Finite Group Databasebeta

← #11788912

Subgroups — #11788912

Order 1152.

924 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 249 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×#536C211yesmaximal, characteristic
2#10596290C211yesmaximal, characteristic
3#10592936C211yesmaximal, characteristic
4C2×#63663C212yesmaximal, complemented
5#10589399C212yesmaximal, complemented
6#71866—34nomaximal
7C2×C32×#41C2211yescharacteristic
8C22×C32×Q8C2211yescharacteristic
9C32×C4⋊Q8—21no—
10C32×#124—21no—
11C32×#123C414yes—
12C32×C22⋊Q8—22no—
13C32×#117C2211yescharacteristic
14#64157—24no—
15C2×#3748—22no—
16C2×#3748C2212yes—
17C2×#3748C2212yes—
18#63663—21no—
19#63663—24no—
20#63660—21no—
21C3×#536D314yes—
22#5819—34no—
23#5237—34no—
24C2×#836—38no—
25#4533—38no—
26C32×#41—22no—
27C32×#41—22no—
28C2×C32×Q8—21no—
29C2×C32×Q8—21no—
30C2×C32×Q8—22no—
31C2×C32×Q8—22no—
32C2×C32×Q8D411yescharacteristic
33C2×C32×D4C2×C412yes—
34C2×C32×D4D411yescharacteristic
35C22×C32×C4C2×C412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C32×C4⋊C4—24no—
39C32×C4⋊C4—21no—
40C32×C22⋊C4—24no—
41C32×C22⋊C4C2×C412yes—
42C32×C42—21no—
43#3748—24no—
44#3748—22no—
45#3748—22no—
46#3748—22no—
47#3748D414yes—
48C2×C3⋊(C3⋊C8)—24no—
49C2×C3⋊(C3⋊C8)—22no—
50C2×C3⋊(C3⋊C8)—22no—
51#1915—91noSylow
52C2×C3×#41C2×D314yes—
53C22×C3×Q8C2×D314yes—
54C3×C4⋊Q8—24no—
55C3×#124—24no—
56C3×#123Dic3116yes—
57C3×C22⋊Q8—28no—
58C3×#117C2×D314yes—
59#948—616no—
60C2×#208—38no—
61C2×#208—38no—
62C2×#208—68no—
63#836—616no—
64#836—64no—
65#833—64no—
66C23×C32C22⋊C412yes—
67C23×C32C22×C411yescharacteristic
68C32×Q8—22no—
69C32×Q8—41no—
70C32×Q8—42no—
71C32×Q8—41no—
72C32×D4—24no—
73C32×D4—22no—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—24no—
79C2×C32×C4C22⋊C412yes—
80C2×C32×C4C2×D412yes—
81C2×C32×C4C2×D411yescharacteristic
82C2×C32×C4C2×D411yescharacteristic
83C2×C32×C4#4112yes—
84C3⋊(C3⋊C8)—24no—
85C3⋊(C3⋊C8)—42no—
86#568—91no—
87#536C3⋊D311yescharacteristic
88#413—91no—
89C2×#101—92no—
90#344—92no—
91C3×#41—28no—
92C3×#41—28no—
93C2×C3×Q8—28no—
94C2×C3×Q8—28no—
95C2×C3×Q8—24no—
96C2×C3×Q8—24no—
97C2×C3×Q8#6714yes—
98C2×C3×D4C2×Dic318yes—
99C2×C3×D4#6714yes—
100C22×C3×C4C2×Dic318yes—
101C22×C3×C4#6718yes—
102C22×C3×C4C22×D314yes—
103C3×C4⋊C4—216no—
104C3×C4⋊C4—24no—
105C3×C22⋊C4—216no—
106C3×C22⋊C4C2×Dic318yes—
107C3×C42—24no—
108#208—316no—
109#208—616no—
110#208—68no—
111#208—68no—
112#208—68no—
113C2×C3⋊C8—616no—
114C2×C3⋊C8—68no—
115C2×C3⋊C8—68no—
116C22×C32—22no—
117C22×C32—41no—
118C22×C32—42no—
119C22×C32C2×C22⋊C411yescharacteristic
120C22×C32C4×D412yes—
121C32×C4—22no—
122C32×C4—42no—
123C32×C4—42no—
124C32×C4#12011yescharacteristic
125C32×C4#12112yes—
126C32×C4#12311yescharacteristic
127C2×#41C2×C3⋊D311yescharacteristic
128C22×Q8C2×C3⋊D311yescharacteristic
129#131—184no—
130C2×M16—92no—
131C2×M16—92no—
132C2×M16—182no—
133C4⋊Q8—21no—
134#124—21no—
135#123C3⋊Dic314yes—
136C22⋊Q8—22no—
137#117C2×C3⋊D311yescharacteristic
138#101—184no—
139#101—181no—
140#100—181no—
141C23×C3#21718yes—
142C23×C3C22×Dic314yes—
143C3×Q8—28no—
144C3×Q8—44no—
145C3×Q8—44no—
146C3×Q8—48no—
147C3×D4—216no—
148C3×D4—28no—
149C2×C3×C4—28no—
150C2×C3×C4—28no—
151C2×C3×C4—216no—
152C2×C3×C4—28no—
153C2×C3×C4—28no—
154C2×C3×C4#21718yes—
155C2×C3×C4D3×D418yes—
156C2×C3×C4#23718yes—
157C2×C3×C4C2×#6714yes—
158C2×C3×C4C2×#6714yes—
159C3⋊C8—616no—
160C3⋊C8—128no—
161C2×C32—21no—
162C2×C32—22no—
163C2×C32—42no—
164C2×C32#38611yescharacteristic
165#41—22no—
166#41—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8#63811yescharacteristic
172C2×D4C2×C3⋊Dic312yes—
173C2×D4#63811yescharacteristic
174C22×C4C2×C3⋊Dic312yes—
175C22×C4#63812yes—
176C22×C4C22×C3⋊D311yescharacteristic
177M16—94no—
178M16—184no—
179M16—182no—
180M16—182no—
181M16—182no—
182C2×C8—184no—
183C2×C8—182no—
184C2×C8—182no—
185C4⋊C4—24no—
186C4⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4C2×C3⋊Dic312yes—
189C42—21no—
190C22×C3—28no—
191C22×C3—44no—
192C22×C3—48no—
193C22×C3D4×Dic318yes—
194C22×C3C2×#21714yes—
195C3×C4—28no—
196C3×C4—48no—
197C3×C4—48no—
198C3×C4#93514yes—
199C3×C4#93714yes—
200C3×C4#93918yes—
201C32#191511yescharacteristic, complemented, Sylow
202C23#375712yes—
203C23C22×C3⋊Dic311yescharacteristic
204Q8—22no—
205Q8—41no—
206Q8—41no—
207Q8—42no—
208D4—24no—
209D4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4#375712yes—
216C2×C4D4×C3⋊D312yes—
217C2×C4#383012yes—
218C2×C4C2×#63811yescharacteristic
219C2×C4C2×#63811yescharacteristic
220C8—184no—
221C8—362no—
222C2×C3—24no—
223C2×C3—28no—
224C2×C3—48no—
225C2×C3#522014yes—
226C22—22no—
227C22—41no—
228C22—42no—
229C22D4×C3⋊Dic312yes—
230C22C2×#375711yescharacteristic
231C4—22no—
232C4—42no—
233C4—42no—
234C4#6414411yescharacteristic
235C4#6414611yescharacteristic
236C4#6414812yes—
237C3#7186614yescomplemented
238C2—21no—
239C2—22no—
240C2—42no—
241C2#1059291911yescharacteristic
subgroupG/H
centerC2#10592919
commutatorC23×C32C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#536C2
socleC2×C32#386
prime radicalC2×C32×#41C22
cosocleC22×C32×C4C23
hypercenter#536C3⋊D3
hypocenterC32#1915
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
4. C2×#63663C2semidirect
5. #10589399C2semidirect
201. C32#1915semidirect
237. C3#71866semidirect

Maximal subgroups

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

subgroupindex
C32×#5362
#105962902
#105929362
C2×#636632
#105893992
#718663