Finite Group Databasebeta

← C2×C3×#64199 #13180696

Subgroups — C2×C3×#64199 #13180696

Order 1728.

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

normal subgroups only

This group has 809 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×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#64199C214yesmaximal, direct product factor
5C2×C3×#986—31nomaximal
6C2×#64199C311yesmaximal, characteristic, direct product factor
7C2×C3×#914—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C2×C3×#626—41nomaximal
10C3×C3⋊(#41⋊C3)C2212yescomplemented
11C3×C3⋊GL(2,𝔽3)C2212yescomplemented
12C32×GL(2,𝔽3)C2212yescomplemented
13C2×C3×#239D311yescharacteristic, complemented
14C2×C3×D3×D4—31no—
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×#986—34no—
20C2×C3×GL(2,𝔽3)—21no—
21C2×C3×GL(2,𝔽3)—31no—
22C2×C3×GL(2,𝔽3)—61no—
23C2×C3×GL(2,𝔽3)D311yescharacteristic, complemented
24C2×C3×GL(2,𝔽3)C2×C311yescharacteristic, complemented
25#64199C2×C314yesdirect product factor
26C2×C32×SD16—31no—
27C2×C3×#215—31no—
28C2×C3×#213—31no—
29C3×#914—34no—
30C3×#914—34no—
31C2×C3×D24—31no—
32C2×C3×ΓL(1,𝔽25)—31no—
33C22×C3×C3⋊D3—41no—
34C22×C32×D3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic, complemented
37C3×#626—44no—
38C2×#986—31no—
39C2×C3×AGL(1,ℤ/8)—91no—
40C2×#914—31no—
41C23×C3×D3—61no—
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×D4—31no—
44C2×C3×#67—61no—
45C3×#239—32no—
46C3×#239C2×D312yescomplemented
47C3×D3×D4—34no—
48C3×D3×D4—62no—
49C2×C3×D12—31no—
50C2×C3×D12—31no—
51C2×C3×D12—121no—
52C2×C3×C4×D3—31no—
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, complemented
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic, complemented
60C2×#626—41no—
61C3⋊(#41⋊C3)—22no—
62C3⋊(#41⋊C3)C22×C312yescomplemented
63C3⋊GL(2,𝔽3)C22×C312yescomplemented
64C3×GL(2,𝔽3)—22no—
65C3×GL(2,𝔽3)—32no—
66C3×GL(2,𝔽3)—62no—
67C3×GL(2,𝔽3)C2×D312yescomplemented
68C3×GL(2,𝔽3)C22×C312yescomplemented
69C32×SD16—32no—
70C32×SD16—32no—
71C2×C32×C8—31no—
72C3×#215—32no—
73C3×#215—32no—
74C3×#213—34no—
75C2×C3×C3⋊C8—31no—
76C3×D24—34no—
77C3×ΓL(1,𝔽25)—34no—
78C22×C33—41no—
79C2×C3×C3⋊D3—42no—
80C2×C3×C3⋊D3—82no—
81C2×C32×D3—42no—
82C2×C32×D3—82no—
83C32×Dic3—42no—
84C2×C3×#41—31no—
85C22×C3×D4—91no—
86C2×#239C3×D311yescharacteristic, complemented
87C2×D3×D4—31no—
88C2×#41⋊C3—31no—
89C2×#41⋊C3—61no—
90#986—34no—
91C2×GL(2,𝔽3)—31no—
92C2×GL(2,𝔽3)—61no—
93C2×GL(2,𝔽3)C3×D311yescharacteristic, complemented
94C3×AGL(1,ℤ/8)—94no—
95C3×AGL(1,ℤ/8)—94no—
96C2×C3×SD16—31no—
97C2×C3×SD16—31no—
98C2×C3×SD16—61no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C2×C3×D8—91no—
102C2×C3×M16—91no—
103C2×#215—31no—
104C2×#213—31no—
105#914—34no—
106#914—34no—
107C2×D24—31no—
108C2×ΓL(1,𝔽25)—31no—
109C23×C32—61no—
110C22×C3⋊D3—41no—
111C22×C3×D3—31no—
112C22×C3×D3—41no—
113C22×C3×D3—41no—
114C22×C3×D3—64no—
115C22×C3×D3—61no—
116C22×C3×D3—81no—
117C22×C3×D3—121no—
118C22×C3×D3—124no—
119C22×C3×D3—241no—
120C32×Q8—31no—
121C32×Q8C22×D311yescharacteristic
122C32×D4—32no—
123C32×D4—61no—
124C2×C32×C4—31no—
125C2×C32×C4—121no—
126C3×#67—64no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—81no—
129C2×C3×Dic3S411yescharacteristic
130C3×D12—32no—
131C3×D12—32no—
132C3×D12—61no—
133C3×D12—61no—
134C3×D12—124no—
135C3×C4×D3—32no—
136C3×C4×D3—32no—
137C3×SL(2,𝔽3)—21no—
138C3×SL(2,𝔽3)—21no—
139C3×SL(2,𝔽3)—21no—
140C3×SL(2,𝔽3)—41no—
141C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
142C3×SL(2,𝔽3)C23×C311yescharacteristic, complemented
143#626—44no—
144C32×C8—32no—
145C3×C3⋊C8—32no—
146C2×AGL(1,ℤ/8)—91noSylow
147C2×C33—42no—
148C2×C33—41no—
149C3×C3⋊D3—82no—
150C32×D3—82no—
151C24×C3—181no—
152C23×D3—61no—
153C3×#41—32no—
154C3×#41—92no—
155C2×C3×Q8—21no—
156C2×C3×Q8D3211yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—31no—
159C2×C3×D4—31no—
160C2×C3×D4—61no—
161C2×C3×D4—91no—
162C2×C3×D4—91no—
163C2×C3×D4—94no—
164C2×C3×D4—181no—
165C2×C3×D4—182no—
166C22×C3×C4—91no—
167C2×#67—61no—
168#239—32no—
169#239C2×C3×D312yescomplemented
170D3×D4—34no—
171D3×D4—62no—
172C2×D12—31no—
173C2×D12—31no—
174C2×D12—121no—
175C2×C4×D3—31no—
176#41⋊C3—32no—
177#41⋊C3—62no—
178C2×SL(2,𝔽3)—21no—
179C2×SL(2,𝔽3)—21no—
180C2×SL(2,𝔽3)—41no—
181C2×SL(2,𝔽3)C2×C3×D311yescharacteristic, complemented
182GL(2,𝔽3)—32no—
183GL(2,𝔽3)—62no—
184GL(2,𝔽3)C2×C3×D312yescomplemented
185C3×SD16—32no—
186C3×SD16—32no—
187C3×SD16—32no—
188C3×SD16—32no—
189C3×SD16—62no—
190C3×SD16—62no—
191C3×SD16—92no—
192C3×SD16—92no—
193C3×D8—94no—
194C3×D8—94no—
195C3×M16—94no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—61no—
199C2×C3×C8—91no—
200#215—32no—
201#215—32no—
202#213—34no—
203C2×C3⋊C8—31no—
204D24—34no—
205ΓL(1,𝔽25)—34no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—62no—
209C22×C32—81no—
210C22×C32—81no—
211C22×C32—81no—
212C22×C32—122no—
213C22×C32—161no—
214C22×C32C2×S411yescharacteristic
215C2×C3⋊D3—42no—
216C2×C3⋊D3—82no—
217C2×C3×D3—32no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—82no—
223C2×C3×D3—82no—
224C2×C3×D3—82no—
225C2×C3×D3—84no—
226C2×C3×D3—122no—
227C2×C3×D3—128no—
228C2×C3×D3—122no—
229C2×C3×D3—242no—
230C2×C3×D3—242no—
231C2×C3×D3—244no—
232C32×C4—31no—
233C32×C4—31no—
234C32×C4—122no—
235C3×Dic3—42no—
236C3×Dic3—82no—
237C3×Dic3C2×S412yes—
238C2×#41—31no—
239C22×D4—91no—
240AGL(1,ℤ/8)—94no—
241AGL(1,ℤ/8)—94no—
242C2×SD16—31no—
243C2×SD16—91no—
244C2×D8—91no—
245C2×D8—91no—
246C2×M16—91no—
247C33—41noSylow
248C23×C3—61no—
249C23×C3—61no—
250C23×C3—91no—
251C23×C3—121no—
252C23×C3—184no—
253C23×C3—181no—
254C23×C3—364no—
255C22×D3—31no—
256C22×D3—41no—
257C22×D3—64no—
258C22×D3—61no—
259C22×D3—121no—
260C22×D3—124no—
261C22×D3—241no—
262C3×Q8—21no—
263C3×Q8—31no—
264C3×Q8—31no—
265C3×Q8—61no—
266C3×Q8C2×D3211yescharacteristic
267C3×Q8C22×C3×D311yescharacteristic
268C3×D4—32no—
269C3×D4—32no—
270C3×D4—62no—
271C3×D4—61no—
272C3×D4—61no—
273C3×D4—92no—
274C3×D4—92no—
275C3×D4—121no—
276C3×D4—184no—
277C3×D4—181no—
278C3×D4—181no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—61no—
283C2×C3×C4—92no—
284C2×C3×C4—92no—
285C2×C3×C4—121no—
286C2×C3×C4—241no—
287#67—64no—
288C2×Dic3C3×S411yescharacteristic
289D12—32no—
290D12—32no—
291D12—61no—
292D12—61no—
293D12—124no—
294C4×D3—32no—
295C4×D3—32no—
296SL(2,𝔽3)—21no—
297SL(2,𝔽3)—21no—
298SL(2,𝔽3)—41no—
299SL(2,𝔽3)C22×C3×D311yescharacteristic, complemented
300C3×C8—32no—
301C3×C8—32no—
302C3×C8—62no—
303C3×C8—92no—
304C3⋊C8—32no—
305C2×C32—42no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—81no—
310C2×C32—82no—
311C2×C32—82no—
312C2×C32—81no—
313C2×C32—81no—
314C2×C32—82no—
315C2×C32—122no—
316C2×C32—161no—
317C2×C32—162no—
318C2×C32#98612yescomplemented
319C2×C32C22×S411yescharacteristic
320C3⋊D3—82no—
321C3×D3—62no—
322C3×D3—82no—
323C3×D3—82no—
324C3×D3—84no—
325C3×D3—122no—
326C3×D3—242no—
327C3×D3—244no—
328C24—181no—
329#41—32no—
330#41—92no—
331C2×Q8C3×D3211yescharacteristic
332C2×D4—31no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—94no—
336C2×D4—181no—
337C2×D4—182no—
338C22×C4—91no—
339SD16—32no—
340SD16—32no—
341SD16—92no—
342SD16—92no—
343D8—94no—
344D8—94no—
345M16—94no—
346C2×C8—31no—
347C2×C8—91no—
348C22×C3—21no—
349C22×C3—41no—
350C22×C3—62no—
351C22×C3—62no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—92no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3—161no—
359C22×C3—182no—
360C22×C3—182no—
361C22×C3—242no—
362C22×C3—368no—
363C22×C3—362no—
364C22×C3D3×S411yescharacteristic
365C22×C3C2×C3×S411yescharacteristic
366C2×D3—32no—
367C2×D3—42no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—82no—
371C2×D3—122no—
372C2×D3—128no—
373C2×D3—122no—
374C2×D3—242no—
375C2×D3—242no—
376C2×D3—244no—
377C3×C4—31no—
378C3×C4—31no—
379C3×C4—32no—
380C3×C4—31no—
381C3×C4—31no—
382C3×C4—61no—
383C3×C4—61no—
384C3×C4—122no—
385C3×C4—242no—
386Dic3C2×C3×S412yes—
387C32—41no—
388C32—41no—
389C32—81no—
390C32—81no—
391C32—81no—
392C32—161no—
393C32C2×#98611yescharacteristic, complemented
394C23—61no—
395C23—91no—
396C23—184no—
397C23—181no—
398C23—364no—
399Q8—31no—
400Q8C2×C3×D3211yescharacteristic
401D4—32no—
402D4—61no—
403D4—92no—
404D4—92no—
405D4—184no—
406D4—181no—
407D4—181no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—92no—
411C2×C4—92no—
412C8—32no—
413C8—92no—
414C2×C3—22no—
415C2×C3—21no—
416C2×C3—42no—
417C2×C3—41no—
418C2×C3—82no—
419C2×C3—82no—
420C2×C3—81no—
421C2×C3—81no—
422C2×C3—122no—
423C2×C3—122no—
424C2×C3—162no—
425C2×C3—161no—
426C2×C3—182no—
427C2×C3—242no—
428C2×C3—362no—
429C2×C3#6419912yesdirect product factor
430C2×C3C3×#98612yescomplemented
431C2×C3C2×D3×S411yescharacteristic
432C2×C3C22×C3×S411yescharacteristic
433D3—62no—
434D3—82no—
435D3—122no—
436D3—242no—
437D3—244no—
438C22—62no—
439C22—92no—
440C22—122no—
441C22—182no—
442C22—182no—
443C22—368no—
444C22—362no—
445C22C3×D3×S411yescharacteristic
446C4—31no—
447C4—31no—
448C4—32no—
449C3—21no—
450C3—41no—
451C3—81no—
452C3—81no—
453C3—161no—
454C3C2×#6419911yescharacteristic, direct product factor
455C3C2×C3×#98611yescharacteristic, complemented
456C2—122no—
457C2—182no—
458C2—362no—
459C2C3×#6419912yesdirect product factor
460C2C2×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×GL(2,𝔽3)C2semidirect
4. C3×#64199C2direct (N × H)
6. C2×#64199C3direct (N × H)
8. C2×C32×SL(2,𝔽3)C22semidirect
10. C3×C3⋊(#41⋊C3)C22semidirect
11. C3×C3⋊GL(2,𝔽3)C22semidirect
12. C32×GL(2,𝔽3)C22semidirect
13. C2×C3×#239D3semidirect
17. C2×C3⋊(#41⋊C3)C2×C3semidirect
18. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
23. C2×C3×GL(2,𝔽3)D3semidirect
24. C2×C3×GL(2,𝔽3)C2×C3semidirect
25. #64199C2×C3direct (N × H)
36. C32×SL(2,𝔽3)C23semidirect
46. C3×#239C2×D3semidirect
58. C2×C3×SL(2,𝔽3)C2×D3semidirect
59. C2×C3×SL(2,𝔽3)C22×C3semidirect
62. C3⋊(#41⋊C3)C22×C3semidirect
63. C3⋊GL(2,𝔽3)C22×C3semidirect
67. C3×GL(2,𝔽3)C2×D3semidirect
68. C3×GL(2,𝔽3)C22×C3semidirect
86. C2×#239C3×D3semidirect
93. C2×GL(2,𝔽3)C3×D3semidirect
141. C3×SL(2,𝔽3)C22×D3semidirect
142. C3×SL(2,𝔽3)C23×C3semidirect
169. #239C2×C3×D3semidirect
181. C2×SL(2,𝔽3)C2×C3×D3semidirect
184. GL(2,𝔽3)C2×C3×D3semidirect
299. SL(2,𝔽3)C22×C3×D3semidirect
318. C2×C32#986semidirect
393. C32C2×#986semidirect
429. C2×C3#64199direct (N × H)
430. C2×C3C3×#986semidirect
454. C3C2×#64199direct (N × H)
455. C3C2×C3×#986semidirect
459. C2C3×#64199direct (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×GL(2,𝔽3)2
C3×#641992
C2×C3×#9863
C2×#641993
C2×C3×#9143
C2×C3×#6264