Finite Group Databasebeta

← C2×C3×#64267 #13181385

Subgroups — C2×C3×#64267 #13181385

Order 1728.

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

normal subgroups only

This group has 975 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×#64267C214yesmaximal, direct product factor
3C2×C3×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
4C2×#64267C311yesmaximal, characteristic, direct product factor
5C2×C3×#986—33nomaximal
6C2×C3×#949—31nomaximal
7C2×C3×C4⋊(C3⋊D3)—41nomaximal
8C32×#41⋊C3C2212yescomplemented
9C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
10C3×C3⋊GL(2,𝔽3)C2214yescomplemented
11C2×C32×#41D311yescharacteristic, complemented
12C22×C3×D12—31no—
13C2×C3×#41⋊C3—21no—
14C2×C3×#41⋊C3—23no—
15C2×C3×#41⋊C3D313yescomplemented
16C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
17#64267C2×C314yesdirect product factor
18C2×C3⋊GL(2,𝔽3)C2×C312yescomplemented
19C3×#986—312no—
20C2×C3×GL(2,𝔽3)—36no—
21C3×#949—34no—
22C3×#949—34no—
23C2×C3×#215—32no—
24C2×C3×#213—32no—
25C2×C3×#208—31no—
26C22×C3×C3⋊D3—42no—
27C2×C33×C4—41no—
28C3×C4⋊(C3⋊D3)—44no—
29C32×SL(2,𝔽3)C2311yescharacteristic, complemented
30C2×#986—33no—
31C2×C3×AGL(1,ℤ/8)—91no—
32C2×#949—31no—
33C23×C3×D3—61no—
34C32×#41—32no—
35C32×#41C2×D312yescomplemented
36C2×C32×Q8C2×D311yescharacteristic
37C2×C32×D4—31no—
38C22×C32×C4—31no—
39C2×C4⋊(C3⋊D3)—41no—
40C2×C3×D12—32no—
41C2×C3×D12—34no—
42C2×C3×D12—62no—
43C2×C3×D12—61no—
44C2×C3×D12—123no—
45C3×#41⋊C3—22no—
46C3×#41⋊C3—26no—
47C3×#41⋊C3C2×D316yescomplemented
48C3×#41⋊C3C22×C312yescomplemented
49C2×C3×SL(2,𝔽3)—21no—
50C2×C3×SL(2,𝔽3)—23no—
51C2×C3×SL(2,𝔽3)C2×D313yescomplemented
52C2×C3×SL(2,𝔽3)C22×C311yescharacteristic, complemented
53C3⋊GL(2,𝔽3)C22×C314yescomplemented
54C3×GL(2,𝔽3)—312no—
55C3×#215—34no—
56C3×#215—34no—
57C3×#213—38no—
58C3×#208—34no—
59C2×C3×C3⋊C8—32no—
60C22×C33—41no—
61C2×C3×C3⋊D3—44no—
62C2×C3×C3⋊D3—84no—
63C33×C4—42no—
64C2×C3×#41—21no—
65C2×C3×#41C3×D311yescharacteristic, complemented
66C2×C3×#41C3⋊D311yescharacteristic, complemented
67C22×C3×D4—91no—
68C22×D12—31no—
69C2×#41⋊C3—23no—
70C2×#41⋊C3C3×D313yescomplemented
71#986—312no—
72C2×GL(2,𝔽3)—36no—
73C3×AGL(1,ℤ/8)—94no—
74C3×AGL(1,ℤ/8)—94no—
75C2×C3×SD16—92no—
76C2×C3×D8—92no—
77C2×C3×M16—91no—
78#949—34no—
79#949—34no—
80C2×#215—32no—
81C2×#213—32no—
82C2×#208—31no—
83C23×C32—31no—
84C22×C3⋊D3—42no—
85C22×C3×D3—64no—
86C22×C3×D3—62no—
87C22×C3×D3—126no—
88C22×C3×D3—124no—
89C32×Q8—31no—
90C32×Q8C22×D311yescharacteristic
91C32×D4—32no—
92C32×D4—61no—
93C2×C32×C4—31no—
94C2×C32×C4—32no—
95C2×C32×C4—32no—
96C2×C32×C4—41no—
97C2×C32×C4—43no—
98C2×C32×C4—83no—
99C2×C32×C4—81no—
100C2×C32×C4S411yescharacteristic
101C4⋊(C3⋊D3)—44no—
102C3×D12—34no—
103C3×D12—62no—
104C3×D12—64no—
105C3×D12—1212no—
106C3×SL(2,𝔽3)—21no—
107C3×SL(2,𝔽3)—23no—
108C3×SL(2,𝔽3)C22×D313yescomplemented
109C3×SL(2,𝔽3)C23×C311yescharacteristic, complemented
110C3×C3⋊C8—34no—
111C2×AGL(1,ℤ/8)—91noSylow
112C2×C33—42no—
113C2×C33—41no—
114C3×C3⋊D3—84no—
115C24×C3—181no—
116C23×D3—61no—
117C3×#41—22no—
118C3×#41—32no—
119C3×#41—32no—
120C3×#41—62no—
121C3×#41C2×C3×D312yescomplemented
122C3×#41C2×C3⋊D312yescomplemented
123C2×C3×Q8—21no—
124C2×C3×Q8C2×C3×D311yescharacteristic
125C2×C3×Q8C2×C3⋊D311yescharacteristic
126C2×C3×D4—31no—
127C2×C3×D4—31no—
128C2×C3×D4—61no—
129C2×C3×D4—92no—
130C2×C3×D4—94no—
131C2×C3×D4—182no—
132C2×C3×D4—181no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—61no—
136C2×D12—32no—
137C2×D12—34no—
138C2×D12—62no—
139C2×D12—61no—
140C2×D12—123no—
141#41⋊C3—26no—
142#41⋊C3C2×C3×D316yescomplemented
143C2×SL(2,𝔽3)—23no—
144C2×SL(2,𝔽3)C2×C3×D313yescomplemented
145GL(2,𝔽3)—312no—
146C3×SD16—94no—
147C3×SD16—94no—
148C3×D8—98no—
149C3×M16—94no—
150C2×C3×C8—92no—
151#215—34no—
152#215—34no—
153#213—38no—
154#208—34no—
155C2×C3⋊C8—32no—
156C22×C32—32no—
157C22×C32—41no—
158C22×C32—43no—
159C22×C32—62no—
160C22×C32—83no—
161C22×C32—81no—
162C22×C32C2×S411yescharacteristic
163C2×C3⋊D3—44no—
164C2×C3⋊D3—84no—
165C2×C3×D3—64no—
166C2×C3×D3—1212no—
167C2×C3×D3—128no—
168C2×C3×D3—124no—
169C2×C3×D3—2412no—
170C32×C4—31no—
171C32×C4—31no—
172C32×C4—42no—
173C32×C4—46no—
174C32×C4—86no—
175C32×C4—82no—
176C32×C4C2×S412yes—
177C2×#41C3×C3⋊D311yescharacteristic, complemented
178C22×D4—91no—
179AGL(1,ℤ/8)—94no—
180AGL(1,ℤ/8)—94no—
181C2×SD16—92no—
182C2×D8—92no—
183C2×M16—91no—
184C33—41noSylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3—184no—
189C23×C3—182no—
190C23×C3—364no—
191C22×D3—64no—
192C22×D3—62no—
193C22×D3—126no—
194C22×D3—124no—
195C3×Q8—21no—
196C3×Q8—31no—
197C3×Q8—31no—
198C3×Q8—61no—
199C3×Q8C22×C3×D311yescharacteristic
200C3×Q8C22×C3⋊D311yescharacteristic
201C3×D4—32no—
202C3×D4—32no—
203C3×D4—62no—
204C3×D4—61no—
205C3×D4—61no—
206C3×D4—62no—
207C3×D4—94no—
208C3×D4—182no—
209C3×D4—184no—
210C2×C3×C4—21no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—43no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—61no—
221C2×C3×C4—83no—
222C2×C3×C4C3×S411yescharacteristic
223C2×C3×C4C3⋊S411yescharacteristic
224D12—34no—
225D12—62no—
226D12—64no—
227D12—1212no—
228SL(2,𝔽3)—23no—
229SL(2,𝔽3)C22×C3×D313yescomplemented
230C3×C8—94no—
231C3⋊C8—34no—
232C2×C32—42no—
233C2×C32—41no—
234C2×C32—46no—
235C2×C32—43no—
236C2×C32—62no—
237C2×C32—83no—
238C2×C32—86no—
239C2×C32—82no—
240C2×C32—81no—
241C2×C32#98612yescomplemented
242C2×C32C22×S411yescharacteristic
243C3⋊D3—84no—
244C3×D3—124no—
245C3×D3—2412no—
246C24—181no—
247#41—32no—
248#41C2×C3×C3⋊D312yescomplemented
249C2×Q8C2×C3×C3⋊D311yescharacteristic
250C2×D4—31no—
251C2×D4—92no—
252C2×D4—94no—
253C2×D4—182no—
254C2×D4—181no—
255C22×C4—31no—
256SD16—94no—
257SD16—94no—
258D8—98no—
259M16—94no—
260C2×C8—92no—
261C22×C3—21no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—43no—
265C22×C3—62no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—83no—
269C22×C3—122no—
270C22×C3—184no—
271C22×C3—368no—
272C22×C3—364no—
273C22×C3C2×C3×S411yescharacteristic
274C22×C3C2×C3⋊S411yescharacteristic
275C2×D3—64no—
276C2×D3—1212no—
277C2×D3—128no—
278C2×D3—124no—
279C2×D3—2412no—
280C3×C4—22no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—46no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4—86no—
289C3×C4C2×C3×S412yes—
290C3×C4C2×C3⋊S412yes—
291C32—41no—
292C32—43no—
293C32—83no—
294C32—81no—
295C32C2×#98611yescharacteristic, complemented
296C23—31no—
297C23—184no—
298C23—182no—
299C23—364no—
300Q8—31no—
301Q8C22×C3×C3⋊D311yescharacteristic
302D4—32no—
303D4—61no—
304D4—94no—
305D4—182no—
306D4—184no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4C3×C3⋊S411yescharacteristic
311C8—94no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—46no—
315C2×C3—43no—
316C2×C3—62no—
317C2×C3—62no—
318C2×C3—86no—
319C2×C3—83no—
320C2×C3—122no—
321C2×C3—364no—
322C2×C3C3×#98612yescomplemented
323C2×C3#6426712yesdirect product factor
324C2×C3C22×C3×S411yescharacteristic
325C2×C3C22×C3⋊S411yescharacteristic
326D3—124no—
327D3—2412no—
328C22—32no—
329C22—62no—
330C22—184no—
331C22—368no—
332C22—364no—
333C22C2×C3×C3⋊S411yescharacteristic
334C4—31no—
335C4—31no—
336C4C2×C3×C3⋊S412yes—
337C3—21no—
338C3—43no—
339C3—83no—
340C3C2×C3×#98611yescharacteristic, complemented
341C3C2×#6426711yescharacteristic, direct product factor
342C2—62no—
343C2—364no—
344C2C3×#6426712yesdirect product factor
345C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3×#64267C2direct (N × H)
3. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
4. C2×#64267C3direct (N × H)
8. C32×#41⋊C3C22semidirect
9. C2×C32×SL(2,𝔽3)C22semidirect
10. C3×C3⋊GL(2,𝔽3)C22semidirect
11. C2×C32×#41D3semidirect
15. C2×C3×#41⋊C3D3semidirect
16. C2×C3×#41⋊C3C2×C3semidirect
17. #64267C2×C3direct (N × H)
18. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
29. C32×SL(2,𝔽3)C23semidirect
35. C32×#41C2×D3semidirect
47. C3×#41⋊C3C2×D3semidirect
48. C3×#41⋊C3C22×C3semidirect
51. C2×C3×SL(2,𝔽3)C2×D3semidirect
52. C2×C3×SL(2,𝔽3)C22×C3semidirect
53. C3⋊GL(2,𝔽3)C22×C3semidirect
65. C2×C3×#41C3×D3semidirect
66. C2×C3×#41C3⋊D3semidirect
70. C2×#41⋊C3C3×D3semidirect
108. C3×SL(2,𝔽3)C22×D3semidirect
109. C3×SL(2,𝔽3)C23×C3semidirect
121. C3×#41C2×C3×D3semidirect
122. C3×#41C2×C3⋊D3semidirect
142. #41⋊C3C2×C3×D3semidirect
144. C2×SL(2,𝔽3)C2×C3×D3semidirect
177. C2×#41C3×C3⋊D3semidirect
229. SL(2,𝔽3)C22×C3×D3semidirect
241. C2×C32#986semidirect
248. #41C2×C3×C3⋊D3semidirect
295. C32C2×#986semidirect
322. C2×C3C3×#986semidirect
323. C2×C3#64267direct (N × H)
340. C3C2×C3×#986semidirect
341. C3C2×#64267direct (N × H)
344. C2C3×#64267direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×#642672
C2×C3×C3⋊GL(2,𝔽3)2
C2×#642673
C2×C3×#9863
C2×C3×#9493
C2×C3×C4⋊(C3⋊D3)4