Finite Group Databasebeta

← #11798708

Subgroups — #11798708

Order 1152.

678 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 93 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#10596143C211yesmaximal, characteristic, complemented
2C32×#455C211yesmaximal, characteristic
3#10592942C212yesmaximal
4#10589407C211yesmaximal, characteristic, complemented
5#10589342C212yesmaximal, complemented
6#70992—34nomaximal
7C32×H(5,𝔽2)C2211yescharacteristic
8C32×#137—21no—
9C32×#124C2211yescharacteristic
10#64153—21no—
11#64144—22no—
12#64144C2212yes—
13#64088—22no—
14#64087—22no—
15#64079—21no—
16C32×#104—21no—
17C32×#100C2211yescharacteristic
18#63661—22no—
19#63661C2212yes—
20#5672—34no—
21C3×#455D314yes—
22#5243—38no—
23#4541—34no—
24#4476—38no—
25C32×#41—21no—
26C32×#41—41no—
27C2×C32×Q8D411yescharacteristic
28C2×C32×D4—22no—
29C2×C32×D4—41no—
30C2×C32×D4C2311yescharacteristic
31C22×C3⋊Dic3D412yes—
32C2×#634—21no—
33C32×Q16—41no—
34C32×SD16—41no—
35C32×M16—21no—
36C32×C22⋊C4—22no—
37C32×C42D411yescharacteristic
38#3757—22no—
39#3757—22no—
40#3757—42no—
41#3757D412yes—
42C4⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊Dic3)—21no—
44#3750—22no—
45#3750—22no—
46#3750—22no—
47#3750—42no—
48C4×C3⋊Dic3—21no—
49#2201—91noSylow
50C3×H(5,𝔽2)C2×D314yes—
51C3×#137—24no—
52C3×#124C2×D314yes—
53#944—64no—
54#935—38no—
55#935—68no—
56#879—68no—
57#878—68no—
58#870—64no—
59C3×#104—24no—
60C3×#100C2×D314yes—
61#834—38no—
62#834—68no—
63C23×C32—22no—
64C23×C32C2×D412yes—
65C32×Q8—21no—
66C32×Q8—41no—
67C32×D4—21no—
68C32×D4—41no—
69C32×D4—42no—
70C32×D4—41no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4C2×D411yescharacteristic
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3—42no—
79C2×C3⋊Dic3—42no—
80#634—81no—
81C32×C8—41no—
82#563—91no—
83#458—92no—
84#455C3⋊D311yescharacteristic
85#354—91no—
86#352—92no—
87C3×#41—24no—
88C3×#41—44no—
89C2×C3×Q8#6714yes—
90C2×C3×D4—28no—
91C2×C3×D4—44no—
92C2×C3×D4C22×D314yes—
93C22×Dic3—38no—
94C2×Dic6—64no—
95C3×Q16—44no—
96C3×SD16—44no—
97C3×M16—24no—
98C3×C22⋊C4—28no—
99C3×C42#6714yes—
100#217—38no—
101#217—68no—
102#217—68no—
103#217—128no—
104C4⋊Dic3—64no—
105C4⋊Dic3—64no—
106Dic3⋊C4—68no—
107Dic3⋊C4—68no—
108Dic3⋊C4—68no—
109Dic3⋊C4—128no—
110C4×Dic3—64no—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32—41no—
114C22×C32—82no—
115C22×C32#12011yescharacteristic
116C32×C4—21no—
117C32×C4—41no—
118C32×C4—41no—
119C32×C4—41no—
120C3⋊Dic3—42no—
121C3⋊Dic3—42no—
122C3⋊Dic3—82no—
123H(5,𝔽2)C2×C3⋊D311yescharacteristic
124#137—21no—
125C4⋊Q8—181no—
126#126—182no—
127#125—181no—
128#124C2×C3⋊D311yescharacteristic
129#123—92no—
130#123—182no—
131C22⋊Q8—182no—
132#104—21no—
133#100C2×C3⋊D311yescharacteristic
134#99—92no—
135#99—182no—
136C23×C3—28no—
137C23×C3D3×D418yes—
138C3×Q8—24no—
139C3×Q8—44no—
140C3×D4—24no—
141C3×D4—44no—
142C3×D4—48no—
143C3×D4—44no—
144C2×C3×C4—28no—
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—44no—
148C2×C3×C4C2×#6714yes—
149C2×Dic3—68no—
150C2×Dic3—68no—
151C2×Dic3—128no—
152C2×Dic3—128no—
153Dic6—244no—
154C3×C8—44no—
155C2×C32—21no—
156C2×C32—42no—
157C2×C32—81no—
158C2×C32B(4,𝔽2)11yescharacteristic
159#41—21no—
160#41—41no—
161C2×Q8—181no—
162C2×Q8#63811yescharacteristic
163C2×D4—22no—
164C2×D4—41no—
165C2×D4C22×C3⋊D311yescharacteristic
166C22×C4—92no—
167Q16—41no—
168SD16—41no—
169M16—21no—
170C4⋊C4—181no—
171C4⋊C4—182no—
172C4⋊C4—182no—
173C4⋊C4—181no—
174C4⋊C4—182no—
175C4⋊C4—362no—
176C22⋊C4—22no—
177C22⋊C4—92no—
178C22⋊C4—182no—
179C22⋊C4—182no—
180C22⋊C4—362no—
181C42—181no—
182C42#63811yescharacteristic
183C22×C3—28no—
184C22×C3—48no—
185C22×C3—44no—
186C22×C3—88no—
187C22×C3#93714yes—
188C3×C4—24no—
189C3×C4—44no—
190C3×C4—44no—
191C3×C4—44no—
192Dic3—128no—
193Dic3—128no—
194Dic3—248no—
195C32#220111yescharacteristic, complemented, Sylow
196C23—22no—
197C23D4×C3⋊D312yes—
198Q8—21no—
199Q8—41no—
200Q8—721no—
201D4—21no—
202D4—41no—
203D4—42no—
204D4—41no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—182no—
210C2×C4—182no—
211C2×C4—362no—
212C2×C4—362no—
213C2×C4C2×#63811yescharacteristic
214C8—41no—
215C2×C3—24no—
216C2×C3—48no—
217C2×C3—84no—
218C2×C3#503214yes—
219C22—22no—
220C22—42no—
221C22—41no—
222C22—82no—
223C22#6414611yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—362no—
229C4—362no—
230C4—722no—
231C3#7099214yescomplemented
232C2—21no—
233C2—42no—
234C2—81no—
235C2#1059273111yescharacteristic
subgroupG/H
centerC2#10592731
commutator, cosocleC2×C32×D4C23
FrattiniC2×D4C22×C3⋊D3
FittingC32×#455C2
socleC2×C32B(4,𝔽2)
prime radicalC32×H(5,𝔽2)C22
hypercenter#455C3⋊D3
hypocenterC32#2201
LCSG ▷ C2×C32×D4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #10596143C2semidirect
4. #10589407C2semidirect
5. #10589342C2semidirect
195. C32#2201semidirect
231. C3#70992semidirect

Maximal subgroups

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

subgroupindex
#105961432
C32×#4552
#105929422
#105894072
#105893422
#709923