Finite Group Databasebeta

← #79933⋊C3 #11921224

Subgroups — #79933⋊C3 #11921224

Order 1152.

227 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#5570⋊C3C211yesmaximal, characteristic
2SL(2,𝔽3)×Dic6C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic
4#79933C311yesmaximal, characteristic, complemented
5#3363⋊C3—31nomaximal
6C3×C4×SL(2,𝔽3)C2211yescharacteristic
7Dic3×SL(2,𝔽3)—21no—
8Dic3×SL(2,𝔽3)—21no—
9C3×#816—41nomaximal
10#5570C2×C311yescharacteristic
11Q8×Dic6C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—31no—
13#557⋊C3—31no—
14C3×C8×Q8C2×C311yescharacteristic
15#4850—31no—
16#4849—31no—
17C4×Dic12—31no—
18#4681—31no—
19C8×SL(2,𝔽3)—21no—
20C8×SL(2,𝔽3)D311yescharacteristic
21C2×C3×Dic6—41no—
22C2×C3×SL(2,𝔽3)D411yescharacteristic
23C2×C32×C8—41no—
24C3×C4⋊Dic3—41no—
25#3363—31noSylow
26C3×C4×Q8C22×C311yescharacteristic
27Q8×Dic3—21no—
28Q8×Dic3—21no—
29C2×Dic12—31no—
30#890—61no—
31#888—61no—
32#869—31no—
33#869—31no—
34C4×Dic6—31no—
35C4×Dic6—31no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)—61no—
38C4×SL(2,𝔽3)—61no—
39C4×SL(2,𝔽3)C2×D311yescharacteristic
40C3×C4⋊C8—31no—
41C3×#103—121no—
42C3×C4×C8—31no—
43#818—31no—
44#816—32no—
45#816—31no—
46#816A411yescharacteristic
47C2×C32×C4—41no—
48C2×C3×Dic3—81no—
49C2×C3×Dic3—81no—
50C3×Dic6—42no—
51C3×Dic6—81no—
52C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
53C32×C8—81no—
54Q82—31no—
55#557—31no—
56#487—91no—
57#477—91no—
58#462—91no—
59C8×Q8C3×D311yescharacteristic
60C4×Q16—91no—
61C2×C3×Q8—121no—
62C2×C3×Q8C3×D411yescharacteristic
63C2×Dic6—31no—
64C2×Dic6—61no—
65C2×Dic6C2×A411yescharacteristic
66C2×SL(2,𝔽3)—21no—
67C2×SL(2,𝔽3)D1211yescharacteristic
68C2×C3×C8—31no—
69C2×C3×C8—41no—
70C2×C3×C8—81no—
71C2×C3×C8C2×A411yescharacteristic
72C3×C4⋊C4—31no—
73C3×C4⋊C4—121no—
74C3×C42—31no—
75C4⋊Dic3—31no—
76C4⋊Dic3—61no—
77C4⋊Dic3—61no—
78C4⋊Dic3—61no—
79C4⋊Dic3C2×A411yescharacteristic
80Dic3⋊C4—61no—
81Dic3⋊C4—61no—
82C4×Dic3—61no—
83C4×Dic3—61no—
84Dic12—62no—
85C22×C32—41no—
86C32×C4—41no—
87C32×C4—41no—
88C3×Dic3—82no—
89C3×Dic3—161no—
90C2×Q16—91no—
91C4⋊Q8—91no—
92C4⋊Q8—91no—
93C4⋊Q8—181no—
94#125—181no—
95C4×Q8—61no—
96C4×Q8—61no—
97C4×Q8—91no—
98C4×Q8—91no—
99C4×Q8C2×C3×D311yescharacteristic
100#107—91no—
101C4⋊C8—31no—
102#103—31no—
103#103—92no—
104#103—91no—
105C4×C8—31no—
106C3×Q8—31no—
107C3×Q8—122no—
108C3×Q8—241no—
109C3×Q8C3×Q1611yescharacteristic
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—41no—
113C2×C3×C4—81no—
114C2×C3×C4—241no—
115C2×C3×C4—241no—
116C2×C3×C4C22×A411yescharacteristic
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—61no—
120C2×Dic3—61no—
121Dic6—21no—
122Dic6—61no—
123Dic6—61no—
124Dic6—241no—
125Dic6#41⋊C312yes—
126SL(2,𝔽3)—21no—
127SL(2,𝔽3)Dic1211yescharacteristic, complemented
128C3×C8—21no—
129C3×C8—61no—
130C3×C8—81no—
131C3×C8—82no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×Q8—31no—
136C2×Q8—91no—
137C2×Q8—181no—
138C2×Q8C3×D1211yescharacteristic
139Q16—182no—
140C2×C8—31no—
141C2×C8D3×A411yescharacteristic
142C4⋊C4—31no—
143C4⋊C4—31no—
144C4⋊C4—91no—
145C4⋊C4—181no—
146C4⋊C4—181no—
147C4⋊C4—181no—
148C4⋊C4—181no—
149C4⋊C4—181no—
150C42—31no—
151C42—181no—
152C42—181no—
153C22×C3—41no—
154C22×C3—81no—
155C22×C3D4×A411yescharacteristic
156C3×C4—31no—
157C3×C4—31no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—61no—
161C3×C4—81no—
162C3×C4—81no—
163C3×C4—242no—
164C3×C4—481no—
165C3×C4C2×#41⋊C311yescharacteristic
166C3×C42−1+4⋊C311yescharacteristic
167Dic3—22no—
168Dic3—41no—
169Dic3—121no—
170Dic3—121no—
171C32—41noSylow
172Q8—31no—
173Q8—32no—
174Q8—61no—
175Q8—181no—
176Q8—181no—
177Q8—721no—
178Q8C3×Dic1211yescharacteristic
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—181no—
184C2×C4—181no—
185C2×C4C2×D3×A411yescharacteristic
186C8—21no—
187C8—61no—
188C2×C3—41no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3—81no—
192C2×C3—81no—
193C2×C3—81no—
194C2×C3#548⋊C311yescharacteristic
195C2×C3A4×Q1611yescharacteristic
196C2×C3#577⋊C311yescharacteristic
197C22A4×D1211yescharacteristic
198C4—31no—
199C4—31no—
200C4—62no—
201C4—61no—
202C4—121no—
203C4—361no—
204C4—361no—
205C4#1007⋊C311yescharacteristic
206C4D3×#41⋊C311yescharacteristic
207C3—41no—
208C3—81no—
209C3#3363⋊C311yescharacteristic, complemented
210C2#5576⋊C311yescharacteristic
211C2A4×Dic1211yescharacteristic
212C2#5751⋊C311yescharacteristic
subgroupG/H
centerC22A4×D12
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×C8×Q8C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×C8D3×A4
hypocenterC3×Q8C3×Q16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #79933C3semidirect
52. C3×SL(2,𝔽3)Q16semidirect
127. SL(2,𝔽3)Dic12semidirect
209. C3#3363⋊C3semidirect

Maximal subgroups

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

subgroupindex
#5570⋊C32
SL(2,𝔽3)×Dic62
C3×C8×SL(2,𝔽3)2
#799333
#3363⋊C33
C3×#8164