Finite Group Databasebeta

← C3×#10593211 #13165887

Subgroups — C3×#10593211 #13165887

Order 1728.

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

normal subgroups only

This group has 430 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
1C2×C3×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4C3×#5419—31nomaximal
5#10593211C311yesmaximal, characteristic, direct product factor
6C3×#4809—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×D3×SL(2,𝔽3)—21no—
9C3×C3⋊#226—21no—
10C3×(C2×D3)⋊Dic3—41nomaximal
11C2×C3×D3×Q8D311yescharacteristic
12C22×C3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)—21no—
14C2×D3×SL(2,𝔽3)C2×C311yescharacteristic
15C2×C3⋊#226C2×C311yescharacteristic, complemented
16C2×C3×#226—31no—
17C2×C3×#226—61no—
18C2×C3×#216—31no—
19C2×C3×Dic12—31no—
20C3×#897—31no—
21C3×#859—21no—
22C3×#859D311yescharacteristic, complemented
23C3×#859C2×C311yescharacteristic, complemented
24C32×#103—31no—
25C3×(C2×D3)⋊C8—31no—
26C3×#810—31no—
27C22×C32×D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic
31#5419—31no—
32C3×#451—91no—
33#4809—31no—
34C2×C32×Q8C2×D311yescharacteristic
35C3×D3×Q8—21no—
36C3×D3×Q8—61no—
37C2×C3×C4×D3—31no—
38C2×C3×Dic6—31no—
39C2×C3×Dic6—31no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—41no—
44C2×C3×SL(2,𝔽3)—61no—
45C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
46C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
47D3×SL(2,𝔽3)—22no—
48D3×SL(2,𝔽3)—21no—
49C3⋊#226—21no—
50C3×#226—62no—
51C3×#226—61no—
52C2×C32×C8—31no—
53C32×C4⋊C4—31no—
54C3×#217—121no—
55C3×#216—61no—
56C3×#216—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—61no—
59C2×C3×C3⋊C8—31no—
60C3×Dic12—62no—
61(C2×D3)⋊Dic3—41no—
62C22×C33—41no—
63C2×C32×D3—42no—
64C2×C32×D3—81no—
65C2×C32×D3—81no—
66C3×C3⋊Dic3—81no—
67C32×Dic3—81no—
68C22×C3×Q8—31no—
69C2×D3×Q8C3×D311yescharacteristic
70C22×SL(2,𝔽3)—31no—
71C22×SL(2,𝔽3)—61no—
72C2×#226—31no—
73C2×#226—61no—
74C2×C3×Q16—91no—
75C2×C3×Q16—91no—
76C3×C22⋊Q8—91no—
77C2×#216—31no—
78C2×Dic12—31no—
79#897—31no—
80#859C3×D311yescharacteristic, complemented
81C3×#103—31no—
82C3×#103—31no—
83C3×#103—61no—
84C3×#103—91no—
85C3×C22⋊C8—91no—
86(C2×D3)⋊C8—31no—
87#810—31no—
88C23×C32—121no—
89C22×C3×D3—41no—
90C22×C3×D3—81no—
91C22×C3×D3S411yescharacteristic
92C32×Q8—31no—
93C32×Q8#6711yescharacteristic
94C2×C32×C4—31no—
95C2×C32×C4—61no—
96C2×C3⋊Dic3—41no—
97C2×C3×Dic3—31no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—81no—
102C2×C3×Dic3—121no—
103C2×C3×Dic3—241no—
104C3×C4×D3—61no—
105C3×C4×D3—61no—
106C3×Dic6—32no—
107C3×Dic6—61no—
108C3×Dic6—61no—
109C3×Dic6—61no—
110C3×SL(2,𝔽3)—21no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)—21no—
113C3×SL(2,𝔽3)—41no—
114C3×SL(2,𝔽3)D1211yescharacteristic
115C3×SL(2,𝔽3)C3×D411yescharacteristic
116C32×C8—61no—
117C3×C3⋊C8—61no—
118#451—91noSylow
119C2×C33—41no—
120C2×C33—41no—
121C2×C33—41no—
122C32×D3—82no—
123C2×C3×Q8—21no—
124C2×C3×Q8—61no—
125C2×C3×Q8—91no—
126C2×C3×Q8—91no—
127C2×C3×Q8—181no—
128C2×C3×Q8D3211yescharacteristic
129C2×C3×Q8C2×C3×D311yescharacteristic
130C22×C3×C4—91no—
131D3×Q8—21no—
132D3×Q8—61no—
133C2×C4×D3—31no—
134C2×Dic6—31no—
135C2×Dic6—31no—
136C2×SL(2,𝔽3)—21no—
137C2×SL(2,𝔽3)—21no—
138C2×SL(2,𝔽3)—41no—
139C2×SL(2,𝔽3)—62no—
140C2×SL(2,𝔽3)—61no—
141C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
142#226—62no—
143#226—61no—
144C3×Q16—182no—
145C3×Q16—181no—
146C3×Q16—181no—
147C2×C3×C8—31no—
148C2×C3×C8—31no—
149C2×C3×C8—61no—
150C2×C3×C8—91no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—181no—
155C3×C22⋊C4—181no—
156#217—121no—
157#216—61no—
158#216—61no—
159(C2×D3)⋊C4—61no—
160C4⋊Dic3—61no—
161C2×C3⋊C8—31no—
162Dic12—62no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—81no—
166C22×C32—81no—
167C22×C32—81no—
168C22×C32—122no—
169C22×C32—161no—
170C22×C32—241no—
171C22×C32—241no—
172C22×C32C2×S411yescharacteristic
173C2×C3×D3—21no—
174C2×C3×D3—21no—
175C2×C3×D3—42no—
176C2×C3×D3—81no—
177C2×C3×D3—81no—
178C2×C3×D3—82no—
179C2×C3×D3—82no—
180C2×C3×D3—82no—
181C2×C3×D3#22612yescomplemented
182C32×C4—31no—
183C32×C4—31no—
184C32×C4—121no—
185C3⋊Dic3—81no—
186C3×Dic3—61no—
187C3×Dic3—81no—
188C3×Dic3—81no—
189C3×Dic3—82no—
190C3×Dic3—121no—
191C3×Dic3—241no—
192C3×Dic3—242no—
193C22×Q8—31no—
194C2×Q16—91no—
195C2×Q16—91no—
196C22⋊Q8—91no—
197#103—31no—
198#103—91no—
199C22⋊C8—91no—
200C33—41noSylow
201C23×C3—31no—
202C23×C3—121no—
203C23×C3—241no—
204C22×D3C3×S411yescharacteristic
205C3×Q8—21no—
206C3×Q8—31no—
207C3×Q8—31no—
208C3×Q8—61no—
209C3×Q8—92no—
210C3×Q8—181no—
211C3×Q8—181no—
212C3×Q8—181no—
213C3×Q8#62611yescharacteristic
214C3×Q8C3×#6711yescharacteristic
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—91no—
221C2×C3×C4—121no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×Dic3—31no—
226C2×Dic3—41no—
227C2×Dic3—61no—
228C2×Dic3—121no—
229C2×Dic3—241no—
230C4×D3—61no—
231C4×D3—61no—
232Dic6—32no—
233Dic6—61no—
234Dic6—61no—
235Dic6—61no—
236SL(2,𝔽3)—21no—
237SL(2,𝔽3)—21no—
238SL(2,𝔽3)—41no—
239SL(2,𝔽3)C3×D1211yescharacteristic
240C3×C8—61no—
241C3×C8—61no—
242C3×C8—121no—
243C3×C8—181no—
244C3⋊C8—61no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32—81no—
252C2×C32—81no—
253C2×C32—81no—
254C2×C32—81no—
255C2×C32—81no—
256C2×C32—81no—
257C2×C32—81no—
258C2×C32—81no—
259C2×C32—81no—
260C2×C32—161no—
261C2×C32—161no—
262C2×C32—161no—
263C2×C32—242no—
264C2×C32C2×#22611yescharacteristic
265C2×C32#98311yescharacteristic
266C2×C32GL(2,ℤ/4)11yescharacteristic
267C3×D3—22no—
268C3×D3—82no—
269C3×D3—84no—
270C2×Q8—61no—
271C2×Q8—91no—
272C2×Q8—91no—
273C2×Q8—181no—
274C2×Q8C3×D3211yescharacteristic
275C22×C4—91no—
276Q16—182no—
277Q16—181no—
278Q16—181no—
279C2×C8—31no—
280C2×C8—91no—
281C4⋊C4—31no—
282C4⋊C4—181no—
283C22⋊C4—181no—
284C22×C3—21no—
285C22×C3—32no—
286C22×C3—41no—
287C22×C3—61no—
288C22×C3—61no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3—122no—
292C22×C3—161no—
293C22×C3—242no—
294C22×C3—241no—
295C22×C3—241no—
296C22×C3—242no—
297C22×C3—242no—
298C22×C3D3×S411yescharacteristic
299C22×C3C2×C3×S411yescharacteristic
300C2×D3—21no—
301C2×D3—21no—
302C2×D3C3×#22612yescomplemented
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—181no—
312C3×C4—241no—
313C3×C4—361no—
314Dic3—61no—
315Dic3—81no—
316Dic3—121no—
317Dic3—241no—
318Dic3—242no—
319C32—41no—
320C32—41no—
321C32—81no—
322C32—81no—
323C32—81no—
324C32—161no—
325C32#541911yescharacteristic, complemented
326C23—31no—
327Q8—31no—
328Q8—92no—
329Q8—181no—
330Q8—181no—
331Q8—181no—
332Q8C3×#62611yescharacteristic
333C2×C4—31no—
334C2×C4—61no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C8—61no—
340C8—181no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—62no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3—81no—
354C2×C3—161no—
355C2×C3—161no—
356C2×C3—161no—
357C2×C3—242no—
358C2×C3—244no—
359C2×C3D3×#22611yescharacteristic
360C2×C3#6420111yescharacteristic
361C2×C3#6421011yescharacteristic
362C2×C3C2×C3×#22611yescharacteristic
363C2×C3C3×#98311yescharacteristic
364C2×C3C3×GL(2,ℤ/4)11yescharacteristic
365D3—22no—
366C22—32no—
367C22—61no—
368C22—61no—
369C22C3×D3×S411yescharacteristic
370C4—31no—
371C4—31no—
372C4—121no—
373C4—181no—
374C4—361no—
375C3—21no—
376C3—41no—
377C3—81no—
378C3—81no—
379C3—161no—
380C3#1059321111yescharacteristic, direct product factor
381C3C3×#541911yescharacteristic, complemented
382C2—62no—
383C2C3×D3×#22611yescharacteristic
384C2C3×#6420111yescharacteristic
385C2C3×#6421011yescharacteristic
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×C3×D3×SL(2,𝔽3)C2
hypocenterC3×SL(2,𝔽3)C3×D4
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
2. C2×C3×C3⋊#226C2semidirect
3. C32×#859C2semidirect
5. #10593211C3direct (N × H)
15. C2×C3⋊#226C2×C3semidirect
22. C3×#859D3semidirect
23. C3×#859C2×C3semidirect
80. #859C3×D3semidirect
181. C2×C3×D3#226semidirect
302. C2×D3C3×#226semidirect
325. C32#5419semidirect
380. C3#10593211direct (N × H)
381. C3C3×#5419semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C2×C3×C3⋊#2262
C32×#8592
C3×#54193
#105932113
C3×#48093
C3×(C2×D3)⋊Dic34