Finite Group Databasebeta

← #11781007

Subgroups — #11781007

Order 1152.

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

normal subgroups only

This group has 498 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×#461C211yesmaximal, characteristic
2#10592779C211yesmaximal, characteristic, complemented
3D4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4#10592707C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589347C211yesmaximal, characteristic, complemented
7#10589346C211yesmaximal, characteristic, complemented
8#71036—34nomaximal
9C2×C32×SD16—21no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4×D4C2211yescharacteristic
12C22×C3⋊(C3⋊C8)—21no—
13#64078C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C32×#103—21no—
16#63659—21no—
17#63657—21no—
18#63635—21no—
19#63635—21no—
20#63635—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C4×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#461D314yes—
25#5080—34no—
26D4×C3⋊C8—34no—
27#5008—34no—
28#4547—34no—
29#4481—34no—
30#4480—34no—
31C2×C32×Q8—21no—
32C2×C32×D4D411yescharacteristic
33C22×C32×C4—21no—
34C2×#634—21no—
35C32×SD16—42no—
36C2×C32×C8—21no—
37C32×C4⋊C4—21no—
38C32×C4⋊C4—21no—
39C32×C4⋊C4D411yescharacteristic
40C32×C22⋊C4—21no—
41C32×C42C2311yescharacteristic
42C4⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊C8)—21no—
45C2×C3⋊(C3⋊C8)—21no—
46C2×C3⋊(C3⋊C8)—42no—
47C2×C3⋊(C3⋊C8)D411yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49#1671—91noSylow
50C2×C3×SD16—24no—
51C3×C4⋊Q8C2×D314yes—
52C3×C4×D4C2×D314yes—
53C22×C3⋊C8—64no—
54#869—34no—
55C3×C4⋊C8C2×D314yes—
56C3×#103—24no—
57#832—64no—
58#830—64no—
59#808—64no—
60#808—64no—
61#808—64no—
62C4⋊(C3⋊C8)—34no—
63C4⋊(C3⋊C8)—34no—
64C4×C3⋊C8—34no—
65C23×C32—21no—
66C32×Q8—41no—
67C32×Q8—41no—
68C32×D4—21no—
69C32×D4SD1612yes—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4C2×D411yescharacteristic
75C2×C32×C4C2×D411yescharacteristic
76C2×C32×C4#4111yescharacteristic
77C2×C3⋊Dic3—41no—
78#634—81no—
79C32×C8—41no—
80C3⋊(C3⋊C8)—21no—
81C3⋊(C3⋊C8)—41no—
82C3⋊(C3⋊C8)—41no—
83C3⋊(C3⋊C8)SD1612yescomplemented
84#498—91no—
85#476—91no—
86#461C3⋊D311yescharacteristic
87C8×D4—91no—
88#334—91no—
89#333—91no—
90#332—91no—
91C2×C3×Q8—24no—
92C2×C3×D4#6714yes—
93C22×C3×C4—24no—
94C2×Dic6—64no—
95C3×SD16—48no—
96C2×C3×C8—24no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4#6714yes—
100C3×C22⋊C4—24no—
101C3×C42C22×D314yes—
102C4⋊Dic3—64no—
103C4⋊Dic3—64no—
104C2×C3⋊C8—34no—
105C2×C3⋊C8—34no—
106C2×C3⋊C8—64no—
107C2×C3⋊C8—64no—
108C2×C3⋊C8—128no—
109C22×C32—22no—
110C22×C32—41no—
111C22×C32—41no—
112C22×C32#12111yescharacteristic
113C32×C4—21no—
114C32×C4—41no—
115C32×C4—81no—
116C32×C4C2×SD1611yescharacteristic
117C32×C4C2×SD1611yescharacteristic
118C32×C4#13511yescharacteristic
119C32×C4#13711yescharacteristic
120C3⋊Dic3—81no—
121C2×SD16—21no—
122C22×C8—181no—
123C4⋊Q8—91no—
124C4⋊Q8C2×C3⋊D311yescharacteristic
125C4×D4C2×C3⋊D311yescharacteristic
126#106—181no—
127#106—181no—
128#106—181no—
129C4⋊C8—91no—
130C4⋊C8—91no—
131C4⋊C8C2×C3⋊D311yescharacteristic
132#103—21no—
133#102—181no—
134C22⋊C8—181no—
135C4×C8—91no—
136C23×C3—24no—
137C3×Q8—44no—
138C3×Q8—44no—
139C3×D4—24no—
140C3×D4#21418yes—
141C2×C3×C4—24no—
142C2×C3×C4—44no—
143C2×C3×C4—44no—
144C2×C3×C4—44no—
145C2×C3×C4D3×D414yes—
146C2×C3×C4#23714yes—
147C2×C3×C4C2×#6714yes—
148C2×Dic3—124no—
149Dic6—244no—
150C3×C8—44no—
151C3⋊C8—38no—
152C3⋊C8—64no—
153C3⋊C8—124no—
154C3⋊C8—124no—
155C2×C32—42no—
156C2×C32#46111yescharacteristic
157C2×C32#46511yescharacteristic
158C2×C32#47111yescharacteristic
159C2×Q8—21no—
160C2×Q8—181no—
161C2×D4#63811yescharacteristic
162C22×C4—21no—
163SD16—42no—
164C2×C8—21no—
165C2×C8—91no—
166C2×C8—91no—
167C2×C8—181no—
168C2×C8—181no—
169C2×C8—362no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—181no—
173C4⋊C4—181no—
174C4⋊C4#63811yescharacteristic
175C22⋊C4—21no—
176C42C22×C3⋊D311yescharacteristic
177C22×C3—28no—
178C22×C3—44no—
179C22×C3—44no—
180C22×C3#93914yes—
181C3×C4—24no—
182C3×C4—44no—
183C3×C4—84no—
184C3×C4D3×SD1614yes—
185C3×C4#91514yes—
186C3×C4C2×#21414yes—
187C3×C4#95014yes—
188Dic3—244no—
189C32#167111yescharacteristic, complemented, Sylow
190C23—21no—
191Q8—41no—
192Q8—41no—
193Q8—721no—
194D4—21no—
195D4#375412yes—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—361no—
201C2×C4D4×C3⋊D311yescharacteristic
202C2×C4#383011yescharacteristic
203C2×C4C2×#63811yescharacteristic
204C8—41no—
205C8—92no—
206C8—181no—
207C8—361no—
208C8—361no—
209C2×C3—48no—
210C2×C3#504114yes—
211C2×C3#516214yes—
212C2×C3#520314yes—
213C22—22no—
214C22—41no—
215C22—41no—
216C22#6414811yescharacteristic
217C4—21no—
218C4—41no—
219C4—81no—
220C4—721no—
221C4SD16×C3⋊D311yescharacteristic
222C4#6412411yescharacteristic
223C4C2×#375411yescharacteristic
224C4#6415911yescharacteristic
225C3#7103614yescomplemented
226C2—42no—
227C2#1059274011yescharacteristic
228C2#1059286111yescharacteristic
229C2#1059290211yescharacteristic
subgroupG/H
centerC22#64148
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#461C2
socleC22×C32#121
prime radicalC2×C32×D4D4
hypercenter#461C3⋊D3
hypocenterC32#1671
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #10592779C2semidirect
5. #10589413C2semidirect
6. #10589347C2semidirect
7. #10589346C2semidirect
83. C3⋊(C3⋊C8)SD16semidirect
189. C32#1671semidirect
225. C3#71036semidirect

Maximal subgroups

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

subgroupindex
C32×#4612
#105927792
D4×C3⋊(C3⋊C8)2
#105927072
#105894132
#105893472
#105893462
#710363