Finite Group Databasebeta

← #11857086

Subgroups — #11857086

Order 1152.

1441 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64097C211yesmaximal, characteristic
2C2×#63919C211yesmaximal, characteristic
3C2×#63840C214yesmaximal
4C3×#4975C212yesmaximal
5#10591349C212yesmaximal
6#10591244C214yesmaximal
7C4×Dic32C211yesmaximal, characteristic
8#74356—32nomaximal
9C22×#634C2212yes—
10C22×#627C2212yes—
11C2×C32×C4⋊C4C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×#3750C2212yes—
14C2×C4×C3⋊Dic3C2211yescharacteristic
15#64097—24no—
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×Dic3⋊C4C2218yes—
18C2×C3×C4×Dic3C2212yes—
19C2×#3724C2212yes—
20C2×Dic3⋊Dic3C2218yes—
21C2×Dic32C2212yes—
22C2×Dic32C2212yes—
23#63919—24no—
24#63840—216no—
25C2×#888—32no—
26C2×#888—32no—
27C2×#888—38no—
28C2×#888—61no—
29C3×#384—32no—
30#4975—38no—
31#4975—32no—
32#4975D312yes—
33#4935—32no—
34C42×Dic3—32no—
35C22×C32×C4C2312yes—
36C22×C32×C4C2311yescharacteristic
37C22×C3⋊Dic3C2312yes—
38C22×C3⋊Dic3C2311yescharacteristic
39C22×C3⋊Dic3C2311yescharacteristic
40C2×#634—24no—
41C2×#634—28no—
42C22×C3×Dic3C2314yes—
43C22×C3×Dic3C2314yes—
44C2×#627—28no—
45C2×#627—24no—
46C32×C4⋊C4—22no—
47C4⋊(C3⋊Dic3)—22no—
48#3750—24no—
49C4×C3⋊Dic3D414yes—
50C3×C4⋊Dic3—24no—
51C3×Dic3⋊C4—216no—
52C3×C4×Dic3Q818yescomplemented
53#3724—24no—
54Dic3⋊Dic3—216no—
55Dic32D418yes—
56Dic32Q818yescomplemented
57#2868—91noSylow
58C22×Dic6—34no—
59C22×Dic6—34no—
60C22×Dic6—62no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—38no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C4⋊C4C2×D312yes—
65C2×C3×C42—32no—
66C2×C4⋊Dic3—32no—
67C2×C4⋊Dic3—38no—
68C2×C4⋊Dic3—61no—
69C2×C4⋊Dic3C2×D312yes—
70C2×Dic3⋊C4—34no—
71C2×Dic3⋊C4—38no—
72C2×Dic3⋊C4—34no—
73C2×Dic3⋊C4—62no—
74C2×Dic3⋊C4C2×D318yes—
75C2×C4×Dic3—32no—
76C2×C4×Dic3—34no—
77C2×C4×Dic3—34no—
78C2×C4×Dic3—61no—
79C2×C4×Dic3C2×D312yes—
80#888—632no—
81#888—68no—
82#888—68no—
83#888—68no—
84C23×C32C2411yescharacteristic
85C2×C32×C4—22no—
86C2×C32×C4—24no—
87C2×C32×C4C2×D412yes—
88C2×C32×C4C2×Q814yes—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3—24no—
91C2×C3⋊Dic3C2×D414yes—
92C2×C3⋊Dic3C2×D412yes—
93C2×C3⋊Dic3C2×D412yes—
94C2×C3⋊Dic3C2×Q814yes—
95#634—48no—
96C2×C3×Dic3—28no—
97C2×C3×Dic3—24no—
98C2×C3×Dic3C2×D418yes—
99C2×C3×Dic3C2×Q818yes—
100C2×C3×Dic3C2×Q818yes—
101#627—48no—
102C2×C4⋊Q8—94no—
103C2×C4⋊Q8—91no—
104C2×C4⋊Q8—91no—
105#384—32no—
106#384—92no—
107#384—94no—
108C43—91no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—34no—
112C22×C3×C4—34no—
113C22×C3×C4C22×D314yes—
114C22×C3×C4C22×D312yes—
115C22×Dic3—32no—
116C22×Dic3—32no—
117C22×Dic3—34no—
118C22×Dic3—61no—
119C22×Dic3—61no—
120C22×Dic3—62no—
121C22×Dic3C22×D314yes—
122C22×Dic3C22×D314yes—
123C2×Dic6—616no—
124C2×Dic6—68no—
125C2×Dic6—68no—
126C2×Dic6—616no—
127C2×Dic6—68no—
128C2×Dic6—128no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—616no—
132C3×C4⋊C4—64no—
133C3×C42—38no—
134C4⋊Dic3—24no—
135C4⋊Dic3—616no—
136C4⋊Dic3—64no—
137C4⋊Dic3—64no—
138Dic3⋊C4—216no—
139Dic3⋊C4—68no—
140Dic3⋊C4—68no—
141Dic3⋊C4—68no—
142Dic3⋊C4—616no—
143C4×Dic3—38no—
144C4×Dic3—316no—
145C4×Dic3—316no—
146C4×Dic3—64no—
147C4×Dic3Dic618yescomplemented
148C22×C32C22×D412yes—
149C22×C32C22×D411yescharacteristic
150C22×C32C22×Q812yes—
151C22×C32C22×Q812yes—
152C32×C4—42no—
153C32×C4C4⋊Q814yescomplemented
154C3⋊Dic3—42no—
155C3⋊Dic3#12714yes—
156C3⋊Dic3C4⋊Q814yescomplemented
157C3×Dic3—44no—
158C3×Dic3C4⋊Q8116yescomplemented
159C22×Q8—92no—
160C22×Q8—92no—
161C4⋊Q8—184no—
162C4⋊Q8—1816no—
163C4⋊Q8—184no—
164C2×C4⋊C4—38no—
165C2×C4⋊C4—32no—
166C2×C4⋊C4—92no—
167C2×C4⋊C4—98no—
168C2×C4⋊C4—92no—
169C2×C4⋊C4—91no—
170C2×C4⋊C4D3211yescharacteristic
171C2×C42—32no—
172C2×C42—92no—
173C2×C42—92no—
174C2×C42—91no—
175C23×C3—21no—
176C23×C3C23×D312yes—
177C2×C3×C4—24no—
178C2×C3×C4—28no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—38no—
183C2×C3×C4—38no—
184C2×C3×C4—38no—
185C2×C3×C4—44no—
186C2×C3×C4—68no—
187C2×C3×C4—64no—
188C2×C3×C4C2×Dic614yes—
189C2×C3×C4D3×D414yes—
190C2×C3×C4D3×Q814yes—
191C2×Dic3—28no—
192C2×Dic3—24no—
193C2×Dic3—34no—
194C2×Dic3—38no—
195C2×Dic3—34no—
196C2×Dic3—38no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—64no—
200C2×Dic3—64no—
201C2×Dic3—64no—
202C2×Dic3—68no—
203C2×Dic3—64no—
204C2×Dic3—124no—
205C2×Dic3C2×Dic618yes—
206C2×Dic3D3×D418yes—
207C2×Dic3D3×Q818yes—
208Dic6—1216no—
209Dic6—1216no—
210Dic6—1216no—
211C2×C32C2×#12711yescharacteristic
212C2×C32C2×C4⋊Q811yescharacteristic
213C2×C32C2×C4⋊Q814yes—
214C2×C32C2×C4⋊Q811yescharacteristic
215C2×Q8—184no—
216C2×Q8—188no—
217C2×Q8—188no—
218C2×Q8—184no—
219C22×C4—34no—
220C22×C4—34no—
221C22×C4—91no—
222C22×C4—91no—
223C22×C4—92no—
224C22×C4C2×D3212yes—
225C22×C4C2×D3211yescharacteristic
226C4⋊C4—22no—
227C4⋊C4—616no—
228C4⋊C4—64no—
229C4⋊C4—184no—
230C4⋊C4—1816no—
231C4⋊C4—184no—
232C4⋊C4—182no—
233C42—38no—
234C42—98no—
235C42—98no—
236C42—94no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—21no—
241C22×C3C22×Dic612yes—
242C22×C3C2×D3×D414yes—
243C22×C3C2×D3×D412yes—
244C22×C3C2×D3×Q812yes—
245C22×C3C2×D3×Q814yes—
246C3×C4—24no—
247C3×C4—316no—
248C3×C4—44no—
249C3×C4—44no—
250C3×C4—124no—
251C3×C4#88818yescomplemented
252Dic3—38no—
253Dic3—38no—
254Dic3—44no—
255Dic3—64no—
256Dic3—64no—
257Dic3—124no—
258Dic3—124no—
259Dic3#888116yescomplemented
260C32#286811yescharacteristic, complemented, Sylow
261C23C22×D3211yescharacteristic
262Q8—368no—
263Q8—368no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—38no—
267C2×C4—38no—
268C2×C4—38no—
269C2×C4—68no—
270C2×C4—64no—
271C2×C4—94no—
272C2×C4—94no—
273C2×C4—92no—
274C2×C4—92no—
275C2×C4—184no—
276C2×C4—182no—
277C2×C4D3×Dic614yes—
278C2×C4#381112yes—
279C2×C3—21no—
280C2×C3—24no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3C2×#88814yes—
284C2×C3C2×#88812yes—
285C2×C3D3×#12712yes—
286C2×C3D3×C4⋊Q812yes—
287C2×C3D3×C4⋊Q814yes—
288C22C2×D3×Dic612yes—
289C22D32×D412yes—
290C22D32×Q812yes—
291C22C2×#381111yescharacteristic
292C4—316no—
293C4—42no—
294C4—94no—
295C4—94no—
296C4—124no—
297C4—362no—
298C4#6384014yescomplemented
299C3—21no—
300C3#7435612yescomplemented
301C2D3×#88814yes—
302C2C2×#6384011yescharacteristic
303C2#1059530011yescharacteristic
304C2#1059536511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, prime radical, cosocleC23×C32C24
FittingC2×C32×C4⋊C4C22
hypercenterC2×C4⋊C4D32
hypocenterC32#2868
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
52. C3×C4×Dic3Q8semidirect
56. Dic32Q8semidirect
147. C4×Dic3Dic6semidirect
153. C32×C4C4⋊Q8semidirect
156. C3⋊Dic3C4⋊Q8semidirect
158. C3×Dic3C4⋊Q8semidirect
251. C3×C4#888semidirect
259. Dic3#888semidirect
260. C32#2868semidirect
298. C4#63840semidirect
300. C3#74356semidirect

Maximal subgroups

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

subgroupindex
C2×#640972
C2×#639192
C2×#638402
C3×#49752
#105913492
#105912442
C4×Dic322
#743563