Finite Group Databasebeta

← C3×#10593226 #13165902

Subgroups — C3×#10593226 #13165902

Order 1728.

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

normal subgroups only

This group has 318 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
2C32×#984C211yesmaximal, characteristic
3C3×C3⋊(#131⋊C3)C211yesmaximal, characteristic
4C3×#5403—31nomaximal
5#10593226C311yesmaximal, characteristic, direct product factor
6C3×#5204—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×C3⋊#226—21no—
9C32×#226—21no—
10C3×#3719—41nomaximal
11#64265C2×C311yescharacteristic
12C3×#984—21no—
13C3×#984—31no—
14C3×#984—61no—
15C3×#984D311yescharacteristic
16C3×#984C2×C311yescharacteristic
17C32×#137—31no—
18C3×#951—31no—
19C3×#948D311yescharacteristic
20C2×C3×#216—31no—
21C3×#131⋊C3—31no—
22C3⋊(#131⋊C3)—21no—
23C3⋊(#131⋊C3)C2×C311yescharacteristic
24C3×#836—31no—
25C3×#824—31no—
26C3×#822—31no—
27C3×#634—41no—
28C32×Dic6—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5403—31no—
32C3×#473—91no—
33#5204—31no—
34C32×#41C2×D311yescharacteristic
35C2×C32×Q8—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×#226—22no—
45C3×#226—21no—
46C3×#226—21no—
47C3×#226—62no—
48C3×#226—61no—
49C32×Q16—61no—
50C32×SD16—61no—
51C32×M16—31no—
52C3×#216—61no—
53C3×#216—61no—
54C3×#216—61no—
55C3×#214—61no—
56C3×#208—31no—
57C3×#208—31no—
58C2×C3×C3⋊C8—31no—
59C3×Dic12—121no—
60#3719—41no—
61C33×C4—41no—
62C3×C3⋊Dic3—81no—
63C32×Dic3—81no—
64#984—31no—
65#984—61no—
66#984C3×D311yescharacteristic
67C3×#137—31no—
68C3×#137—31no—
69C3×#137—61no—
70C3×#137—91no—
71C2×C3×Q16—91no—
72C3×#131—31no—
73#951—31no—
74#948C3×D311yescharacteristic
75C2×#216—31no—
76#131⋊C3—31no—
77#131⋊C3—61no—
78C3×#108—91no—
79C3×#101—91no—
80C3×#101—91no—
81#836—31no—
82#824—31no—
83#822—31no—
84C32×Q8—61no—
85C32×Q8—61no—
86C32×Q8D1211yescharacteristic
87C32×D4—31no—
88C2×C32×C4—31no—
89C2×C32×C4—61no—
90#634—41no—
91C2×C3×Dic3—61no—
92C3×Dic6—41no—
93C3×Dic6—41no—
94C3×Dic6—61no—
95C3×Dic6—61no—
96C3×Dic6—81no—
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#473—91noSylow
113C2×C33—41no—
114C3×#41—21no—
115C3×#41D3211yescharacteristic
116C3×#41C2×C3×D311yescharacteristic
117C2×C3×Q8—31no—
118C2×C3×Q8—31no—
119C2×C3×Q8—61no—
120C2×C3×Q8—91no—
121C2×Dic6—31no—
122#41⋊C3—21no—
123#41⋊C3—21no—
124#41⋊C3—41no—
125#41⋊C3C2×C3×D311yescharacteristic
126#226—21no—
127#226—62no—
128#226—61no—
129C3×Q16—62no—
130C3×Q16—61no—
131C3×Q16—61no—
132C3×Q16—181no—
133C3×Q16—181no—
134C3×Q16—181no—
135C3×SD16—61no—
136C3×SD16—61no—
137C3×SD16—62no—
138C3×SD16—181no—
139C3×M16—31no—
140C3×M16—31no—
141C3×M16—61no—
142C3×M16—91no—
143C3×M16—91no—
144C2×C3×C8—91no—
145#216—61no—
146#216—61no—
147#216—61no—
148#214—61no—
149#208—31no—
150#208—31no—
151C2×C3⋊C8—31no—
152Dic12—121no—
153C22×C32—31no—
154C32×C4—31no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4—81no—
158C32×C4—81no—
159C32×C4—81no—
160C32×C4—121no—
161C32×C4—161no—
162C32×C4C2×S411yescharacteristic
163C3⋊Dic3—81no—
164C3×Dic3—81no—
165C3×Dic3—81no—
166C3×Dic3—82no—
167C3×Dic3—121no—
168C3×Dic3—241no—
169C3×Dic3—242no—
170#137—31no—
171#137—91no—
172C2×Q16—91no—
173#131—31no—
174#108—91no—
175#101—91no—
176#101—91no—
177C33—41noSylow
178C3×Q8—21no—
179C3×Q8—61no—
180C3×Q8—61no—
181C3×Q8—62no—
182C3×Q8—61no—
183C3×Q8—61no—
184C3×Q8—121no—
185C3×Q8—181no—
186C3×Q8—181no—
187C3×Q8#62611yescharacteristic
188C3×Q8C3×D1211yescharacteristic
189C3×D4—31no—
190C3×D4—31no—
191C3×D4—61no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—121no—
198C2×C3×C4—181no—
199C2×Dic3—61no—
200Dic6—41no—
201Dic6—61no—
202Dic6—61no—
203Dic6—121no—
204Dic6—241no—
205SL(2,𝔽3)—21no—
206SL(2,𝔽3)—21no—
207SL(2,𝔽3)—41no—
208SL(2,𝔽3)C3×#6711yescharacteristic
209C3×C8—31no—
210C3×C8—61no—
211C3×C8—61no—
212C3×C8—62no—
213C3×C8—91no—
214C3×C8—121no—
215C3×C8—181no—
216C3×C8—241no—
217C3⋊C8—31no—
218C3⋊C8—61no—
219C3⋊C8C3×S411yescharacteristic
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—61no—
223C2×C32—81no—
224C2×C32—81no—
225C2×C32—81no—
226C2×C32—161no—
227C2×C32C4⋊S411yescharacteristic
228#41C3×D3211yescharacteristic
229C2×Q8—31no—
230C2×Q8—91no—
231Q16—61no—
232Q16—181no—
233Q16—181no—
234Q16—181no—
235SD16—61no—
236SD16—181no—
237M16—31no—
238M16—91no—
239M16—91no—
240C2×C8—91no—
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—61no—
244C3×C4—21no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—41no—
248C3×C4—61no—
249C3×C4—81no—
250C3×C4—81no—
251C3×C4—122no—
252C3×C4—121no—
253C3×C4—121no—
254C3×C4—161no—
255C3×C4—361no—
256C3×C4D3×S411yescharacteristic
257C3×C4C2×C3×S411yescharacteristic
258Dic3—81no—
259Dic3—121no—
260Dic3—241no—
261Dic3—242no—
262C32—41no—
263C32—41no—
264C32—81no—
265C32—81no—
266C32—81no—
267C32—161no—
268C32#540311yescharacteristic, complemented
269Q8—61no—
270Q8—61no—
271Q8—181no—
272Q8—181no—
273Q8C3×#62611yescharacteristic
274D4—31no—
275C2×C4—31no—
276C2×C4—61no—
277C2×C4—181no—
278C8—31no—
279C8—61no—
280C8—91no—
281C8—181no—
282C2×C3—21no—
283C2×C3—41no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—81no—
287C2×C3—81no—
288C2×C3—121no—
289C2×C3—161no—
290C2×C3#6420711yescharacteristic
291C2×C3C3×C4⋊S411yescharacteristic
292C22—31no—
293C4—31no—
294C4—121no—
295C4—361no—
296C4C3×D3×S411yescharacteristic
297C3—21no—
298C3—41no—
299C3—81no—
300C3—81no—
301C3—161no—
302C3#1059322611yescharacteristic, direct product factor
303C3C3×#540311yescharacteristic, complemented
304C2—61no—
305C2C3×#6420711yescharacteristic
subgroupG/H
centerC2×C3#64207
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C4⋊S4
prime radicalC32×#41⋊C3C22
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
5. #10593226C3direct (N × H)
268. C32#5403semidirect
302. C3#10593226direct (N × H)
303. C3C3×#5403semidirect

Maximal subgroups

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

subgroupindex
C3×#642652
C32×#9842
C3×C3⋊(#131⋊C3)2
C3×#54033
#105932263
C3×#52043
C3×#37194