Finite Group Databasebeta

← #11856248

Subgroups — #11856248

Order 1152.

3306 conjugacy classes of subgroups; only the 457 normal subgroups are shown.

normal subgroups only

This group has 3306 conjugacy classes of subgroups — too many to draw. Only its 457 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
1C24×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C32×#120C211yesmaximal, characteristic
3C2×#64144C212yesmaximal, complemented
4#10592946C211yesmaximal, characteristic, complemented
5#10592919C214yesmaximal, complemented
6#10592645C214yesmaximal, complemented
7#10592642C212yesmaximal, complemented
8#82602—34nomaximal
9C25×C32C2211yescharacteristic
10C22×C32×D4C2212yes—
11C22×C32×D4C2211yescharacteristic
12C23×C3⋊Dic3—28no—
13C23×C3⋊Dic3—24no—
14C23×C3⋊Dic3C2214yes—
15C23×C3⋊Dic3C2212yes—
16C32×#120—24no—
17C2×C32×C22⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4C2212yes—
19C2×#3757C2212yes—
20C2×#3757C2214yes—
21C2×#3757C2214yescomplemented
22#64144—28no—
23C2×#3750C2214yescomplemented
24#63658C2218yescomplemented
25C24×Dic3—34no—
26C2×C3×#120D314yes—
27C2×#935—38no—
28#5247—34no—
29#5220—316no—
30#4946—316no—
31#4943—38no—
32C24×C32—24no—
33C24×C32—28no—
34C24×C32D414yes—
35C24×C32C2311yescharacteristic
36C24×C32C2311yescharacteristic
37C24×C32C2312yes—
38C2×C32×D4—24no—
39C2×C32×D4—22no—
40C2×C32×D4—24no—
41C2×C32×D4—28no—
42C22×C32×C4C2311yescharacteristic
43C22×C32×C4C2312yes—
44C22×C3⋊Dic3—232no—
45C22×C3⋊Dic3—28no—
46C22×C3⋊Dic3—24no—
47C22×C3⋊Dic3—24no—
48C22×C3⋊Dic3—216no—
49C22×C3⋊Dic3D418yescomplemented
50C22×C3⋊Dic3C2314yes—
51C22×C3⋊Dic3C2314yes—
52C32×C22⋊C4—22no—
53C32×C22⋊C4—24no—
54#3757—24no—
55#3757—28no—
56#3757—28no—
57#3750—28no—
58#2849—91noSylow
59C25×C3C2×D314yes—
60C22×C3×D4C2×D318yes—
61C22×C3×D4C2×D314yes—
62C23×Dic3—38no—
63C23×Dic3—316no—
64C23×Dic3—632no—
65C23×Dic3—616no—
66C3×#120—216no—
67C2×C3×C22⋊C4C2×D314yes—
68C2×C3×C22⋊C4C2×D318yes—
69C2×#217—38no—
70C2×#217—316no—
71C2×#217—316no—
72#935—632no—
73C2×Dic3⋊C4—316no—
74C3⋊#95—332no—
75C23×C32—232no—
76C23×C32—28no—
77C23×C32—216no—
78C23×C32—24no—
79C23×C32—24no—
80C23×C32—24no—
81C23×C32—22no—
82C23×C32—21no—
83C23×C32—24no—
84C23×C32C2×D414yes—
85C23×C32C2×D412yes—
86C23×C32C2×D414yes—
87C23×C32#4118yes—
88C23×C32C2411yescharacteristic
89C32×D4—44no—
90C32×D4—48no—
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—22no—
94C2×C32×C4—24no—
95C2×C3⋊Dic3—232no—
96C2×C3⋊Dic3—216no—
97C2×C3⋊Dic3—24no—
98C2×C3⋊Dic3—24no—
99C2×C3⋊Dic3—24no—
100C2×C3⋊Dic3—28no—
101C2×C3⋊Dic3C2×D418yes—
102C24×C4—91no—
103C2×#123—92no—
104C2×#120C3⋊D311yescharacteristic
105#386—94no—
106#385—94no—
107#381—92no—
108#379—91no—
109C24×C3—216no—
110C24×C3—232no—
111C24×C3#67116yes—
112C24×C3C22×D314yes—
113C24×C3C22×D314yes—
114C24×C3C22×D318yes—
115C2×C3×D4—216no—
116C2×C3×D4—232no—
117C2×C3×D4—216no—
118C2×C3×D4—28no—
119C22×C3×C4C22×D318yes—
120C22×C3×C4C22×D314yes—
121C22×Dic3—332no—
122C22×Dic3—316no—
123C22×Dic3—316no—
124C22×Dic3—664no—
125C22×Dic3—6128no—
126C22×Dic3—616no—
127C22×Dic3—632no—
128C22×Dic3—616no—
129C3×C22⋊C4—216no—
130C3×C22⋊C4—28no—
131#217—632no—
132#217—616no—
133#217—632no—
134Dic3⋊C4—632no—
135C22×C32—232no—
136C22×C32—24no—
137C22×C32—24no—
138C22×C32—24no—
139C22×C32—28no—
140C22×C32—216no—
141C22×C32—41no—
142C22×C32—42no—
143C22×C32—44no—
144C22×C32#12014yes—
145C22×C32#12318yescomplemented
146C22×C32C22×D412yes—
147C22×C32C22×D411yescharacteristic
148C22×C32C2×#4114yes—
149C32×C4—41no—
150C32×C4—42no—
151C3⋊Dic3—28no—
152C3⋊Dic3—44no—
153C25C2×C3⋊D311yescharacteristic
154C22×D4C2×C3⋊D311yescharacteristic
155C22×D4C2×C3⋊D312yes—
156C23×C4—92no—
157C23×C4—94no—
158C23×C4—188no—
159C23×C4—184no—
160#123—188no—
161#120—24no—
162C2×C4⋊C4—94no—
163C2×C22⋊C4—92no—
164C2×C22⋊C4—94no—
165C2×C22⋊C4—94no—
166C2×C22⋊C4C2×C3⋊D311yescharacteristic
167C2×C22⋊C4C2×C3⋊D312yes—
168#95—98no—
169C23×C3—2128no—
170C23×C3—264no—
171C23×C3—232no—
172C23×C3—216no—
173C23×C3—216no—
174C23×C3—216no—
175C23×C3—28no—
176C23×C3—216no—
177C23×C3—24no—
178C23×C3D3×D4116yes—
179C23×C3#237132yes—
180C23×C3C2×#67116yes—
181C23×C3C2×#6718yes—
182C23×C3C23×D314yes—
183C3×D4—432no—
184C3×D4—416no—
185C2×C3×C4—28no—
186C2×C3×C4—216no—
187C2×C3×C4—28no—
188C2×C3×C4—24no—
189C2×Dic3—332no—
190C2×Dic3—6128no—
191C2×Dic3—664no—
192C2×Dic3—616no—
193C2×Dic3—616no—
194C2×Dic3—616no—
195C2×Dic3—632no—
196C2×C32—24no—
197C2×C32—28no—
198C2×C32—81no—
199C2×C32C2×#12011yescharacteristic
200C2×C32C2×#12312yes—
201C2×C32#52514yes—
202C24—24no—
203C24—28no—
204C24#63814yes—
205C24C22×C3⋊D311yescharacteristic
206C24C22×C3⋊D311yescharacteristic
207C24C22×C3⋊D312yes—
208C2×D4—24no—
209C2×D4—28no—
210C2×D4—24no—
211C2×D4—22no—
212C22×C4—94no—
213C22×C4—98no—
214C22×C4—94no—
215C22×C4—1816no—
216C22×C4—1832no—
217C22×C4—184no—
218C22×C4—188no—
219C22×C4—184no—
220C22×C4C22×C3⋊D312yes—
221C22×C4C22×C3⋊D311yescharacteristic
222C4⋊C4—188no—
223C22⋊C4—24no—
224C22⋊C4—22no—
225C22⋊C4—188no—
226C22⋊C4—184no—
227C22⋊C4—188no—
228C22×C3—2128no—
229C22×C3—216no—
230C22×C3—216no—
231C22×C3—216no—
232C22×C3—232no—
233C22×C3—264no—
234C22×C3—48no—
235C22×C3—44no—
236C22×C3—416no—
237C22×C3#935132yescomplemented
238C22×C3#937116yes—
239C22×C3C2×D3×D418yes—
240C22×C3C2×#237116yes—
241C22×C3C22×#6714yes—
242C3×C4—48no—
243C3×C4—44no—
244Dic3—632no—
245Dic3—1216no—
246C32#284911yescharacteristic, complemented, Sylow
247C23—232no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—216no—
252C23—28no—
253C23—22no—
254C23—24no—
255C23—21no—
256C23D4×C3⋊D314yes—
257C23#383018yes—
258C23C2×#63814yes—
259C23C2×#63812yes—
260C23C23×C3⋊D311yescharacteristic
261D4—48no—
262D4—44no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—98no—
268C2×C4—1832no—
269C2×C4—184no—
270C2×C4—184no—
271C2×C4—1816no—
272C2×C4—184no—
273C2×C4—188no—
274C2×C3—232no—
275C2×C3—216no—
276C2×C3—84no—
277C2×C3#5586116yes—
278C2×C3C2×#93518yes—
279C2×C3C2×#93714yes—
280C22—232no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—28no—
285C22—216no—
286C22—42no—
287C22—41no—
288C22—44no—
289C22#6414418yescomplemented
290C22#6414614yes—
291C22C2×D4×C3⋊D312yes—
292C22C2×#383014yes—
293C22C22×#63811yescharacteristic
294C4—42no—
295C4—41no—
296C4—188no—
297C4—364no—
298C3#8260214yescomplemented
299C2—28no—
300C2—24no—
301C2—81no—
302C2#1059605714yes—
303C2C2×#6414412yes—
304C2C2×#6414611yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutator, socle, cosocleC23×C32C24
Fitting, prime radicalC2×C32×#120C2
hypercenterC2×#120C3⋊D3
hypocenterC32#2849
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊Dic3C2semidirect
3. C2×#64144C2semidirect
4. #10592946C2semidirect
5. #10592919C2semidirect
6. #10592645C2semidirect
7. #10592642C2semidirect
21. C2×#3757C22semidirect
23. C2×#3750C22semidirect
24. #63658C22semidirect
49. C22×C3⋊Dic3D4semidirect
145. C22×C32#123semidirect
237. C22×C3#935semidirect
246. C32#2849semidirect
289. C22#64144semidirect
298. C3#82602semidirect

Maximal subgroups

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

subgroupindex
C24×C3⋊Dic32
C2×C32×#1202
C2×#641442
#105929462
#105929192
#105926452
#105926422
#826023