Finite Group Databasebeta

← C3×#10593212 #13165888

Subgroups — C3×#10593212 #13165888

Order 1728.

346 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 54 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×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×C3⋊#859C211yesmaximal, characteristic
3C32×#859C211yesmaximal, characteristic
4C3×#5386—31nomaximal
5#10593212C311yesmaximal, characteristic, direct product factor
6C3×#4791—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#627—41nomaximal
9C3×Q8×Dic3D311yescharacteristic
10C3×#888—31no—
11C3×C4×SL(2,𝔽3)—31no—
12Dic3×SL(2,𝔽3)—21no—
13Dic3×SL(2,𝔽3)C2×C311yescharacteristic
14C3⋊#859C2×C311yescharacteristic
15C3×#859—21no—
16C3×#859—31no—
17C3×#859—61no—
18C3×#859D311yescharacteristic
19C3×#859C2×C311yescharacteristic
20C32×#103—31no—
21C3×#835—31no—
22C3×#817—31no—
23C3×Dic3⋊C8—31no—
24C3×#808—31no—
25C2×C3×C3⋊Dic3—41no—
26C2×C32×Dic3—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)Q811yescharacteristic, complemented
29C3×#627—44no—
30#5386—31no—
31C3×#475—91no—
32#4791—31no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C3×Dic6—61no—
35C2×C3×Dic6—121no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—41no—
40C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
41C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
42C2×#627—41no—
43C2×C32×C8—31no—
44C32×C4⋊C4—31no—
45C3×C4⋊Dic3—31no—
46C3×C4⋊Dic3—31no—
47C3×Dic3⋊C4—61no—
48C3×C4×Dic3—31no—
49C2×C3×C3⋊C8—31no—
50C22×C33—41no—
51C3×C3⋊Dic3—42no—
52C32×Dic3—42no—
53C32×Dic3—42no—
54C3×C4⋊Q8—91no—
55C3×C4×Q8—31no—
56Q8×Dic3C3×D311yescharacteristic
57#888—31no—
58C4×SL(2,𝔽3)—31no—
59C4×SL(2,𝔽3)—61no—
60#859—31no—
61#859—61no—
62#859C3×D311yescharacteristic
63C3×#106—91no—
64C3×#106—91no—
65C3×C4⋊C8—91no—
66C3×#103—31no—
67C3×#103—31no—
68C3×#103—61no—
69C3×#103—91no—
70#835—31no—
71#817—31no—
72Dic3⋊C8—31no—
73#808—31no—
74C32×Q8—31no—
75C32×Q8Dic611yescharacteristic, complemented
76C2×C32×C4—31no—
77C2×C32×C4—61no—
78C2×C32×C4—121no—
79C2×C3⋊Dic3—41no—
80C2×C3×Dic3—31no—
81C2×C3×Dic3—41no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—81no—
86C2×C3×Dic3—81no—
87C2×C3×Dic3—121no—
88C2×C3×Dic3—241no—
89C2×C3×Dic3S411yescharacteristic
90C3×Dic6—124no—
91C3×Dic6—122no—
92C3×SL(2,𝔽3)—21no—
93C3×SL(2,𝔽3)—21no—
94C3×SL(2,𝔽3)—21no—
95C3×SL(2,𝔽3)—41no—
96C3×SL(2,𝔽3)Dic611yescharacteristic, complemented
97C3×SL(2,𝔽3)C3×Q811yescharacteristic, complemented
98#627—44no—
99C32×C8—61no—
100C3×C3⋊C8—61no—
101#475—91noSylow
102C2×C33—41no—
103C2×C33—41no—
104C2×C33—41no—
105C2×C3×Q8—21no—
106C2×C3×Q8—181no—
107C2×C3×Q8D3211yescharacteristic
108C2×C3×Q8C2×C3×D311yescharacteristic
109C2×Dic6—61no—
110C2×Dic6—121no—
111C2×SL(2,𝔽3)—21no—
112C2×SL(2,𝔽3)—21no—
113C2×SL(2,𝔽3)—41no—
114C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
115C2×C3×C8—31no—
116C2×C3×C8—31no—
117C2×C3×C8—61no—
118C2×C3×C8—91no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—91no—
124C3×C4⋊C4—181no—
125C3×C42—91no—
126C4⋊Dic3—31no—
127C4⋊Dic3—31no—
128Dic3⋊C4—61no—
129C4×Dic3—31no—
130C2×C3⋊C8—31no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32—81no—
134C22×C32—81no—
135C22×C32—81no—
136C22×C32—161no—
137C22×C32C2×S411yescharacteristic
138C32×C4—31no—
139C32×C4—31no—
140C32×C4—122no—
141C32×C4—121no—
142C3⋊Dic3—42no—
143C3×Dic3—42no—
144C3×Dic3—42no—
145C3×Dic3—42no—
146C3×Dic3—61no—
147C3×Dic3—82no—
148C3×Dic3—82no—
149C3×Dic3—122no—
150C3×Dic3—121no—
151C3×Dic3—242no—
152C3×Dic3GL(2,𝔽3)12yes—
153C4⋊Q8—91no—
154C4×Q8—31no—
155#106—91no—
156#106—91no—
157C4⋊C8—91no—
158#103—31no—
159#103—91no—
160C33—41noSylow
161C3×Q8—21no—
162C3×Q8—31no—
163C3×Q8—31no—
164C3×Q8—61no—
165C3×Q8—362no—
166C3×Q8#62711yescharacteristic, complemented
167C3×Q8C3×Dic611yescharacteristic, complemented
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—91no—
175C2×C3×C4—121no—
176C2×C3×C4—121no—
177C2×C3×C4—181no—
178C2×C3×C4—241no—
179C2×Dic3—31no—
180C2×Dic3—41no—
181C2×Dic3—61no—
182C2×Dic3—121no—
183C2×Dic3—241no—
184C2×Dic3C3×S411yescharacteristic
185Dic6—124no—
186Dic6—122no—
187SL(2,𝔽3)—21no—
188SL(2,𝔽3)—21no—
189SL(2,𝔽3)—41no—
190SL(2,𝔽3)C3×Dic611yescharacteristic, complemented
191C3×C8—61no—
192C3×C8—61no—
193C3×C8—121no—
194C3×C8—181no—
195C3⋊C8—61no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—81no—
203C2×C32—81no—
204C2×C32—81no—
205C2×C32—81no—
206C2×C32—81no—
207C2×C32—81no—
208C2×C32—81no—
209C2×C32—81no—
210C2×C32—81no—
211C2×C32—161no—
212C2×C32—161no—
213C2×C32—161no—
214C2×C32A4⋊Q811yescharacteristic
215C2×C32C2×GL(2,𝔽3)11yescharacteristic
216C2×C32#98411yescharacteristic
217C2×Q8—181no—
218C2×Q8C3×D3211yescharacteristic
219C2×C8—31no—
220C2×C8—91no—
221C4⋊C4—31no—
222C4⋊C4—91no—
223C4⋊C4—91no—
224C4⋊C4—181no—
225C42—91no—
226C22×C3—21no—
227C22×C3—41no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3—161no—
231C22×C3D3×S411yescharacteristic
232C22×C3C2×C3×S411yescharacteristic
233C3×C4—32no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—122no—
241C3×C4—121no—
242C3×C4—121no—
243C3×C4—181no—
244C3×C4—242no—
245C3×C4—241no—
246C3×C4—361no—
247Dic3—42no—
248Dic3—61no—
249Dic3—122no—
250Dic3—121no—
251Dic3—242no—
252Dic3C3×GL(2,𝔽3)12yes—
253C32—41no—
254C32—41no—
255C32—81no—
256C32—81no—
257C32—81no—
258C32—161no—
259C32#538611yescharacteristic, complemented
260Q8—31no—
261Q8—362no—
262Q8C3×#62711yescharacteristic, complemented
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—61no—
266C2×C4—91no—
267C2×C4—181no—
268C8—61no—
269C8—181no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—81no—
277C2×C3—81no—
278C2×C3—81no—
279C2×C3—81no—
280C2×C3—81no—
281C2×C3—81no—
282C2×C3—161no—
283C2×C3—161no—
284C2×C3—161no—
285C2×C3#6419611yescharacteristic
286C2×C3D3×GL(2,𝔽3)11yescharacteristic
287C2×C3#6420411yescharacteristic
288C2×C3C3×A4⋊Q811yescharacteristic
289C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
290C2×C3C3×#98411yescharacteristic
291C22C3×D3×S411yescharacteristic
292C4—32no—
293C4—31no—
294C4—31no—
295C4—121no—
296C4—181no—
297C4—361no—
298C3—21no—
299C3—41no—
300C3—81no—
301C3—81no—
302C3—161no—
303C3#1059321211yescharacteristic, direct product factor
304C3C3×#538611yescharacteristic, complemented
305C2C3×#6419611yescharacteristic
306C2C3×D3×GL(2,𝔽3)11yescharacteristic
307C2C3×#6420411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C32×SL(2,𝔽3)C22
hypocenterC3×SL(2,𝔽3)C3×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
5. #10593212C3direct (N × H)
28. C32×SL(2,𝔽3)Q8semidirect
75. C32×Q8Dic6semidirect
96. C3×SL(2,𝔽3)Dic6semidirect
97. C3×SL(2,𝔽3)C3×Q8semidirect
166. C3×Q8#627semidirect
167. C3×Q8C3×Dic6semidirect
190. SL(2,𝔽3)C3×Dic6semidirect
259. C32#5386semidirect
262. Q8C3×#627semidirect
303. C3#10593212direct (N × H)
304. C3C3×#5386semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×SL(2,𝔽3)2
C3×C3⋊#8592
C32×#8592
C3×#53863
#105932123
C3×#47913
C2×C3×#6274