Finite Group Databasebeta

← #11781830

Subgroups — #11781830

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#10592782C211yesmaximal, characteristic
2#10592779C211yesmaximal, characteristic, complemented
3#10592541C211yesmaximal, characteristic
4#10592537C211yesmaximal, characteristic
5C32×#332C211yesmaximal, characteristic, complemented
6#10589347C211yesmaximal, characteristic, complemented
7#10589345C211yesmaximal, characteristic, complemented
8#68450—34nomaximal
9C32×C4⋊Q8C2211yescharacteristic
10C2×#3755—21no—
11C2×#3743—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13#64078C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C32×C4⋊C8C2211yescharacteristic
16#63662—21no—
17#63648—21no—
18#63647—21no—
19#63644—21no—
20#63635—21no—
21#63635—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C4×C3⋊(C3⋊C8)C2211yescharacteristic
24#5083—34no—
25#5080—34no—
26#4842—34no—
27#4838—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#3755—42no—
41#3752—21no—
42C4⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3⋊(C3⋊C8)—21no—
46C2×C3⋊(C3⋊C8)D411yescharacteristic
47C2×C3⋊(C3⋊C8)D411yescharacteristic
48#3743—42no—
49#1691—91noSylow
50C3×C4⋊Q8C2×D314yes—
51C2×#215—64no—
52C2×ΓL(1,𝔽25)—64no—
53C4×D12—34no—
54#869—34no—
55C3×C4⋊C8C2×D314yes—
56C3×C4⋊C8C2×D314yes—
57#835—64no—
58#821—64no—
59(C2×D3)⋊C8—64no—
60#817—64no—
61#808—64no—
62#808—64no—
63C4⋊(C3⋊C8)—34no—
64C4×C3⋊C8—34no—
65C22×C3⋊D3—21no—
66C32×Q8—41no—
67C32×Q8—41no—
68C2×C32×C4—41no—
69C2×C32×C4C2×D411yescharacteristic
70C2×C32×C4C2×D411yescharacteristic
71C2×C32×C4#4111yescharacteristic
72C2×C3⋊Dic3—21no—
73C2×C3⋊Dic3—41no—
74C4⋊(C3⋊D3)—21no—
75C4⋊(C3⋊D3)—21no—
76C4×C3⋊D3—41no—
77C4×C3⋊D3—41no—
78#634—81no—
79C32×C8—41no—
80C32×C8—41no—
81C3⋊(C3⋊C8)—21no—
82C3⋊(C3⋊C8)—41no—
83C3⋊(C3⋊C8)SD1612yescomplemented
84#498—91no—
85#476—91no—
86#461—91no—
87#436—91no—
88#335—91no—
89#333—91no—
90#332C3⋊D311yescharacteristic, complemented
91C2×C3×Q8—24no—
92C2×D12—34no—
93C2×C4×D3—64no—
94C2×Dic6—64no—
95C2×C3×C8—24no—
96C2×C3×C8—24no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—24no—
99C3×C42C22×D314yes—
100#215—128no—
101(C2×D3)⋊C4—64no—
102C4⋊Dic3—34no—
103C4⋊Dic3—64no—
104C4⋊Dic3—64no—
105C2×C3⋊C8—34no—
106C2×C3⋊C8—34no—
107C2×C3⋊C8—64no—
108ΓL(1,𝔽25)—128no—
109C22×C32#12111yescharacteristic
110C2×C3⋊D3—41no—
111C2×C3⋊D3—41no—
112C2×C3⋊D3—41no—
113C32×C4—21no—
114C32×C4—81no—
115C32×C4C2×SD1611yescharacteristic
116C32×C4AGL(1,ℤ/8)11yescharacteristic
117C32×C4#13711yescharacteristic
118C32×C4#13711yescharacteristic
119C3⋊Dic3—41no—
120C3⋊Dic3—81no—
121C2×SD16—181no—
122C2×M16—181no—
123C4⋊Q8—91no—
124C4⋊Q8C2×C3⋊D311yescharacteristic
125C4×D4—91no—
126#106—181no—
127#106—181no—
128#106—181no—
129C4⋊C8—91no—
130C4⋊C8C2×C3⋊D311yescharacteristic
131C4⋊C8C2×C3⋊D311yescharacteristic
132#103—181no—
133#102—181no—
134C22⋊C8—181no—
135C4×C8—91no—
136C22×D3—64no—
137C3×Q8—44no—
138C3×Q8—44no—
139C2×C3×C4—44no—
140C2×C3×C4#23514yes—
141C2×C3×C4D3×D414yes—
142C2×C3×C4D3×D414yes—
143C2×Dic3—64no—
144C2×Dic3—124no—
145D12—64no—
146D12—64no—
147C4×D3—124no—
148C4×D3—124no—
149Dic6—244no—
150C3×C8—44no—
151C3×C8—44no—
152C3⋊C8—38no—
153C3⋊C8—64no—
154C3⋊C8—124no—
155C2×C32#46111yescharacteristic
156C2×C32#46811yescharacteristic
157C2×C32#47111yescharacteristic
158C3⋊D3—81no—
159C2×Q8—21no—
160C2×Q8—181no—
161C2×D4—91no—
162C22×C4—181no—
163SD16—362no—
164M16—362no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—91no—
168C2×C8—91no—
169C2×C8—181no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—91no—
173C4⋊C4—181no—
174C4⋊C4—181no—
175C22⋊C4—181no—
176C42C22×C3⋊D311yescharacteristic
177C22×C3#88414yes—
178C2×D3—124no—
179C2×D3—124no—
180C2×D3—124no—
181C3×C4—24no—
182C3×C4—84no—
183C3×C4#91114yes—
184C3×C4D3×SD1614yes—
185C3×C4#91514yes—
186C3×C4#91814yes—
187Dic3—124no—
188Dic3—244no—
189C32#169111yescharacteristic, complemented, Sylow
190C23—181no—
191Q8—41no—
192Q8—41no—
193Q8—721no—
194D4—181no—
195D4—181no—
196C2×C4—41no—
197C2×C4—181no—
198C2×C4—361no—
199C2×C4—361no—
200C2×C4—361no—
201C2×C4#382811yescharacteristic
202C2×C4D4×C3⋊D311yescharacteristic
203C2×C4D4×C3⋊D311yescharacteristic
204C8—41no—
205C8—41no—
206C8—92no—
207C8—181no—
208C8—361no—
209C2×C3#475414yes—
210C2×C3#478014yes—
211C2×C3#481814yes—
212D3—244no—
213C22—361no—
214C22—361no—
215C22—361no—
216C22#6409311yescharacteristic
217C4—21no—
218C4—81no—
219C4—361no—
220C4—721no—
221C4#6412011yescharacteristic
222C4SD16×C3⋊D311yescharacteristic
223C4#6412411yescharacteristic
224C4#6412711yescharacteristic
225C3#6845014yescomplemented
226C2—721no—
227C2#1059245311yescharacteristic
228C2#1059247911yescharacteristic
229C2#1059251711yescharacteristic
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#1691
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. #10592779C2semidirect
5. C32×#332C2semidirect
6. #10589347C2semidirect
7. #10589345C2semidirect
28. C3×#332D3semidirect
83. C3⋊(C3⋊C8)SD16semidirect
90. #332C3⋊D3semidirect
189. C32#1691semidirect
225. C3#68450semidirect

Maximal subgroups

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

subgroupindex
#105927822
#105927792
#105925412
#105925372
C32×#3322
#105893472
#105893452
#684503