C2×(C3⋊D3)⋊C3semidirect
348. C32#5683semidirect
349. C32C3×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#64156⋊C32
#64146⋊C32
#64094⋊C32
#64091⋊C32
#64084⋊C32
#105961543
C3×#56833