Finite Group Databasebeta

← #11782720

Subgroups — #11782720

Order 1152.

624 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C32×#486C211yesmaximal, characteristic, complemented
2#10592787C211yesmaximal, characteristic, complemented
3#10592778C211yesmaximal, characteristic, complemented
4#10592775C211yesmaximal, characteristic, complemented
5#10592772C211yesmaximal, characteristic
6#10592393C211yesmaximal, characteristic, complemented
7#10589321C211yesmaximal, characteristic, complemented
8#71187—34nomaximal
9C32×C4⋊Q8C2211yescharacteristic
10C32×#127C2211yescharacteristic
11C2×#3754—22no—
12C2×#3753—22no—
13#64083C2211yescharacteristic
14#64078C2211yescharacteristic
15C32×#102—22no—
16C32×C4×C8C2211yescharacteristic
17#63662—22no—
18#63648—22no—
19#63636—22no—
20C4×C3⋊(C3⋊C8)C2211yescharacteristic
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#486D314yescomplemented
23#5088—34no—
24#5079—34no—
25#5076—34no—
26#5073—34no—
27#4694—34no—
28#4455—34no—
29C2×C32×Q8—21no—
30C2×C32×D4—21no—
31C2×C32×D4—21no—
32C2×C32×D4—21no—
33C2×C4⋊(C3⋊D3)—21no—
34C2×C4⋊(C3⋊D3)—21no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×#634—21no—
37C2×C32×C8D412yes—
38C32×C4⋊C4—21no—
39C32×C4⋊C4—21no—
40C32×C42C2311yescharacteristic
41#3754—44no—
42#3753—44no—
43C4⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊C8)D412yes—
46C2×C3⋊(C3⋊C8)D412yes—
47#1717—91noSylow
48C3×C4⋊Q8C2×D314yes—
49C3×#127C2×D314yes—
50C2×#214—68no—
51C2×#213—68no—
52#874—34no—
53#869—34no—
54C3×#102—28no—
55C3×C4×C8C2×D314yes—
56#835—68no—
57#821—68no—
58#809—68no—
59C4×C3⋊C8—34no—
60C4×C3⋊C8—34no—
61C23×C32—41no—
62C22×C3⋊D3—41no—
63C32×Q8—42no—
64C32×D4—42no—
65C32×D4—42no—
66C32×D4—42no—
67C2×C32×C4—41no—
68C2×C32×C4C2×D411yescharacteristic
69C2×C32×C4C2×D411yescharacteristic
70C2×C32×C4C2×D411yescharacteristic
71C2×C3⋊Dic3—41no—
72C4⋊(C3⋊D3)—42no—
73C4⋊(C3⋊D3)—42no—
74C4⋊(C3⋊D3)—42no—
75#634—42no—
76C32×C8—22no—
77C3⋊(C3⋊C8)—22no—
78C3⋊(C3⋊C8)—22no—
79#493—91no—
80#492—91no—
81#487—91no—
82#486—91no—
83#486—91no—
84#486C3⋊D311yescharacteristic, complemented
85#322—91no—
86C2×C3×Q8—24no—
87C2×C3×D4—24no—
88C2×C3×D4—24no—
89C2×C3×D4—24no—
90C2×D12—64no—
91C2×D12—64no—
92C2×D12—64no—
93C2×Dic6—64no—
94C2×C3×C8#6718yes—
95C3×C4⋊C4—24no—
96C3×C4⋊C4—24no—
97C3×C42C22×D314yes—
98#214—1216no—
99#213—1216no—
100C4⋊Dic3—64no—
101C4⋊Dic3—64no—
102C2×C3⋊C8—38no—
103C2×C3⋊C8—38no—
104C22×C32—81no—
105C22×C32—81no—
106C22×C32—81no—
107C22×C32#12711yescharacteristic
108C2×C3⋊D3—81no—
109C2×C3⋊D3—81no—
110C2×C3⋊D3—81no—
111C32×C4—81no—
112C32×C4#13512yes—
113C32×C4AGL(1,ℤ/8)12yes—
114C32×C4AGL(1,ℤ/8)12yes—
115C3⋊Dic3—81no—
116C2×SD16—182no—
117C2×D8—182no—
118C4⋊Q8—91no—
119C4⋊Q8C2×C3⋊D311yescharacteristic
120#127—91no—
121#127C2×C3⋊D311yescharacteristic
122#103—182no—
123#102—22no—
124#102—182no—
125#102—182no—
126C4×C8—91no—
127C4×C8—91no—
128C4×C8C2×C3⋊D311yescharacteristic
129C23×C3—44no—
130C22×D3—124no—
131C3×Q8—48no—
132C3×D4—48no—
133C3×D4—48no—
134C3×D4—48no—
135C2×C3×C4—44no—
136C2×C3×C4D3×D414yes—
137C2×C3×C4D3×D414yes—
138C2×C3×C4C2×#6714yes—
139C2×Dic3—124no—
140D12—128no—
141D12—128no—
142D12—128no—
143Dic6—128no—
144C3×C8—28no—
145C3⋊C8—68no—
146C3⋊C8—68no—
147C2×C32—161no—
148C2×C32#49511yescharacteristic
149C2×C32#49611yescharacteristic
150C2×C32#49611yescharacteristic
151C3⋊D3—161no—
152C2×Q8—21no—
153C2×Q8—181no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—181no—
158C2×D4—181no—
159C2×D4—181no—
160SD16—364no—
161D8—364no—
162C2×C8—92no—
163C2×C8—92no—
164C2×C8#63812yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—181no—
168C4⋊C4—181no—
169C42C22×C3⋊D311yescharacteristic
170C22×C3—84no—
171C22×C3—84no—
172C22×C3—84no—
173C22×C3#94014yes—
174C2×D3—244no—
175C2×D3—244no—
176C2×D3—244no—
177C3×C4—84no—
178C3×C4#91118yes—
179C3×C4#91418yes—
180C3×C4#95018yes—
181Dic3—244no—
182C32#171711yescharacteristic, complemented, Sylow
183C23—41no—
184C23—361no—
185Q8—42no—
186Q8—362no—
187D4—42no—
188D4—42no—
189D4—42no—
190D4—362no—
191D4—362no—
192D4—362no—
193C2×C4—41no—
194C2×C4—361no—
195C2×C4D4×C3⋊D311yescharacteristic
196C2×C4D4×C3⋊D311yescharacteristic
197C2×C4C2×#63811yescharacteristic
198C8—22no—
199C8—182no—
200C8—182no—
201C2×C3—164no—
202C2×C3#505914yes—
203C2×C3#515414yes—
204C2×C3#517614yes—
205D3—484no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—721no—
210C22—721no—
211C22—721no—
212C22#6414911yescharacteristic
213C4—81no—
214C4—721no—
215C4#6412012yes—
216C4#6412312yes—
217C4#6415912yes—
218C3#7118714yescomplemented
219C2—161no—
220C2—1441no—
221C2#1059275811yescharacteristic
222C2#1059285311yescharacteristic
223C2#1059287511yescharacteristic
subgroupG/H
centerC22#64149
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#486C2
socleC22×C32#127
prime radical#10592772C2
hypercenter#486C3⋊D3
hypocenterC32#1717
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. C32×#486C2semidirect
2. #10592787C2semidirect
3. #10592778C2semidirect
4. #10592775C2semidirect
6. #10592393C2semidirect
7. #10589321C2semidirect
22. C3×#486D3semidirect
84. #486C3⋊D3semidirect
182. C32#1717semidirect
218. C3#71187semidirect

Maximal subgroups

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

subgroupindex
C32×#4862
#105927872
#105927782
#105927752
#105927722
#105923932
#105893212
#711873