Finite Group Databasebeta

← C3×#10593214 #13165890

Subgroups — C3×#10593214 #13165890

Order 1728.

458 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 58 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
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4C3×C4×GL(2,𝔽3)—31nomaximal
5#10593214C311yesmaximal, characteristic, direct product factor
6C3×#4788—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#624—41nomaximal
9C3×C3⋊GL(2,𝔽3)C412yescomplemented
10C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic
11C2×C3×GL(2,𝔽3)—31no—
12C2×C3×GL(2,𝔽3)—61no—
13C3×Q8×Dic3D311yescharacteristic, complemented
14C2×C3×#215—31no—
15C3×#893—31no—
16C3×C4×SL(2,𝔽3)—31no—
17Dic3×SL(2,𝔽3)—21no—
18Dic3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19C3×#859—21no—
20C3×#859D311yescharacteristic, complemented
21C3×#859C2×C311yescharacteristic, complemented
22C32×#103—31no—
23C3×#821—31no—
24C3×C8×Dic3—31no—
25C3×#808—31no—
26C22×C3×C3⋊D3—41no—
27C2×C32×Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30C3×#624—44no—
31C4×GL(2,𝔽3)—31no—
32C3×C4×SD16—91no—
33#4788—31no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C3×D12—31no—
36C2×C3×C4×D3—61no—
37C2×C3×C4×D3—121no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—41no—
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
44C2×#624—41no—
45C3⋊GL(2,𝔽3)C3×C412yescomplemented
46C3×GL(2,𝔽3)—32no—
47C3×GL(2,𝔽3)—62no—
48C2×C32×C8—31no—
49C32×C4⋊C4—31no—
50C3×#215—32no—
51C3×#215—32no—
52C3×(C2×D3)⋊C4—61no—
53C3×C4⋊Dic3—31no—
54C3×C4×Dic3—31no—
55C2×C3×C3⋊C8—31no—
56C22×C33—41no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3—42no—
59C2×C3×C3⋊D3—42no—
60C32×Dic3—42no—
61C32×Dic3—42no—
62C2×GL(2,𝔽3)—31no—
63C2×GL(2,𝔽3)—61no—
64C2×C3×SD16—91no—
65C3×C4×Q8—31no—
66C3×C4×D4—91no—
67Q8×Dic3C3×D311yescharacteristic, complemented
68C2×#215—31no—
69#893—31no—
70C4×SL(2,𝔽3)—31no—
71C4×SL(2,𝔽3)—61no—
72#859C3×D311yescharacteristic, complemented
73C3×#106—91no—
74C3×#103—31no—
75C3×#103—31no—
76C3×#103—61no—
77C3×#102—91no—
78C3×C4×C8—91no—
79#821—31no—
80C8×Dic3—31no—
81#808—31no—
82C22×C3⋊D3—41no—
83C22×C3×D3—61no—
84C22×C3×D3—121no—
85C22×C3×D3—241no—
86C32×Q8—31no—
87C32×Q8C4×D311yescharacteristic, complemented
88C2×C32×C4—31no—
89C2×C32×C4—61no—
90C2×C32×C4—121no—
91C2×C3×Dic3—31no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—81no—
96C2×C3×Dic3—81no—
97C2×C3×Dic3S411yescharacteristic
98C3×D12—32no—
99C3×D12—61no—
100C3×C4×D3—124no—
101C3×C4×D3—122no—
102C3×SL(2,𝔽3)—21no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—41no—
106C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
107C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
108#624—44no—
109C32×C8—61no—
110C3×C3⋊C8—32no—
111C4×SD16—91noSylow
112C2×C33—41no—
113C2×C33—41no—
114C2×C33—41no—
115C3×C3⋊D3—44no—
116C2×C3×Q8—21no—
117C2×C3×Q8D3211yescharacteristic
118C2×C3×Q8C2×C3×D311yescharacteristic
119C2×C3×D4—91no—
120C22×C3×C4—181no—
121C2×D12—31no—
122C2×C4×D3—61no—
123C2×C4×D3—121no—
124C2×SL(2,𝔽3)—21no—
125C2×SL(2,𝔽3)—21no—
126C2×SL(2,𝔽3)—41no—
127C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
128GL(2,𝔽3)—32no—
129GL(2,𝔽3)—62no—
130C3×SD16—92no—
131C3×SD16—92no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—61no—
135C2×C3×C8—91no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—91no—
140C3×C22⋊C4—181no—
141C3×C42—91no—
142#215—32no—
143#215—32no—
144(C2×D3)⋊C4—61no—
145C4⋊Dic3—31no—
146C4×Dic3—31no—
147C2×C3⋊C8—31no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—81no—
151C22×C32—81no—
152C22×C32—81no—
153C22×C32—161no—
154C22×C32C2×S411yescharacteristic
155C2×C3⋊D3—42no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—42no—
158C2×C3×D3—62no—
159C2×C3×D3—122no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C2×C3×D3—242no—
165C2×C3×D3—242no—
166C2×C3×D3—242no—
167C32×C4—31no—
168C32×C4—31no—
169C32×C4—122no—
170C32×C4—121no—
171C3×Dic3—42no—
172C3×Dic3—42no—
173C3×Dic3—42no—
174C3×Dic3—61no—
175C3×Dic3—82no—
176C3×Dic3—82no—
177C3×Dic3GL(2,𝔽3)12yescomplemented
178C2×SD16—91no—
179C4×Q8—31no—
180C4×D4—91no—
181#106—91no—
182#103—31no—
183#102—91no—
184C4×C8—91no—
185C33—41noSylow
186C23×C3—181no—
187C22×D3—61no—
188C22×D3—121no—
189C22×D3—241no—
190C3×Q8—21no—
191C3×Q8—31no—
192C3×Q8—31no—
193C3×Q8—61no—
194C3×Q8#62411yescharacteristic, complemented
195C3×Q8C3×C4×D311yescharacteristic, complemented
196C3×D4—92no—
197C3×D4—181no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—121no—
206C2×C3×C4—121no—
207C2×C3×C4—241no—
208C2×C3×C4—362no—
209C2×Dic3—31no—
210C2×Dic3—41no—
211C2×Dic3C3×S411yescharacteristic
212D12—32no—
213D12—61no—
214C4×D3—124no—
215C4×D3—122no—
216SL(2,𝔽3)—21no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)—41no—
219SL(2,𝔽3)C3×C4×D311yescharacteristic, complemented
220C3×C8—61no—
221C3×C8—61no—
222C3×C8—92no—
223C3×C8—121no—
224C3⋊C8—32no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—81no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—81no—
239C2×C32—81no—
240C2×C32—161no—
241C2×C32—161no—
242C2×C32—161no—
243C2×C32C4×S411yescharacteristic
244C2×C32C2×GL(2,𝔽3)11yescharacteristic
245C2×C32#98511yescharacteristic
246C3⋊D3—44no—
247C3×D3—124no—
248C3×D3—122no—
249C3×D3—244no—
250C2×Q8C3×D3211yescharacteristic
251C2×D4—91no—
252C22×C4—181no—
253SD16—92no—
254SD16—92no—
255C2×C8—31no—
256C2×C8—91no—
257C4⋊C4—31no—
258C4⋊C4—91no—
259C22⋊C4—181no—
260C42—91no—
261C22×C3—21no—
262C22×C3—41no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3—161no—
266C22×C3—182no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3D3×S411yescharacteristic
270C22×C3C2×C3×S411yescharacteristic
271C2×D3—62no—
272C2×D3—122no—
273C2×D3—122no—
274C2×D3—122no—
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—242no—
278C2×D3—242no—
279C2×D3—242no—
280C3×C4—32no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—122no—
288C3×C4—121no—
289C3×C4—121no—
290C3×C4—181no—
291C3×C4—242no—
292C3×C4—241no—
293Dic3—42no—
294Dic3—61no—
295Dic3C3×GL(2,𝔽3)12yescomplemented
296C32—41no—
297C32—41no—
298C32—81no—
299C32—81no—
300C32—81no—
301C32—161no—
302C32C4×GL(2,𝔽3)11yescharacteristic, complemented
303C23—181no—
304Q8—31no—
305Q8C3×#62411yescharacteristic, complemented
306D4—92no—
307D4—181no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—61no—
311C2×C4—91no—
312C2×C4—362no—
313C8—61no—
314C8—92no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—81no—
322C2×C3—81no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3—81no—
326C2×C3—81no—
327C2×C3—161no—
328C2×C3—161no—
329C2×C3—161no—
330C2×C3—362no—
331C2×C3#6419811yescharacteristic
332C2×C3D3×GL(2,𝔽3)11yescharacteristic
333C2×C3(C3⋊S4)⋊C411yescharacteristic
334C2×C3C3×C4×S411yescharacteristic
335C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
336C2×C3C3×#98511yescharacteristic
337D3—124no—
338D3—122no—
339D3—244no—
340C22—182no—
341C22—361no—
342C22—361no—
343C22C3×D3×S411yescharacteristic
344C4—32no—
345C4—31no—
346C4—31no—
347C4—121no—
348C4—181no—
349C3—21no—
350C3—41no—
351C3—81no—
352C3—81no—
353C3—161no—
354C3#1059321411yescharacteristic, direct product factor
355C3C3×C4×GL(2,𝔽3)11yescharacteristic, complemented
356C2—362no—
357C2C3×#6419811yescharacteristic
358C2C3×D3×GL(2,𝔽3)11yescharacteristic
359C2C3×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×SL(2,𝔽3)C2semidirect
3. C32×#859C2semidirect
5. #10593214C3direct (N × H)
9. C3×C3⋊GL(2,𝔽3)C4semidirect
13. C3×Q8×Dic3D3semidirect
18. Dic3×SL(2,𝔽3)C2×C3semidirect
20. C3×#859D3semidirect
21. C3×#859C2×C3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
45. C3⋊GL(2,𝔽3)C3×C4semidirect
67. Q8×Dic3C3×D3semidirect
72. #859C3×D3semidirect
87. C32×Q8C4×D3semidirect
106. C3×SL(2,𝔽3)C4×D3semidirect
107. C3×SL(2,𝔽3)C2×C3×C4semidirect
177. C3×Dic3GL(2,𝔽3)semidirect
194. C3×Q8#624semidirect
195. C3×Q8C3×C4×D3semidirect
219. SL(2,𝔽3)C3×C4×D3semidirect
295. Dic3C3×GL(2,𝔽3)semidirect
302. C32C4×GL(2,𝔽3)semidirect
305. Q8C3×#624semidirect
354. C3#10593214direct (N × H)
355. C3C3×C4×GL(2,𝔽3)semidirect

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
C3×Dic3×SL(2,𝔽3)2
C32×#8592
C3×C4×GL(2,𝔽3)3
#105932143
C3×#47883
C2×C3×#6244