Finite Group Databasebeta

← C2×C3×#64264 #13181380

Subgroups — C2×C3×#64264 #13181380

Order 1728.

971 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#64264C214yesmaximal, direct product factor
3C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
5C2×#64264C311yesmaximal, characteristic, direct product factor
6C2×C3×#983—33nomaximal
7C2×C3×#942—31nomaximal
8C2×C3×#638—41nomaximal
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C2×C32×SL(2,𝔽3)C2212yescomplemented
11C3×C3⋊GL(2,𝔽3)C2212yescomplemented
12C3×C3⋊#226C2212yescomplemented
13C22×C32×Q8D311yescharacteristic, complemented
14C2×C3×#235—31no—
15C22×C3×SL(2,𝔽3)—21no—
16C22×C3×SL(2,𝔽3)—23no—
17C22×C3×SL(2,𝔽3)D313yescomplemented
18C22×C3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19#64264C2×C314yesdirect product factor
20C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
21C2×C3⋊#226C2×C311yescharacteristic, complemented
22C3×#983—312no—
23C2×C3×GL(2,𝔽3)—33no—
24C2×C3×#226—33no—
25C2×C3×#216—31no—
26C2×C3×#216—31no—
27C3×#942—34no—
28C3×#942—34no—
29C2×C3×#215—31no—
30C2×C3×#215—31no—
31C2×C3×#208—31no—
32C23×C33—41no—
33C22×C3×C3⋊D3—41no—
34C3×#638—44no—
35C2×C3×C3⋊Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C2×#983—33no—
38C2×C3×#137—91no—
39C2×#942—31no—
40C2×C32×Q8—31no—
41C2×C32×Q8—32no—
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×Q8C2×D312yescomplemented
44C22×C32×C4—31no—
45C2×#638—41no—
46C2×C3×#67—61no—
47C2×C3×#67—123no—
48C3×#235—34no—
49C3×#235—62no—
50C2×C3×D12—31no—
51C2×C3×C4×D3—61no—
52C2×C3×Dic6—31no—
53C2×C3×SL(2,𝔽3)—21no—
54C2×C3×SL(2,𝔽3)—22no—
55C2×C3×SL(2,𝔽3)—23no—
56C2×C3×SL(2,𝔽3)—26no—
57C2×C3×SL(2,𝔽3)C2×D313yes—
58C2×C3×SL(2,𝔽3)C2×D316yescomplemented
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
60C2×C3×SL(2,𝔽3)C22×C312yescomplemented
61C3⋊GL(2,𝔽3)C22×C312yescomplemented
62C3⋊#226C22×C312yescomplemented
63C3×GL(2,𝔽3)—36no—
64C3×#226—36no—
65C3×#216—32no—
66C3×#216—34no—
67C3×#216—32no—
68C3×#215—32no—
69C3×#215—34no—
70C3×#215—32no—
71C3×#208—34no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C22×C33—41no—
75C22×C33—42no—
76C22×C33—82no—
77C2×C3×C3⋊D3—42no—
78C2×C3×C3⋊D3—82no—
79C3×C3⋊Dic3—42no—
80C2×C3×#41—91no—
81C22×C3×Q8—21no—
82C22×C3×Q8C3×D311yescharacteristic, complemented
83C22×C3×Q8C3⋊D311yescharacteristic, complemented
84C2×#235—31no—
85C22×SL(2,𝔽3)—23no—
86C22×SL(2,𝔽3)C3×D313yescomplemented
87#983—312no—
88C2×GL(2,𝔽3)—33no—
89C2×#226—33no—
90C3×#137—94no—
91C3×#137—94no—
92C2×C3×Q16—91no—
93C2×C3×Q16—91no—
94C2×C3×SD16—91no—
95C2×C3×SD16—91no—
96C2×C3×M16—91no—
97C2×#216—31no—
98C2×#216—31no—
99#942—34no—
100#942—34no—
101C2×#215—31no—
102C2×#215—31no—
103C2×#208—31no—
104C23×C32—41no—
105C23×C32—43no—
106C23×C32—83no—
107C23×C32—81no—
108C23×C32S411yescharacteristic
109C22×C3⋊D3—41no—
110C22×C3×D3—61no—
111C22×C3×D3—123no—
112C32×Q8—31no—
113C32×Q8—32no—
114C32×Q8—61no—
115C32×Q8C22×D311yescharacteristic
116C2×C32×C4—31no—
117C2×C32×C4—32no—
118C2×C32×C4—31no—
119C2×C32×C4—32no—
120#638—44no—
121C2×C3⋊Dic3—41no—
122C3×#67—64no—
123C3×#67—1212no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—123no—
126C3×D12—32no—
127C3×D12—61no—
128C3×C4×D3—64no—
129C3×Dic6—32no—
130C3×Dic6—61no—
131C3×SL(2,𝔽3)—21no—
132C3×SL(2,𝔽3)—23no—
133C3×SL(2,𝔽3)C22×D313yes—
134C3×SL(2,𝔽3)C23×C311yescharacteristic
135C3×C3⋊C8—32no—
136C3×C3⋊C8—32no—
137C2×#137—91noSylow
138C2×C33—42no—
139C2×C33—41no—
140C2×C33—82no—
141C3×C3⋊D3—82no—
142C3×#41—94no—
143C3×#41—182no—
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—31no—
147C2×C3×Q8—32no—
148C2×C3×Q8—31no—
149C2×C3×Q8—32no—
150C2×C3×Q8—61no—
151C2×C3×Q8—62no—
152C2×C3×Q8—91no—
153C2×C3×Q8C2×C3×D311yescharacteristic
154C2×C3×Q8C2×C3×D312yescomplemented
155C2×C3×Q8C2×C3⋊D311yescharacteristic
156C2×C3×Q8C2×C3⋊D312yescomplemented
157C2×C3×D4—91no—
158C2×C3×D4—181no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—181no—
163C2×#67—61no—
164C2×#67—123no—
165#235—34no—
166#235—62no—
167C2×D12—31no—
168C2×C4×D3—61no—
169C2×Dic6—31no—
170C2×SL(2,𝔽3)—23no—
171C2×SL(2,𝔽3)—26no—
172C2×SL(2,𝔽3)C2×C3×D313yes—
173C2×SL(2,𝔽3)C2×C3×D316yescomplemented
174GL(2,𝔽3)—36no—
175#226—36no—
176C3×Q16—92no—
177C3×Q16—94no—
178C3×Q16—92no—
179C3×SD16—92no—
180C3×SD16—94no—
181C3×SD16—92no—
182C3×M16—94no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185#216—32no—
186#216—34no—
187#216—32no—
188#215—32no—
189#215—34no—
190#215—32no—
191#208—34no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C22×C32—22no—
195C22×C32—41no—
196C22×C32—42no—
197C22×C32—46no—
198C22×C32—43no—
199C22×C32—83no—
200C22×C32—86no—
201C22×C32—86no—
202C22×C32—82no—
203C22×C32—812no—
204C22×C32—84no—
205C22×C32—82no—
206C22×C32—81no—
207C22×C32C2×S412yes—
208C22×C32C2×S411yescharacteristic
209C2×C3⋊D3—42no—
210C2×C3⋊D3—82no—
211C2×C3×D3—62no—
212C2×C3×D3—126no—
213C2×C3×D3—122no—
214C2×C3×D3—246no—
215C32×C4—31no—
216C32×C4—31no—
217C32×C4—32no—
218C3⋊Dic3—42no—
219C3×Dic3—62no—
220C3×Dic3—126no—
221C2×#41—91no—
222C22×Q8C3×C3⋊D311yescharacteristic, complemented
223#137—94no—
224#137—94no—
225C2×Q16—91no—
226C2×Q16—91no—
227C2×SD16—91no—
228C2×SD16—91no—
229C2×M16—91no—
230C33—41noSylow
231C23×C3—21no—
232C23×C3—43no—
233C23×C3—83no—
234C23×C3—181no—
235C23×C3C3×S411yescharacteristic
236C23×C3C3⋊S411yescharacteristic
237C22×D3—61no—
238C22×D3—123no—
239C3×Q8—21no—
240C3×Q8—31no—
241C3×Q8—32no—
242C3×Q8—31no—
243C3×Q8—32no—
244C3×Q8—61no—
245C3×Q8—62no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—62no—
249C3×Q8—92no—
250C3×Q8—181no—
251C3×Q8C22×C3×D311yescharacteristic
252C3×Q8C22×C3⋊D311yescharacteristic
253C3×D4—92no—
254C3×D4—181no—
255C3×D4—184no—
256C2×C3×C4—31no—
257C2×C3×C4—32no—
258C2×C3×C4—31no—
259C2×C3×C4—32no—
260C2×C3×C4—32no—
261C2×C3×C4—32no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—184no—
269C2×C3×C4—181no—
270#67—64no—
271#67—1212no—
272C2×Dic3—61no—
273C2×Dic3—123no—
274D12—32no—
275D12—61no—
276C4×D3—64no—
277Dic6—32no—
278Dic6—61no—
279SL(2,𝔽3)—23no—
280SL(2,𝔽3)C22×C3×D313yes—
281C3×C8—92no—
282C3×C8—92no—
283C3⋊C8—32no—
284C3⋊C8—32no—
285C2×C32—22no—
286C2×C32—41no—
287C2×C32—42no—
288C2×C32—46no—
289C2×C32—43no—
290C2×C32—83no—
291C2×C32—86no—
292C2×C32—82no—
293C2×C32—86no—
294C2×C32—812no—
295C2×C32—84no—
296C2×C32—81no—
297C2×C32—82no—
298C2×C32#98312yescomplemented
299C2×C32C22×S411yescharacteristic
300C3⋊D3—82no—
301C3×D3—122no—
302C3×D3—246no—
303#41—94no—
304#41—182no—
305C2×Q8—31no—
306C2×Q8—32no—
307C2×Q8—91no—
308C2×Q8C2×C3×C3⋊D311yescharacteristic
309C2×Q8C2×C3×C3⋊D312yescomplemented
310C2×D4—91no—
311C2×D4—181no—
312C22×C4—31no—
313C22×C4—181no—
314Q16—92no—
315Q16—94no—
316Q16—92no—
317SD16—92no—
318SD16—94no—
319SD16—92no—
320M16—94no—
321C2×C8—91no—
322C2×C8—91no—
323C22×C3—24no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—43no—
329C22×C3—46no—
330C22×C3—86no—
331C22×C3—812no—
332C22×C3—86no—
333C22×C3—83no—
334C22×C3—182no—
335C22×C3—362no—
336C22×C3C2×C3×S412yes—
337C22×C3C2×C3×S411yescharacteristic
338C22×C3C2×C3⋊S412yes—
339C22×C3C2×C3⋊S411yescharacteristic
340C2×D3—62no—
341C2×D3—126no—
342C2×D3—122no—
343C2×D3—246no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—32no—
347C3×C4—31no—
348C3×C4—32no—
349C3×C4—31no—
350C3×C4—61no—
351C3×C4—62no—
352C3×C4—61no—
353C3×C4—182no—
354Dic3—62no—
355Dic3—126no—
356C32—41no—
357C32—43no—
358C32—83no—
359C32—81no—
360C32C2×#98311yescharacteristic, complemented
361C23—181no—
362C23C3×C3⋊S411yescharacteristic
363Q8—31no—
364Q8—32no—
365Q8—61no—
366Q8—92no—
367Q8—181no—
368Q8C22×C3×C3⋊D311yescharacteristic
369D4—92no—
370D4—181no—
371D4—184no—
372C2×C4—31no—
373C2×C4—32no—
374C2×C4—31no—
375C2×C4—32no—
376C2×C4—184no—
377C2×C4—181no—
378C8—92no—
379C8—92no—
380C2×C3—24no—
381C2×C3—22no—
382C2×C3—22no—
383C2×C3—22no—
384C2×C3—21no—
385C2×C3—46no—
386C2×C3—43no—
387C2×C3—86no—
388C2×C3—812no—
389C2×C3—86no—
390C2×C3—83no—
391C2×C3—362no—
392C2×C3C3×#98312yescomplemented
393C2×C3#6426412yesdirect product factor
394C2×C3C22×C3×S411yescharacteristic
395C2×C3C22×C3⋊S411yescharacteristic
396D3—122no—
397D3—246no—
398C22—22no—
399C22—182no—
400C22—362no—
401C22C2×C3×C3⋊S412yes—
402C22C2×C3×C3⋊S411yescharacteristic
403C4—31no—
404C4—32no—
405C4—31no—
406C4—182no—
407C3—21no—
408C3—43no—
409C3—83no—
410C3C2×C3×#98311yescharacteristic, complemented
411C3C2×#6426411yescharacteristic, direct product factor
412C2—22no—
413C2—362no—
414C2C3×#6426412yesdirect product factor
415C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
hypercenterC23×C3C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C3×#64264C2direct (N × H)
3. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
4. C2×C3×C3⋊#226C2semidirect
5. C2×#64264C3direct (N × H)
10. C2×C32×SL(2,𝔽3)C22semidirect
11. C3×C3⋊GL(2,𝔽3)C22semidirect
12. C3×C3⋊#226C22semidirect
13. C22×C32×Q8D3semidirect
17. C22×C3×SL(2,𝔽3)D3semidirect
18. C22×C3×SL(2,𝔽3)C2×C3semidirect
19. #64264C2×C3direct (N × H)
20. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
21. C2×C3⋊#226C2×C3semidirect
43. C2×C32×Q8C2×D3semidirect
58. C2×C3×SL(2,𝔽3)C2×D3semidirect
60. C2×C3×SL(2,𝔽3)C22×C3semidirect
61. C3⋊GL(2,𝔽3)C22×C3semidirect
62. C3⋊#226C22×C3semidirect
82. C22×C3×Q8C3×D3semidirect
83. C22×C3×Q8C3⋊D3semidirect
86. C22×SL(2,𝔽3)C3×D3semidirect
154. C2×C3×Q8C2×C3×D3semidirect
156. C2×C3×Q8C2×C3⋊D3semidirect
173. C2×SL(2,𝔽3)C2×C3×D3semidirect
222. C22×Q8C3×C3⋊D3semidirect
298. C2×C32#983semidirect
309. C2×Q8C2×C3×C3⋊D3semidirect
360. C32C2×#983semidirect
392. C2×C3C3×#983semidirect
393. C2×C3#64264direct (N × H)
410. C3C2×C3×#983semidirect
411. C3C2×#64264direct (N × H)
414. C2C3×#64264direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C3×#642642
C2×C3×C3⋊GL(2,𝔽3)2
C2×C3×C3⋊#2262
C2×#642643
C2×C3×#9833
C2×C3×#9423
C2×C3×#6384