Finite Group Databasebeta

← #11771739

Subgroups — #11771739

Order 1152.

255 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C3×#5137C211yesmaximal, characteristic
2#10589423C211yesmaximal, characteristic
3C3×#4553C211yesmaximal, characteristic
4#67780—31nomaximal
5#67536—31nomaximal
6C2×C32×M16C2211yescharacteristic
7#64135—21no—
8C3×#924—21no—
9C2×C3×Dic12C411yescharacteristic, complemented
10C3×#817C411yescharacteristic, complemented
11C3×#816—21no—
12C3×#812—21no—
13C3×#470—31no—
14#5137D311yescharacteristic
15C3×#349—31no—
16#4557—31no—
17#4557—31no—
18#4557—61no—
19#4553D311yescharacteristic
20C22×C32×C4D411yescharacteristic
21C2×C3×Dic6—21no—
22C32×M16—41no—
23C2×C32×C8C2×C411yescharacteristic
24C2×C32×C8D411yescharacteristic
25#3757—41no—
26C4⋊(C3⋊Dic3)—41no—
27C4×C3⋊Dic3—22no—
28C3×#217—41no—
29C3×C4⋊Dic3—21no—
30C3×Dic3⋊C4—41no—
31C3×Dic12—41no—
32C3×C3⋊C16—41no—
33#1340—91noSylow
34C2×C3×Q16—31no—
35C2×C3×M16—21no—
36C2×C3×M16C2×D311yescharacteristic
37C2×C3×M16C2×D311yescharacteristic
38C3×C22⋊Q8—61no—
39#926—62no—
40#926—61no—
41#926—61no—
42#924—21no—
43C2×Dic12Dic311yescharacteristic, complemented
44C3×M32—61no—
45C3×#106—31no—
46C3×#103—61no—
47#817Dic311yescharacteristic, complemented
48#816—21no—
49#812—21no—
50C23×C32D811yescharacteristic
51C2×C32×C4—21no—
52C2×C32×C4—21no—
53C2×C32×C4C22⋊C411yescharacteristic
54C2×C32×C4SD1611yescharacteristic
55C2×C3⋊Dic3—42no—
56C2×C3×Dic3—41no—
57C2×C3×Dic3—41no—
58C3×Dic6—41no—
59C3×Dic6—41no—
60C32×C8—21no—
61C32×C8—21no—
62#470—31no—
63#349—31no—
64H(3,ℤ/4)—91no—
65C2×C3×Q8—61no—
66C22×C3×C4—21no—
67C22×C3×C4D1211yescharacteristic
68C22×C3×C4#6711yescharacteristic
69C2×Dic6—21no—
70C3×Q16—121no—
71C3×M16—41no—
72C3×M16—41no—
73C3×M16—42no—
74C2×C3×C8—21no—
75C2×C3×C8—21no—
76C2×C3×C8C4×D311yescharacteristic
77C2×C3×C8C2×Dic311yescharacteristic
78C2×C3×C8#6711yescharacteristic
79C2×C3×C8#6711yescharacteristic
80C3×C4⋊C4—61no—
81C3×C4⋊C4—121no—
82C3×C22⋊C4—121no—
83#217—41no—
84#217—122no—
85#217—121no—
86#217—121no—
87C4⋊Dic3—21no—
88C4⋊Dic3—122no—
89C4⋊Dic3—121no—
90C4⋊Dic3—121no—
91Dic3⋊C4—41no—
92C4×Dic3—64no—
93C4×Dic3—62no—
94C4×Dic3—62no—
95Dic12—41no—
96C3×C16—121no—
97C3⋊C16—41no—
98C22×C32—21no—
99C22×C32—41no—
100C22×C32#10211yescharacteristic
101C32×C4—21no—
102C32×C4#9911yescharacteristic
103C32×C4#10411yescharacteristic
104C3⋊Dic3—82no—
105C3×Dic3—81no—
106C3×Dic3—81no—
107C2×Q16—31no—
108C2×M16D3211yescharacteristic
109C22⋊Q8—61no—
110#117—181no—
111M32—61no—
112#106—31no—
113#103—61no—
114C23×C3—21no—
115C23×C3D2411yescharacteristic
116C23×C3#21311yescharacteristic
117C3×Q8—121no—
118C3×Q8—121no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—121no—
128C2×C3×C4—121no—
129C2×C3×C4#20411yescharacteristic
130C2×C3×C4(C2×D3)⋊C411yescharacteristic
131C2×C3×C4#21411yescharacteristic
132C2×C3×C4#21711yescharacteristic
133C2×Dic3—41no—
134C2×Dic3—41no—
135C2×Dic3—124no—
136C2×Dic3—122no—
137C2×Dic3—122no—
138Dic6—41no—
139Dic6—41no—
140C3×C8—21no—
141C3×C8—21no—
142C3×C8—21no—
143C3×C8—21no—
144C3×C8—41no—
145C3×C8—41no—
146C2×C32—21no—
147C2×C32—41no—
148C2×C32#32711yescharacteristic
149C2×Q8—61no—
150C22×C4#62611yescharacteristic
151Q16—121no—
152M16—41no—
153C2×C8D3×Dic311yescharacteristic
154C2×C8#62511yescharacteristic
155C4⋊C4—61no—
156C4⋊C4—121no—
157C4⋊C4—361no—
158C22⋊C4—121no—
159C22⋊C4—361no—
160C42—182no—
161C16—121no—
162C22×C3—22no—
163C22×C3—21no—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—81no—
169C22×C3#82111yescharacteristic
170C22×C3#83211yescharacteristic
171C3×C4—22no—
172C3×C4—21no—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—241no—
177C3×C4—241no—
178C3×C4#80611yescharacteristic
179C3×C4#82511yescharacteristic
180C3×C4#83411yescharacteristic
181C3×C4#83711yescharacteristic
182Dic3—81no—
183Dic3—81no—
184Dic3—244no—
185Dic3—242no—
186Dic3—242no—
187C32#134011yescharacteristic, complemented, Sylow
188C23#371411yescharacteristic
189Q8—121no—
190Q8—121no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—121no—
194C2×C4—121no—
195C2×C4—362no—
196C2×C4#371611yescharacteristic
197C2×C4(C2×D3)⋊Dic311yescharacteristic
198C8—21no—
199C8—21no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—41no—
204C2×C3—42no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3#447011yescharacteristic
208C2×C3#453711yescharacteristic
209C22—21no—
210C22—41no—
211C22#6356311yescharacteristic
212C4—21no—
213C4—241no—
214C4—241no—
215C4—722no—
216C4#6356811yescharacteristic
217C4#6358011yescharacteristic
218C3—21no—
219C3#6753611yescharacteristic, complemented
220C3#6778011yescharacteristic, complemented
221C2—21no—
222C2—41no—
223C2#1058915011yescharacteristic
subgroupG/H
centerC2#10589150
commutatorC2×C32×C8C2×C4
Frattini, hypercenterC2×M16D32
Fitting, cosocleC2×C32×M16C22
socleC2×C32#327
prime radicalC23×C32D8
hypocenterC32#1340
LCSG ▷ C2×C32×C8 ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
9. C2×C3×Dic12C4semidirect
10. C3×#817C4semidirect
43. C2×Dic12Dic3semidirect
47. #817Dic3semidirect
187. C32#1340semidirect
219. C3#67536semidirect
220. C3#67780semidirect

Maximal subgroups

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

subgroupindex
C3×#51372
#105894232
C3×#45532
#677803
#675363