Finite Group Databasebeta

← C3×#73024 #11920802

Subgroups — C3×#73024 #11920802

Order 1152.

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

normal subgroups only

This group has 229 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×C4⋊#226C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73024C311yesmaximal, characteristic, direct product factor
5C3×#1682—31nomaximal
6C2×C3×GL(2,𝔽3)—21no—
7C2×C3×#226—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#821—41nomaximal
10C4⋊GL(2,𝔽3)C2×C311yescharacteristic
11C4⋊#226C2×C311yescharacteristic, complemented
12C3×#486—31no—
13C3×#462—31no—
14C3×#460—31no—
15C3×C8×Q8D311yescharacteristic, complemented
16C8×SL(2,𝔽3)—21no—
17C8×SL(2,𝔽3)C2×C311yescharacteristic, complemented
18C3×#335—31no—
19C3×#332—31no—
20C3×#331—31no—
21C2×C3×D12—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C3×GL(2,𝔽3)—41no—
24C3×#226—41no—
25C2×C32×C8—41no—
26C3×C4⋊Dic3—41no—
27#1682—31noSylow
28C2×GL(2,𝔽3)—21no—
29C2×#226—21no—
30C2×C3×Q16—61no—
31C2×C3×SD16—61no—
32C3×C4⋊Q8—31no—
33C3×#127—31no—
34C3×C4×Q8C2×D311yescharacteristic
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)C22×C311yescharacteristic
37C3×C4⋊C8—31no—
38C3×C4⋊C8—31no—
39C3×C4⋊C8—31no—
40C3×#103—61no—
41C3×#102—61no—
42C3×#102—61no—
43C3×#102—61no—
44C3×C4×C8—31no—
45#821—41no—
46C22×C3×D3—81no—
47C2×C32×C4—41no—
48C2×C3×Dic3—81no—
49C3×D12—42no—
50C3×D12—81no—
51C3×SL(2,𝔽3)D811yescharacteristic
52C32×C8—81no—
53#486—31no—
54#462—31no—
55#460—31no—
56C8×Q8C3×D311yescharacteristic, complemented
57#335—31no—
58#332—31no—
59#331—31no—
60C2×C3×Q8—61no—
61C2×C3×Q8D1211yescharacteristic
62C2×C3×D4—61no—
63C2×C3×D4—61no—
64C2×C3×D4—61no—
65C2×D12—41no—
66C2×SL(2,𝔽3)—21no—
67C2×SL(2,𝔽3)C3×D411yescharacteristic
68GL(2,𝔽3)—41no—
69#226—41no—
70C3×Q16—121no—
71C3×Q16—121no—
72C3×SD16—121no—
73C3×SD16—121no—
74C2×C3×C8—31no—
75C2×C3×C8—41no—
76C2×C3×C8—61no—
77C2×C3×C8—61no—
78C2×C3×C8—81no—
79C2×C3×C8S411yescharacteristic
80C3×C4⋊C4—31no—
81C3×C4⋊C4—61no—
82C3×C4⋊C4—61no—
83C3×C42—31no—
84C4⋊Dic3—41no—
85C22×C32—41no—
86C2×C3×D3—82no—
87C2×C3×D3—161no—
88C2×C3×D3—161no—
89C32×C4—41no—
90C32×C4—41no—
91C3×Dic3—161no—
92C2×Q16—61no—
93C2×SD16—61no—
94C4⋊Q8—31no—
95#127—31no—
96C4×Q8C2×C3×D311yescharacteristic
97C4⋊C8—31no—
98C4⋊C8—31no—
99C4⋊C8—31no—
100#103—61no—
101#102—61no—
102#102—61no—
103#102—61no—
104C4×C8—31no—
105C23×C3—121no—
106C22×D3—81no—
107C3×Q8—31no—
108C3×Q8—121no—
109C3×Q8—121no—
110C3×Q8D2411yescharacteristic
111C3×D4—121no—
112C3×D4—121no—
113C3×D4—121no—
114C3×D4—121no—
115C3×D4—241no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—41no—
119C2×C3×C4—81no—
120C2×C3×C4—121no—
121C2×C3×C4C2×S411yescharacteristic
122C2×Dic3—81no—
123D12—42no—
124D12—81no—
125SL(2,𝔽3)—21no—
126SL(2,𝔽3)C3×D811yescharacteristic
127C3×C8—21no—
128C3×C8—61no—
129C3×C8—81no—
130C3×C8—82no—
131C3×C8—121no—
132C3×C8—121no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—41no—
136C3×D3—162no—
137C2×Q8—61no—
138C2×Q8C3×D1211yescharacteristic
139C2×D4—61no—
140C2×D4—61no—
141C2×D4—61no—
142Q16—121no—
143Q16—121no—
144SD16—121no—
145SD16—121no—
146C2×C8—31no—
147C2×C8—61no—
148C2×C8—61no—
149C2×C8C3×S411yescharacteristic
150C4⋊C4—31no—
151C4⋊C4—61no—
152C4⋊C4—61no—
153C42—31no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3—241no—
157C22×C3—241no—
158C22×C3—241no—
159C22×C3C4⋊S411yescharacteristic
160C2×D3—82no—
161C2×D3—161no—
162C2×D3—161no—
163C3×C4—31no—
164C3×C4—31no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—61no—
168C3×C4—81no—
169C3×C4—81no—
170C3×C4—241no—
171C3×C4#98311yescharacteristic
172C3×C4#98511yescharacteristic
173Dic3—161no—
174C32—41noSylow
175C23—121no—
176Q8—31no—
177Q8—121no—
178Q8—121no—
179Q8C3×D2411yescharacteristic
180D4—121no—
181D4—121no—
182D4—121no—
183D4—121no—
184D4—241no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—121no—
188C2×C4C2×C3×S411yescharacteristic
189C8—21no—
190C8—61no—
191C8—121no—
192C8—121no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—41no—
196C2×C3—81no—
197C2×C3—81no—
198C2×C3—81no—
199C2×C3—481no—
200C2×C3#539511yescharacteristic
201C2×C3#540211yescharacteristic
202C2×C3#540611yescharacteristic
203D3—162no—
204C22—241no—
205C22—241no—
206C22—241no—
207C22C3×C4⋊S411yescharacteristic
208C4—31no—
209C4—31no—
210C4—61no—
211C4—241no—
212C4C3×#98311yescharacteristic
213C4C3×#98511yescharacteristic
214C3—41no—
215C3—81no—
216C3#7302411yescharacteristic, direct product factor
217C2—481no—
218C2C3×#539511yescharacteristic
219C2C3×#540211yescharacteristic
220C2C3×#540611yescharacteristic
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×D8
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×C4⋊#226C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
4. #73024C3direct (N × H)
11. C4⋊#226C2×C3semidirect
15. C3×C8×Q8D3semidirect
17. C8×SL(2,𝔽3)C2×C3semidirect
56. C8×Q8C3×D3semidirect
216. C3#73024direct (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×C4⋊#2262
C3×C8×SL(2,𝔽3)2
#730243
C3×#16823
C3×#8214