Finite Group Databasebeta

← C3×#10596623 #13181451

Subgroups — C3×#10596623 #13181451

Order 1728.

1036 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 172 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×#1010⋊C3C213yesmaximal, complemented
3C3×D3×#41⋊C3C213yesmaximal, complemented
4C3×#5966C311yesmaximal, characteristic, complemented
5C3×#5945—31nomaximal
6#10596623C313yesmaximal, direct product factor
7C32×#239—41nomaximal
8C32×#41⋊C3C2213yescomplemented
9C3×D3×SL(2,𝔽3)C2213yescomplemented
10C3×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
11C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
12C3×#1010—31no—
13C3×#1010C2×C313yescomplemented
14C3×D3×#41—31no—
15C3×D3×#41C2×C313yescomplemented
16C3×#1004—33no—
17C2×C3×#237—33no—
18C3×2−1+4⋊C3—33no—
19C3×#994—21no—
20C3×#994—23no—
21C3×#994D311yescharacteristic, complemented
22C3×#994C2×C313yescomplemented
23C2×C3×#41⋊C3—33no—
24#1010⋊C3C2×C319yescomplemented
25D3×#41⋊C3C2×C319yescomplemented
26C33×Q8—41no—
27C32×D12—43no—
28C32×C4×D3—43no—
29C32×SL(2,𝔽3)C2311yescharacteristic
30C3×#585—31no—
31#5966C3211yescharacteristic, complemented
32#5945—33no—
33C32×#41—31no—
34C32×#41—121no—
35C32×#41C22×C313yes—
36C2×C32×D4—33no—
37C2×C3×#67—36no—
38C22×C3×Dic3—32no—
39C3×#239—43no—
40C3×#239A411yescharacteristic
41C3×#239C22×C311yescharacteristic
42C3×D3×Q8—31no—
43C3×D3×Q8C22×C313yes—
44C3×#237—36no—
45C3×#237—36no—
46C3×#237—33no—
47C3×#237—33no—
48C3×D3×D4—33no—
49C3×D3×D4—33no—
50C3×#235—33no—
51C3×#235—33no—
52C2×C3×C4×D3—33no—
53C2×C3×Dic6—33no—
54C3×#41⋊C3—23no—
55C3×#41⋊C3—29no—
56C3×#41⋊C3—31no—
57C3×#41⋊C3C2×D313yescomplemented
58C3×#41⋊C3C22×C319yescomplemented
59C2×C3×SL(2,𝔽3)—33no—
60D3×SL(2,𝔽3)C22×C319yescomplemented
61C3⋊(#41⋊C3)C22×C313yescomplemented
62C2×C32×D3—43no—
63C33×C4—43no—
64C32×Dic3—41no—
65C3×2−1+4—33no—
66C3×2−1+4—91no—
67C3×H(5,𝔽2)—21no—
68C3×H(5,𝔽2)—93no—
69C3×H(5,𝔽2)C3×D311yescharacteristic, complemented
70C3×H(5,𝔽2)C2×C3211yescharacteristic, complemented
71C2×C3×#41—33no—
72C2×C3×#41—93no—
73C2×C3×#41—91no—
74#1010—31no—
75#1010C2×C3213yescomplemented
76D3×#41—31no—
77D3×#41C2×C3213yescomplemented
78#1004—33no—
79C2×#237—33no—
802−1+4⋊C3—39no—
81#994—23no—
82#994C3×D313yescomplemented
83C2×#41⋊C3—39no—
84C23×C32—32no—
85C22×C3×D3—33no—
86C32×Q8—41no—
87C32×Q8—43no—
88C32×Q8—83no—
89C32×Q8—81no—
90C32×Q8C2×A411yescharacteristic
91C32×Q8C23×C311yescharacteristic
92C32×D4—33no—
93C32×D4—33no—
94C32×D4—123no—
95C2×C32×C4—33no—
96C2×C32×C4—123no—
97C3×#67—36no—
98C3×#67—36no—
99C2×C3×Dic3—36no—
100C2×C3×Dic3—33no—
101C3×D12—31no—
102C3×D12—49no—
103C3×D12C2×A413yes—
104C3×C4×D3—33no—
105C3×C4×D3—33no—
106C3×C4×D3—31no—
107C3×C4×D3—49no—
108C3×C4×D3C2×A413yes—
109C3×Dic6—33no—
110C3×Dic6—33no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)—23no—
113C3×SL(2,𝔽3)C22×D311yescharacteristic
114C3×SL(2,𝔽3)C23×C313yes—
115#585—31noSylow
116C2×C33—41no—
117C32×D3—83no—
118C3×#41—23no—
119C3×#41—31no—
120C3×#41—31no—
121C3×#41—31no—
122C3×#41—31no—
123C3×#41—61no—
124C3×#41—93no—
125C3×#41—93no—
126C3×#41—93no—
127C3×#41—96no—
128C3×#41—96no—
129C3×#41—93no—
130C3×#41—123no—
131C3×#41C2×C3×D313yes—
132C3×#41C22×C3213yes—
133C2×C3×Q8—33no—
134C2×C3×Q8—93no—
135C2×C3×Q8—91no—
136C2×C3×D4—33no—
137C2×C3×D4—33no—
138C2×C3×D4—63no—
139C2×C3×D4—93no—
140C2×C3×D4—93no—
141C2×C3×D4—96no—
142C22×C3×C4—93no—
143C22×C3×C4—92no—
144C2×#67—36no—
145C22×Dic3—32no—
146#239C3×A411yescharacteristic
147#239C22×C3211yescharacteristic
148D3×Q8—31no—
149D3×Q8C22×C3213yes—
150#237—36no—
151#237—36no—
152#237—33no—
153#237—33no—
154D3×D4—33no—
155D3×D4—33no—
156#235—33no—
157#235—33no—
158C2×C4×D3—33no—
159C2×Dic6—33no—
160#41⋊C3—29no—
161#41⋊C3—33no—
162#41⋊C3C2×C3×D319yescomplemented
163C2×SL(2,𝔽3)—39no—
164C22×C32—33no—
165C22×C32—123no—
166C22×C32—122no—
167C2×C3×D3—31no—
168C2×C3×D3—49no—
169C2×C3×D3—123no—
170C2×C3×D3C22×A413yes—
171C32×C4—31no—
172C32×C4—43no—
173C32×C4—49no—
174C32×C4—89no—
175C32×C4—83no—
176C32×C4—121no—
177C32×C4C22×A413yes—
178C3×Dic3—33no—
179C3×Dic3—43no—
180C3×Dic3C22×A411yescharacteristic
1812−1+4—33no—
1822−1+4—91no—
183H(5,𝔽2)—93no—
184H(5,𝔽2)C32×D311yescharacteristic, complemented
185C2×#41—33no—
186C2×#41—93no—
187C2×#41—91no—
188C33—41noSylow
189C23×C3—32no—
190C23×C3—32no—
191C23×C3—62no—
192C23×C3—93no—
193C22×D3—33no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—43no—
197C3×Q8—83no—
198C3×Q8—93no—
199C3×Q8—93no—
200C3×Q8D3×A411yescharacteristic
201C3×Q8C2×C3×A411yescharacteristic
202C3×Q8C22×C3×D311yescharacteristic
203C3×Q8C23×C3211yescharacteristic
204C3×D4—33no—
205C3×D4—33no—
206C3×D4—33no—
207C3×D4—33no—
208C3×D4—33no—
209C3×D4—63no—
210C3×D4—63no—
211C3×D4—91no—
212C3×D4—96no—
213C3×D4—96no—
214C3×D4—129no—
215C2×C3×C4—33no—
216C2×C3×C4—33no—
217C2×C3×C4—33no—
218C2×C3×C4—63no—
219C2×C3×C4—91no—
220C2×C3×C4—93no—
221C2×C3×C4—93no—
222C2×C3×C4—96no—
223C2×C3×C4—93no—
224C2×C3×C4—129no—
225#67—36no—
226#67—36no—
227C2×Dic3—36no—
228C2×Dic3—33no—
229D12—31no—
230D12C2×C3×A413yes—
231C4×D3—33no—
232C4×D3—33no—
233C4×D3—31no—
234C4×D3C2×C3×A413yes—
235Dic6—33no—
236Dic6—33no—
237SL(2,𝔽3)—23no—
238SL(2,𝔽3)C22×C3×D313yes—
239C2×C32—41no—
240C2×C32—43no—
241C2×C32—63no—
242C2×C32—83no—
243C2×C32—81no—
244C2×C32—243no—
245C2×C32C23×A411yescharacteristic
246C3×D3—23no—
247C3×D3—61no—
248C3×D3—89no—
249#41—31no—
250#41—31no—
251#41—31no—
252#41—93no—
253#41—93no—
254#41—93no—
255#41—96no—
256#41—96no—
257#41—93no—
258#41C2×C32×D313yes—
259C2×Q8—33no—
260C2×Q8—93no—
261C2×Q8—91no—
262C2×D4—33no—
263C2×D4—93no—
264C2×D4—93no—
265C2×D4—96no—
266C22×C4—93no—
267C22×C4—92no—
268C22×C3—33no—
269C22×C3—33no—
270C22×C3—33no—
271C22×C3—63no—
272C22×C3—91no—
273C22×C3—129no—
274C22×C3—122no—
275C22×C3—122no—
276C22×C3—242no—
277C22×C3—363no—
278C2×D3—31no—
279C2×D3—123no—
280C2×D3C22×C3×A413yes—
281C3×C4—23no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—49no—
286C3×C4—61no—
287C3×C4—89no—
288C3×C4—93no—
289C3×C4—123no—
290C3×C4C2×D3×A413yes—
291C3×C4C22×C3×A413yes—
292Dic3—33no—
293Dic3C22×C3×A411yescharacteristic
294C32—41no—
295C32—43no—
296C32—83no—
297C32—81no—
298C32#594511yescharacteristic, complemented
299C23—32no—
300C23—93no—
301Q8—93no—
302Q8—93no—
303Q8C3×D3×A411yescharacteristic
304Q8C22×C32×D311yescharacteristic
305D4—33no—
306D4—33no—
307D4—33no—
308D4—91no—
309D4—96no—
310D4—96no—
311C2×C4—33no—
312C2×C4—33no—
313C2×C4—91no—
314C2×C4—93no—
315C2×C4—93no—
316C2×C4—96no—
317C2×C4—93no—
318C2×C3—21no—
319C2×C3—43no—
320C2×C3—63no—
321C2×C3—63no—
322C2×C3—63no—
323C2×C3—83no—
324C2×C3—123no—
325C2×C3—181no—
326C2×C3—249no—
327C2×C3C22×D3×A411yescharacteristic
328C2×C3C23×C3×A411yescharacteristic
329D3—23no—
330D3—61no—
331C22—33no—
332C22—33no—
333C22—91no—
334C22—122no—
335C22—363no—
336C4—31no—
337C4—31no—
338C4—93no—
339C4C2×C3×D3×A413yes—
340C3—21no—
341C3—43no—
342C3—83no—
343C3#1059662311yescharacteristic, direct product factor
344C3C3×#594511yescharacteristic, complemented
345C2—63no—
346C2—63no—
347C2—181no—
348C2C22×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×#1010⋊C3C2semidirect
3. C3×D3×#41⋊C3C2semidirect
4. C3×#5966C3semidirect
6. #10596623C3direct (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×#1010C2×C3semidirect
15. C3×D3×#41C2×C3semidirect
21. C3×#994D3semidirect
22. C3×#994C2×C3semidirect
24. #1010⋊C3C2×C3semidirect
25. D3×#41⋊C3C2×C3semidirect
31. #5966C32semidirect
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. #1010C2×C32semidirect
77. D3×#41C2×C32semidirect
82. #994C3×D3semidirect
162. #41⋊C3C2×C3×D3semidirect
184. H(5,𝔽2)C32×D3semidirect
298. C32#5945semidirect
343. C3#10596623direct (N × H)
344. C3C3×#5945semidirect

Maximal subgroups

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

subgroupindex
C32×#9942
C3×#1010⋊C32
C3×D3×#41⋊C32
C3×#59663
C3×#59453
#105966233
C32×#2394