Finite Group Databasebeta

← #11781515

Subgroups — #11781515

Order 1152.

480 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 119 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#10592786C211yesmaximal, characteristic
2#10592778C211yesmaximal, characteristic, complemented
3#10592541C211yesmaximal, characteristic
4#10592533C211yesmaximal, characteristic
5C32×#332C211yesmaximal, characteristic, complemented
6#10589347C211yesmaximal, characteristic, complemented
7#10589345C211yesmaximal, characteristic, complemented
8#68447—34nomaximal
9C32×C4⋊Q8C2211yescharacteristic
10C2×C8×C3⋊D3—21no—
11C4×C4⋊(C3⋊D3)C2211yescharacteristic
12#64078C2211yescharacteristic
13C32×C4⋊C8C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15#63662—21no—
16#63662—21no—
17#63648—21no—
18#63647—21no—
19#63644—21no—
20#63636—21no—
21#63634—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C4×C3⋊(C3⋊C8)C2211yescharacteristic
24#5087—34no—
25#5079—34no—
26#4842—34no—
27#4834—34no—
28C3×#332D314yescomplemented
29#4481—34no—
30#4479—34no—
31C2×C32×Q8—21no—
32C2×C4⋊(C3⋊D3)D411yescharacteristic
33C2×C4×C3⋊D3—21no—
34C2×#634—21no—
35C2×C32×C8—21no—
36C2×C32×C8—21no—
37C32×C4⋊C4—21no—
38C32×C4⋊C4—21no—
39C32×C42C2311yescharacteristic
40#3752—21no—
41C4⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊C8)—21no—
45C2×C3⋊(C3⋊C8)D411yescharacteristic
46C2×C3⋊(C3⋊C8)D411yescharacteristic
47C8×C3⋊D3—42no—
48#1683—91noSylow
49C3×C4⋊Q8C2×D314yes—
50C2×D3×C8—64no—
51C4×D12—34no—
52#869—34no—
53C3×C4⋊C8C2×D314yes—
54C3×C4⋊C8C2×D314yes—
55#835—64no—
56#835—64no—
57#821—64no—
58(C2×D3)⋊C8—64no—
59#817—64no—
60#809—64no—
61#807—64no—
62C4⋊(C3⋊C8)—34no—
63C4×C3⋊C8—34no—
64C22×C3⋊D3—21no—
65C32×Q8—41no—
66C32×Q8—41no—
67C2×C32×C4—41no—
68C2×C32×C4C2×D411yescharacteristic
69C2×C32×C4C2×D411yescharacteristic
70C2×C32×C4#4111yescharacteristic
71C2×C3⋊Dic3—21no—
72C2×C3⋊Dic3—41no—
73C4⋊(C3⋊D3)—21no—
74C4⋊(C3⋊D3)Q1612yes—
75C4×C3⋊D3—41no—
76C4×C3⋊D3—41no—
77#634—81no—
78C32×C8—41no—
79C32×C8—41no—
80C3⋊(C3⋊C8)—21no—
81C3⋊(C3⋊C8)—21no—
82C3⋊(C3⋊C8)—41no—
83#487—91no—
84#476—91no—
85#474—91no—
86C8×D4—91no—
87#335—91no—
88#333—91no—
89#332C3⋊D311yescharacteristic, complemented
90C2×C3×Q8—24no—
91C2×D12—34no—
92C2×C4×D3—64no—
93C2×Dic6—64no—
94C2×C3×C8—24no—
95C2×C3×C8—24no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—24no—
98C3×C42C22×D314yes—
99(C2×D3)⋊C4—64no—
100C4⋊Dic3—34no—
101C4⋊Dic3—64no—
102C4⋊Dic3—64no—
103C2×C3⋊C8—34no—
104C2×C3⋊C8—34no—
105C2×C3⋊C8—64no—
106D3×C8—128no—
107C22×C32#12111yescharacteristic
108C2×C3⋊D3—22no—
109C2×C3⋊D3—41no—
110C2×C3⋊D3—41no—
111C32×C4—21no—
112C32×C4—81no—
113C32×C4C2×Q1611yescharacteristic
114C32×C4#13511yescharacteristic
115C32×C4#13511yescharacteristic
116C32×C4AGL(1,ℤ/8)11yescharacteristic
117C3⋊Dic3—41no—
118C3⋊Dic3—81no—
119C22×C8—181no—
120C4⋊Q8—91no—
121C4⋊Q8C2×C3⋊D311yescharacteristic
122C4×D4—91no—
123#107—181no—
124#106—181no—
125C4⋊C8—91no—
126C4⋊C8C2×C3⋊D311yescharacteristic
127C4⋊C8C2×C3⋊D311yescharacteristic
128#103—181no—
129#103—181no—
130#102—181no—
131#102—181no—
132C22⋊C8—181no—
133C4×C8—91no—
134C22×D3—64no—
135C3×Q8—44no—
136C3×Q8—44no—
137C2×C3×C4—44no—
138C2×C3×C4#23514yes—
139C2×C3×C4D3×D414yes—
140C2×C3×C4D3×D414yes—
141C2×Dic3—64no—
142C2×Dic3—124no—
143D12—38no—
144D12—64no—
145C4×D3—124no—
146C4×D3—124no—
147Dic6—244no—
148C3×C8—44no—
149C3×C8—44no—
150C3⋊C8—64no—
151C3⋊C8—64no—
152C3⋊C8—124no—
153C2×C32#46311yescharacteristic
154C2×C32#46711yescharacteristic
155C2×C32#47111yescharacteristic
156C3⋊D3—42no—
157C2×Q8—21no—
158C2×Q8—181no—
159C2×D4—91no—
160C22×C4—181no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—91no—
164C2×C8—91no—
165C2×C8—181no—
166C2×C8—362no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—91no—
170C4⋊C4—181no—
171C4⋊C4—181no—
172C22⋊C4—181no—
173C42C22×C3⋊D311yescharacteristic
174C22×C3#88414yes—
175C2×D3—68no—
176C2×D3—124no—
177C2×D3—124no—
178C3×C4—24no—
179C3×C4—84no—
180C3×C4#91114yes—
181C3×C4#91614yes—
182C3×C4#91614yes—
183C3×C4D3×Q1614yes—
184Dic3—124no—
185Dic3—244no—
186C32#168311yescharacteristic, complemented, Sylow
187C23—181no—
188Q8—41no—
189Q8—41no—
190Q8—721no—
191D4—92no—
192D4—181no—
193C2×C4—41no—
194C2×C4—181no—
195C2×C4—361no—
196C2×C4—361no—
197C2×C4—361no—
198C2×C4#382811yescharacteristic
199C2×C4D4×C3⋊D311yescharacteristic
200C2×C4D4×C3⋊D311yescharacteristic
201C8—41no—
202C8—41no—
203C8—181no—
204C8—181no—
205C8—361no—
206C2×C3#475414yes—
207C2×C3#478714yes—
208C2×C3#481314yes—
209D3—128no—
210C22—182no—
211C22—361no—
212C22—361no—
213C22#6409311yescharacteristic
214C4—21no—
215C4—81no—
216C4—361no—
217C4—721no—
218C4#6412011yescharacteristic
219C4#6412511yescharacteristic
220C4#6412511yescharacteristic
221C4Q16×C3⋊D311yescharacteristic
222C3#6844714yescomplemented
223C2—362no—
224C2#1059245311yescharacteristic
225C2#1059248611yescharacteristic
226C2#1059251211yescharacteristic
subgroupG/H
centerC22#64093
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#332C2
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenter#332C3⋊D3
hypocenterC32#1683
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #10592778C2semidirect
5. C32×#332C2semidirect
6. #10589347C2semidirect
7. #10589345C2semidirect
28. C3×#332D3semidirect
89. #332C3⋊D3semidirect
186. C32#1683semidirect
222. C3#68447semidirect

Maximal subgroups

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

subgroupindex
#105927862
#105927782
#105925412
#105925332
C32×#3322
#105893472
#105893452
#684473