Finite Group Databasebeta

← C22×C3×#859 #11922784

Subgroups — C22×C3×#859 #11922784

Order 1152.

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

normal subgroups only

This group has 961 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×#859C216yesmaximal, direct product factor
3C22×#859C311yesmaximal, characteristic, direct product factor
4C22×C3×#103—31nomaximal
5C23×C3×Dic3—41nomaximal
6C22×C3×SL(2,𝔽3)C414yescomplemented
7C22×C3×SL(2,𝔽3)C2213yes—
8C3×#859C2214yesdirect product factor
9C23×C3×Q8D311yescharacteristic
10C23×SL(2,𝔽3)—21no—
11C23×SL(2,𝔽3)C2×C311yescharacteristic
12C23×C3×C8—31no—
13C22×C3×C4⋊C4—31no—
14C2×#859C2×C316yesdirect product factor
15C2×C3×#103—36no—
16C2×C3×#103—36no—
17C24×C32—41no—
18C22×C3×Dic3—46no—
19C22×C3×Dic3—48no—
20C2×C3×SL(2,𝔽3)C2×C416yescomplemented
21C2×C3×SL(2,𝔽3)C2311yescharacteristic
22C22×#103—31noSylow
23C22×C3×Q8—34no—
24C22×C3×Q8—33no—
25C22×C3×Q8Dic314yescomplemented
26C22×C3×Q8C2×D313yes—
27C23×C3×C4—31no—
28C23×C3×C4—61no—
29C23×Dic3—41no—
30C22×SL(2,𝔽3)—24no—
31C22×SL(2,𝔽3)—23no—
32C22×SL(2,𝔽3)C3×C414yescomplemented
33C22×SL(2,𝔽3)C22×C313yes—
34C22×C3×C8—36no—
35C22×C3×C8—64no—
36C2×C3×C4⋊C4—36no—
37C2×C3×C4⋊C4—63no—
38#859C22×C314yesdirect product factor
39C3×#103—312no—
40C3×#103—34no—
41C23×C32—44no—
42C23×C32—44no—
43C23×C32—44no—
44C23×C32—43no—
45C2×C3×Dic3—44no—
46C2×C3×Dic3—424no—
47C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
48C23×Q8C3×D311yescharacteristic
49C23×C8—31no—
50C22×C4⋊C4—31no—
51C2×#103—36no—
52C2×#103—36no—
53C24×C3—41no—
54C24×C3—81no—
55C24×C3S411yescharacteristic
56C2×C3×Q8—33no—
57C2×C3×Q8—312no—
58C2×C3×Q8—36no—
59C2×C3×Q8—66no—
60C2×C3×Q8—61no—
61C2×C3×Q8C2×Dic316yescomplemented
62C2×C3×Q8C22×D311yescharacteristic
63C22×C3×C4—34no—
64C22×C3×C4—34no—
65C22×C3×C4—33no—
66C22×C3×C4—33no—
67C22×C3×C4—66no—
68C22×C3×C4—124no—
69C22×Dic3—46no—
70C22×Dic3—48no—
71C2×SL(2,𝔽3)—26no—
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)C2×C3×C416yescomplemented
74C2×SL(2,𝔽3)C23×C311yescharacteristic
75C2×C3×C8—34no—
76C2×C3×C8—612no—
77C3×C4⋊C4—34no—
78C3×C4⋊C4—66no—
79C22×C32—416no—
80C22×C32—46no—
81C22×C32—46no—
82C22×C32—46no—
83C22×C32—41no—
84C3×Dic3—48no—
85C22×Q8—33no—
86C22×Q8—34no—
87C22×Q8C3×Dic314yescomplemented
88C22×Q8C2×C3×D313yes—
89C23×C4—31no—
90C23×C4—61no—
91C22×C8—36no—
92C22×C8—64no—
93C2×C4⋊C4—36no—
94C2×C4⋊C4—63no—
95#103—312no—
96#103—34no—
97C23×C3—43no—
98C23×C3—44no—
99C23×C3—44no—
100C23×C3—44no—
101C23×C3—83no—
102C23×C3—84no—
103C23×C3—84no—
104C23×C3—84no—
105C23×C3#22614yes—
106C23×C3GL(2,𝔽3)14yes—
107C23×C3SL(2,ℤ/4)14yes—
108C23×C3C2×S413yes—
109C3×Q8—36no—
110C3×Q8—31no—
111C3×Q8—63no—
112C3×Q8—64no—
113C3×Q8C22×Dic311yescharacteristic, complemented
114C2×C3×C4—36no—
115C2×C3×C4—36no—
116C2×C3×C4—312no—
117C2×C3×C4—31no—
118C2×C3×C4—33no—
119C2×C3×C4—64no—
120C2×C3×C4—1212no—
121C2×Dic3—44no—
122C2×Dic3—424no—
123SL(2,𝔽3)—21no—
124SL(2,𝔽3)C22×C3×C411yescharacteristic, complemented
125C3×C8—64no—
126C2×C32—41no—
127C2×C32—41no—
128C2×C32—41no—
129C2×C32—412no—
130C24C3×S411yescharacteristic
131C2×Q8—33no—
132C2×Q8—312no—
133C2×Q8—36no—
134C2×Q8—66no—
135C2×Q8—61no—
136C2×Q8C2×C3×Dic316yescomplemented
137C2×Q8C22×C3×D311yescharacteristic
138C22×C4—33no—
139C22×C4—33no—
140C22×C4—34no—
141C22×C4—34no—
142C22×C4—66no—
143C22×C4—124no—
144C2×C8—34no—
145C2×C8—612no—
146C4⋊C4—34no—
147C4⋊C4—66no—
148C22×C3—46no—
149C22×C3—416no—
150C22×C3—41no—
151C22×C3—46no—
152C22×C3—46no—
153C22×C3—86no—
154C22×C3—816no—
155C22×C3—81no—
156C22×C3—86no—
157C22×C3—86no—
158C22×C3#859116yesdirect product factor
159C22×C3C2×#22616yes—
160C22×C3C2×GL(2,𝔽3)16yes—
161C22×C3C2×SL(2,ℤ/4)16yes—
162C22×C3C22×S411yescharacteristic
163C3×C4—31no—
164C3×C4—31no—
165C3×C4—36no—
166C3×C4—124no—
167Dic3—48no—
168C32—41noSylow
169C23C3×#22614yes—
170C23C3×GL(2,𝔽3)14yes—
171C23C3×SL(2,ℤ/4)14yes—
172C23C2×C3×S413yes—
173Q8—36no—
174Q8—31no—
175Q8—63no—
176Q8—64no—
177Q8C22×C3×Dic311yescharacteristic, complemented
178C2×C4—36no—
179C2×C4—36no—
180C2×C4—312no—
181C2×C4—31no—
182C2×C4—33no—
183C2×C4—64no—
184C2×C4—1212no—
185C8—64no—
186C2×C3—412no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—41no—
190C2×C3—812no—
191C2×C3—81no—
192C2×C3—81no—
193C2×C3—81no—
194C2×C3C2×#859112yesdirect product factor
195C2×C3C22×#22611yescharacteristic
196C2×C3C22×GL(2,𝔽3)11yescharacteristic
197C2×C3C22×SL(2,ℤ/4)11yescharacteristic
198C22C3×#859116yesdirect product factor
199C22C2×C3×#22616yes—
200C22C2×C3×GL(2,𝔽3)16yes—
201C22C2×C3×SL(2,ℤ/4)16yes—
202C22C22×C3×S411yescharacteristic
203C4—31no—
204C4—31no—
205C4—36no—
206C4—124no—
207C3—41no—
208C3—81no—
209C3C22×#85911yescharacteristic, direct product factor
210C2C2×C3×#859112yesdirect product factor
211C2C22×C3×#22611yescharacteristic
212C2C22×C3×GL(2,𝔽3)11yescharacteristic
213C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socle, hypercenterC24×C3S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C23×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#859C2direct (N × H)
3. C22×#859C3direct (N × H)
6. C22×C3×SL(2,𝔽3)C4semidirect
8. C3×#859C22direct (N × H)
14. C2×#859C2×C3direct (N × H)
20. C2×C3×SL(2,𝔽3)C2×C4semidirect
25. C22×C3×Q8Dic3semidirect
32. C22×SL(2,𝔽3)C3×C4semidirect
38. #859C22×C3direct (N × H)
47. C3×SL(2,𝔽3)C22×C4semidirect
61. C2×C3×Q8C2×Dic3semidirect
73. C2×SL(2,𝔽3)C2×C3×C4semidirect
87. C22×Q8C3×Dic3semidirect
113. C3×Q8C22×Dic3semidirect
124. SL(2,𝔽3)C22×C3×C4semidirect
136. C2×Q8C2×C3×Dic3semidirect
158. C22×C3#859direct (N × H)
177. Q8C22×C3×Dic3semidirect
194. C2×C3C2×#859direct (N × H)
198. C22C3×#859direct (N × H)
209. C3C22×#859direct (N × H)
210. C2C2×C3×#859direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×C3×#8592
C22×#8593
C22×C3×#1033
C23×C3×Dic34