Finite Group Databasebeta

← #11788335

Subgroups — #11788335

Order 1152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic
2C2×#64133C211yesmaximal, characteristic
3#10592906C211yesmaximal, characteristic
4C2×#63651C212yesmaximal, complemented
5#10589420C212yesmaximal, complemented
6#72003—34nomaximal
7C23×C32×C4C2211yescharacteristic
8C22×#634C2211yescharacteristic
9C2×C32×M16—21no—
10C2×C32×M16—24no—
11C2×C32×M16C2212yes—
12C22×C32×C8—21no—
13C2×#3757C412yes—
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15#64133—21no—
16#64133—21no—
17#64133—21no—
18#64133—21no—
19C2×#3750C412yes—
20C2×#3748C2212yes—
21#63657—21no—
22#63657—21no—
23#63651—24no—
24C22×C3×M16D314yes—
25C2×#924—34no—
26#5207—34no—
27C2×#824—38no—
28#4554—38no—
29C24×C32C2×C411yescharacteristic
30C22×C32×C4—22no—
31C22×C32×C4—22no—
32C22×C32×C4C2×C411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4D411yescharacteristic
37C22×C32×C4C2311yescharacteristic
38C22×C3⋊Dic3C2×C412yes—
39C22×C3⋊Dic3C2×C412yes—
40C2×#634—21no—
41C2×#634—21no—
42C2×#634—22no—
43C2×#634—22no—
44C32×M16—24no—
45C32×M16—22no—
46C32×M16D414yes—
47C2×C32×C8—24no—
48C2×C32×C8—22no—
49#3757—22no—
50#3757—22no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53#3750—22no—
54#3750—22no—
55#3748—22no—
56#3748—22no—
57C2×C3⋊(C3⋊C8)—22no—
58#1901—91noSylow
59C23×C3×C4C2×D314yes—
60C22×Dic6—34no—
61C2×C3×M16—24no—
62C2×C3×M16—216no—
63C2×C3×M16C2×D318yes—
64C22×C3×C8—24no—
65C2×#217—38no—
66C2×C4⋊Dic3—34no—
67#924—64no—
68#924—64no—
69#924—64no—
70#924—64no—
71C2×Dic3⋊C4—38no—
72C2×#208—38no—
73#830—64no—
74#830—64no—
75#824—616no—
76C23×C32—22no—
77C23×C32—22no—
78C23×C32—41no—
79C23×C32C22⋊C411yescharacteristic
80C23×C32C22⋊C411yescharacteristic
81C23×C32C22×C411yescharacteristic
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4C22⋊C411yescharacteristic
90C2×C32×C4C22⋊C411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D412yes—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4#4112yes—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—24no—
98C2×C3⋊Dic3—42no—
99#634—41no—
100#634—42no—
101#634—41no—
102C32×C8—24no—
103C3⋊(C3⋊C8)—42no—
104C22×M16C3⋊D311yescharacteristic
105C2×C22⋊Q8—91no—
106C2×#101—92no—
107#407—91no—
108#343—92no—
109C24×C3C4×D314yes—
110C22×C3×C4—28no—
111C22×C3×C4—28no—
112C22×C3×C4C4×D314yes—
113C22×C3×C4D1214yes—
114C22×C3×C4D1214yes—
115C22×C3×C4#6714yes—
116C22×C3×C4#6714yes—
117C22×C3×C4C22×D314yes—
118C22×Dic3—38no—
119C22×Dic3—38no—
120C2×Dic6—64no—
121C2×Dic6—68no—
122C2×Dic6—68no—
123C2×Dic6—64no—
124C3×M16—216no—
125C3×M16—28no—
126C3×M16#67116yes—
127C2×C3×C8—216no—
128C2×C3×C8—28no—
129#217—68no—
130#217—68no—
131C4⋊Dic3—64no—
132C4⋊Dic3—64no—
133Dic3⋊C4—68no—
134Dic3⋊C4—68no—
135#208—68no—
136#208—68no—
137C2×C3⋊C8—68no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—44no—
146C22×C32#10112yescomplemented
147C22×C32C2×C22⋊C411yescharacteristic
148C22×C32C4×D412yes—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4#12011yescharacteristic
152C32×C4#12112yes—
153C32×C4#12311yescharacteristic
154C3⋊Dic3—42no—
155C3⋊Dic3—42no—
156C22×Q8—91no—
157C23×C4C2×C3⋊D311yescharacteristic
158C2×M16—21no—
159C2×M16—24no—
160C2×M16—92no—
161C2×M16C2×C3⋊D312yes—
162C22×C8—21no—
163C22⋊Q8—181no—
164C22⋊Q8—181no—
165C22⋊Q8—181no—
166C22⋊Q8—181no—
167C2×C4⋊C4—92no—
168C2×C4⋊C4—91no—
169C2×C22⋊C4—92no—
170#101—184no—
171C22⋊C8—181no—
172C22⋊C8—181no—
173C23×C3—28no—
174C23×C3—28no—
175C23×C3—44no—
176C23×C3(C2×D3)⋊C414yes—
177C23×C3(C2×D3)⋊C414yes—
178C23×C3C2×C4×D314yes—
179C2×C3×C4—28no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—28no—
183C2×C3×C4—28no—
184C2×C3×C4—24no—
185C2×C3×C4—24no—
186C2×C3×C4(C2×D3)⋊C414yes—
187C2×C3×C4(C2×D3)⋊C414yes—
188C2×C3×C4C2×D1214yes—
189C2×C3×C4#23518yes—
190C2×C3×C4C2×#6714yes—
191C2×C3×C4C2×#6718yes—
192C2×Dic3—68no—
193C2×Dic3—68no—
194C2×Dic3—616no—
195C2×Dic3—128no—
196Dic6—124no—
197Dic6—128no—
198Dic6—124no—
199C3×C8—216no—
200C3⋊C8—128no—
201C2×C32—22no—
202C2×C32—22no—
203C2×C32—41no—
204C2×C32#38611yescharacteristic
205C2×C32C2×#10111yescharacteristic
206C2×C32#41311yescharacteristic
207C24C4×C3⋊D311yescharacteristic
208C2×Q8—181no—
209C2×Q8—182no—
210C2×Q8—182no—
211C2×Q8—181no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—92no—
215C22×C4—92no—
216C22×C4C4×C3⋊D311yescharacteristic
217C22×C4C4⋊(C3⋊D3)11yescharacteristic
218C22×C4C4⋊(C3⋊D3)11yescharacteristic
219C22×C4#63811yescharacteristic
220C22×C4#63811yescharacteristic
221C22×C4C22×C3⋊D311yescharacteristic
222M16—24no—
223M16—22no—
224M16—182no—
225M16—182no—
226M16#63814yes—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—182no—
230C4⋊C4—182no—
231C4⋊C4—182no—
232C4⋊C4—181no—
233C4⋊C4—181no—
234C22⋊C4—182no—
235C22⋊C4—182no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—24no—
241C22×C3—24no—
242C22×C3—24no—
243C22×C3—416no—
244C22×C3#82418yescomplemented
245C22×C3C2×(C2×D3)⋊C414yes—
246C22×C3C4×#6718yes—
247C3×C4—24no—
248C3×C4—24no—
249C3×C4#92914yes—
250C3×C4#93018yes—
251C3×C4#95314yes—
252Dic3—128no—
253Dic3—128no—
254C32#190111yescharacteristic, complemented, Sylow
255C23—22no—
256C23—22no—
257C23—41no—
258C23#375211yescharacteristic
259C23#375211yescharacteristic
260C23C2×C4×C3⋊D311yescharacteristic
261Q8—361no—
262Q8—362no—
263Q8—361no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4—184no—
274C2×C4—362no—
275C2×C4#375211yescharacteristic
276C2×C4#375211yescharacteristic
277C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
278C2×C4#382812yes—
279C2×C4C2×#63811yescharacteristic
280C2×C4C2×#63812yes—
281C8—24no—
282C8—362no—
283C2×C3—28no—
284C2×C3—28no—
285C2×C3—44no—
286C2×C3#513214yes—
287C2×C3C2×#82414yes—
288C2×C3#521314yes—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22#6365112yescomplemented
298C22C2×#375211yescharacteristic
299C22C4×#63812yes—
300C4—21no—
301C4—21no—
302C4—362no—
303C4—362no—
304C4#6413811yescharacteristic
305C4#6413912yes—
306C4#6416211yescharacteristic
307C3#7200314yescomplemented
308C2—22no—
309C2—22no—
310C2—41no—
311C2#1059283111yescharacteristic
312C2C2×#6365111yescharacteristic
313C2#1059291211yescharacteristic
subgroupG/H
centerC22C2×#3752
commutatorC23×C32C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC22×C32×M16C2
socleC22×C32C2×C22⋊C4
prime radicalC24×C32C2×C4
cosocleC22×C32×C4C23
hypercenterC22×M16C3⋊D3
hypocenterC32#1901
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
4. C2×#63651C2semidirect
5. #10589420C2semidirect
146. C22×C32#101semidirect
244. C22×C3#824semidirect
254. C32#1901semidirect
297. C22#63651semidirect
307. C3#72003semidirect

Maximal subgroups

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

subgroupindex
C22×C32×M162
C2×#641332
#105929062
C2×#636512
#105894202
#720033