Finite Group Databasebeta

← #84017⋊C3 #11921351

Subgroups — #84017⋊C3 #11921351

Order 1152.

219 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 35 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×#578⋊C3C211yesmaximal, characteristic, complemented
2#5753⋊C3C211yesmaximal, characteristic, complemented
3C3⋊(#504⋊C3)C211yesmaximal, characteristic, complemented
4#84017C311yesmaximal, characteristic, complemented
5#3417⋊C3—31nomaximal
6C3×2−1+4⋊C3—21no—
7#1010⋊C3—21no—
8C3×#131⋊C3C2211yescharacteristic
9C3×#826—41nomaximal
10C3×#578C2×C311yescharacteristic, complemented
11#5753C2×C311yescharacteristic, complemented
12#578⋊C3—21no—
13#578⋊C3—31no—
14#578⋊C3D311yescharacteristic, complemented
15#5195—31no—
16#5194—31no—
17C2×#829—31no—
18#5179—31no—
19#5140C2×C311yescharacteristic, complemented
20#504⋊C3—31no—
21C3×#41⋊C3D411yescharacteristic
22C2×C3×SL(2,𝔽3)—41no—
23D3×SL(2,𝔽3)—41no—
24C32×D8—41no—
25C3×D24—41no—
26C3×C3⋊C16—41no—
27#3417—31noSylow
28C3×2−1+4—21no—
29#1010—21no—
302−1+4⋊C3—22no—
312−1+4⋊C3—21no—
322−1+4⋊C3—61no—
33C3×#137—61no—
34C3×#135—31no—
35C2×C3×Q16—31no—
36C3×#131C22×C311yescharacteristic
37#909—61no—
38C2×Dic12—31no—
39#904—31no—
40#131⋊C3—21no—
41#131⋊C3C2×D311yescharacteristic
42C3×D16—121no—
43#829—32no—
44#829—31no—
45#828—31no—
46#827—31no—
47#826A411yescharacteristic
48#812—31no—
49C2×C3⋊C16—31no—
50C32×D4—81no—
51C3×D12—81no—
52C3×SL(2,𝔽3)D811yescharacteristic
53C32×C8—41no—
54#578—31no—
55#578C3×D311yescharacteristic, complemented
56#510—91no—
57#510—91no—
58#508—91no—
59C2×Q32—91no—
60#504—31no—
61C3×#41—61no—
62C3×#41—121no—
63C3×#41C3×D411yescharacteristic
64C2×C3×Q8—41no—
65C2×C3×Q8—61no—
66D3×Q8—41no—
67#237—121no—
68#235—61no—
69C2×Dic6—61no—
70#41⋊C3—21no—
71#41⋊C3#6711yescharacteristic
72C2×SL(2,𝔽3)—42no—
73C2×SL(2,𝔽3)—41no—
74C2×SL(2,𝔽3)—121no—
75C3×Q16—31no—
76C3×Q16—61no—
77C3×SD16—61no—
78C3×D8—41no—
79C3×D8—81no—
80C3×D8—121no—
81C3×D8C2×A411yescharacteristic
82C3×M16—31no—
83C2×C3×C8—31no—
84Dic12—31no—
85Dic12—61no—
86D24C2×A411yescharacteristic
87#204—61no—
88C3×C16—121no—
89C3⋊C16—31no—
90C3⋊C16C2×A411yescharacteristic
91C22×C32—161no—
92C2×C3×D3—161no—
93C32×C4—41no—
942−1+4—21no—
952−1+4—61no—
96#137—61no—
97#137—181no—
98#135—31no—
99#135—91no—
100C2×Q16—31no—
101C2×Q16—91no—
102#131C2×C3×D311yescharacteristic
103Q32—91no—
104Q32—92no—
105SD32—91no—
106SD32—91no—
107D16—31no—
108M32—91no—
109C2×C16—91no—
110C3×Q8—61no—
111C3×Q8—121no—
112C3×Q8C3×D811yescharacteristic
113C3×D4—21no—
114C3×D4—31no—
115C3×D4—81no—
116C3×D4—82no—
117C3×D4—121no—
118C3×D4—241no—
119C2×C3×C4—31no—
120C2×C3×C4—121no—
121C2×C3×C4—121no—
122#67—121no—
123C2×Dic3—121no—
124D12—21no—
125C4×D3—121no—
126Dic6—61no—
127Dic6—121no—
128SL(2,𝔽3)—21no—
129SL(2,𝔽3)#21311yescharacteristic
130C3×C8—31no—
131C3×C8—41no—
132C3×C8—81no—
133C3×C8C22×A411yescharacteristic
134C2×C32—41no—
135C2×C32—321no—
136C3×D3—321no—
137#41—61no—
138#41—121no—
139#41—181no—
140#41—361no—
141#41C3×#6711yescharacteristic
142C2×Q8—41no—
143C2×Q8—61no—
144C2×Q8—121no—
145C2×Q8—181no—
146Q16—31no—
147Q16—61no—
148Q16—91no—
149Q16—181no—
150SD16—61no—
151SD16—181no—
152D8—31no—
153D8D3×A411yescharacteristic
154M16—31no—
155C2×C8—31no—
156C16—31no—
157C16—91no—
158C22×C3—31no—
159C22×C3—41no—
160C22×C3—161no—
161C22×C3—162no—
162C22×C3—481no—
163C2×D3—41no—
164C3×C4—31no—
165C3×C4—41no—
166C3×C4—81no—
167C3×C4—121no—
168C3×C4D4×A411yescharacteristic
169Dic3—121no—
170C32—41noSylow
171Q8—61no—
172Q8—121no—
173Q8—181no—
174Q8—361no—
175Q8C3×#21311yescharacteristic
176D4—21no—
177D4—31no—
178D4—61no—
179D4—121no—
180D4—361no—
181C2×C4—31no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—361no—
185C2×C4—361no—
186C8—31no—
187C8C2×D3×A411yescharacteristic
188C2×C3—41no—
189C2×C3—61no—
190C2×C3—81no—
191C2×C3—81no—
192C2×C3—321no—
193C2×C3—322no—
194C2×C3—961no—
195C2×C3A4×D811yescharacteristic
196D3—81no—
197C22—31no—
198C22—41no—
199C22—121no—
200C4—31no—
201C4—121no—
202C4—361no—
203C4A4×#6711yescharacteristic
204C3—41no—
205C3—81no—
206C3#3417⋊C311yescharacteristic, complemented
207C2—61no—
208C2—81no—
209C2—241no—
210C2A4×#21311yescharacteristic
subgroupG/H
centerC2A4×#213
commutator, cosocleC3×#131C22×C3
FrattiniC8C2×D3×A4
FittingC3×#578C2×C3
socleC2×C3A4×D8
hypercenterD8D3×A4
hypocenterC3×Q8C3×D8
LCSG ▷ C3×#131 ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
1. C3×#578⋊C3C2semidirect
2. #5753⋊C3C2semidirect
3. C3⋊(#504⋊C3)C2semidirect
4. #84017C3semidirect
10. C3×#578C2×C3semidirect
11. #5753C2×C3semidirect
14. #578⋊C3D3semidirect
19. #5140C2×C3semidirect
55. #578C3×D3semidirect
206. C3#3417⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#578⋊C32
#5753⋊C32
C3⋊(#504⋊C3)2
#840173
#3417⋊C33
C3×#8264