Finite Group Databasebeta

← #11788201

Subgroups — #11788201

Order 1152.

1074 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 261 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×#525C211yesmaximal, characteristic
2C2×#64135C211yesmaximal, characteristic, complemented
3C2×#63664C212yesmaximal, complemented
4#10592906C211yesmaximal, characteristic, complemented
5#10589398C212yesmaximal, complemented
6#71712—34nomaximal
7C2×C32×#41C2211yescharacteristic
8C23×C32×C4C2211yescharacteristic
9C32×#123—21no—
10C32×C22⋊Q8C412yescomplemented
11C32×#121C412yescomplemented
12C32×#120—21no—
13C32×C4×D4—22no—
14C32×#117C2211yescharacteristic
15C2×#3757—21no—
16#64135—24no—
17C2×C4⋊(C3⋊Dic3)—21no—
18C2×C4×C3⋊Dic3C2212yes—
19C2×#3748C2212yes—
20#63664—24no—
21#63657—22no—
22C3×#525D314yes—
23C2×#926—34no—
24C2×#837—38no—
25#5207—34no—
26#4532—38no—
27C24×C32D411yescharacteristic
28C32×#41—22no—
29C32×#41—22no—
30C2×C32×Q8C2×C411yescharacteristic
31C2×C32×D4—22no—
32C2×C32×D4—21no—
33C2×C32×D4C2×C411yescharacteristic
34C22×C32×C4—22no—
35C22×C32×C4—22no—
36C22×C32×C4—21no—
37C22×C32×C4C2×C412yes—
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C22×C3⋊Dic3—22no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4C2×C412yes—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—21no—
46C32×C22⋊C4—22no—
47C32×C42—21no—
48#3757—24no—
49C4⋊(C3⋊Dic3)—24no—
50C4×C3⋊Dic3—22no—
51C4×C3⋊Dic3D414yescomplemented
52#3748—22no—
53#3748—22no—
54C2×C3⋊(C3⋊C8)—22no—
55#1897—91noSylow
56C2×C3×#41C2×D314yes—
57C23×C3×C4C2×D314yes—
58C3×#123—24no—
59C3×C22⋊Q8Dic318yescomplemented
60C3×#121Dic318yescomplemented
61C3×#120—24no—
62C3×C4×D4—28no—
63C3×#117C2×D314yes—
64C2×#217—64no—
65#926—616no—
66C2×C4⋊Dic3—64no—
67C2×C4×Dic3—38no—
68C2×#208—38no—
69#837—616no—
70#830—68no—
71C23×C32—21no—
72C23×C32—21no—
73C23×C32—21no—
74C23×C32—42no—
75C23×C32C22⋊C412yes—
76C23×C32C2×D411yescharacteristic
77C32×Q8—22no—
78C32×D4—22no—
79C32×D4—44no—
80C32×D4—41no—
81C32×D4—41no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—42no—
89C2×C32×C4C22⋊C412yes—
90C2×C32×C4C22×C411yescharacteristic
91C2×C32×C4C2×D412yes—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4#4112yes—
94C2×C3⋊Dic3—24no—
95C2×C3⋊Dic3—44no—
96C3⋊(C3⋊C8)—42no—
97#525C3⋊D311yescharacteristic
98C2×#117—91no—
99C2×#104—92no—
100#407—91no—
101#339—92no—
102C24×C3#6714yes—
103C3×#41—28no—
104C3×#41—28no—
105C2×C3×Q8C2×Dic314yes—
106C2×C3×D4—28no—
107C2×C3×D4—24no—
108C2×C3×D4C2×Dic314yes—
109C22×C3×C4—28no—
110C22×C3×C4—28no—
111C22×C3×C4—24no—
112C22×C3×C4C2×Dic318yes—
113C22×C3×C4#6714yes—
114C22×C3×C4#6718yes—
115C22×C3×C4C22×D314yes—
116C22×Dic3—68no—
117C3×C4⋊C4—28no—
118C3×C4⋊C4C2×Dic318yes—
119C3×C22⋊C4—28no—
120C3×C22⋊C4—24no—
121C3×C22⋊C4—28no—
122C3×C42—24no—
123#217—616no—
124C4⋊Dic3—616no—
125C4×Dic3—316no—
126C4×Dic3—68no—
127#208—68no—
128#208—68no—
129C2×C3⋊C8—68no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—42no—
135C22×C32—42no—
136C22×C32—41no—
137C22×C32—42no—
138C22×C32C2×C22⋊C411yescharacteristic
139C22×C32#12011yescharacteristic
140C22×C32#12311yescharacteristic
141C32×C4—22no—
142C32×C4—41no—
143C32×C4—42no—
144C32×C4C4×D412yes—
145C32×C4#12112yes—
146C3⋊Dic3—44no—
147C2×#41C2×C3⋊D311yescharacteristic
148C23×C4C2×C3⋊D311yescharacteristic
149C2×M16—92no—
150#123—21no—
151C22⋊Q8C3⋊Dic312yescomplemented
152#121C3⋊Dic312yescomplemented
153#120—21no—
154C4×D4—22no—
155#117—184no—
156#117C2×C3⋊D311yescharacteristic
157C2×C4⋊C4—181no—
158C2×C22⋊C4—181no—
159C2×C42—92no—
160#104—184no—
161C22⋊C8—182no—
162C23×C3—24no—
163C23×C3—24no—
164C23×C3—24no—
165C23×C3—48no—
166C23×C3#21718yes—
167C23×C3C2×#6714yes—
168C3×Q8—28no—
169C3×D4—28no—
170C3×D4—416no—
171C3×D4—44no—
172C3×D4—44no—
173C2×C3×C4—24no—
174C2×C3×C4—24no—
175C2×C3×C4—216no—
176C2×C3×C4—216no—
177C2×C3×C4—28no—
178C2×C3×C4—24no—
179C2×C3×C4—48no—
180C2×C3×C4#21718yes—
181C2×C3×C4D3×D418yes—
182C2×C3×C4#23718yes—
183C2×C3×C4C22×Dic314yes—
184C2×C3×C4C2×#6714yes—
185C2×Dic3—616no—
186C2×Dic3—1216no—
187C3⋊C8—128no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—42no—
191C2×C32—81no—
192C2×C32#38611yescharacteristic
193C2×C32#42112yes—
194C24#63811yescharacteristic
195#41—22no—
196#41—22no—
197C2×Q8C2×C3⋊Dic311yescharacteristic
198C2×D4—22no—
199C2×D4—21no—
200C2×D4C2×C3⋊Dic311yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—182no—
205C22×C4C2×C3⋊Dic312yes—
206C22×C4#63811yescharacteristic
207C22×C4#63812yes—
208C22×C4C22×C3⋊D311yescharacteristic
209M16—182no—
210M16—182no—
211C2×C8—182no—
212C4⋊C4—22no—
213C4⋊C4—184no—
214C4⋊C4C2×C3⋊Dic312yes—
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—22no—
218C22⋊C4—184no—
219C42—21no—
220C42—94no—
221C42—182no—
222C22×C3—28no—
223C22×C3—28no—
224C22×C3—28no—
225C22×C3—28no—
226C22×C3—48no—
227C22×C3—48no—
228C22×C3—44no—
229C22×C3—48no—
230C22×C3#93514yes—
231C22×C3#93714yes—
232C22×C3C2×#21714yes—
233C3×C4—28no—
234C3×C4—44no—
235C3×C4—48no—
236C3×C4D4×Dic318yes—
237C3×C4#93918yes—
238Dic3—1216no—
239C32#189711yescharacteristic, complemented, Sylow
240C23—21no—
241C23—21no—
242C23—21no—
243C23—42no—
244C23#375712yes—
245C23C2×#63811yescharacteristic
246Q8—22no—
247D4—22no—
248D4—44no—
249D4—41no—
250D4—41no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—184no—
259C2×C4—364no—
260C2×C4#375712yes—
261C2×C4D4×C3⋊D312yes—
262C2×C4#383012yes—
263C2×C4C22×C3⋊Dic311yescharacteristic
264C2×C4C2×#63811yescharacteristic
265C8—362no—
266C2×C3—24no—
267C2×C3—24no—
268C2×C3—48no—
269C2×C3—84no—
270C2×C3#522014yes—
271C2×C3#523618yes—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22#6414411yescharacteristic
281C22#6414611yescharacteristic
282C22C2×#375711yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—364no—
287C4D4×C3⋊Dic312yes—
288C4#6414812yes—
289C3#7171214yescomplemented
290C2—21no—
291C2—21no—
292C2—42no—
293C2—81no—
294C2#1059291911yescharacteristic
295C2#1059293512yes—
subgroupG/H
centerC2×C4C2×#638
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
Fitting, prime radicalC32×#525C2
socleC22×C32C2×C22⋊C4
cosocleC22×C32×C4C23
hypercenter#525C3⋊D3
hypocenterC32#1897
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
2. C2×#64135C2semidirect
3. C2×#63664C2semidirect
4. #10592906C2semidirect
5. #10589398C2semidirect
10. C32×C22⋊Q8C4semidirect
11. C32×#121C4semidirect
51. C4×C3⋊Dic3D4semidirect
59. C3×C22⋊Q8Dic3semidirect
60. C3×#121Dic3semidirect
151. C22⋊Q8C3⋊Dic3semidirect
152. #121C3⋊Dic3semidirect
239. C32#1897semidirect
289. C3#71712semidirect

Maximal subgroups

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

subgroupindex
C32×#5252
C2×#641352
C2×#636642
#105929062
#105893982
#717123