Finite Group Databasebeta

← #11800037

Subgroups — #11800037

Order 1152.

480 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 111 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#10592850C211yesmaximal, characteristic
2#10592573C211yesmaximal, characteristic, complemented
3C32×#357C211yesmaximal, characteristic, complemented
4#10589428C211yesmaximal, characteristic, complemented
5#10589375C211yesmaximal, characteristic, complemented
6C16×C3⋊Dic3C211yesmaximal, characteristic, complemented
7#10589357C211yesmaximal, characteristic, complemented
8#69183—34nomaximal
9C2×C32×D8C2211yescharacteristic
10#64149—21no—
11C2×#3753—21no—
12C2×#3745C2211yescharacteristic
13#64097—21no—
14C2×C32×C16C2211yescharacteristic
15C32×#107C2211yescharacteristic
16#63645C2211yescharacteristic
17C8×C3⋊Dic3C2211yescharacteristic
18C2×C3⋊(C3⋊C16)C2211yescharacteristic
19#63634—21no—
20#5151—34no—
21#4874—34no—
22C3×#357D314yescomplemented
23#4562—34no—
24#4509—34no—
25Dic3×C16—34no—
26#4491—34no—
27C2×C32×D4—21no—
28C2×#638—41no—
29C2×#638—41no—
30C2×C4⋊(C3⋊D3)—21no—
31C2×#634—41no—
32C32×D8—21no—
33C32×D8—21no—
34C2×C32×C8C2311yescharacteristic
35C32×C4⋊C4—21no—
36#3753—42no—
37C4⋊(C3⋊Dic3)—21no—
38#3750—41no—
39C4×C3⋊Dic3D411yescharacteristic
40C2×C3⋊(C3⋊C8)D411yescharacteristic
41#3745—21no—
42#3745—21no—
43C32×C16—21no—
44C3⋊(C3⋊C16)—21no—
45#2240—91noSylow
46C2×C3×D8C2×D314yes—
47#940—64no—
48C2×#213—64no—
49C2×D24—34no—
50#888—64no—
51C2×C3×C16C2×D314yes—
52C3×#107C2×D314yes—
53#818—34no—
54C8×Dic3—34no—
55C2×C3⋊C16—34no—
56#807—64no—
57C23×C32—41no—
58C22×C3⋊D3—41no—
59C32×D4—41no—
60C32×D4—41no—
61C2×C32×C4—41no—
62C2×C32×C4C2×D411yescharacteristic
63#638—82no—
64#638—82no—
65C2×C3⋊Dic3—41no—
66C2×C3⋊Dic3D811yescharacteristic
67C2×C3⋊Dic3D811yescharacteristic
68C4⋊(C3⋊D3)—41no—
69C4⋊(C3⋊D3)—41no—
70#634—82no—
71C32×C8#4111yescharacteristic
72C32×C8#4111yescharacteristic
73C3⋊(C3⋊C8)SD1612yes—
74#500—91no—
75#493—91no—
76#357—91no—
77#357—91no—
78#357—91no—
79#357C3⋊D311yescharacteristic, complemented
80C4×C16—91no—
81C2×C3×D4—24no—
82C2×#67—124no—
83C2×#67—124no—
84C2×D12—64no—
85C2×Dic6—124no—
86C3×D8—24no—
87C3×D8—24no—
88C2×C3×C8C22×D314yes—
89C3×C4⋊C4—24no—
90#213—128no—
91C4⋊Dic3—64no—
92Dic3⋊C4—124no—
93C4×Dic3—34no—
94C2×C3⋊C8—34no—
95D24—64no—
96D24—64no—
97C3×C16—24no—
98C3⋊C16—64no—
99C22×C32—81no—
100C22×C32—81no—
101C22×C32—81no—
102C22×C32C2×D811yescharacteristic
103C2×C3⋊D3—81no—
104C2×C3⋊D3—81no—
105C2×C3⋊D3—81no—
106C32×C4—81no—
107C32×C4#12411yescharacteristic
108C32×C4C2×SD1611yescharacteristic
109C3⋊Dic3—81no—
110C3⋊Dic3D1612yes—
111C3⋊Dic3SD3212yes—
112C2×D8—91no—
113C2×D8—181no—
114C2×D8C2×C3⋊D311yescharacteristic
115C4⋊Q8—181no—
116#127—181no—
117C2×C16—91no—
118C2×C16C2×C3⋊D311yescharacteristic
119#107—91no—
120#107—181no—
121#107C2×C3⋊D311yescharacteristic
122C4×C8—91no—
123C23×C3—44no—
124C22×D3—124no—
125C3×D4—44no—
126C3×D4—44no—
127C2×C3×C4—44no—
128C2×C3×C4D3×D414yes—
129#67—248no—
130#67—248no—
131C2×Dic3—34no—
132C2×Dic3—34no—
133C2×Dic3—124no—
134D12—124no—
135D12—124no—
136Dic6—248no—
137C3×C8#23514yes—
138C3×C8#23714yes—
139C3⋊C8—38no—
140C2×C32—161no—
141C2×C32#48611yescharacteristic
142C2×C32C2×D1611yescharacteristic
143C2×C32C2×SD3211yescharacteristic
144C3⋊D3—161no—
145C2×Q8—361no—
146C2×D4—21no—
147C2×D4—181no—
148C2×D4—361no—
149C2×D4—361no—
150D8—21no—
151D8—21no—
152D8—181no—
153D8—181no—
154D8—362no—
155C2×C8—91no—
156C2×C8C22×C3⋊D311yescharacteristic
157C4⋊C4—21no—
158C4⋊C4—181no—
159C4⋊C4—361no—
160C42—91no—
161C16—21no—
162C16—181no—
163C22×C3—84no—
164C22×C3—84no—
165C22×C3—84no—
166C22×C3D3×D814yes—
167C2×D3—244no—
168C2×D3—244no—
169C2×D3—244no—
170C3×C4—84no—
171C3×C4#88514yes—
172C3×C4D3×SD1614yes—
173Dic3—38no—
174Dic3—38no—
175Dic3—244no—
176C32#224011yescharacteristic, complemented, Sylow
177C23—41no—
178C23—361no—
179Q8—722no—
180D4—41no—
181D4—41no—
182D4—361no—
183D4—361no—
184D4—722no—
185D4—722no—
186C2×C4—41no—
187C2×C4—91no—
188C2×C4—91no—
189C2×C4—361no—
190C2×C4D4×C3⋊D311yescharacteristic
191C8—92no—
192C8#382811yescharacteristic
193C8#383011yescharacteristic
194C2×C3—164no—
195C2×C3#476014yes—
196C2×C3D3×D1614yes—
197C2×C3D3×SD3214yes—
198D3—484no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—721no—
203C22—721no—
204C22—721no—
205C22D8×C3⋊D311yescharacteristic
206C4—81no—
207C4—92no—
208C4—92no—
209C4—721no—
210C4#6409411yescharacteristic
211C4SD16×C3⋊D311yescharacteristic
212C3#6918314yescomplemented
213C2—161no—
214C2—1441no—
215C2#1059245911yescharacteristic
216C2C3⋊D3×D1611yescharacteristic
217C2C3⋊D3×SD3211yescharacteristic
subgroupG/H
centerC22D8×C3⋊D3
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C22×C3⋊D3
FittingC32×#357C2
socleC22×C32C2×D8
prime radical#10592850C2
hypercenter#357C3⋊D3
hypocenterC32#2240
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #357

Complements

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

NH ≅ G/Ntype
2. #10592573C2semidirect
3. C32×#357C2semidirect
4. #10589428C2semidirect
5. #10589375C2semidirect
6. C16×C3⋊Dic3C2semidirect
7. #10589357C2semidirect
22. C3×#357D3semidirect
79. #357C3⋊D3semidirect
176. C32#2240semidirect
212. C3#69183semidirect

Maximal subgroups

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

subgroupindex
#105928502
#105925732
C32×#3572
#105894282
#105893752
C16×C3⋊Dic32
#105893572
#691833