Finite Group Databasebeta

← C2×C3×#64197 #13180693

Subgroups — C2×C3×#64197 #13180693

Order 1728.

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

normal subgroups only

This group has 729 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⋊#226C211yesmaximal, characteristic, complemented
3C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#64197C214yesmaximal, direct product factor
5C2×C3×#985—31nomaximal
6C2×#64197C311yesmaximal, characteristic, direct product factor
7C2×C3×#916—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×D3×Dic3—41nomaximal
10C3×C3⋊(#41⋊C3)C2212yescomplemented
11C3×C3⋊#226C2212yescomplemented
12C32×GL(2,𝔽3)C2212yescomplemented
13C2×C3×#239D311yescharacteristic, complemented
14C2×C3×#237—31no—
15C2×C3×#41⋊C3—31no—
16C2×C3⋊(#41⋊C3)—21no—
17C2×C3⋊(#41⋊C3)C2×C311yescharacteristic, complemented
18C2×C3⋊#226C2×C311yescharacteristic, complemented
19C3×#985—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#64197C2×C314yesdirect product factor
26C2×C32×SD16—31no—
27C2×C3×#216—31no—
28C2×C3×#213—31no—
29C3×#916—34no—
30C3×#916—34no—
31C2×C3×#204—31no—
32C2×C3×D3×C8—31no—
33C22×C32×D3—41no—
34C2×C3×C3⋊Dic3—41no—
35C2×C32×Dic3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37C3×D3×Dic3—44no—
38C2×#985—31no—
39C2×C3×#135—91no—
40C2×#916—31no—
41C2×C32×Q8C2×D311yescharacteristic
42C2×C32×D4—31no—
43C2×C3×#67—61no—
44C22×C3×Dic3—61no—
45C3×#239—32no—
46C3×#239C2×D312yescomplemented
47C3×#237—34no—
48C3×#237—62no—
49C2×C3×D12—31no—
50C2×C3×C4×D3—31no—
51C2×C3×C4×D3—121no—
52C2×C3×Dic6—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
59C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
60C2×D3×Dic3—41no—
61C3⋊(#41⋊C3)—22no—
62C3⋊(#41⋊C3)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×#213—34no—
75C2×C3×C3⋊C8—31no—
76C3×#204—34no—
77C3×D3×C8—34no—
78C22×C33—41no—
79C2×C32×D3—42no—
80C2×C32×D3—44no—
81C3×C3⋊Dic3—42no—
82C32×Dic3—42no—
83C2×C3×#41—31no—
84C2×C3×#41—91no—
85C2×#239C3×D311yescharacteristic, complemented
86C2×#237—31no—
87C2×#41⋊C3—31no—
88C2×#41⋊C3—61no—
89#985—34no—
90C2×GL(2,𝔽3)C3×D311yescharacteristic, complemented
91C2×#226—31no—
92C2×#226—61no—
93C3×#135—94no—
94C3×#135—94no—
95C2×C3×Q16—91no—
96C2×C3×SD16—31no—
97C2×C3×SD16—31no—
98C2×C3×SD16—61no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C22×C3×C8—91no—
102C2×#216—31no—
103C2×#213—31no—
104#916—34no—
105#916—34no—
106C2×#204—31no—
107C2×D3×C8—31no—
108C23×C32—61no—
109C22×C3×D3—31no—
110C22×C3×D3—41no—
111C22×C3×D3—41no—
112C22×C3×D3—81no—
113C32×Q8—31no—
114C32×Q8C22×D311yescharacteristic
115C32×D4—32no—
116C32×D4—61no—
117C2×C32×C4—31no—
118C2×C32×C4—121no—
119C2×C3⋊Dic3—41no—
120C3×#67—64no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—81no—
125C2×C3×Dic3—121no—
126C2×C3×Dic3—241no—
127C2×C3×Dic3S411yescharacteristic
128C3×D12—32no—
129C3×D12—61no—
130C3×C4×D3—32no—
131C3×C4×D3—32no—
132C3×C4×D3—124no—
133C3×Dic6—32no—
134C3×Dic6—61no—
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
141D3×Dic3—44no—
142C32×C8—32no—
143C3×C3⋊C8—32no—
144C2×#135—91noSylow
145C2×C33—42no—
146C2×C33—41no—
147C32×D3—44no—
148C3×#41—32no—
149C3×#41—92no—
150C3×#41—94no—
151C3×#41—182no—
152C2×C3×Q8—21no—
153C2×C3×Q8—91no—
154C2×C3×Q8D3211yescharacteristic
155C2×C3×Q8C2×C3×D311yescharacteristic
156C2×C3×D4—31no—
157C2×C3×D4—31no—
158C2×C3×D4—61no—
159C2×C3×D4—91no—
160C2×C3×D4—181no—
161C22×C3×C4—91no—
162C22×C3×C4—181no—
163C2×#67—61no—
164C22×Dic3—61no—
165#239—32no—
166#239C2×C3×D312yescomplemented
167#237—34no—
168#237—62no—
169C2×D12—31no—
170C2×C4×D3—31no—
171C2×C4×D3—121no—
172C2×Dic6—31no—
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)C2×C3×D312yescomplemented
180#226—32no—
181#226—62no—
182C3×Q16—92no—
183C3×Q16—92no—
184C3×SD16—32no—
185C3×SD16—32no—
186C3×SD16—32no—
187C3×SD16—32no—
188C3×SD16—62no—
189C3×SD16—62no—
190C3×SD16—94no—
191C3×D8—94no—
192C2×C3×C8—31no—
193C2×C3×C8—31no—
194C2×C3×C8—61no—
195C2×C3×C8—94no—
196C2×C3×C8—91no—
197#216—32no—
198#216—32no—
199#213—34no—
200C2×C3⋊C8—31no—
201#204—34no—
202D3×C8—34no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—62no—
206C22×C32—81no—
207C22×C32—81no—
208C22×C32—81no—
209C22×C32—122no—
210C22×C32—161no—
211C22×C32C2×S411yescharacteristic
212C2×C3×D3—32no—
213C2×C3×D3—42no—
214C2×C3×D3—42no—
215C2×C3×D3—44no—
216C2×C3×D3—44no—
217C2×C3×D3—62no—
218C2×C3×D3—82no—
219C2×C3×D3—84no—
220C32×C4—31no—
221C32×C4—31no—
222C32×C4—122no—
223C3⋊Dic3—42no—
224C3×Dic3—42no—
225C3×Dic3—62no—
226C3×Dic3—82no—
227C3×Dic3—122no—
228C3×Dic3—242no—
229C3×Dic3C2×S412yes—
230C2×#41—31no—
231C2×#41—91no—
232#135—94no—
233#135—94no—
234C2×Q16—91no—
235C2×SD16—31no—
236C2×SD16—91no—
237C2×D8—91no—
238C22×C8—91no—
239C33—41noSylow
240C23×C3—61no—
241C23×C3—61no—
242C23×C3—91no—
243C23×C3—121no—
244C22×D3—31no—
245C22×D3—41no—
246C3×Q8—21no—
247C3×Q8—31no—
248C3×Q8—31no—
249C3×Q8—61no—
250C3×Q8—92no—
251C3×Q8—181no—
252C3×Q8C2×D3211yescharacteristic
253C3×Q8C22×C3×D311yescharacteristic
254C3×D4—32no—
255C3×D4—32no—
256C3×D4—62no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—92no—
260C3×D4—121no—
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—92no—
268C2×C3×C4—92no—
269C2×C3×C4—121no—
270C2×C3×C4—184no—
271C2×C3×C4—181no—
272C2×C3×C4—241no—
273#67—64no—
274C2×Dic3—61no—
275C2×Dic3—64no—
276C2×Dic3—121no—
277C2×Dic3—241no—
278C2×Dic3C3×S411yescharacteristic
279D12—32no—
280D12—61no—
281C4×D3—32no—
282C4×D3—32no—
283C4×D3—124no—
284Dic6—32no—
285Dic6—61no—
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—122no—
306C2×C32—161no—
307C2×C32—162no—
308C2×C32#98512yescomplemented
309C2×C32C22×S411yescharacteristic
310C3×D3—44no—
311C3×D3—44no—
312C3×D3—62no—
313C3×D3—84no—
314#41—32no—
315#41—92no—
316#41—94no—
317#41—182no—
318C2×Q8—91no—
319C2×Q8C3×D3211yescharacteristic
320C2×D4—31no—
321C2×D4—91no—
322C2×D4—181no—
323C22×C4—91no—
324C22×C4—181no—
325Q16—92no—
326Q16—92no—
327SD16—32no—
328SD16—32no—
329SD16—94no—
330D8—94no—
331C2×C8—31no—
332C2×C8—94no—
333C2×C8—91no—
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—62no—
337C22×C3—62no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—92no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—122no—
344C22×C3—161no—
345C22×C3—182no—
346C22×C3—242no—
347C22×C3D3×S411yescharacteristic
348C22×C3C2×C3×S411yescharacteristic
349C2×D3—32no—
350C2×D3—42no—
351C2×D3—44no—
352C2×D3—62no—
353C3×C4—31no—
354C3×C4—32no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—122no—
361C3×C4—182no—
362C3×C4—242no—
363Dic3—62no—
364Dic3—122no—
365Dic3—242no—
366Dic3C2×C3×S412yes—
367C32—41no—
368C32—41no—
369C32—81no—
370C32—81no—
371C32—81no—
372C32—161no—
373C32C2×#98511yescharacteristic, complemented
374C23—61no—
375C23—91no—
376Q8—31no—
377Q8—92no—
378Q8—181no—
379Q8C2×C3×D3211yescharacteristic
380D4—32no—
381D4—61no—
382D4—92no—
383D4—181no—
384D4—184no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—184no—
390C2×C4—181no—
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—122no—
402C2×C3—122no—
403C2×C3—162no—
404C2×C3—161no—
405C2×C3—182no—
406C2×C3—242no—
407C2×C3#6419712yesdirect product factor
408C2×C3C3×#98512yescomplemented
409C2×C3C2×D3×S411yescharacteristic
410C2×C3C22×C3×S411yescharacteristic
411D3—44no—
412D3—62no—
413C22—62no—
414C22—92no—
415C22—122no—
416C22—182no—
417C22C3×D3×S411yescharacteristic
418C4—31no—
419C4—32no—
420C4—31no—
421C4—182no—
422C3—21no—
423C3—41no—
424C3—81no—
425C3—81no—
426C3—161no—
427C3C2×#6419711yescharacteristic, direct product factor
428C3C2×C3×#98511yescharacteristic, complemented
429C2—122no—
430C2—182no—
431C2C3×#6419712yesdirect 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⋊#226C2semidirect
3. C2×C32×GL(2,𝔽3)C2semidirect
4. C3×#64197C2direct (N × H)
6. C2×#64197C3direct (N × H)
10. C3×C3⋊(#41⋊C3)C22semidirect
11. C3×C3⋊#226C22semidirect
12. C32×GL(2,𝔽3)C22semidirect
13. C2×C3×#239D3semidirect
17. C2×C3⋊(#41⋊C3)C2×C3semidirect
18. C2×C3⋊#226C2×C3semidirect
21. C2×C3×GL(2,𝔽3)D3semidirect
22. C2×C3×GL(2,𝔽3)C2×C3semidirect
25. #64197C2×C3direct (N × H)
46. C3×#239C2×D3semidirect
62. C3⋊(#41⋊C3)C22×C3semidirect
63. C3⋊#226C22×C3semidirect
65. C3×GL(2,𝔽3)C2×D3semidirect
66. C3×GL(2,𝔽3)C22×C3semidirect
85. C2×#239C3×D3semidirect
90. C2×GL(2,𝔽3)C3×D3semidirect
166. #239C2×C3×D3semidirect
179. GL(2,𝔽3)C2×C3×D3semidirect
308. C2×C32#985semidirect
373. C32C2×#985semidirect
407. C2×C3#64197direct (N × H)
408. C2×C3C3×#985semidirect
427. C3C2×#64197direct (N × H)
428. C3C2×C3×#985semidirect
431. C2C3×#64197direct (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⋊#2262
C2×C32×GL(2,𝔽3)2
C3×#641972
C2×C3×#9853
C2×#641973
C2×C3×#9163
C2×C3×D3×Dic34