Finite Group Databasebeta

← C3×#73096 #11920874

Subgroups — C3×#73096 #11920874

Order 1152.

245 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 40 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73096C311yesmaximal, characteristic, direct product factor
5C3×#1626—31nomaximal
6C2×C3×GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#809—41nomaximal
10C3×#557D311yescharacteristic, complemented
11#557⋊C3—21no—
12#557⋊C3C2×C311yescharacteristic, complemented
13C4⋊GL(2,𝔽3)C2×C311yescharacteristic
14C3×#486—31no—
15C3×#460—31no—
16C3×#460—31no—
17SL(2,𝔽3)⋊C8C2×C311yescharacteristic, complemented
18C3×#332—31no—
19C3×Q8⋊C8—31no—
20C3×Q8⋊C8—31no—
21C2×C3×D12—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3×GL(2,𝔽3)—41no—
24C32×C4⋊C4—41no—
25C2×C3×C3⋊C8—41no—
26#1626—31noSylow
27C2×GL(2,𝔽3)—21no—
28C2×C3×SD16—61no—
29C2×C3×SD16—61no—
30C3×C4⋊Q8—31no—
31C3×#127—31no—
32C3×#125—61no—
33C3×C4×Q8—21no—
34C3×C4×Q8—31no—
35C3×C4×Q8C2×D311yescharacteristic
36C4×SL(2,𝔽3)—22no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)C22×C311yescharacteristic
40C3×C4⋊C8—31no—
41C3×C4⋊C8—31no—
42C3×#102—61no—
43C3×#102—62no—
44C3×#102—61no—
45C3×C4×C8—31no—
46#809—41no—
47C22×C3×D3—81no—
48C2×C32×C4—41no—
49C2×C32×C4—81no—
50C3×D12—42no—
51C3×D12—81no—
52C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
53C3×C3⋊C8—81no—
54#557C3×D311yescharacteristic, complemented
55#486—31no—
56#460—31no—
57#460—31no—
58#332—31no—
59Q8⋊C8—31no—
60Q8⋊C8—31no—
61C2×C3×Q8—31no—
62C2×C3×Q8#6711yescharacteristic
63C2×C3×D4—61no—
64C2×C3×D4—61no—
65C2×C3×D4—61no—
66C2×D12—41no—
67C2×SL(2,𝔽3)—21no—
68C2×SL(2,𝔽3)C3×D411yescharacteristic
69GL(2,𝔽3)—41no—
70C3×SD16—121no—
71C3×SD16—122no—
72C3×SD16—121no—
73C2×C3×C8—61no—
74C2×C3×C8—61no—
75C2×C3×C8—61no—
76C3×C4⋊C4—31no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—41no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—61no—
81C3×C4⋊C4—81no—
82C3×C4⋊C4S411yescharacteristic
83C3×C42—31no—
84C3×C42—61no—
85C2×C3⋊C8—41no—
86C22×C32—41no—
87C2×C3×D3—82no—
88C2×C3×D3—161no—
89C2×C3×D3—161no—
90C32×C4—41no—
91C32×C4—41no—
92C32×C4—161no—
93C2×SD16—61no—
94C2×SD16—61no—
95C4⋊Q8—31no—
96#127—31no—
97#125—61no—
98C4×Q8—21no—
99C4×Q8—31no—
100C4×Q8C2×C3×D311yescharacteristic
101C4⋊C8—31no—
102C4⋊C8—31no—
103#102—61no—
104#102—62no—
105#102—61no—
106C4×C8—31no—
107C23×C3—121no—
108C22×D3—81no—
109C3×Q8—31no—
110C3×Q8—61no—
111C3×Q8—61no—
112C3×Q8#21511yescharacteristic
113C3×D4—121no—
114C3×D4—121no—
115C3×D4—122no—
116C3×D4—121no—
117C3×D4—121no—
118C2×C3×C4—21no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—41no—
122C2×C3×C4—61no—
123C2×C3×C4—81no—
124C2×C3×C4—82no—
125C2×C3×C4—81no—
126C2×C3×C4C2×S411yescharacteristic
127D12—42no—
128D12—81no—
129SL(2,𝔽3)—21no—
130SL(2,𝔽3)C3×SD1611yescharacteristic, complemented
131C3×C8—121no—
132C3×C8—121no—
133C3×C8—121no—
134C3⋊C8—81no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—41no—
138C3×D3—162no—
139C2×Q8—31no—
140C2×Q8C3×#6711yescharacteristic
141C2×D4—61no—
142C2×D4—61no—
143C2×D4—61no—
144SD16—121no—
145SD16—122no—
146SD16—121no—
147C2×C8—61no—
148C2×C8—61no—
149C2×C8—61no—
150C4⋊C4—31no—
151C4⋊C4—32no—
152C4⋊C4—61no—
153C4⋊C4—61no—
154C4⋊C4C3×S411yescharacteristic
155C42—31no—
156C42—61no—
157C22×C3—41no—
158C22×C3—81no—
159C22×C3—241no—
160C22×C3—241no—
161C22×C3—241no—
162C22×C3GL(2,ℤ/4)11yescharacteristic
163C2×D3—82no—
164C2×D3—161no—
165C2×D3—161no—
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—41no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—61no—
172C3×C4—81no—
173C3×C4—81no—
174C3×C4—121no—
175C3×C4—161no—
176C3×C4—162no—
177C3×C4#98511yescharacteristic
178C3×C4#98611yescharacteristic
179C32—41noSylow
180C23—121no—
181Q8—31no—
182Q8—61no—
183Q8—61no—
184Q8C3×#21511yescharacteristic
185D4—121no—
186D4—121no—
187D4—122no—
188D4—121no—
189D4—121no—
190C2×C4—21no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—61no—
194C2×C4C2×C3×S411yescharacteristic
195C8—121no—
196C8—121no—
197C8—121no—
198C2×C3—41no—
199C2×C3—41no—
200C2×C3—41no—
201C2×C3—81no—
202C2×C3—81no—
203C2×C3—81no—
204C2×C3—481no—
205C2×C3#541711yescharacteristic
206C2×C3#542711yescharacteristic
207C2×C3#543111yescharacteristic
208D3—162no—
209C22—241no—
210C22—241no—
211C22—241no—
212C22C3×GL(2,ℤ/4)11yescharacteristic
213C4—31no—
214C4—31no—
215C4—41no—
216C4—61no—
217C4—121no—
218C4C3×#98511yescharacteristic
219C4C3×#98611yescharacteristic
220C3—41no—
221C3—81no—
222C3#7309611yescharacteristic, direct product factor
223C2—481no—
224C2C3×#541711yescharacteristic
225C2C3×#542711yescharacteristic
226C2C3×#543111yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×#557D3
prime radicalC3×C4⋊GL(2,𝔽3)C2
hypercenterC3×C4⋊C4S4
hypocenterSL(2,𝔽3)C3×SD16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
4. #73096C3direct (N × H)
10. C3×#557D3semidirect
12. #557⋊C3C2×C3semidirect
17. SL(2,𝔽3)⋊C8C2×C3semidirect
52. C3×SL(2,𝔽3)SD16semidirect
54. #557C3×D3semidirect
130. SL(2,𝔽3)C3×SD16semidirect
222. C3#73096direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C3×C4⋊GL(2,𝔽3)2
C3×SL(2,𝔽3)⋊C82
#730963
C3×#16263
C3×#8094