Finite Group Databasebeta

← C3×#10593210 #13165886

Subgroups — C3×#10593210 #13165886

Order 1728.

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

normal subgroups only

This group has 437 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×C32×#226C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4C3×#5419—31nomaximal
5#10593210C311yesmaximal, characteristic, direct product factor
6C3×#5186—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×D3×SL(2,𝔽3)—21no—
9C32×#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×#226—21no—
16C2×C3×#226D311yescharacteristic, complemented
17C2×C3×#226C2×C311yescharacteristic, complemented
18C2×C32×Q16—31no—
19C3×#946—31no—
20C2×C3×#216—31no—
21C3⋊#859C2×C311yescharacteristic, complemented
22C3×#859—31no—
23C3×#859—61no—
24C3×#835—31no—
25C3×(C2×D3)⋊C8—31no—
26C3×#816—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#5186—31no—
34C2×C32×Q8—31no—
35C2×C32×Q8C2×D311yescharacteristic
36C3×D3×Q8—21no—
37C3×D3×Q8—61no—
38C2×C3×C4×D3—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—22no—
50C3×#226—21no—
51C3×#226—21no—
52C32×Q16—61no—
53C32×Q16—61no—
54C2×C32×C8—31no—
55C3×#217—121no—
56C3×#216—62no—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C2×C3×C3⋊C8—31no—
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×#226C3×D311yescharacteristic, complemented
73C2×C3×Q16—31no—
74C2×C3×Q16—31no—
75C2×C3×Q16—61no—
76C2×C3×Q16—91no—
77C3×C22⋊Q8—91no—
78#946—31no—
79C2×#216—31no—
80#859—31no—
81#859—61no—
82C3×#103—91no—
83C3×#103—91no—
84C3×C22⋊C8—91no—
85#835—31no—
86(C2×D3)⋊C8—31no—
87#816—31no—
88C23×C32—121no—
89C22×C3×D3—41no—
90C22×C3×D3—81no—
91C22×C3×D3S411yescharacteristic
92C32×Q8—31no—
93C32×Q8—61no—
94C32×Q8—61no—
95C32×Q8#6711yescharacteristic
96C2×C32×C4—31no—
97C2×C32×C4—61no—
98C2×C3⋊Dic3—41no—
99C2×C3×Dic3—31no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—81no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—241no—
106C3×C4×D3—61no—
107C3×C4×D3—61no—
108C3×Dic6—32no—
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)#6711yescharacteristic
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—31no—
125C2×C3×Q8—31no—
126C2×C3×Q8—61no—
127C2×C3×Q8—61no—
128C2×C3×Q8—91no—
129C2×C3×Q8—181no—
130C2×C3×Q8D3211yescharacteristic
131C2×C3×Q8C2×C3×D311yescharacteristic
132C22×C3×C4—91no—
133D3×Q8—21no—
134D3×Q8—61no—
135C2×C4×D3—31no—
136C2×Dic6—31no—
137C2×SL(2,𝔽3)—21no—
138C2×SL(2,𝔽3)—21no—
139C2×SL(2,𝔽3)—41no—
140C2×SL(2,𝔽3)—62no—
141C2×SL(2,𝔽3)—61no—
142C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
143#226—21no—
144C3×Q16—62no—
145C3×Q16—61no—
146C3×Q16—61no—
147C3×Q16—62no—
148C3×Q16—61no—
149C3×Q16—61no—
150C3×Q16—182no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—181no—
157C3×C22⋊C4—181no—
158#217—121no—
159#216—62no—
160(C2×D3)⋊C4—61no—
161C4⋊Dic3—31no—
162Dic3⋊C4—61no—
163C2×C3⋊C8—31no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—81no—
167C22×C32—81no—
168C22×C32—81no—
169C22×C32—122no—
170C22×C32—161no—
171C22×C32—241no—
172C22×C32—241no—
173C22×C32C2×S411yescharacteristic
174C2×C3×D3—21no—
175C2×C3×D3—21no—
176C2×C3×D3—42no—
177C2×C3×D3—81no—
178C2×C3×D3—81no—
179C2×C3×D3—82no—
180C2×C3×D3—82no—
181C2×C3×D3—82no—
182C2×C3×D3#22612yescomplemented
183C32×C4—31no—
184C32×C4—31no—
185C32×C4—121no—
186C3⋊Dic3—81no—
187C3×Dic3—61no—
188C3×Dic3—81no—
189C3×Dic3—81no—
190C3×Dic3—82no—
191C3×Dic3—121no—
192C3×Dic3—241no—
193C3×Dic3—242no—
194C22×Q8—31no—
195C2×Q16—31no—
196C2×Q16—91no—
197C22⋊Q8—91no—
198#103—91no—
199#103—91no—
200C22⋊C8—91no—
201C33—41noSylow
202C23×C3—31no—
203C23×C3—121no—
204C23×C3—241no—
205C22×D3C3×S411yescharacteristic
206C3×Q8—21no—
207C3×Q8—31no—
208C3×Q8—31no—
209C3×Q8—61no—
210C3×Q8—61no—
211C3×Q8—61no—
212C3×Q8—61no—
213C3×Q8—61no—
214C3×Q8—62no—
215C3×Q8—92no—
216C3×Q8—121no—
217C3×Q8—181no—
218C3×Q8#62511yescharacteristic
219C3×Q8C3×#6711yescharacteristic
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—121no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×Dic3—31no—
231C2×Dic3—41no—
232C2×Dic3—61no—
233C2×Dic3—121no—
234C2×Dic3—241no—
235C4×D3—61no—
236C4×D3—61no—
237Dic6—32no—
238Dic6—61no—
239SL(2,𝔽3)—21no—
240SL(2,𝔽3)—21no—
241SL(2,𝔽3)—41no—
242SL(2,𝔽3)C3×#6711yescharacteristic
243C3×C8—61no—
244C3×C8—61no—
245C3×C8—62no—
246C3×C8—181no—
247C3⋊C8—61no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—81no—
255C2×C32—81no—
256C2×C32—81no—
257C2×C32—81no—
258C2×C32—81no—
259C2×C32—81no—
260C2×C32—81no—
261C2×C32—81no—
262C2×C32—81no—
263C2×C32—161no—
264C2×C32—161no—
265C2×C32—161no—
266C2×C32—242no—
267C2×C32C2×#22611yescharacteristic
268C2×C32#98311yescharacteristic
269C2×C32GL(2,ℤ/4)11yescharacteristic
270C3×D3—22no—
271C3×D3—82no—
272C3×D3—84no—
273C2×Q8—31no—
274C2×Q8—61no—
275C2×Q8—91no—
276C2×Q8—181no—
277C2×Q8C3×D3211yescharacteristic
278C22×C4—91no—
279Q16—61no—
280Q16—61no—
281Q16—182no—
282C2×C8—31no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—181no—
286C22⋊C4—181no—
287C22×C3—21no—
288C22×C3—32no—
289C22×C3—41no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—122no—
295C22×C3—161no—
296C22×C3—242no—
297C22×C3—241no—
298C22×C3—241no—
299C22×C3—242no—
300C22×C3—242no—
301C22×C3D3×S411yescharacteristic
302C22×C3C2×C3×S411yescharacteristic
303C2×D3—21no—
304C2×D3—21no—
305C2×D3C3×#22612yescomplemented
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—61no—
311C3×C4—61no—
312C3×C4—122no—
313C3×C4—121no—
314C3×C4—121no—
315C3×C4—181no—
316C3×C4—361no—
317Dic3—61no—
318Dic3—81no—
319Dic3—121no—
320Dic3—241no—
321Dic3—242no—
322C32—41no—
323C32—41no—
324C32—81no—
325C32—81no—
326C32—81no—
327C32—161no—
328C32#541911yescharacteristic, complemented
329C23—31no—
330Q8—31no—
331Q8—61no—
332Q8—61no—
333Q8—92no—
334Q8—181no—
335Q8C3×#62511yescharacteristic
336C2×C4—31no—
337C2×C4—61no—
338C2×C4—91no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C8—61no—
343C8—181no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—62no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3—161no—
358C2×C3—161no—
359C2×C3—161no—
360C2×C3—242no—
361C2×C3—244no—
362C2×C3D3×#22611yescharacteristic
363C2×C3#6420211yescharacteristic
364C2×C3#6420911yescharacteristic
365C2×C3C2×C3×#22611yescharacteristic
366C2×C3C3×#98311yescharacteristic
367C2×C3C3×GL(2,ℤ/4)11yescharacteristic
368D3—22no—
369C22—32no—
370C22—61no—
371C22—61no—
372C22C3×D3×S411yescharacteristic
373C4—31no—
374C4—31no—
375C4—121no—
376C4—181no—
377C4—361no—
378C3—21no—
379C3—41no—
380C3—81no—
381C3—81no—
382C3—161no—
383C3#1059321011yescharacteristic, direct product factor
384C3C3×#541911yescharacteristic, complemented
385C2—62no—
386C2C3×D3×#22611yescharacteristic
387C2C3×#6420211yescharacteristic
388C2C3×#6420911yescharacteristic
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×C32×#226C2semidirect
3. C3×C3⋊#859C2semidirect
5. #10593210C3direct (N × H)
16. C2×C3×#226D3semidirect
17. C2×C3×#226C2×C3semidirect
21. C3⋊#859C2×C3semidirect
72. C2×#226C3×D3semidirect
182. C2×C3×D3#226semidirect
305. C2×D3C3×#226semidirect
328. C32#5419semidirect
383. C3#10593210direct (N × H)
384. 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×C32×#2262
C3×C3⋊#8592
C3×#54193
#105932103
C3×#51863
C3×(C2×D3)⋊Dic34