Finite Group Databasebeta

← #11787278

Subgroups — #11787278

Order 1152.

1080 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic
2C2×#64135C211yesmaximal, characteristic
3C2×#63662C212yesmaximal, complemented
4C2×#63657C211yesmaximal, characteristic
5#10589395C212yesmaximal, complemented
6#71693—34nomaximal
7C22×C32×Q8C2211yescharacteristic
8C23×C32×C4C2211yescharacteristic
9C32×C22⋊Q8—22no—
10C32×C22⋊Q8C414yescomplemented
11C2×C32×C4⋊C4—21no—
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C32×C22⋊C4—21no—
14C2×#3757—21no—
15#64135—24no—
16C2×C4⋊(C3⋊Dic3)C2212yes—
17C2×C4×C3⋊Dic3—21no—
18C22×C3⋊(C3⋊C8)C2212yes—
19#63662—24no—
20#63657—22no—
21C2×C3×C22⋊Q8D314yes—
22C2×#926—34no—
23C2×#835—38no—
24C2×#830—34no—
25#4529—38no—
26C24×C32D411yescharacteristic
27C2×C32×Q8—21no—
28C2×C32×Q8—22no—
29C2×C32×Q8—22no—
30C2×C32×Q8C2×C412yes—
31C22×C32×C4—22no—
32C22×C32×C4—22no—
33C22×C32×C4—21no—
34C22×C32×C4—21no—
35C22×C32×C4C2×C412yes—
36C22×C32×C4D411yescharacteristic
37C22×C32×C4D412yes—
38C22×C32×C4C2311yescharacteristic
39C22×C3⋊Dic3—22no—
40C32×C4⋊C4—24no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4C2×C412yes—
43C32×C22⋊C4—24no—
44#3757—24no—
45C4⋊(C3⋊Dic3)—22no—
46C4⋊(C3⋊Dic3)D414yes—
47C4×C3⋊Dic3—24no—
48C2×C3⋊(C3⋊C8)—22no—
49C2×C3⋊(C3⋊C8)—22no—
50C2×C3⋊(C3⋊C8)—22no—
51#1873—91noSylow
52C22×C3×Q8C2×D314yes—
53C23×C3×C4C2×D314yes—
54C3×C22⋊Q8—28no—
55C3×C22⋊Q8Dic3116yescomplemented
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4C2×D314yes—
58C2×C3×C22⋊C4—24no—
59C2×#217—64no—
60#926—616no—
61C2×C4⋊Dic3—38no—
62C2×C4×Dic3—64no—
63C22×C3⋊C8—38no—
64#835—616no—
65#830—68no—
66C23×C32—21no—
67C23×C32—21no—
68C23×C32—22no—
69C23×C32—22no—
70C23×C32C22⋊C412yes—
71C23×C32C2×D411yescharacteristic
72C32×Q8—24no—
73C32×Q8—42no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—24no—
77C2×C32×C4—24no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C32×C4C22⋊C412yes—
84C2×C32×C4C22×C411yescharacteristic
85C2×C32×C4C2×D412yes—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4#4112yes—
88C2×C3⋊Dic3—24no—
89C2×C3⋊Dic3—42no—
90C2×C3⋊Dic3—42no—
91C3⋊(C3⋊C8)—42no—
92C2×C22⋊Q8C3⋊D311yescharacteristic
93C2×#117—91no—
94C2×#103—92no—
95C2×C22⋊C8—91no—
96#340—92no—
97C24×C3#6714yes—
98C2×C3×Q8—24no—
99C2×C3×Q8—28no—
100C2×C3×Q8—28no—
101C2×C3×Q8C2×Dic318yes—
102C22×C3×C4—28no—
103C22×C3×C4—28no—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4C2×Dic318yes—
107C22×C3×C4#6714yes—
108C22×C3×C4#6718yes—
109C22×C3×C4C22×D314yes—
110C22×Dic3—68no—
111C3×C4⋊C4—216no—
112C3×C4⋊C4—24no—
113C3×C4⋊C4C2×Dic318yes—
114C3×C22⋊C4—216no—
115#217—616no—
116C4⋊Dic3—316no—
117C4⋊Dic3—68no—
118C4×Dic3—616no—
119C2×C3⋊C8—68no—
120C2×C3⋊C8—68no—
121C2×C3⋊C8—68no—
122C22×C32—22no—
123C22×C32—22no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32—42no—
127C22×C32C2×C22⋊C411yescharacteristic
128C22×C32#12011yescharacteristic
129C22×C32#12311yescharacteristic
130C22×C32#13512yes—
131C22×C32#13712yes—
132C32×C4—22no—
133C32×C4—42no—
134C32×C4—42no—
135C32×C4C4×D412yes—
136C32×C4#12112yes—
137C3⋊Dic3—44no—
138C22×Q8C2×C3⋊D311yescharacteristic
139C23×C4C2×C3⋊D311yescharacteristic
140C22×C8—92no—
141C22⋊Q8—22no—
142C22⋊Q8C3⋊Dic314yescomplemented
143#117—184no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—92no—
146C2×C4⋊C4C2×C3⋊D311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—181no—
149C2×C42—181no—
150#103—184no—
151C22⋊C8—182no—
152C23×C3—24no—
153C23×C3—24no—
154C23×C3—28no—
155C23×C3—28no—
156C23×C3#21718yes—
157C23×C3C2×#6714yes—
158C3×Q8—216no—
159C3×Q8—48no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—216no—
163C2×C3×C4—216no—
164C2×C3×C4—28no—
165C2×C3×C4—28no—
166C2×C3×C4—44no—
167C2×C3×C4—44no—
168C2×C3×C4—48no—
169C2×C3×C4#21718yes—
170C2×C3×C4D3×D418yes—
171C2×C3×C4#23718yes—
172C2×C3×C4C22×Dic314yes—
173C2×C3×C4C2×#6714yes—
174C2×Dic3—616no—
175C2×Dic3—128no—
176C2×Dic3—128no—
177C3⋊C8—128no—
178C2×C32—42no—
179C2×C32#38611yescharacteristic
180C2×C32#41611yescharacteristic
181C2×C32#41911yescharacteristic
182C2×C32#45212yes—
183C2×C32#48512yes—
184C24#63811yescharacteristic
185C2×Q8—21no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8C2×C3⋊Dic312yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—182no—
194C22×C4C2×C3⋊Dic312yes—
195C22×C4#63811yescharacteristic
196C22×C4#63812yes—
197C22×C4C22×C3⋊D311yescharacteristic
198C2×C8—182no—
199C2×C8—182no—
200C2×C8—182no—
201C4⋊C4—24no—
202C4⋊C4—21no—
203C4⋊C4—94no—
204C4⋊C4—182no—
205C4⋊C4C2×C3⋊Dic312yes—
206C22⋊C4—24no—
207C22⋊C4—184no—
208C42—184no—
209C22×C3—28no—
210C22×C3—28no—
211C22×C3—28no—
212C22×C3—48no—
213C22×C3—48no—
214C22×C3#93514yes—
215C22×C3#93714yes—
216C22×C3#94218yes—
217C22×C3#95018yes—
218C22×C3C2×#21714yes—
219C3×C4—28no—
220C3×C4—48no—
221C3×C4—48no—
222C3×C4D4×Dic318yes—
223C3×C4#93918yes—
224Dic3—1216no—
225C32#187311yescharacteristic, complemented, Sylow
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23#375712yes—
231C23C2×#63811yescharacteristic
232Q8—24no—
233Q8—42no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—184no—
244C2×C4—362no—
245C2×C4—362no—
246C2×C4#375712yes—
247C2×C4D4×C3⋊D312yes—
248C2×C4#383012yes—
249C2×C4C22×C3⋊Dic311yescharacteristic
250C2×C4C2×#63811yescharacteristic
251C8—362no—
252C2×C3—48no—
253C2×C3#504518yes—
254C2×C3#504718yes—
255C2×C3#522014yes—
256C2×C3#522514yes—
257C2×C3#523314yes—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—42no—
262C22—42no—
263C22#6414411yescharacteristic
264C22#6414611yescharacteristic
265C22#6415112yes—
266C22#6415912yes—
267C22C2×#375711yescharacteristic
268C4—22no—
269C4—42no—
270C4—42no—
271C4—364no—
272C4D4×C3⋊Dic312yes—
273C4#6414812yes—
274C3#7169314yescomplemented
275C2—42no—
276C2#1059274412yes—
277C2#1059274612yes—
278C2#1059291911yescharacteristic
279C2#1059292411yescharacteristic
280C2#1059293211yescharacteristic
subgroupG/H
centerC23C2×#638
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C2×D4
prime radicalC24×C32D4
cosocleC22×C32×C4C23
hypercenterC2×C22⋊Q8C3⋊D3
hypocenterC32#1873
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#63662C2semidirect
5. #10589395C2semidirect
10. C32×C22⋊Q8C4semidirect
55. C3×C22⋊Q8Dic3semidirect
142. C22⋊Q8C3⋊Dic3semidirect
225. C32#1873semidirect
274. C3#71693semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C2×#641352
C2×#636622
C2×#636572
#105893952
#716933