Finite Group Databasebeta

← #11781022

Subgroups — #11781022

Order 1152.

498 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 121 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#10592783C211yesmaximal, characteristic
2#10592779C211yesmaximal, characteristic, complemented
3#10592540C211yesmaximal, characteristic
4#10592533C211yesmaximal, characteristic
5C32×#333C211yesmaximal, characteristic, complemented
6#10589352C211yesmaximal, characteristic, complemented
7#10589346C211yesmaximal, characteristic, complemented
8#68496—34nomaximal
9C32×C4⋊Q8C2211yescharacteristic
10C2×#3744—21no—
11C2×C8×C3⋊D3—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13#64078C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C32×C4⋊C8C2211yescharacteristic
16#63647—21no—
17#63643—21no—
18#63636—21no—
19#63635—21no—
20#63635—21no—
21#63635—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C4×C3⋊(C3⋊C8)C2211yescharacteristic
24#5084—34no—
25#5080—34no—
26#4841—34no—
27#4834—34no—
28C3×#333D314yescomplemented
29#4486—34no—
30#4480—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
47#3744—42no—
48C8×C3⋊D3—42no—
49#1671—91noSylow
50C3×C4⋊Q8C2×D314yes—
51C2×#204—64no—
52C2×D3×C8—64no—
53C4×D12—34no—
54#869—34no—
55C3×C4⋊C8C2×D314yes—
56C3×C4⋊C8C2×D314yes—
57(C2×D3)⋊C8—64no—
58#816—64no—
59#809—64no—
60#808—64no—
61#808—64no—
62#808—64no—
63C4⋊(C3⋊C8)—34no—
64C4×C3⋊C8—34no—
65C22×C3⋊D3—21no—
66C32×Q8—81no—
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)SD1612yes—
75C4×C3⋊D3—41no—
76C4×C3⋊D3—41no—
77#634—41no—
78#634—41no—
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—
87C8×D4—91no—
88#334—91no—
89#333C3⋊D311yescharacteristic, complemented
90#332—91no—
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(C2×D3)⋊C4—64no—
101C4⋊Dic3—34no—
102C4⋊Dic3—64no—
103C4⋊Dic3—64no—
104C2×C3⋊C8—34no—
105C2×C3⋊C8—34no—
106C2×C3⋊C8—64no—
107#204—128no—
108D3×C8—128no—
109C22×C32#12111yescharacteristic
110C2×C3⋊D3—22no—
111C2×C3⋊D3—41no—
112C2×C3⋊D3—41no—
113C32×C4—21no—
114C32×C4—81no—
115C32×C4C2×SD1611yescharacteristic
116C32×C4C2×SD1611yescharacteristic
117C32×C4#13511yescharacteristic
118C32×C4#13711yescharacteristic
119C3⋊Dic3—41no—
120C3⋊Dic3—81no—
121C2×SD16—181no—
122C22×C8—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—84no—
138C2×C3×C4—44no—
139C2×C3×C4#23514yes—
140C2×C3×C4D3×D414yes—
141C2×C3×C4D3×D414yes—
142C2×Dic3—64no—
143C2×Dic3—124no—
144D12—38no—
145D12—64no—
146C4×D3—124no—
147C4×D3—124no—
148Dic6—124no—
149Dic6—124no—
150C3×C8—44no—
151C3×C8—44no—
152C3⋊C8—38no—
153C3⋊C8—64no—
154C3⋊C8—124no—
155C2×C32#46111yescharacteristic
156C2×C32#46511yescharacteristic
157C2×C32#47111yescharacteristic
158C3⋊D3—42no—
159C2×Q8—21no—
160C2×Q8—181no—
161C2×D4—91no—
162C22×C4—181no—
163SD16—362no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—91no—
167C2×C8—91no—
168C2×C8—181no—
169C2×C8—362no—
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—68no—
179C2×D3—124no—
180C2×D3—124no—
181C3×C4—24no—
182C3×C4—84no—
183C3×C4D3×SD1614yes—
184C3×C4D3×SD1614yes—
185C3×C4#91814yes—
186C3×C4#91914yes—
187Dic3—124no—
188Dic3—244no—
189C32#167111yescharacteristic, complemented, Sylow
190C23—181no—
191Q8—81no—
192Q8—361no—
193Q8—361no—
194D4—92no—
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#474814yes—
210C2×C3#480714yes—
211C2×C3#481814yes—
212D3—128no—
213C22—182no—
214C22—361no—
215C22—361no—
216C22#6409311yescharacteristic
217C4—21no—
218C4—81no—
219C4—361no—
220C4—721no—
221C4SD16×C3⋊D311yescharacteristic
222C4SD16×C3⋊D311yescharacteristic
223C4#6412711yescharacteristic
224C4#6412811yescharacteristic
225C3#6849614yescomplemented
226C2—362no—
227C2#1059244711yescharacteristic
228C2#1059250611yescharacteristic
229C2#1059251711yescharacteristic
subgroupG/H
centerC22#64093
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#333C2
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenter#333C3⋊D3
hypocenterC32#1671
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #10592779C2semidirect
5. C32×#333C2semidirect
6. #10589352C2semidirect
7. #10589346C2semidirect
28. C3×#333D3semidirect
83. C3⋊(C3⋊C8)SD16semidirect
89. #333C3⋊D3semidirect
189. C32#1671semidirect
225. C3#68496semidirect

Maximal subgroups

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

subgroupindex
#105927832
#105927792
#105925402
#105925332
C32×#3332
#105893522
#105893462
#684963