Finite Group Databasebeta

← C3×#10596582 #13181382

Subgroups — C3×#10596582 #13181382

Order 1728.

930 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 118 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×Q8×A4C211yesmaximal, characteristic, complemented
2C3×C4⋊(C3⋊S4)C213yesmaximal, complemented
3C3×C4×C3⋊S4C213yesmaximal, complemented
4#10596582C311yesmaximal, characteristic, direct product factor
5C3×#5919—33nomaximal
6C3×#5817—31nomaximal
7C2×C3×C3⋊S4C2213yes—
8C3×#3832—41nomaximal
9C32×C4×A4C2213yes—
10C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic
11C22×C32×Q8D311yescharacteristic, complemented
12C2×C3×#239—31no—
13C3×Q8×A4—21no—
14C3×Q8×A4—23no—
15C3×Q8×A4D313yescomplemented
16C3×Q8×A4C2×C311yescharacteristic, complemented
17C4⋊(C3⋊S4)C2×C313yescomplemented
18C4×C3⋊S4C2×C313yescomplemented
19C3×C4⋊S4—39no—
20C3×C4×S4—39no—
21C3×#947—33no—
22C3×#946—33no—
23C3×Q8×Dic3—31no—
24C3×#930—33no—
25C3×C4×#67—33no—
26C2×C32×A4C2311yescharacteristic
27C3×C3⋊S4—23no—
28C33×Q8—41no—
29C3×C4⋊(C3⋊D3)—43no—
30C3×C4×C3⋊D3—43no—
31#5919—33no—
32C3×#548—91no—
33#5817—31no—
34C2×C3⋊S4C22×C313yes—
35C2×C3×S4—39no—
36C2×C32×Q8—31no—
37C2×C32×Q8—33no—
38C22×C32×C4C2×D313yes—
39#3832—41no—
40C2×C3×#67—33no—
41C3×#239—63no—
42C3×#239—61no—
43C3×#239—123no—
44C2×C3×D12—33no—
45C2×C3×C4×D3—33no—
46C3×C4×A4—23no—
47C3×C4×A4—29no—
48C3×C4×A4C2×D319yes—
49C3×C4×A4C22×C313yes—
50C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
51C3×SL(2,ℤ/4)—33no—
52C3×#217—31no—
53C3×(C2×D3)⋊C4—33no—
54C3×(C2×D3)⋊C4—36no—
55C3×C4⋊Dic3—33no—
56C3×Dic3⋊C4—33no—
57C3×C4×Dic3—33no—
58C2×C3×C3⋊D3—43no—
59C32×A4#4111yescharacteristic, complemented
60C33×C4—43no—
61C3×C3⋊Dic3—41no—
62C2×C3×#41—91no—
63C22×C3×Q8—21no—
64C22×C3×Q8C3×D311yescharacteristic, complemented
65C22×C3×Q8C3⋊D311yescharacteristic, complemented
66C2×#239—31no—
67Q8×A4—23no—
68Q8×A4C3×D313yescomplemented
69C4⋊S4—39no—
70C4×S4—39no—
71C3×#124—93no—
72C3×C22⋊Q8—93no—
73C3×#121—93no—
74C3×C4×Q8—91no—
75C3×C4×D4—93no—
76#947—33no—
77#946—33no—
78Q8×Dic3—31no—
79#930—33no—
80C4×#67—33no—
81C23×C32C22×D311yescharacteristic
82C22×C3×D3—33no—
83C2×C3×A4—21no—
84C2×C3×A4—23no—
85C2×C3×A4C22×D313yes—
86C2×C3×A4C23×C311yescharacteristic
87C3⋊S4—23no—
88C3×S4—69no—
89C32×Q8—33no—
90C32×Q8—41no—
91C32×Q8—43no—
92C32×Q8—61no—
93C32×Q8—83no—
94C32×Q8—81no—
95C32×Q8S411yescharacteristic, complemented
96C2×C32×C4—33no—
97C2×C32×C4—33no—
98C4⋊(C3⋊D3)—43no—
99C4×C3⋊D3—43no—
100C3×#67—63no—
101C3×#67—63no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—33no—
104C2×C3×Dic3—31no—
105C3×D12—63no—
106C3×D12—63no—
107C3×D12—129no—
108C3×C4×D3—63no—
109C3×C4×D3—63no—
110C3×C4×D3—129no—
111#548—91noSylow
112C2×C33—41no—
113C3×C3⋊D3—83no—
114C2×S4—39no—
115C3×#41—183no—
116C3×#41—181no—
117C2×C3×Q8—31no—
118C2×C3×Q8—31no—
119C2×C3×Q8—33no—
120C2×C3×Q8—33no—
121C2×C3×Q8—61no—
122C2×C3×Q8—63no—
123C2×C3×D4—93no—
124C2×C3×D4—93no—
125C22×C3×C4—23no—
126C22×C3×C4—93no—
127C22×C3×C4C2×C3×D313yes—
128C22×C3×C4C2×C3⋊D313yes—
129C2×#67—33no—
130#239—63no—
131#239—61no—
132#239—123no—
133C2×D12—33no—
134C2×C4×D3—33no—
135C4×A4—29no—
136C4×A4C2×C3×D319yes—
137SL(2,ℤ/4)—33no—
138C3×C4⋊C4—93no—
139C3×C4⋊C4—93no—
140C3×C22⋊C4—93no—
141C3×C22⋊C4—96no—
142C3×C22⋊C4—91no—
143C3×C42—93no—
144#217—31no—
145(C2×D3)⋊C4—33no—
146(C2×D3)⋊C4—36no—
147C4⋊Dic3—33no—
148Dic3⋊C4—33no—
149C4×Dic3—33no—
150C22×C32—31no—
151C22×C32—31no—
152C22×C32#23911yescharacteristic, complemented
153C2×C3⋊D3—43no—
154C2×C3×D3—63no—
155C2×C3×D3—63no—
156C2×C3×D3—63no—
157C2×C3×D3—129no—
158C3×A4—21no—
159C3×A4—23no—
160C3×A4#23913yescomplemented
161C3×A4C3×#4111yescharacteristic, complemented
162C32×C4—33no—
163C32×C4—43no—
164C32×C4—49no—
165C32×C4—89no—
166C32×C4—83no—
167C32×C4C2×S413yes—
168C3⋊Dic3—41no—
169C3×Dic3—61no—
170C3×Dic3—63no—
171C3×Dic3—61no—
172C3×Dic3—123no—
173C2×#41—91no—
174C22×Q8C3×C3⋊D311yescharacteristic, complemented
175#124—93no—
176C22⋊Q8—93no—
177#121—93no—
178C4×Q8—91no—
179C4×D4—93no—
180C33—41noSylow
181C23×C3—21no—
182C23×C3—93no—
183C23×C3C22×C3×D311yescharacteristic
184C23×C3C22×C3⋊D311yescharacteristic
185C22×D3—33no—
186C2×A4—23no—
187C2×A4C22×C3×D313yes—
188S4—69no—
189C3×Q8—21no—
190C3×Q8—33no—
191C3×Q8—33no—
192C3×Q8—43no—
193C3×Q8—63no—
194C3×Q8—61no—
195C3×Q8—61no—
196C3×Q8—62no—
197C3×Q8—83no—
198C3×Q8C3×S411yescharacteristic, complemented
199C3×Q8C3⋊S411yescharacteristic, complemented
200C3×D4—183no—
201C3×D4—183no—
202C3×D4—183no—
203C3×D4—183no—
204C2×C3×C4—33no—
205C2×C3×C4—33no—
206C2×C3×C4—33no—
207C2×C3×C4—33no—
208C2×C3×C4—63no—
209C2×C3×C4—63no—
210C2×C3×C4—91no—
211C2×C3×C4—93no—
212C2×C3×C4—91no—
213C2×C3×C4—183no—
214C2×C3×C4—183no—
215#67—63no—
216#67—63no—
217C2×Dic3—31no—
218C2×Dic3—33no—
219C2×Dic3—31no—
220D12—63no—
221D12—63no—
222D12—129no—
223C4×D3—63no—
224C4×D3—63no—
225C4×D3—129no—
226C2×C32—31no—
227C2×C32—31no—
228C2×C32—41no—
229C2×C32—43no—
230C2×C32—83no—
231C2×C32—81no—
232C2×C32C22×S411yescharacteristic
233C3⋊D3—83no—
234C3×D3—123no—
235C3×D3—249no—
236#41—183no—
237#41—181no—
238C2×Q8—31no—
239C2×Q8—33no—
240C2×D4—93no—
241C2×D4—93no—
242C22×C4—93no—
243C22×C4C2×C3×C3⋊D313yes—
244C4⋊C4—93no—
245C4⋊C4—93no—
246C22⋊C4—93no—
247C22⋊C4—96no—
248C22⋊C4—91no—
249C42—93no—
250C22×C3—21no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—183no—
258C22×C3—183no—
259C22×C3—183no—
260C22×C3C3×#23911yescharacteristic, complemented
261C22×C3#383211yescharacteristic, complemented
262C2×D3—63no—
263C2×D3—63no—
264C2×D3—63no—
265C2×D3—129no—
266A4—23no—
267A4C3×#23913yescomplemented
268C3×C4—23no—
269C3×C4—33no—
270C3×C4—33no—
271C3×C4—49no—
272C3×C4—63no—
273C3×C4—89no—
274C3×C4—183no—
275C3×C4—181no—
276C3×C4—181no—
277C3×C4C2×C3×S413yes—
278C3×C4C2×C3⋊S413yes—
279Dic3—63no—
280Dic3—61no—
281Dic3—61no—
282Dic3—123no—
283C32—41no—
284C32—43no—
285C32—83no—
286C32—81no—
287C32#591911yescharacteristic, complemented
288C23—93no—
289C23C22×C3×C3⋊D311yescharacteristic
290Q8—33no—
291Q8—61no—
292Q8C3×C3⋊S411yescharacteristic, complemented
293D4—183no—
294D4—183no—
295D4—183no—
296D4—183no—
297C2×C4—33no—
298C2×C4—33no—
299C2×C4—91no—
300C2×C4—93no—
301C2×C4—91no—
302C2×C4—183no—
303C2×C4—183no—
304C2×C3—21no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—43no—
310C2×C3—61no—
311C2×C3—61no—
312C2×C3—83no—
313C2×C3—363no—
314C2×C3C22×C3×S411yescharacteristic
315C2×C3C22×C3⋊S411yescharacteristic
316D3—123no—
317D3—249no—
318C22—31no—
319C22—31no—
320C22—183no—
321C22—183no—
322C22—183no—
323C22C3×#383211yescharacteristic, complemented
324C4—33no—
325C4—183no—
326C4—181no—
327C4—181no—
328C4C2×C3×C3⋊S413yes—
329C3—21no—
330C3—43no—
331C3—83no—
332C3C3×#591911yescharacteristic, complemented
333C3#1059658211yescharacteristic, direct product factor
334C2—31no—
335C2—31no—
336C2—363no—
337C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C22×C3⋊S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC22×C32×Q8D3
socleC23×C32C22×D3
hypercenterC3×Q8C3⋊S4
hypocenterC3×A4C3×#41
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
2. C3×C4⋊(C3⋊S4)C2semidirect
3. C3×C4×C3⋊S4C2semidirect
4. #10596582C3direct (N × H)
11. C22×C32×Q8D3semidirect
15. C3×Q8×A4D3semidirect
16. C3×Q8×A4C2×C3semidirect
17. C4⋊(C3⋊S4)C2×C3semidirect
18. C4×C3⋊S4C2×C3semidirect
59. C32×A4#41semidirect
64. C22×C3×Q8C3×D3semidirect
65. C22×C3×Q8C3⋊D3semidirect
68. Q8×A4C3×D3semidirect
95. C32×Q8S4semidirect
152. C22×C32#239semidirect
160. C3×A4#239semidirect
161. C3×A4C3×#41semidirect
174. C22×Q8C3×C3⋊D3semidirect
198. C3×Q8C3×S4semidirect
199. C3×Q8C3⋊S4semidirect
260. C22×C3C3×#239semidirect
261. C22×C3#3832semidirect
267. A4C3×#239semidirect
287. C32#5919semidirect
292. Q8C3×C3⋊S4semidirect
323. C22C3×#3832semidirect
332. C3C3×#5919semidirect
333. C3#10596582direct (N × H)

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C3×C4⋊(C3⋊S4)2
C3×C4×C3⋊S42
#105965823
C3×#59193
C3×#58173
C3×#38324