Finite Group Databasebeta

← C2×C3×#393⋊C3 #11919434

Subgroups — C2×C3×#393⋊C3 #11919434

Order 1152.

564 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 72 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#393⋊C3C212yesmaximal, direct product factor
3C2×C3×#393C311yesmaximal, characteristic, complemented
4C2×#393⋊C3C313yesmaximal, direct product factor
5C23×C3×A4—41nomaximal
6C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
7C23×C3×Q8C2×C311yescharacteristic, complemented
8C2×C22⋊SL(2,𝔽3)C2×C313yescomplemented
9C22×C3×C22⋊C4—31no—
10C3×#393—32no—
11C3×#393C2×C312yescomplemented
12C2×C3×#95—31no—
13#393⋊C3C2×C316yesdirect product factor
14C22×C3×A4—44no—
15C22×C3×A4—41no—
16C22×C3×A4—42no—
17C2×C3×SL(2,𝔽3)—82no—
18C2×#393C3211yescharacteristic, complemented, Sylow
19C25×C3A411yescharacteristic
20C23×A4—43no—
21C22×C3×Q8—31no—
22C22×C3×Q8—62no—
23C22×C3×Q8—62no—
24C22×C3×Q8C22×C311yescharacteristic, complemented
25C23×C3×C4—31no—
26C23×C3×C4—31no—
27C22⋊SL(2,𝔽3)C22×C313yescomplemented
28C2×C3×C22⋊C4—32no—
29C2×C3×C22⋊C4—32no—
30C2×C3×C22⋊C4—62no—
31C2×C3×C22⋊C4—62no—
32C3×#95—32no—
33C3×#95—32no—
34C23×C32—161no—
35C2×C3×A4—42no—
36C2×C3×A4—44no—
37C2×C3×A4—41no—
38C3×SL(2,𝔽3)—82no—
39C23×Q8C2×C3211yescharacteristic, complemented
40C22×C22⋊C4—31no—
41#393—32no—
42#393C2×C3212yescomplemented
43C2×#95—31no—
44C24×C3—61no—
45C24×C3—62no—
46C24×C3—61no—
47C24×C3SL(2,𝔽3)14yescomplemented
48C24×C3C2×A411yescharacteristic
49C24×C3C2×A412yes—
50C22×A4—412no—
51C22×A4—46no—
52C22×A4—43no—
53C2×C3×Q8—22no—
54C2×C3×Q8—62no—
55C2×C3×Q8—62no—
56C2×C3×Q8—64no—
57C2×C3×Q8—62no—
58C2×C3×Q8—62no—
59C2×C3×Q8—64no—
60C2×C3×Q8—62no—
61C22×C3×C4—32no—
62C22×C3×C4—32no—
63C22×C3×C4—31no—
64C22×C3×C4—31no—
65C22×C3×C4—32no—
66C22×C3×C4—62no—
67C22×C3×C4—62no—
68C22×C3×C4—61no—
69C22×C3×C4—61no—
70C22×C3×C4—64no—
71C2×SL(2,𝔽3)—86no—
72C3×C22⋊C4—64no—
73C3×C22⋊C4—64no—
74C22×C32—164no—
75C22×C32—162no—
76C22×C32—161no—
77C3×A4—41no—
78C25C3×A411yescharacteristic
79C22×Q8—31no—
80C22×Q8—62no—
81C22×Q8—62no—
82C22×Q8C22×C3211yescharacteristic, complemented
83C23×C4—31no—
84C23×C4—31no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—62no—
88C2×C22⋊C4—62no—
89#95—32no—
90#95—32no—
91C23×C3—31no—
92C23×C3—32no—
93C23×C3—31no—
94C23×C3—41no—
95C23×C3—62no—
96C23×C3—64no—
97C23×C3—64no—
98C23×C3—62no—
99C23×C3—122no—
100C23×C3—121no—
101C23×C3—122no—
102C23×C3—163no—
103C23×C3C2×SL(2,𝔽3)12yes—
104C23×C3C2×SL(2,𝔽3)14yescomplemented
105C23×C3C22×A411yescharacteristic
106C2×A4—43no—
107C2×A4—46no—
108C2×A4—412no—
109C3×Q8—22no—
110C3×Q8—62no—
111C3×Q8—62no—
112C3×Q8—62no—
113C3×Q8—64no—
114C2×C3×C4—32no—
115C2×C3×C4—32no—
116C2×C3×C4—62no—
117C2×C3×C4—64no—
118C2×C3×C4—62no—
119C2×C3×C4—62no—
120C2×C3×C4—62no—
121C2×C3×C4—62no—
122C2×C3×C4—64no—
123C2×C3×C4—122no—
124C2×C3×C4—122no—
125SL(2,𝔽3)—86no—
126C2×C32—164no—
127C2×C32—162no—
128C2×C32—161no—
129C24—61no—
130C24—61no—
131C24—62no—
132C24C3×SL(2,𝔽3)14yescomplemented
133C24C2×C3×A411yescharacteristic
134C24C2×C3×A412yes—
135C2×Q8—22no—
136C2×Q8—62no—
137C2×Q8—62no—
138C2×Q8—64no—
139C2×Q8—62no—
140C2×Q8—62no—
141C2×Q8—64no—
142C2×Q8—62no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—32no—
148C22×C4—62no—
149C22×C4—62no—
150C22×C4—61no—
151C22×C4—61no—
152C22×C4—64no—
153C22⋊C4—64no—
154C22⋊C4—64no—
155C22×C3—31no—
156C22×C3—32no—
157C22×C3—32no—
158C22×C3—32no—
159C22×C3—32no—
160C22×C3—31no—
161C22×C3—31no—
162C22×C3—44no—
163C22×C3—42no—
164C22×C3—64no—
165C22×C3—122no—
166C22×C3—124no—
167C22×C3—166no—
168C22×C3—1612no—
169C22×C3—163no—
170C22×C3#120⋊C311yescharacteristic
171C22×C3C22×SL(2,𝔽3)11yescharacteristic
172A4—43no—
173C3×C4—62no—
174C3×C4—62no—
175C3×C4—122no—
176C32—161noSylow
177C23—31no—
178C23—32no—
179C23—31no—
180C23—41no—
181C23—62no—
182C23—64no—
183C23—64no—
184C23—62no—
185C23—121no—
186C23—122no—
187C23—122no—
188C23C2×C3×SL(2,𝔽3)12yes—
189C23C2×C3×SL(2,𝔽3)14yescomplemented
190C23C22×C3×A411yescharacteristic
191Q8—22no—
192Q8—62no—
193Q8—62no—
194Q8—62no—
195Q8—64no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—62no—
199C2×C4—64no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—64no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—32no—
210C2×C3—44no—
211C2×C3—1612no—
212C2×C3—163no—
213C2×C3—166no—
214C2×C3#393⋊C312yesdirect product factor
215C2×C3C2×#120⋊C311yescharacteristic
216C22—31no—
217C22—32no—
218C22—32no—
219C22—31no—
220C22—31no—
221C22—32no—
222C22—32no—
223C22—42no—
224C22—44no—
225C22—64no—
226C22—122no—
227C22—124no—
228C22C3×#120⋊C311yescharacteristic
229C22C22×C3×SL(2,𝔽3)11yescharacteristic
230C4—62no—
231C4—62no—
232C4—122no—
233C3—163no—
234C3C2×#393⋊C311yescharacteristic, direct product factor
235C2—31no—
236C2—31no—
237C2—32no—
238C2—44no—
239C2C3×#393⋊C312yesdirect product factor
240C2C2×C3×#120⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22×C3#120⋊C3
commutator, cosocle, hypocenterC22×Q8C22×C32
FrattiniC23C22×C3×A4
FittingC2×C3×#393C3
socleC24×C3C2×A4
LCSG ▷ C22×Q8
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊SL(2,𝔽3)C2semidirect
2. C3×#393⋊C3C2direct (N × H)
3. C2×C3×#393C3semidirect
4. C2×#393⋊C3C3direct (N × H)
6. C3×C22⋊SL(2,𝔽3)C22semidirect
7. C23×C3×Q8C2×C3semidirect
8. C2×C22⋊SL(2,𝔽3)C2×C3semidirect
11. C3×#393C2×C3semidirect
13. #393⋊C3C2×C3direct (N × H)
18. C2×#393C32semidirect
24. C22×C3×Q8C22×C3semidirect
27. C22⋊SL(2,𝔽3)C22×C3semidirect
39. C23×Q8C2×C32semidirect
42. #393C2×C32semidirect
47. C24×C3SL(2,𝔽3)semidirect
82. C22×Q8C22×C32semidirect
104. C23×C3C2×SL(2,𝔽3)semidirect
132. C24C3×SL(2,𝔽3)semidirect
189. C23C2×C3×SL(2,𝔽3)semidirect
214. C2×C3#393⋊C3direct (N × H)
234. C3C2×#393⋊C3direct (N × H)
239. C2C3×#393⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3×#393⋊C32
C2×C3×#3933
C2×#393⋊C33
C23×C3×A44