Finite Group Databasebeta

← #11798296

Subgroups — #11798296

Order 1152.

648 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 117 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#10596219C211yesmaximal, characteristic
2C32×#470C211yesmaximal, characteristic, complemented
3#10592878C212yesmaximal, complemented
4#10589419C211yesmaximal, characteristic, complemented
5#10589354C212yesmaximal, complemented
6#71091—34nomaximal
7C2×#3828—21no—
8C2×C32×Q16C2211yescharacteristic, complemented
9C2×C32×M16C2211yescharacteristic
10C32×C22⋊Q8—21no—
11#64118—22no—
12#64117—22no—
13#64113—21no—
14#64113C2212yes—
15C32×#106C2211yescharacteristic, complemented
16C32×#103—21no—
17#63656—22no—
18#63655—22no—
19#63639C2212yescomplemented
20#5748—34no—
21C3×#470D314yescomplemented
22#5179—38no—
23#4553—34no—
24#4488—38no—
25C2×C32×Q8—21no—
26C22×C32×C4D411yescharacteristic
27C2×#638—41no—
28#3828—22no—
29#3828—22no—
30#3828—41no—
31#3828—41no—
32C2×C4⋊(C3⋊D3)—21no—
33C2×C4×C3⋊D3—41no—
34C2×#634—21no—
35C32×Q16—22no—
36C32×M16—22no—
37C2×C32×C8D411yescharacteristic
38C2×C32×C8C2311yescharacteristic
39C32×C4⋊C4—21no—
40C32×C4⋊C4—41no—
41C32×C22⋊C4—41no—
42#3746—21no—
43#3746D412yes—
44#3745—21no—
45#3745D412yes—
46#3744—22no—
47#3744—22no—
48C3⋊(C3⋊C16)—22no—
49#2191—91noSylow
50C2×#235—64no—
51C2×C3×Q16C2×D314yescomplemented
52C2×C3×M16C2×D314yes—
53C3×C22⋊Q8—24no—
54#909—68no—
55#908—68no—
56#904—38no—
57#904—64no—
58C3×#106C2×D314yescomplemented
59C3×#103—24no—
60#829—68no—
61#828—68no—
62#812—38no—
63C23×C32D811yescharacteristic
64C22×C3⋊D3—41no—
65C32×Q8—41no—
66C32×Q8—41no—
67C2×C32×C4—21no—
68C2×C32×C4—21no—
69C2×C32×C4—41no—
70C2×C32×C4—41no—
71C2×C32×C4D811yescharacteristic
72C2×C32×C4C2×D411yescharacteristic
73#638—42no—
74#638—42no—
75C2×C3⋊Dic3—41no—
76C4⋊(C3⋊D3)—22no—
77C4⋊(C3⋊D3)—41no—
78C4×C3⋊D3—42no—
79C4×C3⋊D3—42no—
80#634—22no—
81#634—41no—
82C32×C8—21no—
83C32×C8C2×D412yes—
84#575—91no—
85#510—92no—
86#470C3⋊D311yescharacteristic, complemented
87#360—92no—
88#349—91no—
89C2×C3×Q8—24no—
90C22×C3×C4#6714yes—
91C2×#67—124no—
92#235—68no—
93#235—68no—
94#235—124no—
95#235—124no—
96C2×D12—64no—
97C2×C4×D3—124no—
98C2×Dic6—64no—
99C3×Q16—28no—
100C3×M16—28no—
101C2×C3×C8#6714yes—
102C2×C3×C8C22×D314yes—
103C3×C4⋊C4—24no—
104C3×C4⋊C4—44no—
105C3×C22⋊C4—44no—
106Dic12—38no—
107Dic12—64no—
108D24—38no—
109D24—64no—
110#204—68no—
111#204—68no—
112C3⋊C16—68no—
113C22×C32—21no—
114C22×C32—41no—
115C22×C32C2×D811yescharacteristic
116C2×C3⋊D3—42no—
117C2×C3⋊D3—82no—
118C32×C4—21no—
119C32×C4—81no—
120C32×C4—81no—
121C32×C4#12011yescharacteristic
122C32×C4AGL(1,ℤ/8)11yescharacteristic
123C3⋊Dic3—42no—
124C2×#41—181no—
125#137—182no—
126AGL(1,ℤ/8)—182no—
127#135—92no—
128#135—181no—
129C2×Q16C2×C3⋊D311yescharacteristic, complemented
130C2×M16C2×C3⋊D311yescharacteristic
131C22⋊Q8—21no—
132Q32—182no—
133SD32—182no—
134M32—92no—
135#106C2×C3⋊D311yescharacteristic, complemented
136#103—21no—
137C23×C3#21314yes—
138C22×D3—124no—
139C3×Q8—44no—
140C3×Q8—44no—
141C2×C3×C4—24no—
142C2×C3×C4—24no—
143C2×C3×C4—44no—
144C2×C3×C4—44no—
145C2×C3×C4#21314yes—
146C2×C3×C4C2×#6714yes—
147#67—128no—
148#67—128no—
149C2×Dic3—124no—
150D12—68no—
151D12—124no—
152C4×D3—128no—
153C4×D3—128no—
154Dic6—68no—
155Dic6—124no—
156C3×C8—24no—
157C3×C8D3×D418yes—
158C2×C32—21no—
159C2×C32—41no—
160C2×C32#44711yescharacteristic
161C3⋊D3—82no—
162#41—182no—
163#41—182no—
164#41—361no—
165#41—361no—
166C2×Q8—21no—
167C2×Q8—181no—
168C2×D4—181no—
169C2×D4—361no—
170C22×C4—361no—
171C22×C4#63811yescharacteristic
172Q16—22no—
173Q16—92no—
174Q16—181no—
175SD16—182no—
176SD16—182no—
177D8—92no—
178D8—181no—
179M16—22no—
180C2×C8#63811yescharacteristic
181C2×C8C22×C3⋊D311yescharacteristic
182C4⋊C4—21no—
183C4⋊C4—41no—
184C22⋊C4—41no—
185C16—182no—
186C22×C3—24no—
187C22×C3—44no—
188C22×C3C2×#21314yes—
189C2×D3—128no—
190C2×D3—248no—
191C3×C4—24no—
192C3×C4—84no—
193C3×C4—84no—
194C3×C4#93714yes—
195C3×C4#94914yes—
196Dic3—128no—
197C32#219111yescharacteristic, complemented, Sylow
198C23—361no—
199C23#375311yescharacteristic
200Q8—41no—
201Q8—41no—
202Q8—182no—
203Q8—361no—
204D4—182no—
205D4—361no—
206D4—362no—
207D4—362no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—362no—
213C2×C4—361no—
214C2×C4—362no—
215C2×C4#375311yescharacteristic
216C2×C4C2×#63811yescharacteristic
217C8—21no—
218C8D4×C3⋊D312yes—
219C2×C3—24no—
220C2×C3—44no—
221C2×C3#503614yes—
222D3—248no—
223C22—21no—
224C22—41no—
225C22—362no—
226C22—722no—
227C22C2×#375311yescharacteristic
228C4—21no—
229C4—81no—
230C4—81no—
231C4—362no—
232C4#6414611yescharacteristic
233C4#6415811yescharacteristic
234C3#7109114yescomplemented
235C2—21no—
236C2—41no—
237C2—722no—
238C2#1059273511yescharacteristic
subgroupG/H
centerC2#10592735
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C22×C3⋊D3
FittingC32×#470C2
socleC2×C32#447
prime radical#10596219C2
hypercenter#470C3⋊D3
hypocenterC32#2191
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #470

Complements

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

NH ≅ G/Ntype
2. C32×#470C2semidirect
3. #10592878C2semidirect
4. #10589419C2semidirect
5. #10589354C2semidirect
8. C2×C32×Q16C22semidirect
15. C32×#106C22semidirect
19. #63639C22semidirect
21. C3×#470D3semidirect
51. C2×C3×Q16C2×D3semidirect
58. C3×#106C2×D3semidirect
86. #470C3⋊D3semidirect
129. C2×Q16C2×C3⋊D3semidirect
135. #106C2×C3⋊D3semidirect
197. C32#2191semidirect
234. C3#71091semidirect

Maximal subgroups

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

subgroupindex
#105962192
C32×#4702
#105928782
#105894192
#105893542
#710913