Finite Group Databasebeta

← C3×D3×#994 #13181450

Subgroups — C3×D3×#994 #13181450

Order 1728.

1084 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#994C211yesmaximal, characteristic, complemented
2C3×D3×#41⋊C3C213yesmaximal, complemented
3C3×C3⋊#994C213yesmaximal, complemented
4C3×D3×H(5,𝔽2)C311yesmaximal, characteristic, complemented
5C2×C3×#994—31nomaximal
6D3×#994C313yesmaximal, direct product factor
7C32×D3×Q8—41nomaximal
8C32×#41⋊C3C2213yescomplemented
9C3×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
10C3×C3⋊(#41⋊C3)C2213yescomplemented
11C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
12C3×#1009—31no—
13C3×#1009C2×C313yescomplemented
14C3×D3×#41—31no—
15C3×D3×#41C2×C313yes—
16C3×#1004—33no—
17C2×C3×D3×D4—33no—
18C3×#994—21no—
19C3×#994—23no—
20C3×#994—33no—
21C3×#994D311yescharacteristic, direct product factor
22C3×#994C2×C313yescomplemented
23C2×C3×#41⋊C3—33no—
24D3×#41⋊C3C2×C319yescomplemented
25C3⋊#994C2×C319yescomplemented
26C33×Q8—41no—
27C32×C4×D3—43no—
28C32×Dic6—43no—
29C32×SL(2,𝔽3)C2311yescharacteristic, complemented
30C2×C3×H(5,𝔽2)—31no—
31D3×H(5,𝔽2)C3211yescharacteristic, complemented
32C2×#994—33no—
33C23×C3×D3—32no—
34C32×#41—31no—
35C32×#41C22×C313yes—
36C2×C32×Q8—121no—
37C2×C32×D4—33no—
38C2×C3×#67—36no—
39C3×#239—31no—
40C3×#239C22×C313yes—
41C3×D3×Q8—43no—
42C3×D3×Q8A411yescharacteristic
43C3×D3×Q8C22×C311yescharacteristic
44C3×#237—33no—
45C3×#237—33no—
46C3×D3×D4—36no—
47C3×D3×D4—33no—
48C3×D3×D4—36no—
49C3×D3×D4—33no—
50C3×#235—33no—
51C3×#235—33no—
52C2×C3×D12—33no—
53C2×C3×C4×D3—33no—
54C3×#41⋊C3—23no—
55C3×#41⋊C3—29no—
56C3×#41⋊C3—33no—
57C3×#41⋊C3C2×D313yescomplemented
58C3×#41⋊C3C22×C319yescomplemented
59C2×C3×SL(2,𝔽3)—31no—
60D3×SL(2,𝔽3)C22×C313yescomplemented
61C3⋊(#41⋊C3)C22×C319yescomplemented
62C2×C32×D3—41no—
63C33×C4—43no—
64C32×Dic3—43no—
65C3×H(5,𝔽2)—21no—
66C3×H(5,𝔽2)—33no—
67C3×H(5,𝔽2)—93no—
68C3×H(5,𝔽2)—91no—
69C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
70C3×H(5,𝔽2)C2×C3211yescharacteristic, complemented
71C2×C3×#41—33no—
72C2×C3×#41—91no—
73C22×C3×D4—93no—
74#1009—31no—
75#1009C2×C3213yescomplemented
76D3×#41—31no—
77D3×#41C2×C3213yes—
78#1004—33no—
79C2×D3×D4—33no—
80#994—23no—
81#994—39no—
82#994C3×D313yesdirect product factor
83C2×#41⋊C3—39no—
84C23×C32—32no—
85C22×C3×D3—36no—
86C22×C3×D3—33no—
87C22×C3×D3—124no—
88C32×Q8—41no—
89C32×Q8—43no—
90C32×Q8—83no—
91C32×Q8—81no—
92C32×Q8—123no—
93C32×Q8C2×A411yescharacteristic
94C32×Q8C23×C311yescharacteristic
95C32×D4—33no—
96C32×D4—33no—
97C2×C32×C4—33no—
98C2×C32×C4—123no—
99C3×#67—36no—
100C3×#67—36no—
101C2×C3×Dic3—33no—
102C3×D12—33no—
103C3×D12—33no—
104C3×C4×D3—31no—
105C3×C4×D3—33no—
106C3×C4×D3—33no—
107C3×C4×D3—49no—
108C3×C4×D3C2×A413yes—
109C3×Dic6—31no—
110C3×Dic6—49no—
111C3×Dic6C2×A413yes—
112C3×SL(2,𝔽3)—21no—
113C3×SL(2,𝔽3)—23no—
114C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
115C3×SL(2,𝔽3)C23×C313yescomplemented
116C2×H(5,𝔽2)—31noSylow
117C2×C33—41no—
118C32×D3—42no—
119C24×C3—92no—
120C23×D3—32no—
121C3×#41—23no—
122C3×#41—31no—
123C3×#41—31no—
124C3×#41—33no—
125C3×#41—61no—
126C3×#41—93no—
127C3×#41—93no—
128C3×#41—93no—
129C3×#41—91no—
130C3×#41—93no—
131C3×#41C2×C3×D313yes—
132C3×#41C22×C3213yes—
133C2×C3×Q8—31no—
134C2×C3×Q8—31no—
135C2×C3×Q8—123no—
136C2×C3×D4—33no—
137C2×C3×D4—33no—
138C2×C3×D4—63no—
139C2×C3×D4—93no—
140C2×C3×D4—93no—
141C2×C3×D4—93no—
142C2×C3×D4—96no—
143C2×C3×D4—96no—
144C2×C3×D4—96no—
145C22×C3×C4—93no—
146C2×#67—36no—
147#239—31no—
148#239C22×C3213yes—
149D3×Q8C3×A411yescharacteristic
150D3×Q8C22×C3211yescharacteristic
151#237—33no—
152#237—33no—
153D3×D4—33no—
154D3×D4—36no—
155D3×D4—36no—
156D3×D4—33no—
157#235—33no—
158#235—33no—
159C2×D12—33no—
160C2×C4×D3—33no—
161#41⋊C3—29no—
162#41⋊C3—39no—
163#41⋊C3C2×C3×D319yescomplemented
164C2×SL(2,𝔽3)—33no—
165C22×C32—33no—
166C22×C32—121no—
167C22×C32—122no—
168C2×C3×D3—33no—
169C2×C3×D3—43no—
170C2×C3×D3—66no—
171C2×C3×D3—126no—
172C2×C3×D3C22×A411yescharacteristic
173C32×C4—31no—
174C32×C4—43no—
175C32×C4—49no—
176C32×C4—89no—
177C32×C4—83no—
178C32×C4—123no—
179C32×C4C22×A413yes—
180C3×Dic3—31no—
181C3×Dic3—49no—
182C3×Dic3C22×A413yes—
183H(5,𝔽2)—33no—
184H(5,𝔽2)—93no—
185H(5,𝔽2)—91no—
186H(5,𝔽2)C32×D311yescharacteristic, complemented
187C2×#41—33no—
188C2×#41—91no—
189C22×D4—93no—
190C33—41noSylow
191C23×C3—32no—
192C23×C3—32no—
193C23×C3—62no—
194C23×C3—93no—
195C23×C3—96no—
196C23×C3—364no—
197C22×D3—36no—
198C22×D3—33no—
199C22×D3—124no—
200C3×Q8—21no—
201C3×Q8—21no—
202C3×Q8—33no—
203C3×Q8—43no—
204C3×Q8—83no—
205C3×Q8—91no—
206C3×Q8—129no—
207C3×Q8D3×A411yescharacteristic
208C3×Q8C2×C3×A411yescharacteristic
209C3×Q8C22×C3×D311yescharacteristic
210C3×Q8C23×C3211yescharacteristic
211C3×D4—33no—
212C3×D4—33no—
213C3×D4—33no—
214C3×D4—33no—
215C3×D4—63no—
216C3×D4—63no—
217C3×D4—96no—
218C3×D4—93no—
219C3×D4—96no—
220C3×D4—93no—
221C2×C3×C4—33no—
222C2×C3×C4—33no—
223C2×C3×C4—33no—
224C2×C3×C4—63no—
225C2×C3×C4—93no—
226C2×C3×C4—91no—
227C2×C3×C4—93no—
228C2×C3×C4—93no—
229C2×C3×C4—129no—
230#67—36no—
231#67—36no—
232C2×Dic3—33no—
233D12—33no—
234D12—33no—
235C4×D3—31no—
236C4×D3—33no—
237C4×D3—33no—
238C4×D3C2×C3×A413yes—
239Dic6—31no—
240Dic6C2×C3×A413yes—
241SL(2,𝔽3)—23no—
242SL(2,𝔽3)C22×C3×D313yescomplemented
243C2×C32—41no—
244C2×C32—43no—
245C2×C32—63no—
246C2×C32—83no—
247C2×C32—81no—
248C2×C32—122no—
249C2×C32C23×A411yescharacteristic
250C3×D3—46no—
251C3×D3—63no—
252C3×D3#99412yesdirect product factor
253C24—92no—
254#41—31no—
255#41—33no—
256#41—93no—
257#41—93no—
258#41—93no—
259#41—91no—
260#41—93no—
261#41C2×C32×D313yes—
262C2×Q8—31no—
263C2×Q8—31no—
264C2×D4—33no—
265C2×D4—93no—
266C2×D4—93no—
267C2×D4—93no—
268C2×D4—96no—
269C2×D4—96no—
270C2×D4—96no—
271C22×C4—93no—
272C22×C3—31no—
273C22×C3—33no—
274C22×C3—33no—
275C22×C3—63no—
276C22×C3—93no—
277C22×C3—123no—
278C22×C3—122no—
279C22×C3—122no—
280C22×C3—186no—
281C22×C3—242no—
282C22×C3—366no—
283C2×D3—33no—
284C2×D3—66no—
285C2×D3—126no—
286C2×D3C22×C3×A411yescharacteristic
287C3×C4—23no—
288C3×C4—31no—
289C3×C4—33no—
290C3×C4—31no—
291C3×C4—49no—
292C3×C4—61no—
293C3×C4—89no—
294C3×C4—91no—
295C3×C4—129no—
296C3×C4C2×D3×A413yes—
297C3×C4C22×C3×A413yes—
298Dic3—31no—
299Dic3C22×C3×A413yes—
300C32—41no—
301C32—43no—
302C32—83no—
303C32—81no—
304C32C2×#99411yescharacteristic, complemented
305C23—32no—
306C23—93no—
307C23—96no—
308C23—364no—
309Q8—33no—
310Q8—91no—
311Q8C3×D3×A411yescharacteristic
312Q8C22×C32×D311yescharacteristic
313D4—33no—
314D4—33no—
315D4—96no—
316D4—93no—
317D4—96no—
318D4—93no—
319C2×C4—33no—
320C2×C4—33no—
321C2×C4—93no—
322C2×C4—91no—
323C2×C4—93no—
324C2×C4—93no—
325C2×C3—21no—
326C2×C3—32no—
327C2×C3—43no—
328C2×C3—63no—
329C2×C3—63no—
330C2×C3—83no—
331C2×C3—126no—
332C2×C3—123no—
333C2×C3—183no—
334C2×C3C22×D3×A411yescharacteristic
335C2×C3C23×C3×A411yescharacteristic
336D3—63no—
337D3C3×#99412yesdirect product factor
338C22—31no—
339C22—33no—
340C22—93no—
341C22—122no—
342C22—186no—
343C22—366no—
344C4—31no—
345C4—33no—
346C4—91no—
347C4C2×C3×D3×A413yes—
348C3—21no—
349C3—43no—
350C3—83no—
351C3D3×#99411yescharacteristic, direct product factor
352C3C2×C3×#99411yescharacteristic, complemented
353C2—32no—
354C2—63no—
355C2—183no—
356C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC2×C3C22×D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C32
FrattiniC2C22×C3×D3×A4
FittingC32×H(5,𝔽2)C2×C3
socleC2×C32C23×A4
hypercenterC3×Q8D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×#994C2semidirect
2. C3×D3×#41⋊C3C2semidirect
3. C3×C3⋊#994C2semidirect
4. C3×D3×H(5,𝔽2)C3semidirect
6. D3×#994C3direct (N × H)
8. C32×#41⋊C3C22semidirect
9. C3×D3×SL(2,𝔽3)C22semidirect
10. C3×C3⋊(#41⋊C3)C22semidirect
11. C32×H(5,𝔽2)C2×C3semidirect
13. C3×#1009C2×C3semidirect
21. C3×#994D3direct (N × H)
22. C3×#994C2×C3semidirect
24. D3×#41⋊C3C2×C3semidirect
25. C3⋊#994C2×C3semidirect
29. C32×SL(2,𝔽3)C23semidirect
31. D3×H(5,𝔽2)C32semidirect
57. C3×#41⋊C3C2×D3semidirect
58. C3×#41⋊C3C22×C3semidirect
60. D3×SL(2,𝔽3)C22×C3semidirect
61. C3⋊(#41⋊C3)C22×C3semidirect
69. C3×H(5,𝔽2)C3×D3semidirect
70. C3×H(5,𝔽2)C2×C32semidirect
75. #1009C2×C32semidirect
82. #994C3×D3direct (N × H)
114. C3×SL(2,𝔽3)C22×D3semidirect
115. C3×SL(2,𝔽3)C23×C3semidirect
163. #41⋊C3C2×C3×D3semidirect
186. H(5,𝔽2)C32×D3semidirect
242. SL(2,𝔽3)C22×C3×D3semidirect
252. C3×D3#994direct (N × H)
304. C32C2×#994semidirect
337. D3C3×#994direct (N × H)
351. C3D3×#994direct (N × H)
352. C3C2×C3×#994semidirect

Maximal subgroups

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

subgroupindex
C32×#9942
C3×D3×#41⋊C32
C3×C3⋊#9942
C3×D3×H(5,𝔽2)3
C2×C3×#9943
D3×#9943
C32×D3×Q84