Finite Group Databasebeta

← C22×C3×#984 #11923695

Subgroups — C22×C3×#984 #11923695

Order 1152.

1529 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1529 conjugacy classes of subgroups — too many to draw. Only its 220 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3×#984C2112yesmaximal, direct product factor
3C22×C3×#226C212yesmaximal, complemented
4C22×#984C311yesmaximal, characteristic, direct product factor
5C22×C3×#137—31nomaximal
6C22×C3×Dic6—41nomaximal
7C2×C3×#41⋊C3C2216yes—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C3×#984C22116yesdirect product factor
10C2×C3×#226C22112yescomplemented
11C22×C3×#41D311yescharacteristic
12C23×C3×Q8—31no—
13C22×#41⋊C3—21no—
14C22×#41⋊C3C2×C311yescharacteristic
15C2×#984C2×C3112yesdirect product factor
16C22×#226C2×C312yescomplemented
17C2×C3×#137—312no—
18C2×C3×#137—312no—
19C22×C3×Q16—32no—
20C22×C3×SD16—32no—
21C22×C3×M16—31no—
22C22×C32×C4—41no—
23C22×C3×Dic3—42no—
24C2×C3×Dic6—412no—
25C3×#41⋊C3C2314yes—
26C2×C3×SL(2,𝔽3)C2313yes—
27C3×#226C2318yescomplemented
28C22×#137—31noSylow
29C2×C3×#41—36no—
30C2×C3×#41C2×D316yes—
31C22×C3×Q8—32no—
32C22×C3×Q8—312no—
33C22×C3×Q8—66no—
34C22×C3×Q8—61no—
35C22×C3×Q8C2×D311yescharacteristic
36C22×C3×D4—31no—
37C23×C3×C4—31no—
38C23×C3×C4—61no—
39C22×Dic6—41no—
40C2×#41⋊C3—26no—
41C2×#41⋊C3C22×C316yes—
42C22×SL(2,𝔽3)—21no—
43C22×SL(2,𝔽3)C22×C311yescharacteristic
44#984C22×C3116yesdirect product factor
45C2×#226C22×C3112yescomplemented
46C3×#137—348no—
47C3×#137—316no—
48C2×C3×Q16—312no—
49C2×C3×Q16—312no—
50C2×C3×SD16—324no—
51C2×C3×M16—312no—
52C22×C3×C8—32no—
53C23×C32—41no—
54C2×C32×C4—46no—
55C2×C3×Dic3—412no—
56C3×Dic6—416no—
57C3×SL(2,𝔽3)C2411yescharacteristic
58C22×#41C3×D311yescharacteristic
59C23×Q8—31no—
60C2×#137—312no—
61C2×#137—312no—
62C22×Q16—32no—
63C22×SD16—32no—
64C22×M16—31no—
65C24×C3—31no—
66C3×#41—312no—
67C3×#41—64no—
68C3×#41C22×D314yes—
69C2×C3×Q8—33no—
70C2×C3×Q8—312no—
71C2×C3×Q8—316no—
72C2×C3×Q8—66no—
73C2×C3×Q8—624no—
74C2×C3×Q8—612no—
75C2×C3×Q8C22×D313yes—
76C2×C3×D4—36no—
77C2×C3×D4—63no—
78C22×C3×C4—31no—
79C22×C3×C4—36no—
80C22×C3×C4—36no—
81C22×C3×C4—41no—
82C22×C3×C4—612no—
83C22×C3×C4—62no—
84C22×C3×C4—81no—
85C22×C3×C4S411yescharacteristic
86C22×Dic3—42no—
87C2×Dic6—412no—
88#41⋊C3—24no—
89#41⋊C3C23×C314yes—
90C2×SL(2,𝔽3)—23no—
91C2×SL(2,𝔽3)C23×C313yes—
92#226C23×C318yescomplemented
93C3×Q16—324no—
94C3×Q16—38no—
95C3×SD16—332no—
96C3×M16—316no—
97C2×C3×C8—312no—
98C22×C32—44no—
99C22×C32—43no—
100C32×C4—44no—
101C3×Dic3—48no—
102C2×#41—36no—
103C2×#41C2×C3×D316yes—
104C22×Q8—32no—
105C22×Q8—312no—
106C22×Q8—66no—
107C22×Q8—61no—
108C22×Q8C2×C3×D311yescharacteristic
109C22×D4—31no—
110C23×C4—31no—
111C23×C4—61no—
112#137—348no—
113#137—316no—
114C2×Q16—312no—
115C2×Q16—312no—
116C2×SD16—324no—
117C2×M16—312no—
118C22×C8—32no—
119C23×C3—36no—
120C23×C3—41no—
121C23×C3—64no—
122C23×C3—81no—
123C23×C3C2×S411yescharacteristic
124C3×Q8—33no—
125C3×Q8—38no—
126C3×Q8—61no—
127C3×Q8—612no—
128C3×Q8—616no—
129C3×Q8C23×D311yescharacteristic
130C3×D4—34no—
131C3×D4—66no—
132C2×C3×C4—33no—
133C2×C3×C4—33no—
134C2×C3×C4—34no—
135C2×C3×C4—312no—
136C2×C3×C4—46no—
137C2×C3×C4—616no—
138C2×C3×C4—612no—
139C2×C3×C4—86no—
140C2×C3×C4C2×S416yes—
141C2×Dic3—412no—
142Dic6—416no—
143SL(2,𝔽3)—21no—
144SL(2,𝔽3)C24×C311yescharacteristic
145C3×C8—38no—
146C2×C32—46no—
147C2×C32—41no—
148C24—31no—
149#41—312no—
150#41—64no—
151#41C22×C3×D314yes—
152C2×Q8—33no—
153C2×Q8—312no—
154C2×Q8—316no—
155C2×Q8—66no—
156C2×Q8—624no—
157C2×Q8—612no—
158C2×Q8C22×C3×D313yes—
159C2×D4—36no—
160C2×D4—63no—
161C22×C4—31no—
162C22×C4—36no—
163C22×C4—36no—
164C22×C4—612no—
165C22×C4—62no—
166C22×C4C3×S411yescharacteristic
167Q16—324no—
168Q16—38no—
169SD16—332no—
170M16—316no—
171C2×C8—312no—
172C22×C3—34no—
173C22×C3—43no—
174C22×C3—44no—
175C22×C3—612no—
176C22×C3—83no—
177C22×C3—84no—
178C22×C3#98414yesdirect product factor
179C22×C3C22×S413yes—
180C3×C4—31no—
181C3×C4—33no—
182C3×C4—44no—
183C3×C4—68no—
184C3×C4—84no—
185C3×C4C22×S414yes—
186Dic3—48no—
187C32—41noSylow
188C23—36no—
189C23—64no—
190C23C2×C3×S411yescharacteristic
191Q8—33no—
192Q8—38no—
193Q8—61no—
194Q8—612no—
195Q8—616no—
196Q8C23×C3×D311yescharacteristic
197D4—34no—
198D4—66no—
199C2×C4—33no—
200C2×C4—33no—
201C2×C4—34no—
202C2×C4—312no—
203C2×C4—616no—
204C2×C4—612no—
205C2×C4C2×C3×S416yes—
206C8—38no—
207C2×C3—41no—
208C2×C3—46no—
209C2×C3—64no—
210C2×C3—81no—
211C2×C3—86no—
212C2×C3C2×#98416yesdirect product factor
213C2×C3C23×S411yescharacteristic
214C22—34no—
215C22—612no—
216C22C3×#98414yesdirect product factor
217C22C22×C3×S413yes—
218C4—31no—
219C4—33no—
220C4—68no—
221C4C22×C3×S414yes—
222C3—41no—
223C3—81no—
224C3C22×#98411yescharacteristic, direct product factor
225C2—64no—
226C2C2×C3×#98416yesdirect product factor
227C2C23×C3×S411yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24×C3
FrattiniC2C23×C3×S4
FittingC22×C3×#41D3
prime radicalC22×C3×#41⋊C3C2
hypercenterC22×C3×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#984C2direct (N × H)
3. C22×C3×#226C2semidirect
4. C22×#984C3direct (N × H)
9. C3×#984C22direct (N × H)
10. C2×C3×#226C22semidirect
15. C2×#984C2×C3direct (N × H)
16. C22×#226C2×C3semidirect
27. C3×#226C23semidirect
44. #984C22×C3direct (N × H)
45. C2×#226C22×C3semidirect
92. #226C23×C3semidirect
178. C22×C3#984direct (N × H)
212. C2×C3C2×#984direct (N × H)
216. C22C3×#984direct (N × H)
224. C3C22×#984direct (N × H)
226. C2C2×C3×#984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3×#9842
C22×C3×#2262
C22×#9843
C22×C3×#1373
C22×C3×Dic64