Finite Group Databasebeta

← C3×#10593232 #13165908

Subgroups — C3×#10593232 #13165908

Order 1728.

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

normal subgroups only

This group has 358 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×#1010⋊C3C211yesmaximal, characteristic
2C3×#64265C211yesmaximal, characteristic, complemented
3C32×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#5430—31nomaximal
5#10593232C311yesmaximal, characteristic, direct product factor
6C3×#4826—31nomaximal
7C32×#41⋊C3C2211yescharacteristic
8C3×D3×SL(2,𝔽3)—21no—
9C3×C3⋊#226—21no—
10C3×#3716—41nomaximal
11C3×#1010D311yescharacteristic
12C3×2−1+4⋊C3—31no—
13#1010⋊C3—21no—
14#1010⋊C3C2×C311yescharacteristic
15#64265C2×C311yescharacteristic, complemented
16C3×#984—31no—
17C3×#984—61no—
18C3×#951—31no—
19C3×#909—31no—
20C3×#869—31no—
21C3×GU(2,𝔽3)—21no—
22C3×GU(2,𝔽3)D311yescharacteristic, complemented
23C3×GU(2,𝔽3)C2×C311yescharacteristic, complemented
24C32×#104—31no—
25C3×#824—31no—
26C3×#805—31no—
27C3×#634—41no—
28C32×D12—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30C32×C3⋊C8—41no—
31#5430—31no—
32C3×#456—91no—
33#4826—31no—
34C32×#41C2×D311yescharacteristic
35C3×D3×Q8—21no—
36C3×#237—61no—
37C3×#235—31no—
38C2×C3×Dic6—31no—
39C2×C3×Dic6—31no—
40C3×#41⋊C3—21no—
41C3×#41⋊C3—21no—
42C3×#41⋊C3—21no—
43C3×#41⋊C3—41no—
44C3×#41⋊C3C2×D311yescharacteristic
45C3×#41⋊C3C22×C311yescharacteristic
46C2×C3×SL(2,𝔽3)—61no—
47D3×SL(2,𝔽3)—22no—
48D3×SL(2,𝔽3)—21no—
49C3⋊#226—21no—
50C3×#226—62no—
51C3×#226—61no—
52C32×M16—31no—
53C32×C42—31no—
54C3×#216—61no—
55C3×#214—61no—
56C3×#214—121no—
57C3×C4⋊Dic3—62no—
58C3×#208—31no—
59C3×Dic12—61no—
60C3×#204—61no—
61#3716—41no—
62C2×C32×D3—81no—
63C33×C4—41no—
64C3×C3⋊Dic3—81no—
65C3×2−1+4—31no—
66#1010C3×D311yescharacteristic
672−1+4⋊C3—31no—
682−1+4⋊C3—61no—
69#984—31no—
70#984—61no—
71C3×#137—91no—
72C3×#137—91no—
73C3×C4⋊Q8—91no—
74#951—31no—
75#909—31no—
76#869—31no—
77GU(2,𝔽3)C3×D311yescharacteristic, complemented
78C3×#104—31no—
79C3×#104—31no—
80C3×#104—61no—
81C3×#104—91no—
82C3×#101—91no—
83#824—31no—
84#805—31no—
85C32×Q8#6711yescharacteristic
86C32×D4—31no—
87C32×D4—121no—
88C2×C32×C4—31no—
89C2×C32×C4—61no—
90#634—41no—
91C3×#67—61no—
92C2×C3×Dic3—32no—
93C2×C3×Dic3—61no—
94C3×D12—41no—
95C3×D12—81no—
96C3×D12S411yescharacteristic
97C3×C4×D3—61no—
98C3×Dic6—31no—
99C3×Dic6—61no—
100C3×Dic6—61no—
101C3×Dic6—61no—
102C3×Dic6—121no—
103C3×Dic6—241no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—21no—
106C3×SL(2,𝔽3)—21no—
107C3×SL(2,𝔽3)—41no—
108C3×SL(2,𝔽3)D1211yescharacteristic
109C3×SL(2,𝔽3)C3×D411yescharacteristic
110C32×C8—61no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—81no—
115#456—91noSylow
116C2×C33—41no—
117C32×D3—161no—
118C3×#41—21no—
119C3×#41—91no—
120C3×#41—181no—
121C3×#41D3211yescharacteristic
122C3×#41C2×C3×D311yescharacteristic
123C2×C3×Q8—61no—
124C2×C3×Q8—91no—
125C2×C3×Q8—91no—
126D3×Q8—21no—
127#237—61no—
128#235—31no—
129C2×Dic6—31no—
130C2×Dic6—31no—
131#41⋊C3—21no—
132#41⋊C3—21no—
133#41⋊C3—41no—
134#41⋊C3C2×C3×D311yescharacteristic
135C2×SL(2,𝔽3)—62no—
136C2×SL(2,𝔽3)—61no—
137#226—62no—
138#226—61no—
139C3×Q16—181no—
140C3×Q16—181no—
141C3×SD16—181no—
142C3×SD16—181no—
143C3×M16—31no—
144C3×M16—31no—
145C3×M16—61no—
146C3×M16—91no—
147C3×C4⋊C4—182no—
148C3×C42—31no—
149C3×C42—31no—
150C3×C42—61no—
151#216—61no—
152#214—61no—
153#214—121no—
154C4⋊Dic3—62no—
155#208—31no—
156Dic12—61no—
157#204—61no—
158C22×C32—31no—
159C22×C32—241no—
160C2×C3×D3—21no—
161C2×C3×D3—81no—
162C2×C3×D3—82no—
163C32×C4—31no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4—62no—
167C32×C4—81no—
168C32×C4—81no—
169C32×C4—81no—
170C32×C4—161no—
171C32×C4C2×S411yescharacteristic
172C3⋊Dic3—81no—
173C3×Dic3—61no—
174C3×Dic3—121no—
175C3×Dic3—241no—
176C3×Dic3—242no—
1772−1+4—31no—
178#137—91no—
179#137—91no—
180C4⋊Q8—91no—
181#104—31no—
182#104—91no—
183#101—91no—
184C33—41noSylow
185C3×Q8—21no—
186C3×Q8—91no—
187C3×Q8—181no—
188C3×Q8—181no—
189C3×Q8—181no—
190C3×Q8#62611yescharacteristic
191C3×Q8C3×#6711yescharacteristic
192C3×D4—31no—
193C3×D4—31no—
194C3×D4—31no—
195C3×D4—61no—
196C3×D4—121no—
197C3×D4—181no—
198C3×D4—241no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—92no—
205C2×C3×C4—121no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208#67—61no—
209C2×Dic3—32no—
210C2×Dic3—61no—
211D12C3×S411yescharacteristic
212C4×D3—61no—
213Dic6—31no—
214Dic6—61no—
215Dic6—61no—
216Dic6—61no—
217Dic6—121no—
218Dic6—241no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)—21no—
221SL(2,𝔽3)—41no—
222SL(2,𝔽3)C3×D1211yescharacteristic
223C3×C8—61no—
224C3×C8—61no—
225C3×C8—121no—
226C3×C8—181no—
227C3⋊C8—41no—
228C3⋊C8—61no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—61no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—161no—
236C2×C32—481no—
237C2×C32GL(2,ℤ/4)11yescharacteristic
238C3×D3—41no—
239C3×D3—161no—
240C3×D3—162no—
241#41—91no—
242#41—181no—
243#41C3×D3211yescharacteristic
244C2×Q8—61no—
245C2×Q8—91no—
246C2×Q8—91no—
247Q16—181no—
248Q16—181no—
249SD16—181no—
250SD16—181no—
251M16—31no—
252M16—91no—
253C4⋊C4—182no—
254C42—31no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—241no—
260C22×C3—242no—
261C2×D3—21no—
262C3×C4—21no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—41no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—62no—
269C3×C4—81no—
270C3×C4—81no—
271C3×C4—122no—
272C3×C4—161no—
273C3×C4—181no—
274C3×C4—361no—
275C3×C4D3×S411yescharacteristic
276C3×C4C2×C3×S411yescharacteristic
277Dic3—61no—
278Dic3—121no—
279Dic3—241no—
280Dic3—242no—
281C32—41no—
282C32—41no—
283C32—81no—
284C32—81no—
285C32—81no—
286C32—161no—
287C32#543011yescharacteristic, complemented
288Q8—91no—
289Q8—181no—
290Q8—181no—
291Q8—181no—
292Q8C3×#62611yescharacteristic
293D4—31no—
294D4—31no—
295D4—181no—
296C2×C4—31no—
297C2×C4—61no—
298C2×C4—92no—
299C2×C4—181no—
300C2×C4—181no—
301C8—61no—
302C8—181no—
303C2×C3—21no—
304C2×C3—41no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—81no—
308C2×C3—81no—
309C2×C3—121no—
310C2×C3—121no—
311C2×C3—161no—
312C2×C3—481no—
313C2×C3—482no—
314C2×C3#6421011yescharacteristic
315C2×C3C3×GL(2,ℤ/4)11yescharacteristic
316D3—41no—
317C22—31no—
318C22—61no—
319C4—31no—
320C4—62no—
321C4—181no—
322C4—361no—
323C4C3×D3×S411yescharacteristic
324C3—21no—
325C3—41no—
326C3—81no—
327C3—81no—
328C3—161no—
329C3#1059323211yescharacteristic, direct product factor
330C3C3×#543011yescharacteristic, complemented
331C2—61no—
332C2—121no—
333C2C3×#6421011yescharacteristic
subgroupG/H
centerC2×C3#64210
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC4C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32GL(2,ℤ/4)
prime radicalC3×#1010⋊C3C2
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
2. C3×#64265C2semidirect
3. C32×GU(2,𝔽3)C2semidirect
5. #10593232C3direct (N × H)
15. #64265C2×C3semidirect
22. C3×GU(2,𝔽3)D3semidirect
23. C3×GU(2,𝔽3)C2×C3semidirect
77. GU(2,𝔽3)C3×D3semidirect
287. C32#5430semidirect
329. C3#10593232direct (N × H)
330. C3C3×#5430semidirect

Maximal subgroups

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

subgroupindex
C3×#1010⋊C32
C3×#642652
C32×GU(2,𝔽3)2
C3×#54303
#105932323
C3×#48263
C3×#37164