Finite Group Databasebeta

← C3×#73023 #11920801

Subgroups — C3×#73023 #11920801

Order 1152.

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

normal subgroups only

This group has 221 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×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5391C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73023C311yesmaximal, characteristic, direct product factor
5C3×#1678—31nomaximal
6C2×C3×GL(2,𝔽3)—21no—
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×#859—21no—
9C3×#821—41nomaximal
10C4⋊GL(2,𝔽3)C2×C311yescharacteristic
11#5391C2×C311yescharacteristic, complemented
12C3×#486—31no—
13C3×#477—31no—
14C3×#460—31no—
15C3×C8×Q8D311yescharacteristic, complemented
16C8×SL(2,𝔽3)—21no—
17C8×SL(2,𝔽3)C2×C311yescharacteristic, complemented
18C3×#334—31no—
19C3×#333—31no—
20C3×#331—31no—
21C2×C3×D12—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3×GL(2,𝔽3)—41no—
24C2×C32×C8—41no—
25C3×C4⋊Dic3—41no—
26#1678—31noSylow
27C2×GL(2,𝔽3)—21no—
28C2×C3×SD16—61no—
29C3×C4⋊Q8—31no—
30C3×#127—31no—
31C3×C4×Q8C2×D311yescharacteristic
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)C22×C311yescharacteristic
34#859—21no—
35C3×#107—61no—
36C3×C4⋊C8—31no—
37C3×C4⋊C8—31no—
38C3×C4⋊C8—31no—
39C3×#103—61no—
40C3×#102—61no—
41C3×#102—61no—
42C3×#102—61no—
43C3×C4×C8—31no—
44#821—41no—
45C22×C3×D3—81no—
46C2×C32×C4—41no—
47C2×C3×Dic3—81no—
48C3×D12—42no—
49C3×D12—81no—
50C3×SL(2,𝔽3)SD1611yescharacteristic
51C32×C8—81no—
52#486—31no—
53#477—31no—
54#460—31no—
55C8×Q8C3×D311yescharacteristic, complemented
56#334—31no—
57#333—31no—
58#331—31no—
59C2×C3×Q8—61no—
60C2×C3×Q8D1211yescharacteristic
61C2×C3×D4—61no—
62C2×C3×D4—61no—
63C2×C3×D4—61no—
64C2×D12—41no—
65C2×SL(2,𝔽3)—21no—
66C2×SL(2,𝔽3)C3×D411yescharacteristic
67GL(2,𝔽3)—41no—
68C3×SD16—121no—
69C3×SD16—121no—
70C2×C3×C8—31no—
71C2×C3×C8—41no—
72C2×C3×C8—61no—
73C2×C3×C8—61no—
74C2×C3×C8—81no—
75C2×C3×C8S411yescharacteristic
76C3×C4⋊C4—31no—
77C3×C4⋊C4—61no—
78C3×C4⋊C4—61no—
79C3×C42—31no—
80C4⋊Dic3—41no—
81C22×C32—41no—
82C2×C3×D3—82no—
83C2×C3×D3—161no—
84C2×C3×D3—161no—
85C32×C4—41no—
86C32×C4—41no—
87C3×Dic3—161no—
88C2×SD16—61no—
89C4⋊Q8—31no—
90#127—31no—
91C4×Q8C2×C3×D311yescharacteristic
92#107—61no—
93C4⋊C8—31no—
94C4⋊C8—31no—
95C4⋊C8—31no—
96#103—61no—
97#102—61no—
98#102—61no—
99#102—61no—
100C4×C8—31no—
101C23×C3—121no—
102C22×D3—81no—
103C3×Q8—31no—
104C3×Q8—241no—
105C3×Q8#20411yescharacteristic
106C3×D4—121no—
107C3×D4—121no—
108C3×D4—121no—
109C3×D4—121no—
110C3×D4—241no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—41no—
114C2×C3×C4—81no—
115C2×C3×C4—121no—
116C2×C3×C4C2×S411yescharacteristic
117C2×Dic3—81no—
118D12—42no—
119D12—81no—
120SL(2,𝔽3)—21no—
121SL(2,𝔽3)C3×SD1611yescharacteristic
122C3×C8—21no—
123C3×C8—61no—
124C3×C8—81no—
125C3×C8—82no—
126C3×C8—121no—
127C3×C8—121no—
128C2×C32—41no—
129C2×C32—41no—
130C2×C32—41no—
131C3×D3—162no—
132C2×Q8—61no—
133C2×Q8C3×D1211yescharacteristic
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137SD16—121no—
138SD16—121no—
139C2×C8—31no—
140C2×C8—61no—
141C2×C8—61no—
142C2×C8C3×S411yescharacteristic
143C4⋊C4—31no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C42—31no—
147C22×C3—41no—
148C22×C3—81no—
149C22×C3—241no—
150C22×C3—241no—
151C22×C3—241no—
152C22×C3C4⋊S411yescharacteristic
153C2×D3—82no—
154C2×D3—161no—
155C2×D3—161no—
156C3×C4—31no—
157C3×C4—31no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—61no—
161C3×C4—81no—
162C3×C4—81no—
163C3×C4—241no—
164C3×C4#98311yescharacteristic
165C3×C4#98511yescharacteristic
166Dic3—161no—
167C32—41noSylow
168C23—121no—
169Q8—31no—
170Q8—241no—
171Q8C3×#20411yescharacteristic
172D4—121no—
173D4—121no—
174D4—121no—
175D4—121no—
176D4—241no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—121no—
180C2×C4C2×C3×S411yescharacteristic
181C8—21no—
182C8—61no—
183C8—121no—
184C8—121no—
185C2×C3—41no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—81no—
189C2×C3—81no—
190C2×C3—81no—
191C2×C3—481no—
192C2×C3#539511yescharacteristic
193C2×C3#540111yescharacteristic
194C2×C3#540711yescharacteristic
195D3—162no—
196C22—241no—
197C22—241no—
198C22—241no—
199C22C3×C4⋊S411yescharacteristic
200C4—31no—
201C4—31no—
202C4—61no—
203C4—241no—
204C4C3×#98311yescharacteristic
205C4C3×#98511yescharacteristic
206C3—41no—
207C3—81no—
208C3#7302311yescharacteristic, direct product factor
209C2—481no—
210C2C3×#539511yescharacteristic
211C2C3×#540111yescharacteristic
212C2C3×#540711yescharacteristic
subgroupG/H
center, socleC22×C3C4⋊S4
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×C8×Q8D3
prime radicalC3×C4⋊GL(2,𝔽3)C2
hypercenterC2×C3×C8S4
hypocenterSL(2,𝔽3)C3×SD16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#5391C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
4. #73023C3direct (N × H)
11. #5391C2×C3semidirect
15. C3×C8×Q8D3semidirect
17. C8×SL(2,𝔽3)C2×C3semidirect
55. C8×Q8C3×D3semidirect
208. C3#73023direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4⋊GL(2,𝔽3)2
C3×#53912
C3×C8×SL(2,𝔽3)2
#730233
C3×#16783
C3×#8214