Finite Group Databasebeta

← C2×C3×#64198 #13180695

Subgroups — C2×C3×#64198 #13180695

Order 1728.

733 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 82 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⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C32×#226C211yesmaximal, characteristic, complemented
4C3×#64198C214yesmaximal, direct product factor
5C2×C3×#985—31nomaximal
6C2×#64198C311yesmaximal, characteristic, direct product factor
7C2×C3×#919—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×#624—41nomaximal
10C3×C3⋊(#41⋊C3)C2212yescomplemented
11C3×C3⋊GL(2,𝔽3)C2212yescomplemented
12C32×#226C2212yescomplemented
13C2×C3×#239—31no—
14C2×C3×#239D311yescharacteristic, complemented
15C2×C3×#41⋊C3—31no—
16C2×C3⋊(#41⋊C3)—21no—
17C2×C3⋊(#41⋊C3)C2×C311yescharacteristic, complemented
18C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
19C3×#985—34no—
20C2×C3×GL(2,𝔽3)—31no—
21C2×C3×GL(2,𝔽3)—61no—
22C2×C3×#226—21no—
23C2×C3×#226D311yescharacteristic, complemented
24C2×C3×#226C2×C311yescharacteristic, complemented
25#64198C2×C314yesdirect product factor
26C2×C32×Q16—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C3×#919—34no—
30C3×#919—34no—
31C2×C3×D24—31no—
32C2×C3×D3×C8—31no—
33C22×C3×C3⋊D3—41no—
34C2×C32×Dic3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×#624—44no—
38C2×#985—31no—
39C2×C3×#135—91no—
40C2×#919—31no—
41C2×C32×Q8—31no—
42C2×C32×Q8C2×D311yescharacteristic
43C3×#239—32no—
44C3×#239—34no—
45C3×#239—62no—
46C3×#239C2×D312yescomplemented
47C2×C3×D12—31no—
48C2×C3×D12—31no—
49C2×C3×D12—61no—
50C2×C3×C4×D3—31no—
51C2×C3×C4×D3—61no—
52C2×C3×C4×D3—121no—
53C3×#41⋊C3—32no—
54C2×C3×SL(2,𝔽3)—21no—
55C2×C3×SL(2,𝔽3)—21no—
56C2×C3×SL(2,𝔽3)—21no—
57C2×C3×SL(2,𝔽3)—41no—
58C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
60C2×#624—41no—
61C3⋊(#41⋊C3)—22no—
62C3⋊(#41⋊C3)C22×C312yescomplemented
63C3⋊GL(2,𝔽3)C22×C312yescomplemented
64C3×GL(2,𝔽3)—32no—
65C3×GL(2,𝔽3)—62no—
66C3×#226—22no—
67C3×#226C2×D312yescomplemented
68C3×#226C22×C312yescomplemented
69C32×Q16—32no—
70C32×Q16—32no—
71C2×C32×C8—31no—
72C3×#215—34no—
73C3×#215—32no—
74C3×#215—32no—
75C2×C3×C3⋊C8—31no—
76C3×D24—34no—
77C3×D3×C8—34no—
78C22×C33—41no—
79C2×C3×C3⋊D3—42no—
80C2×C3×C3⋊D3—44no—
81C32×Dic3—42no—
82C32×Dic3—42no—
83C2×C3×#41—31no—
84C2×C3×#41—91no—
85C2×#239—31no—
86C2×#239C3×D311yescharacteristic, complemented
87C2×#41⋊C3—31no—
88C2×#41⋊C3—61no—
89#985—34no—
90C2×GL(2,𝔽3)—31no—
91C2×GL(2,𝔽3)—61no—
92C2×#226C3×D311yescharacteristic, complemented
93C3×#135—94no—
94C3×#135—94no—
95C2×C3×Q16—31no—
96C2×C3×Q16—31no—
97C2×C3×Q16—61no—
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C22×C3×C8—91no—
102C2×#215—31no—
103C2×#215—31no—
104#919—34no—
105#919—34no—
106C2×D24—31no—
107C2×D3×C8—31no—
108C22×C3⋊D3—41no—
109C22×C3×D3—31no—
110C22×C3×D3—61no—
111C22×C3×D3—121no—
112C22×C3×D3—241no—
113C32×Q8—31no—
114C32×Q8—32no—
115C32×Q8—61no—
116C32×Q8C22×D311yescharacteristic
117C2×C32×C4—31no—
118C2×C32×C4—61no—
119C2×C32×C4—121no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—81no—
124C2×C3×Dic3—81no—
125C2×C3×Dic3S411yescharacteristic
126C3×D12—32no—
127C3×D12—32no—
128C3×D12—61no—
129C3×D12—61no—
130C3×D12—64no—
131C3×C4×D3—32no—
132C3×C4×D3—32no—
133C3×C4×D3—64no—
134C3×C4×D3—124no—
135C3×SL(2,𝔽3)—21no—
136C3×SL(2,𝔽3)—21no—
137C3×SL(2,𝔽3)—21no—
138C3×SL(2,𝔽3)—41no—
139C3×SL(2,𝔽3)C22×D311yescharacteristic
140C3×SL(2,𝔽3)C23×C311yescharacteristic
141#624—44no—
142C32×C8—32no—
143C3×C3⋊C8—32no—
144C2×#135—91noSylow
145C2×C33—42no—
146C2×C33—41no—
147C3×C3⋊D3—44no—
148C3×#41—32no—
149C3×#41—92no—
150C3×#41—94no—
151C3×#41—182no—
152C2×C3×Q8—21no—
153C2×C3×Q8—31no—
154C2×C3×Q8—31no—
155C2×C3×Q8—61no—
156C2×C3×Q8D3211yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—91no—
159C2×C3×D4—91no—
160C2×C3×D4—181no—
161C22×C3×C4—91no—
162C22×C3×C4—181no—
163#239—32no—
164#239—34no—
165#239—62no—
166#239C2×C3×D312yescomplemented
167C2×D12—31no—
168C2×D12—31no—
169C2×D12—61no—
170C2×C4×D3—31no—
171C2×C4×D3—61no—
172C2×C4×D3—121no—
173#41⋊C3—32no—
174#41⋊C3—62no—
175C2×SL(2,𝔽3)—21no—
176C2×SL(2,𝔽3)—21no—
177C2×SL(2,𝔽3)—41no—
178C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
179GL(2,𝔽3)—32no—
180GL(2,𝔽3)—62no—
181#226C2×C3×D312yescomplemented
182C3×Q16—32no—
183C3×Q16—32no—
184C3×Q16—32no—
185C3×Q16—32no—
186C3×Q16—62no—
187C3×Q16—62no—
188C3×SD16—94no—
189C3×SD16—92no—
190C3×SD16—92no—
191C3×D8—94no—
192C2×C3×C8—31no—
193C2×C3×C8—31no—
194C2×C3×C8—61no—
195C2×C3×C8—94no—
196C2×C3×C8—91no—
197#215—34no—
198#215—32no—
199#215—32no—
200C2×C3⋊C8—31no—
201D24—34no—
202D3×C8—34no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32—161no—
209C22×C32C2×S411yescharacteristic
210C2×C3⋊D3—42no—
211C2×C3⋊D3—44no—
212C2×C3×D3—32no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—122no—
216C2×C3×D3—124no—
217C2×C3×D3—122no—
218C2×C3×D3—242no—
219C2×C3×D3—244no—
220C32×C4—31no—
221C32×C4—31no—
222C32×C4—62no—
223C32×C4—122no—
224C3×Dic3—42no—
225C3×Dic3—42no—
226C3×Dic3—42no—
227C3×Dic3—82no—
228C3×Dic3—82no—
229C3×Dic3C2×S412yes—
230C2×#41—31no—
231C2×#41—91no—
232#135—94no—
233#135—94no—
234C2×Q16—31no—
235C2×SD16—91no—
236C2×SD16—91no—
237C2×D8—91no—
238C22×C8—91no—
239C33—41noSylow
240C23×C3—91no—
241C23×C3—181no—
242C22×D3—31no—
243C22×D3—61no—
244C22×D3—121no—
245C22×D3—241no—
246C3×Q8—21no—
247C3×Q8—31no—
248C3×Q8—31no—
249C3×Q8—32no—
250C3×Q8—32no—
251C3×Q8—61no—
252C3×Q8—62no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—121no—
256C3×Q8C2×D3211yescharacteristic
257C3×Q8C22×C3×D311yescharacteristic
258C3×D4—92no—
259C3×D4—92no—
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—184no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—121no—
272C2×C3×C4—121no—
273C2×C3×C4—184no—
274C2×C3×C4—241no—
275C2×Dic3—41no—
276C2×Dic3C3×S411yescharacteristic
277D12—32no—
278D12—32no—
279D12—61no—
280D12—61no—
281D12—64no—
282C4×D3—32no—
283C4×D3—32no—
284C4×D3—64no—
285C4×D3—124no—
286SL(2,𝔽3)—21no—
287SL(2,𝔽3)—21no—
288SL(2,𝔽3)—41no—
289SL(2,𝔽3)C22×C3×D311yescharacteristic
290C3×C8—32no—
291C3×C8—32no—
292C3×C8—62no—
293C3×C8—92no—
294C3⋊C8—32no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—42no—
298C2×C32—42no—
299C2×C32—81no—
300C2×C32—82no—
301C2×C32—81no—
302C2×C32—81no—
303C2×C32—82no—
304C2×C32—82no—
305C2×C32—161no—
306C2×C32—162no—
307C2×C32#98512yescomplemented
308C2×C32C22×S411yescharacteristic
309C3⋊D3—44no—
310C3×D3—62no—
311C3×D3—124no—
312C3×D3—122no—
313C3×D3—244no—
314#41—32no—
315#41—92no—
316#41—94no—
317#41—182no—
318C2×Q8—31no—
319C2×Q8C3×D3211yescharacteristic
320C2×D4—91no—
321C2×D4—91no—
322C2×D4—181no—
323C22×C4—91no—
324C22×C4—181no—
325Q16—32no—
326Q16—32no—
327SD16—94no—
328SD16—92no—
329SD16—92no—
330D8—94no—
331C2×C8—31no—
332C2×C8—94no—
333C2×C8—91no—
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—92no—
339C22×C3—161no—
340C22×C3—182no—
341C22×C3—182no—
342C22×C3—362no—
343C22×C3D3×S411yescharacteristic
344C22×C3C2×C3×S411yescharacteristic
345C2×D3—32no—
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—122no—
349C2×D3—124no—
350C2×D3—122no—
351C2×D3—242no—
352C2×D3—244no—
353C3×C4—31no—
354C3×C4—32no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—62no—
361C3×C4—62no—
362C3×C4—122no—
363C3×C4—122no—
364C3×C4—242no—
365Dic3—42no—
366Dic3C2×C3×S412yes—
367C32—41no—
368C32—41no—
369C32—81no—
370C32—81no—
371C32—81no—
372C32—161no—
373C32C2×#98511yescharacteristic, complemented
374C23—91no—
375C23—181no—
376Q8—31no—
377Q8—32no—
378Q8—61no—
379Q8C2×C3×D3211yescharacteristic
380D4—92no—
381D4—92no—
382D4—181no—
383D4—181no—
384D4—184no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—61no—
388C2×C4—92no—
389C2×C4—92no—
390C2×C4—184no—
391C8—32no—
392C8—92no—
393C2×C3—22no—
394C2×C3—21no—
395C2×C3—42no—
396C2×C3—41no—
397C2×C3—82no—
398C2×C3—82no—
399C2×C3—81no—
400C2×C3—81no—
401C2×C3—162no—
402C2×C3—161no—
403C2×C3—182no—
404C2×C3—362no—
405C2×C3#6419812yesdirect product factor
406C2×C3C3×#98512yescomplemented
407C2×C3C2×D3×S411yescharacteristic
408C2×C3C22×C3×S411yescharacteristic
409D3—62no—
410D3—124no—
411D3—122no—
412D3—244no—
413C22—92no—
414C22—182no—
415C22—182no—
416C22—362no—
417C22C3×D3×S411yescharacteristic
418C4—31no—
419C4—32no—
420C4—31no—
421C4—62no—
422C3—21no—
423C3—41no—
424C3—81no—
425C3—81no—
426C3—161no—
427C3C2×#6419811yescharacteristic, direct product factor
428C3C2×C3×#98511yescharacteristic, complemented
429C2—182no—
430C2—362no—
431C2C3×#6419812yesdirect product factor
432C2C2×C3×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C2×C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊(#41⋊C3)C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C2×C32×#226C2semidirect
4. C3×#64198C2direct (N × H)
6. C2×#64198C3direct (N × H)
10. C3×C3⋊(#41⋊C3)C22semidirect
11. C3×C3⋊GL(2,𝔽3)C22semidirect
12. C32×#226C22semidirect
14. C2×C3×#239D3semidirect
17. C2×C3⋊(#41⋊C3)C2×C3semidirect
18. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
23. C2×C3×#226D3semidirect
24. C2×C3×#226C2×C3semidirect
25. #64198C2×C3direct (N × H)
46. C3×#239C2×D3semidirect
62. C3⋊(#41⋊C3)C22×C3semidirect
63. C3⋊GL(2,𝔽3)C22×C3semidirect
67. C3×#226C2×D3semidirect
68. C3×#226C22×C3semidirect
86. C2×#239C3×D3semidirect
92. C2×#226C3×D3semidirect
166. #239C2×C3×D3semidirect
181. #226C2×C3×D3semidirect
307. C2×C32#985semidirect
373. C32C2×#985semidirect
405. C2×C3#64198direct (N × H)
406. C2×C3C3×#985semidirect
427. C3C2×#64198direct (N × H)
428. C3C2×C3×#985semidirect
431. C2C3×#64198direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3×C3⋊GL(2,𝔽3)2
C2×C32×#2262
C3×#641982
C2×C3×#9853
C2×#641983
C2×C3×#9193
C2×C3×#6244