Finite Group Databasebeta

← C3×#10593223 #13165899

Subgroups — C3×#10593223 #13165899

Order 1728.

336 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C3×#64265C211yesmaximal, characteristic, complemented
2C32×#986C211yesmaximal, characteristic
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4C3×#5406—31nomaximal
5#10593223C311yesmaximal, characteristic, direct product factor
6C3×#5200—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊#226—21no—
9C32×GL(2,𝔽3)—21no—
10C3×#3716—41nomaximal
11#64265C2×C311yescharacteristic, complemented
12C3×#986—21no—
13C3×#986D311yescharacteristic
14C3×#986C2×C311yescharacteristic
15C3×#984—31no—
16C3×#984—61no—
17C32×AGL(1,ℤ/8)—31no—
18C3×#951—31no—
19C3×#948D311yescharacteristic, complemented
20C2×C3×#214—31no—
21C3×#131⋊C3—31no—
22C3⋊(#131⋊C3)—21no—
23C3⋊(#131⋊C3)C2×C311yescharacteristic, complemented
24C3×#833—31no—
25C3×#824—31no—
26C3×#822—31no—
27C3×#634—41no—
28C32×D12—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5406—31no—
32C3×#471—91no—
33#5200—31no—
34C32×#41C2×D311yescharacteristic
35C2×C32×D4—31no—
36C2×C3×Dic6—31no—
37C3×#41⋊C3—21no—
38C3×#41⋊C3—21no—
39C3×#41⋊C3—21no—
40C3×#41⋊C3—41no—
41C3×#41⋊C3C2×D311yescharacteristic
42C3×#41⋊C3C22×C311yescharacteristic
43C3⋊#226—21no—
44C3×GL(2,𝔽3)—22no—
45C3×GL(2,𝔽3)—21no—
46C3×GL(2,𝔽3)—21no—
47C3×#226—62no—
48C3×#226—61no—
49C32×SD16—61no—
50C32×D8—61no—
51C32×M16—31no—
52C3×#216—61no—
53C3×#214—61no—
54C3×#214—61no—
55C3×#214—61no—
56C3×#208—31no—
57C3×#208—31no—
58C2×C3×C3⋊C8—31no—
59C3×#204—121no—
60#3716—41no—
61C2×C32×D3—81no—
62C33×C4—41no—
63C3×C3⋊Dic3—81no—
64#986C3×D311yescharacteristic
65#984—31no—
66#984—61no—
67C3×#137—91no—
68C3×AGL(1,ℤ/8)—31no—
69C3×AGL(1,ℤ/8)—31no—
70C3×AGL(1,ℤ/8)—61no—
71C2×C3×SD16—91no—
72C3×#131—31no—
73#951—31no—
74#948C3×D311yescharacteristic, complemented
75C2×#214—31no—
76#131⋊C3—31no—
77#131⋊C3—61no—
78C3×#108—91no—
79C3×#101—91no—
80C3×#100—91no—
81#833—31no—
82#824—31no—
83#822—31no—
84C23×C32—61no—
85C32×Q8D1211yescharacteristic
86C32×D4—31no—
87C32×D4—61no—
88C32×D4—61no—
89C2×C32×C4—31no—
90#634—41no—
91C2×C3×Dic3—61no—
92C3×D12—41no—
93C3×D12—41no—
94C3×D12—81no—
95C3×Dic6—61no—
96C3×Dic6—61no—
97C3×Dic6—121no—
98C3×Dic6—241no—
99C3×SL(2,𝔽3)—21no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—41no—
103C3×SL(2,𝔽3)#6711yescharacteristic
104C3×SL(2,𝔽3)C3×D411yescharacteristic
105C32×C8—61no—
106C32×C8—121no—
107C3×C3⋊C8—31no—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—61no—
110C3×C3⋊C8—81no—
111C3×C3⋊C8S411yescharacteristic
112#471—91noSylow
113C2×C33—41no—
114C32×D3—161no—
115C3×#41—21no—
116C3×#41D3211yescharacteristic
117C3×#41C2×C3×D311yescharacteristic
118C2×C3×Q8—91no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—61no—
122C2×Dic6—31no—
123#41⋊C3—21no—
124#41⋊C3—21no—
125#41⋊C3—41no—
126#41⋊C3C2×C3×D311yescharacteristic
127GL(2,𝔽3)—21no—
128#226—62no—
129#226—61no—
130C3×Q16—181no—
131C3×SD16—62no—
132C3×SD16—61no—
133C3×SD16—61no—
134C3×SD16—181no—
135C3×SD16—181no—
136C3×SD16—181no—
137C3×D8—61no—
138C3×D8—61no—
139C3×D8—62no—
140C3×M16—31no—
141C3×M16—31no—
142C3×M16—61no—
143C3×M16—91no—
144C3×M16—91no—
145C2×C3×C8—91no—
146#216—61no—
147#214—61no—
148#214—61no—
149#214—61no—
150#208—31no—
151#208—31no—
152C2×C3⋊C8—31no—
153#204—121no—
154C22×C32—31no—
155C22×C32—121no—
156C22×C32—241no—
157C2×C3×D3—81no—
158C2×C3×D3—81no—
159C2×C3×D3—82no—
160C32×C4—31no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—81no—
164C32×C4—81no—
165C32×C4—81no—
166C32×C4—161no—
167C32×C4C2×S411yescharacteristic
168C3⋊Dic3—81no—
169C3×Dic3—121no—
170C3×Dic3—241no—
171C3×Dic3—242no—
172#137—91no—
173AGL(1,ℤ/8)—31no—
174C2×SD16—91no—
175#131—31no—
176#108—91no—
177#101—91no—
178#100—91no—
179C33—41noSylow
180C23×C3—61no—
181C23×C3—61no—
182C23×C3—121no—
183C3×Q8—21no—
184C3×Q8—181no—
185C3×Q8—181no—
186C3×Q8#62611yescharacteristic
187C3×Q8C3×D1211yescharacteristic
188C3×D4—31no—
189C3×D4—31no—
190C3×D4—61no—
191C3×D4—61no—
192C3×D4—61no—
193C3×D4—62no—
194C3×D4—61no—
195C3×D4—61no—
196C3×D4—121no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—61no—
200C2×C3×C4—181no—
201C2×Dic3—61no—
202D12—41no—
203Dic6—61no—
204Dic6—61no—
205Dic6—121no—
206Dic6—241no—
207SL(2,𝔽3)—21no—
208SL(2,𝔽3)—21no—
209SL(2,𝔽3)—41no—
210SL(2,𝔽3)C3×#6711yescharacteristic
211C3×C8—31no—
212C3×C8—61no—
213C3×C8—61no—
214C3×C8—62no—
215C3×C8—91no—
216C3×C8—121no—
217C3×C8—181no—
218C3×C8—241no—
219C3⋊C8—31no—
220C3⋊C8—61no—
221C3⋊C8C3×S411yescharacteristic
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—61no—
225C2×C32—81no—
226C2×C32—81no—
227C2×C32—81no—
228C2×C32—161no—
229C2×C32—241no—
230C2×C32C4⋊S411yescharacteristic
231C3×D3—161no—
232C3×D3—161no—
233C3×D3—162no—
234#41C3×D3211yescharacteristic
235C2×Q8—91no—
236C2×D4—31no—
237Q16—181no—
238SD16—61no—
239SD16—181no—
240SD16—181no—
241SD16—181no—
242D8—61no—
243M16—31no—
244M16—91no—
245M16—91no—
246C2×C8—91no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—61no—
250C22×C3—122no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—241no—
254C22×C3—241no—
255C22×C3—242no—
256C2×D3—81no—
257C3×C4—21no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—41no—
261C3×C4—61no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—161no—
265C3×C4—361no—
266C3×C4D3×S411yescharacteristic
267C3×C4C2×C3×S411yescharacteristic
268Dic3—121no—
269Dic3—241no—
270Dic3—242no—
271C32—41no—
272C32—41no—
273C32—81no—
274C32—81no—
275C32—81no—
276C32—161no—
277C32#540611yescharacteristic, complemented
278C23—61no—
279Q8—181no—
280Q8—181no—
281Q8C3×#62611yescharacteristic
282D4—31no—
283D4—61no—
284D4—61no—
285C2×C4—31no—
286C2×C4—181no—
287C8—31no—
288C8—61no—
289C8—91no—
290C8—181no—
291C2×C3—21no—
292C2×C3—41no—
293C2×C3—61no—
294C2×C3—61no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3—121no—
298C2×C3—161no—
299C2×C3—242no—
300C2×C3—241no—
301C2×C3—241no—
302C2×C3#6420711yescharacteristic
303C2×C3C3×C4⋊S411yescharacteristic
304D3—161no—
305C22—31no—
306C22—121no—
307C22—241no—
308C4—31no—
309C4—361no—
310C4C3×D3×S411yescharacteristic
311C3—21no—
312C3—41no—
313C3—81no—
314C3—81no—
315C3—161no—
316C3#1059322311yescharacteristic, direct product factor
317C3C3×#540611yescharacteristic, complemented
318C2—61no—
319C2—241no—
320C2C3×#6420711yescharacteristic
subgroupG/H
centerC2×C3#64207
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
prime radicalC32×#986C2
hypercenterC3×C4D3×S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64265C2semidirect
3. C3×C3⋊(#131⋊C3)C2semidirect
5. #10593223C3direct (N × H)
11. #64265C2×C3semidirect
19. C3×#948D3semidirect
23. C3⋊(#131⋊C3)C2×C3semidirect
74. #948C3×D3semidirect
277. C32#5406semidirect
316. C3#10593223direct (N × H)
317. C3C3×#5406semidirect

Maximal subgroups

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

subgroupindex
C3×#642652
C32×#9862
C3×C3⋊(#131⋊C3)2
C3×#54063
#105932233
C3×#52003
C3×#37164