Finite Group Databasebeta

← C3×#10593628 #13172996

Subgroups — C3×#10593628 #13172996

Order 1728.

504 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593628C311yesmaximal, characteristic, direct product factor
5C3×#5394—33nomaximal
6C3×#5026—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)C412yescomplemented
9C3×#3752—41nomaximal
10C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic
11C2×C3×GL(2,𝔽3)—33no—
12C32×C4×Q8D311yescharacteristic, complemented
13C2×C3×#215—31no—
14C3×C4×D12—31no—
15C3×C4×SL(2,𝔽3)—21no—
16C3×C4×SL(2,𝔽3)—23no—
17C3×C4×SL(2,𝔽3)D313yescomplemented
18C3×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19C3⋊#859C2×C311yescharacteristic, complemented
20C3×#859—33no—
21C3×#835—31no—
22C3×#809—31no—
23C3×#807—31no—
24C3×#803—31no—
25C22×C3×C3⋊D3—41no—
26C2×C33×C4—41no—
27C2×C3×C3⋊Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
29#5394—33no—
30C3×#440—91no—
31#5026—31no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×C3×D12—31no—
34C2×C3×C4×D3—61no—
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)—23no—
37C2×C3×SL(2,𝔽3)C2×D313yes—
38C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
39C3⋊GL(2,𝔽3)C3×C412yescomplemented
40C3×GL(2,𝔽3)—36no—
41C32×C4⋊C4—31no—
42C32×C42—31no—
43#3752—41no—
44C3×#215—32no—
45C3×#215—32no—
46C3×(C2×D3)⋊C4—61no—
47C3×(C2×D3)⋊C4—123no—
48C3×C4⋊Dic3—31no—
49C2×C3×C3⋊C8—31no—
50C2×C3×C3⋊C8—31no—
51C22×C33—41no—
52C2×C3×C3⋊D3—42no—
53C2×C3×C3⋊D3—81no—
54C2×C3×C3⋊D3—81no—
55C33×C4—81no—
56C3×C3⋊Dic3—81no—
57C2×GL(2,𝔽3)—33no—
58C2×C3×SD16—91no—
59C3×C4×Q8—21no—
60C3×C4×Q8C3×D311yescharacteristic, complemented
61C3×C4×Q8C3⋊D311yescharacteristic, complemented
62C3×C4×D4—91no—
63C2×#215—31no—
64C4×D12—31no—
65C4×SL(2,𝔽3)—23no—
66C4×SL(2,𝔽3)C3×D313yescomplemented
67#859—33no—
68C3×#107—91no—
69C3×#103—91no—
70C3×#102—91no—
71C3×#97—91no—
72#835—31no—
73#809—31no—
74#807—31no—
75#803—31no—
76C22×C3⋊D3—41no—
77C22×C3×D3—61no—
78C22×C3×D3—123no—
79C32×Q8—31no—
80C32×Q8C4×D311yescharacteristic
81C2×C32×C4—31no—
82C2×C32×C4—31no—
83C2×C32×C4—41no—
84C2×C32×C4—43no—
85C2×C32×C4—83no—
86C2×C32×C4—81no—
87C2×C32×C4S411yescharacteristic
88C2×C3⋊Dic3—41no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—123no—
91C3×D12—32no—
92C3×D12—61no—
93C3×C4×D3—122no—
94C3×SL(2,𝔽3)—21no—
95C3×SL(2,𝔽3)—23no—
96C3×SL(2,𝔽3)C4×D313yescomplemented
97C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
98C3×C3⋊C8—32no—
99C3×C3⋊C8—61no—
100#440—91noSylow
101C2×C33—41no—
102C2×C33—41no—
103C2×C33—41no—
104C3×C3⋊D3—82no—
105C2×C3×Q8—21no—
106C2×C3×Q8C2×C3×D311yescharacteristic
107C2×C3×Q8C2×C3⋊D311yescharacteristic
108C2×C3×D4—91no—
109C22×C3×C4—181no—
110C2×D12—31no—
111C2×C4×D3—61no—
112C2×SL(2,𝔽3)—23no—
113C2×SL(2,𝔽3)C2×C3×D313yes—
114GL(2,𝔽3)—36no—
115C3×SD16—92no—
116C3×SD16—92no—
117C2×C3×C8—91no—
118C2×C3×C8—91no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—91no—
123C3×C22⋊C4—181no—
124C3×C42—31no—
125C3×C42—31no—
126C3×C42—61no—
127#215—32no—
128#215—32no—
129(C2×D3)⋊C4—61no—
130(C2×D3)⋊C4—123no—
131C4⋊Dic3—31no—
132C2×C3⋊C8—31no—
133C2×C3⋊C8—31no—
134C22×C32—41no—
135C22×C32—43no—
136C22×C32—83no—
137C22×C32—81no—
138C22×C32C2×S411yescharacteristic
139C2×C3⋊D3—42no—
140C2×C3⋊D3—81no—
141C2×C3⋊D3—81no—
142C2×C3×D3—62no—
143C2×C3×D3—126no—
144C2×C3×D3—121no—
145C2×C3×D3—121no—
146C2×C3×D3—243no—
147C2×C3×D3—243no—
148C32×C4—21no—
149C32×C4—31no—
150C32×C4—31no—
151C32×C4—61no—
152C32×C4—83no—
153C32×C4—81no—
154C32×C4—86no—
155C32×C4—82no—
156C3⋊Dic3—81no—
157C3×Dic3—121no—
158C3×Dic3—243no—
159C2×SD16—91no—
160C4×Q8C3×C3⋊D311yescharacteristic, complemented
161C4×D4—91no—
162#107—91no—
163#103—91no—
164#102—91no—
165#97—91no—
166C33—41noSylow
167C23×C3—181no—
168C22×D3—61no—
169C22×D3—123no—
170C3×Q8—21no—
171C3×Q8—31no—
172C3×Q8—31no—
173C3×Q8—61no—
174C3×Q8C3×C4×D311yescharacteristic
175C3×Q8C4×C3⋊D311yescharacteristic
176C3×D4—92no—
177C3×D4—181no—
178C2×C3×C4—21no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—43no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—83no—
187C2×C3×C4—181no—
188C2×C3×C4—362no—
189C2×C3×C4C3×S411yescharacteristic
190C2×C3×C4C3⋊S411yescharacteristic
191C2×Dic3—61no—
192C2×Dic3—123no—
193D12—32no—
194D12—61no—
195C4×D3—122no—
196SL(2,𝔽3)—23no—
197SL(2,𝔽3)C3×C4×D313yescomplemented
198C3×C8—92no—
199C3×C8—181no—
200C3⋊C8—32no—
201C3⋊C8—61no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—43no—
206C2×C32—43no—
207C2×C32—43no—
208C2×C32—83no—
209C2×C32—83no—
210C2×C32—83no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32—81no—
214C2×C32C4×S411yescharacteristic
215C2×C32#98311yescharacteristic
216C2×C32#98611yescharacteristic
217C3⋊D3—82no—
218C3×D3—122no—
219C3×D3—246no—
220C2×Q8C2×C3×C3⋊D311yescharacteristic
221C2×D4—91no—
222C22×C4—181no—
223SD16—92no—
224SD16—92no—
225C2×C8—91no—
226C2×C8—91no—
227C4⋊C4—31no—
228C4⋊C4—91no—
229C22⋊C4—181no—
230C42—31no—
231C22×C3—21no—
232C22×C3—43no—
233C22×C3—83no—
234C22×C3—182no—
235C22×C3—361no—
236C22×C3—361no—
237C22×C3C2×C3×S411yescharacteristic
238C22×C3C2×C3⋊S411yescharacteristic
239C2×D3—62no—
240C2×D3—126no—
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—243no—
244C2×D3—243no—
245C3×C4—22no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—83no—
257C3×C4—86no—
258C3×C4—121no—
259C3×C4—361no—
260Dic3—121no—
261Dic3—243no—
262C32—41no—
263C32—43no—
264C32—83no—
265C32—81no—
266C32#539411yescharacteristic, complemented
267C23—181no—
268Q8—31no—
269Q8C3×C4×C3⋊D311yescharacteristic
270D4—92no—
271D4—181no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—181no—
275C2×C4—362no—
276C2×C4C3×C3⋊S411yescharacteristic
277C8—92no—
278C8—181no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—43no—
283C2×C3—43no—
284C2×C3—43no—
285C2×C3—83no—
286C2×C3—83no—
287C2×C3—83no—
288C2×C3—362no—
289C2×C3C3×C4×S411yescharacteristic
290C2×C3C3×#98311yescharacteristic
291C2×C3C3×#98611yescharacteristic
292C2×C3C4×C3⋊S411yescharacteristic
293C2×C3#6426411yescharacteristic
294C2×C3#6426711yescharacteristic
295D3—122no—
296D3—246no—
297C22—182no—
298C22—361no—
299C22—361no—
300C22C2×C3×C3⋊S411yescharacteristic
301C4—21no—
302C4—31no—
303C4—31no—
304C4—61no—
305C4—361no—
306C3—21no—
307C3—43no—
308C3—83no—
309C3C3×#539411yescharacteristic, complemented
310C3#1059362811yescharacteristic, direct product factor
311C2—362no—
312C2C3×C4×C3⋊S411yescharacteristic
313C2C3×#6426411yescharacteristic
314C2C3×#6426711yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×SL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
4. #10593628C3direct (N × H)
8. C3×C3⋊GL(2,𝔽3)C4semidirect
12. C32×C4×Q8D3semidirect
17. C3×C4×SL(2,𝔽3)D3semidirect
18. C3×C4×SL(2,𝔽3)C2×C3semidirect
19. C3⋊#859C2×C3semidirect
28. C32×SL(2,𝔽3)C2×C4semidirect
39. C3⋊GL(2,𝔽3)C3×C4semidirect
60. C3×C4×Q8C3×D3semidirect
61. C3×C4×Q8C3⋊D3semidirect
66. C4×SL(2,𝔽3)C3×D3semidirect
96. C3×SL(2,𝔽3)C4×D3semidirect
97. C3×SL(2,𝔽3)C2×C3×C4semidirect
160. C4×Q8C3×C3⋊D3semidirect
197. SL(2,𝔽3)C3×C4×D3semidirect
266. C32#5394semidirect
309. C3C3×#5394semidirect
310. C3#10593628direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
C32×C4×SL(2,𝔽3)2
C3×C3⋊#8592
#105936283
C3×#53943
C3×#50263
C3×#37524