Finite Group Databasebeta

← C2×C3×#4622 #11920762

Subgroups — C2×C3×#4622 #11920762

Order 1152.

372 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 34 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×#95⋊C3C211yesmaximal, characteristic, complemented
2C3×#4622C212yesmaximal, direct product factor
3C2×#4622C311yesmaximal, characteristic, direct product factor
4C2×C3×#327—31nomaximal
5C22×C3×S4—41nomaximal
6C3×#95⋊C3C2211yescharacteristic, complemented
7C2×C3×#121—31no—
8C2×C3×C22⋊C8—31no—
9C2×C3×#95D311yescharacteristic, complemented
10C2×#95⋊C3—21no—
11C2×#95⋊C3C2×C311yescharacteristic, complemented
12#4622C2×C312yesdirect product factor
13C3×#327—32no—
14C3×#327—32no—
15C22×C3×A4—41no—
16C2×C3×S4—42no—
17C2×C3×S4—44no—
18C2×#327—31noSylow
19C22×S4—41no—
20C22×C3×D4—31no—
21C22×C3×D4—61no—
22C23×C3×C4—31no—
23C22×C3×C8—61no—
24C3×#121—32no—
25C3×#121—32no—
26C3×#121—61no—
27C3×#121—61no—
28C2×C3×C4⋊C4—31no—
29C2×C3×C22⋊C4—61no—
30C3×C22⋊C8—32no—
31C3×C22⋊C8—61no—
32C3×#95—31no—
33C3×#95C2×D311yescharacteristic, complemented
34#95⋊C3—21no—
35#95⋊C3C22×C311yescharacteristic, complemented
36C22×C3×D3—161no—
37C2×C3×A4—42no—
38C2×C3×A4—41no—
39C3×S4—44no—
40C2×#121—31no—
41C2×C22⋊C8—31no—
42C2×#95C3×D311yescharacteristic, complemented
43#327—32no—
44#327—32no—
45C24×C3—61no—
46C24×C3S411yescharacteristic
47C22×A4—41no—
48C22×A4—81no—
49C2×S4—42no—
50C2×S4—44no—
51C2×C3×D4—32no—
52C2×C3×D4—61no—
53C2×C3×D4—62no—
54C2×C3×D4—62no—
55C2×C3×D4—122no—
56C2×C3×D4—122no—
57C2×C3×D4—122no—
58C22×C3×C4—31no—
59C22×C3×C4—31no—
60C22×C3×C4—31no—
61C22×C3×C4—31no—
62C22×C3×C4—31no—
63C22×C3×C4—31no—
64C22×C3×C4—61no—
65C22×C3×C4—122no—
66C2×C3×C8—62no—
67C2×C3×C8—122no—
68C3×C4⋊C4—32no—
69C3×C4⋊C4—61no—
70C3×C22⋊C4—62no—
71C3×C22⋊C4—62no—
72C22×C32—161no—
73C2×C3×D3—162no—
74C2×C3×D3—164no—
75C3×A4—41no—
76C22×D4—31no—
77C22×D4—61no—
78C23×C4—31no—
79C22×C8—61no—
80#121—32no—
81#121—32no—
82#121—61no—
83#121—61no—
84C2×C4⋊C4—31no—
85C2×C22⋊C4—61no—
86C22⋊C8—32no—
87C22⋊C8—61no—
88#95—31no—
89#95C2×C3×D311yescharacteristic, complemented
90C23×C3—31no—
91C23×C3—31no—
92C23×C3—32no—
93C23×C3—62no—
94C23×C3—122no—
95C23×C3—122no—
96C23×C3—121no—
97C23×C3—121no—
98C23×C3GL(2,𝔽3)12yes—
99C23×C3C2×S411yescharacteristic
100C22×D3—161no—
101C2×A4—41no—
102C2×A4—42no—
103C2×A4—81no—
104C2×A4—82no—
105S4—44no—
106C3×D4—122no—
107C3×D4—122no—
108C3×D4—122no—
109C3×D4—124no—
110C3×D4—122no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—62no—
116C2×C3×C4—62no—
117C2×C3×C4—62no—
118C2×C3×C4—121no—
119C2×C3×C4—121no—
120C2×C3×C4—121no—
121C2×C3×C4—121no—
122C2×C3×C4—122no—
123C3×C8—122no—
124C2×C32—161no—
125C2×C32—162no—
126C3×D3—164no—
127C24—61no—
128C24C3×S411yescharacteristic
129C2×D4—32no—
130C2×D4—61no—
131C2×D4—62no—
132C2×D4—62no—
133C2×D4—122no—
134C2×D4—122no—
135C2×D4—122no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—61no—
143C22×C4—122no—
144C2×C8—62no—
145C2×C8—122no—
146C4⋊C4—32no—
147C4⋊C4—61no—
148C22⋊C4—62no—
149C22⋊C4—62no—
150C22×C3—31no—
151C22×C3—32no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—32no—
155C22×C3—32no—
156C22×C3—61no—
157C22×C3—122no—
158C22×C3—122no—
159C22×C3—122no—
160C22×C3—161no—
161C22×C3—242no—
162C22×C3—242no—
163C22×C3—321no—
164C22×C3#85711yescharacteristic
165C22×C3C2×GL(2,𝔽3)11yescharacteristic
166C2×D3—162no—
167C2×D3—164no—
168A4—41no—
169A4—81no—
170C3×C4—61no—
171C3×C4—61no—
172C3×C4—61no—
173C3×C4—61no—
174C3×C4—122no—
175C32—161noSylow
176C23—31no—
177C23—31no—
178C23—32no—
179C23—62no—
180C23—122no—
181C23—122no—
182C23—121no—
183C23—121no—
184C23C3×GL(2,𝔽3)12yes—
185C23C2×C3×S411yescharacteristic
186D4—122no—
187D4—122no—
188D4—122no—
189D4—124no—
190D4—122no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4—122no—
203C8—122no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—32no—
207C2×C3—161no—
208C2×C3—162no—
209C2×C3—242no—
210C2×C3—321no—
211C2×C3—322no—
212C2×C3#462212yesdirect product factor
213C2×C3C2×#85711yescharacteristic
214D3—164no—
215C22—31no—
216C22—32no—
217C22—31no—
218C22—31no—
219C22—32no—
220C22—32no—
221C22—61no—
222C22—122no—
223C22—122no—
224C22—122no—
225C22—242no—
226C22—242no—
227C22C3×#85711yescharacteristic
228C22C2×C3×GL(2,𝔽3)11yescharacteristic
229C4—61no—
230C4—61no—
231C4—61no—
232C4—61no—
233C4—122no—
234C3—161no—
235C3—321no—
236C3C2×#462211yescharacteristic, direct product factor
237C2—31no—
238C2—31no—
239C2—32no—
240C2—242no—
241C2C3×#462212yesdirect product factor
242C2C2×C3×#85711yescharacteristic
subgroupG/H
center, hypercenterC22×C3#857
commutator, cosocle, hypocenter#95⋊C3C22×C3
FrattiniC23C2×C3×S4
FittingC2×C3×#95D3
socleC24×C3S4
LCSG ▷ #95⋊C3
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#95⋊C3C2semidirect
2. C3×#4622C2direct (N × H)
3. C2×#4622C3direct (N × H)
6. C3×#95⋊C3C22semidirect
9. C2×C3×#95D3semidirect
11. C2×#95⋊C3C2×C3semidirect
12. #4622C2×C3direct (N × H)
33. C3×#95C2×D3semidirect
35. #95⋊C3C22×C3semidirect
42. C2×#95C3×D3semidirect
89. #95C2×C3×D3semidirect
212. C2×C3#4622direct (N × H)
236. C3C2×#4622direct (N × H)
241. C2C3×#4622direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#95⋊C32
C3×#46222
C2×#46223
C2×C3×#3273
C22×C3×S44