Finite Group Databasebeta

← C2×C3×#5948 #11919894

Subgroups — C2×C3×#5948 #11919894

Order 1152.

2392 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 2392 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C3×#5948C212yesmaximal, direct product factor
2C2×C3×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C23×C3×#41C311yesmaximal, characteristic, complemented
4C2×#5948C313yesmaximal, direct product factor
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
6C2×C3×#41⋊C3—44nomaximal
7C2×C3×C4×A4—41nomaximal
8C22×C3×#41—38no—
9C22×C3×#41—34no—
10C22×C3×#41—34no—
11C22×C3×#41—32no—
12C22×C3×#41C2×C312yescomplemented
13C23×C3×Q8C2×C311yescharacteristic
14C23×C3×D4—31no—
15C24×C3×C4—31no—
16#5948C2×C316yesdirect product factor
17C2×C22⋊SL(2,𝔽3)C2×C313yescomplemented
18C22×C3×A4—41no—
19C3×#41⋊C3—48no—
20C2×C3×SL(2,𝔽3)—44no—
21C3×C4×A4—42no—
22C23×#41C3211yescharacteristic, complemented, Sylow
23C25×C3—31no—
24C2×C3×#41—38no—
25C2×C3×#41—312no—
26C2×C3×#41—324no—
27C2×C3×#41—324no—
28C2×C3×#41—324no—
29C2×C3×#41—324no—
30C2×C3×#41—38no—
31C2×C3×#41—34no—
32C2×C3×#41—38no—
33C2×C3×#41—38no—
34C2×C3×#41—34no—
35C2×C3×#41A414yescomplemented
36C22×C3×Q8—34no—
37C22×C3×Q8—34no—
38C22×C3×Q8—31no—
39C22×C3×Q8C22×C311yescharacteristic
40C22×C3×D4—32no—
41C22×C3×D4—312no—
42C22×C3×D4—312no—
43C22×C3×D4—32no—
44C23×C3×C4—312no—
45C23×C3×C4—36no—
46C23×C3×C4—36no—
47C23×C3×C4—31no—
48C23×C3×C4—32no—
49C23×C3×C4—32no—
50C23×C3×C4A411yescharacteristic
51C22⋊SL(2,𝔽3)C22×C313yescomplemented
52C2×#41⋊C3—412no—
53C2×C4×A4—43no—
54C2×C3×A4—42no—
55C2×C3×A4—41no—
56C2×C32×C4—161no—
57C3×SL(2,𝔽3)—44no—
58C22×#41—38no—
59C22×#41—34no—
60C22×#41—34no—
61C22×#41—32no—
62C22×#41C2×C3212yescomplemented
63C23×Q8C2×C3211yescharacteristic
64C23×D4—31no—
65C24×C4—31no—
66C24×C3—36no—
67C24×C3—36no—
68C24×C3—32no—
69C24×C3—66no—
70C24×C3—62no—
71C24×C3C2×A411yescharacteristic
72C22×A4—43no—
73C3×#41—324no—
74C3×#41—324no—
75C3×#41—324no—
76C3×#41—348no—
77C3×#41—324no—
78C3×#41—38no—
79C3×#41—38no—
80C3×#41—38no—
81C3×#41C2×A418yescomplemented
82C2×C3×Q8—312no—
83C2×C3×Q8—312no—
84C2×C3×Q8—34no—
85C2×C3×Q8—34no—
86C2×C3×Q8—34no—
87C2×C3×Q8C2×A414yes—
88C2×C3×D4—34no—
89C2×C3×D4—324no—
90C2×C3×D4—324no—
91C2×C3×D4—324no—
92C2×C3×D4—324no—
93C2×C3×D4—312no—
94C22×C3×C4—36no—
95C22×C3×C4—36no—
96C22×C3×C4—324no—
97C22×C3×C4—312no—
98C22×C3×C4—312no—
99C22×C3×C4—324no—
100C22×C3×C4—312no—
101C22×C3×C4—312no—
102C22×C3×C4—312no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—34no—
107C22×C3×C4—31no—
108C22×C3×C4—32no—
109C22×C3×C4C2×A412yes—
110#41⋊C3—424no—
111C2×SL(2,𝔽3)—412no—
112C4×A4—46no—
113C22×C32—161no—
114C3×A4—41no—
115C32×C4—162no—
116C25—31no—
117C2×#41—38no—
118C2×#41—312no—
119C2×#41—324no—
120C2×#41—324no—
121C2×#41—324no—
122C2×#41—324no—
123C2×#41—38no—
124C2×#41—34no—
125C2×#41—38no—
126C2×#41—38no—
127C2×#41—34no—
128C2×#41C3×A414yescomplemented
129C22×Q8—34no—
130C22×Q8—34no—
131C22×Q8—31no—
132C22×Q8C22×C3211yescharacteristic
133C22×D4—32no—
134C22×D4—312no—
135C22×D4—312no—
136C22×D4—32no—
137C23×C4—312no—
138C23×C4—36no—
139C23×C4—36no—
140C23×C4—31no—
141C23×C4—32no—
142C23×C4—32no—
143C23×C4C3×A411yescharacteristic
144C23×C3—31no—
145C23×C3—32no—
146C23×C3—31no—
147C23×C3—312no—
148C23×C3—312no—
149C23×C3—34no—
150C23×C3—612no—
151C23×C3—66no—
152C23×C3—612no—
153C23×C3—612no—
154C23×C3—612no—
155C23×C3—62no—
156C23×C3#41⋊C312yescomplemented
157C23×C3C22×A411yescharacteristic
158C2×A4—43no—
159C2×A4—46no—
160C3×Q8—34no—
161C3×Q8—312no—
162C3×Q8—34no—
163C3×Q8C22×A414yes—
164C3×D4—38no—
165C3×D4—38no—
166C3×D4—324no—
167C3×D4—324no—
168C2×C3×C4—32no—
169C2×C3×C4—36no—
170C2×C3×C4—312no—
171C2×C3×C4—36no—
172C2×C3×C4—324no—
173C2×C3×C4—324no—
174C2×C3×C4—31no—
175C2×C3×C4—34no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—38no—
179C2×C3×C4—38no—
180C2×C3×C4—32no—
181C2×C3×C4—163no—
182C2×C3×C4C22⋊A411yescharacteristic
183SL(2,𝔽3)—412no—
184C2×C32—162no—
185C2×C32—161no—
186C24—36no—
187C24—36no—
188C24—32no—
189C24—66no—
190C24—62no—
191C24C2×C3×A411yescharacteristic
192#41—324no—
193#41—324no—
194#41—324no—
195#41—348no—
196#41—38no—
197#41—324no—
198#41—38no—
199#41—38no—
200#41C2×C3×A418yescomplemented
201C2×Q8—312no—
202C2×Q8—312no—
203C2×Q8—34no—
204C2×Q8—34no—
205C2×Q8—34no—
206C2×Q8C2×C3×A414yes—
207C2×D4—34no—
208C2×D4—324no—
209C2×D4—324no—
210C2×D4—324no—
211C2×D4—324no—
212C2×D4—312no—
213C22×C4—36no—
214C22×C4—36no—
215C22×C4—324no—
216C22×C4—312no—
217C22×C4—312no—
218C22×C4—312no—
219C22×C4—312no—
220C22×C4—324no—
221C22×C4—312no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—34no—
226C22×C4—31no—
227C22×C4—32no—
228C22×C4C2×C3×A412yes—
229C22×C3—32no—
230C22×C3—31no—
231C22×C3—32no—
232C22×C3—32no—
233C22×C3—32no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—38no—
237C22×C3—624no—
238C22×C3—612no—
239C22×C3—612no—
240C22×C3—68no—
241C22×C3—163no—
242C22×C3C2×#41⋊C311yescharacteristic, complemented
243C22×C3C2×C22⋊A411yescharacteristic
244A4—43no—
245C3×C4—32no—
246C3×C4—34no—
247C3×C4—34no—
248C3×C4—166no—
249C3×C4C2×C22⋊A412yes—
250C32—161noSylow
251C23—31no—
252C23—32no—
253C23—31no—
254C23—312no—
255C23—312no—
256C23—34no—
257C23—66no—
258C23—612no—
259C23—612no—
260C23—612no—
261C23—612no—
262C23—62no—
263C23C3×#41⋊C312yescomplemented
264C23C22×C3×A411yescharacteristic
265Q8—34no—
266Q8—34no—
267Q8—312no—
268Q8C22×C3×A414yes—
269D4—38no—
270D4—324no—
271D4—38no—
272D4—324no—
273C2×C4—32no—
274C2×C4—36no—
275C2×C4—312no—
276C2×C4—36no—
277C2×C4—324no—
278C2×C4—324no—
279C2×C4—31no—
280C2×C4—34no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—38no—
284C2×C4—38no—
285C2×C4—32no—
286C2×C4C3×C22⋊A411yescharacteristic
287C2×C3—32no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—68no—
291C2×C3—166no—
292C2×C3—163no—
293C2×C3#594812yesdirect product factor
294C2×C3C22×C22⋊A411yescharacteristic
295C22—31no—
296C22—32no—
297C22—32no—
298C22—32no—
299C22—32no—
300C22—31no—
301C22—31no—
302C22—38no—
303C22—612no—
304C22—612no—
305C22—624no—
306C22—68no—
307C22C2×C3×#41⋊C311yescharacteristic, complemented
308C22C2×C3×C22⋊A411yescharacteristic
309C4—32no—
310C4—34no—
311C4—34no—
312C4C2×C3×C22⋊A412yes—
313C3—163no—
314C3C2×#594811yescharacteristic, direct product factor
315C2—31no—
316C2—31no—
317C2—32no—
318C2—68no—
319C2C3×#594812yesdirect product factor
320C2C22×C3×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C22⋊A4
commutator, cosocle, hypocenterC22×Q8C22×C32
FrattiniC2C22×C3×C22⋊A4
FittingC23×C3×#41C3
socleC24×C3C2×A4
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#5948C2direct (N × H)
2. C2×C3×C22⋊SL(2,𝔽3)C2semidirect
3. C23×C3×#41C3semidirect
4. C2×#5948C3direct (N × H)
5. C3×C22⋊SL(2,𝔽3)C22semidirect
12. C22×C3×#41C2×C3semidirect
16. #5948C2×C3direct (N × H)
17. C2×C22⋊SL(2,𝔽3)C2×C3semidirect
22. C23×#41C32semidirect
35. C2×C3×#41A4semidirect
51. C22⋊SL(2,𝔽3)C22×C3semidirect
62. C22×#41C2×C32semidirect
81. C3×#41C2×A4semidirect
128. C2×#41C3×A4semidirect
156. C23×C3#41⋊C3semidirect
200. #41C2×C3×A4semidirect
242. C22×C3C2×#41⋊C3semidirect
263. C23C3×#41⋊C3semidirect
293. C2×C3#5948direct (N × H)
307. C22C2×C3×#41⋊C3semidirect
314. C3C2×#5948direct (N × H)
319. C2C3×#5948direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#59482
C2×C3×C22⋊SL(2,𝔽3)2
C23×C3×#413
C2×#59483
C2×C3×#41⋊C34
C2×C3×C4×A44