Finite Group Databasebeta

← C2×C3×#64201 #13180700

Subgroups — C2×C3×#64201 #13180700

Order 1728.

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

normal subgroups only

This group has 745 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#64201C214yesmaximal, direct product factor
5C2×C3×#983—31nomaximal
6C2×#64201C311yesmaximal, characteristic, direct product factor
7C2×C3×#915—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×#625—41nomaximal
10C3×D3×SL(2,𝔽3)C2212yescomplemented
11C3×C3⋊#226C2212yescomplemented
12C32×GL(2,𝔽3)C2212yescomplemented
13C2×C3×D3×Q8D311yescharacteristic, complemented
14C2×C3×#237—31no—
15C22×C3×SL(2,𝔽3)—31no—
16C2×D3×SL(2,𝔽3)—21no—
17C2×D3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×C3⋊#226C2×C311yescharacteristic, complemented
19C3×#983—34no—
20C2×C3×GL(2,𝔽3)—21no—
21C2×C3×GL(2,𝔽3)D311yescharacteristic, complemented
22C2×C3×GL(2,𝔽3)C2×C311yescharacteristic, complemented
23C2×C3×#226—31no—
24C2×C3×#226—61no—
25#64201C2×C314yesdirect product factor
26C2×C32×SD16—31no—
27C2×C3×#216—31no—
28C2×C3×#214—31no—
29C3×#915—34no—
30C3×#915—34no—
31C2×C3×Dic12—31no—
32C2×C3×ΓL(1,𝔽25)—31no—
33C22×C32×D3—41no—
34C22×C32×D3—41no—
35C2×C3×C3⋊Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×#625—44no—
38C2×#983—31no—
39C2×C3×#137—91no—
40C2×#915—31no—
41C2×C32×Q8C2×D311yescharacteristic
42C2×C32×D4—31no—
43C2×C3×#67—61no—
44C2×C3×#67—121no—
45C22×C3×Dic3—61no—
46C3×D3×Q8—32no—
47C3×D3×Q8C2×D312yescomplemented
48C3×#237—34no—
49C3×#237—62no—
50C2×C3×C4×D3—31no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—31no—
53C2×C3×SL(2,𝔽3)—21no—
54C2×C3×SL(2,𝔽3)—21no—
55C2×C3×SL(2,𝔽3)—21no—
56C2×C3×SL(2,𝔽3)—32no—
57C2×C3×SL(2,𝔽3)—41no—
58C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
60C2×#625—41no—
61D3×SL(2,𝔽3)—22no—
62D3×SL(2,𝔽3)C22×C312yescomplemented
63C3⋊#226C22×C312yescomplemented
64C3×GL(2,𝔽3)—22no—
65C3×GL(2,𝔽3)C2×D312yescomplemented
66C3×GL(2,𝔽3)C22×C312yescomplemented
67C3×#226—32no—
68C3×#226—62no—
69C32×SD16—32no—
70C32×SD16—32no—
71C2×C32×C8—31no—
72C3×#216—32no—
73C3×#216—32no—
74C3×#214—34no—
75C2×C3×C3⋊C8—31no—
76C3×Dic12—34no—
77C3×ΓL(1,𝔽25)—34no—
78C22×C33—41no—
79C2×C32×D3—42no—
80C2×C32×D3—42no—
81C2×C32×D3—82no—
82C2×C32×D3—82no—
83C3×C3⋊Dic3—42no—
84C2×C3×#41—91no—
85C22×C3×Q8—31no—
86C2×D3×Q8C3×D311yescharacteristic, complemented
87C2×#237—31no—
88C22×SL(2,𝔽3)—31no—
89C22×SL(2,𝔽3)—61no—
90#983—34no—
91C2×GL(2,𝔽3)C3×D311yescharacteristic, complemented
92C2×#226—31no—
93C2×#226—61no—
94C3×#137—94no—
95C3×#137—94no—
96C2×C3×Q16—91no—
97C2×C3×Q16—91no—
98C2×C3×SD16—31no—
99C2×C3×SD16—31no—
100C2×C3×SD16—61no—
101C2×C3×SD16—91no—
102C2×C3×M16—91no—
103C2×#216—31no—
104C2×#214—31no—
105#915—34no—
106#915—34no—
107C2×Dic12—31no—
108C2×ΓL(1,𝔽25)—31no—
109C23×C32—61no—
110C23×C32—121no—
111C22×C3×D3—41no—
112C22×C3×D3—41no—
113C22×C3×D3—41no—
114C22×C3×D3—81no—
115C22×C3×D3—81no—
116C22×C3×D3S411yescharacteristic
117C32×Q8—31no—
118C32×Q8C22×D311yescharacteristic
119C32×D4—32no—
120C32×D4—61no—
121C2×C32×C4—31no—
122C2×C3⋊Dic3—41no—
123C3×#67—64no—
124C3×#67—124no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—121no—
129C2×C3×Dic3—241no—
130C3×C4×D3—32no—
131C3×C4×D3—32no—
132C3×Dic6—32no—
133C3×Dic6—32no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3×SL(2,𝔽3)—21no—
137C3×SL(2,𝔽3)—21no—
138C3×SL(2,𝔽3)—21no—
139C3×SL(2,𝔽3)—41no—
140C3×SL(2,𝔽3)C22×D311yescharacteristic
141C3×SL(2,𝔽3)C23×C311yescharacteristic
142#625—44no—
143C32×C8—32no—
144C3×C3⋊C8—32no—
145C2×#137—91noSylow
146C2×C33—42no—
147C2×C33—41no—
148C32×D3—82no—
149C32×D3—82no—
150C3×#41—94no—
151C3×#41—182no—
152C2×C3×Q8—21no—
153C2×C3×Q8—32no—
154C2×C3×Q8—91no—
155C2×C3×Q8—91no—
156C2×C3×Q8—92no—
157C2×C3×Q8D3211yescharacteristic
158C2×C3×Q8C2×C3×D311yescharacteristic
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—181no—
163C22×C3×C4—91no—
164C22×C3×C4—181no—
165C2×#67—61no—
166C2×#67—121no—
167C22×Dic3—61no—
168D3×Q8—32no—
169D3×Q8C2×C3×D312yescomplemented
170#237—34no—
171#237—62no—
172C2×C4×D3—31no—
173C2×Dic6—31no—
174C2×Dic6—31no—
175C2×SL(2,𝔽3)—21no—
176C2×SL(2,𝔽3)—21no—
177C2×SL(2,𝔽3)—32no—
178C2×SL(2,𝔽3)—41no—
179C2×SL(2,𝔽3)—62no—
180C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
181GL(2,𝔽3)C2×C3×D312yescomplemented
182#226—32no—
183#226—62no—
184C3×Q16—94no—
185C3×Q16—92no—
186C3×Q16—92no—
187C3×SD16—32no—
188C3×SD16—32no—
189C3×SD16—32no—
190C3×SD16—32no—
191C3×SD16—62no—
192C3×SD16—62no—
193C3×SD16—94no—
194C3×M16—94no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197C2×C3×C8—61no—
198C2×C3×C8—91no—
199#216—32no—
200#216—32no—
201#214—34no—
202C2×C3⋊C8—31no—
203Dic12—34no—
204ΓL(1,𝔽25)—34no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—62no—
208C22×C32—81no—
209C22×C32—81no—
210C22×C32—81no—
211C22×C32—122no—
212C22×C32—122no—
213C22×C32—161no—
214C22×C32—242no—
215C22×C32C2×S411yescharacteristic
216C2×C3×D3—22no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—82no—
221C2×C3×D3—84no—
222C2×C3×D3—82no—
223C2×C3×D3—82no—
224C2×C3×D3—82no—
225C2×C3×D3—84no—
226C2×C3×D3—82no—
227C2×C3×D3C2×S412yes—
228C32×C4—31no—
229C32×C4—31no—
230C3⋊Dic3—42no—
231C3×Dic3—32no—
232C3×Dic3—62no—
233C3×Dic3—122no—
234C3×Dic3—242no—
235C2×#41—91no—
236C22×Q8—31no—
237#137—94no—
238#137—94no—
239C2×Q16—91no—
240C2×Q16—91no—
241C2×SD16—31no—
242C2×SD16—91no—
243C2×M16—91no—
244C33—41noSylow
245C23×C3—31no—
246C23×C3—61no—
247C23×C3—61no—
248C23×C3—121no—
249C23×C3—121no—
250C23×C3—241no—
251C22×D3—41no—
252C22×D3C3×S411yescharacteristic
253C3×Q8—21no—
254C3×Q8—31no—
255C3×Q8—31no—
256C3×Q8—61no—
257C3×Q8—92no—
258C3×Q8—92no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×Q8C2×D3211yescharacteristic
262C3×Q8C22×C3×D311yescharacteristic
263C3×D4—32no—
264C3×D4—32no—
265C3×D4—62no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—121no—
269C3×D4—184no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—61no—
273C2×C3×C4—92no—
274C2×C3×C4—92no—
275C2×C3×C4—91no—
276C2×C3×C4—184no—
277C2×C3×C4—181no—
278#67—64no—
279#67—124no—
280C2×Dic3—31no—
281C2×Dic3—64no—
282C2×Dic3—61no—
283C2×Dic3—121no—
284C2×Dic3—241no—
285C4×D3—32no—
286C4×D3—32no—
287Dic6—32no—
288Dic6—32no—
289Dic6—61no—
290Dic6—61no—
291SL(2,𝔽3)—21no—
292SL(2,𝔽3)—21no—
293SL(2,𝔽3)—41no—
294SL(2,𝔽3)C22×C3×D311yescharacteristic
295C3×C8—32no—
296C3×C8—32no—
297C3×C8—62no—
298C3×C8—92no—
299C3⋊C8—32no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—42no—
303C2×C32—42no—
304C2×C32—81no—
305C2×C32—82no—
306C2×C32—81no—
307C2×C32—81no—
308C2×C32—82no—
309C2×C32—82no—
310C2×C32—122no—
311C2×C32—161no—
312C2×C32—162no—
313C2×C32—242no—
314C2×C32#98312yescomplemented
315C2×C32C22×S411yescharacteristic
316C3×D3—22no—
317C3×D3—82no—
318C3×D3—84no—
319C3×D3—82no—
320C3×D3—82no—
321C3×D3—84no—
322#41—94no—
323#41—182no—
324C2×Q8—32no—
325C2×Q8—91no—
326C2×Q8—91no—
327C2×Q8—92no—
328C2×Q8C3×D3211yescharacteristic
329C2×D4—31no—
330C2×D4—181no—
331C22×C4—91no—
332C22×C4—181no—
333Q16—94no—
334Q16—92no—
335Q16—92no—
336SD16—32no—
337SD16—32no—
338SD16—94no—
339M16—94no—
340C2×C8—31no—
341C2×C8—91no—
342C22×C3—21no—
343C22×C3—32no—
344C22×C3—41no—
345C22×C3—62no—
346C22×C3—62no—
347C22×C3—62no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—122no—
353C22×C3—122no—
354C22×C3—161no—
355C22×C3—242no—
356C22×C3—242no—
357C22×C3—244no—
358C22×C3—242no—
359C22×C3D3×S411yescharacteristic
360C22×C3C2×C3×S411yescharacteristic
361C2×D3—22no—
362C2×D3—42no—
363C2×D3—82no—
364C2×D3C2×C3×S412yes—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—92no—
372C3×C4—182no—
373Dic3—32no—
374Dic3—62no—
375Dic3—122no—
376Dic3—242no—
377C32—41no—
378C32—41no—
379C32—81no—
380C32—81no—
381C32—81no—
382C32—161no—
383C32C2×#98311yescharacteristic, complemented
384C23—31no—
385C23—61no—
386Q8—31no—
387Q8—92no—
388Q8—92no—
389Q8—181no—
390Q8—181no—
391Q8C2×C3×D3211yescharacteristic
392D4—32no—
393D4—61no—
394D4—184no—
395C2×C4—31no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—91no—
399C2×C4—184no—
400C2×C4—181no—
401C8—32no—
402C8—92no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—42no—
406C2×C3—41no—
407C2×C3—62no—
408C2×C3—82no—
409C2×C3—82no—
410C2×C3—81no—
411C2×C3—81no—
412C2×C3—122no—
413C2×C3—122no—
414C2×C3—162no—
415C2×C3—161no—
416C2×C3—242no—
417C2×C3—244no—
418C2×C3—242no—
419C2×C3#6420112yesdirect product factor
420C2×C3C3×#98312yescomplemented
421C2×C3C2×D3×S411yescharacteristic
422C2×C3C22×C3×S411yescharacteristic
423D3—22no—
424D3—82no—
425C22—32no—
426C22—62no—
427C22—62no—
428C22—122no—
429C22C3×D3×S411yescharacteristic
430C4—31no—
431C4—31no—
432C4—92no—
433C4—182no—
434C3—21no—
435C3—41no—
436C3—81no—
437C3—81no—
438C3—161no—
439C3C2×#6420111yescharacteristic, direct product factor
440C3C2×C3×#98311yescharacteristic, complemented
441C2—62no—
442C2—122no—
443C2C3×#6420112yesdirect product factor
444C2C2×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×D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊#226C2semidirect
3. C2×C32×GL(2,𝔽3)C2semidirect
4. C3×#64201C2direct (N × H)
6. C2×#64201C3direct (N × H)
10. C3×D3×SL(2,𝔽3)C22semidirect
11. C3×C3⋊#226C22semidirect
12. C32×GL(2,𝔽3)C22semidirect
13. C2×C3×D3×Q8D3semidirect
17. C2×D3×SL(2,𝔽3)C2×C3semidirect
18. C2×C3⋊#226C2×C3semidirect
21. C2×C3×GL(2,𝔽3)D3semidirect
22. C2×C3×GL(2,𝔽3)C2×C3semidirect
25. #64201C2×C3direct (N × H)
47. C3×D3×Q8C2×D3semidirect
62. D3×SL(2,𝔽3)C22×C3semidirect
63. C3⋊#226C22×C3semidirect
65. C3×GL(2,𝔽3)C2×D3semidirect
66. C3×GL(2,𝔽3)C22×C3semidirect
86. C2×D3×Q8C3×D3semidirect
91. C2×GL(2,𝔽3)C3×D3semidirect
169. D3×Q8C2×C3×D3semidirect
181. GL(2,𝔽3)C2×C3×D3semidirect
314. C2×C32#983semidirect
383. C32C2×#983semidirect
419. C2×C3#64201direct (N × H)
420. C2×C3C3×#983semidirect
439. C3C2×#64201direct (N × H)
440. C3C2×C3×#983semidirect
443. C2C3×#64201direct (N × H)

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
C2×C32×GL(2,𝔽3)2
C3×#642012
C2×C3×#9833
C2×#642013
C2×C3×#9153
C2×C3×#6254