Finite Group Databasebeta

← #11801969

Subgroups — #11801969

Order 1152.

3672 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 3672 conjugacy classes of subgroups — too many to draw. Only its 509 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×#64162C211yesmaximal, characteristic
3C2×#64090C213yesmaximal, complemented
4C32×#379C211yesmaximal, characteristic, complemented
5#10592657C213yesmaximal, complemented
6#10592645C213yesmaximal, complemented
7C22⋊C4×C3⋊Dic3C213yesmaximal, complemented
8#73818—34nomaximal
9C25×C32C2211yescharacteristic
10C22×#638C2213yes—
11C23×C3⋊Dic3—212no—
12C23×C3⋊Dic3C2213yes—
13C23×C3⋊Dic3C2213yes—
14C22×C4×C3⋊D3C2211yescharacteristic, complemented
15C2×C32×C22⋊C4C2213yes—
16C2×C32×C22⋊C4C2213yes—
17#64162C418yescomplemented
18C2×#3757C2213yes—
19C2×#3752C2213yescomplemented
20C2×#3750C2213yescomplemented
21C2×C4×C3⋊Dic3C2213yescomplemented
22#64090—212no—
23#63658C2216yescomplemented
24C24×Dic3—34no—
25C2×#953—34no—
26C2×#67⋊C4—312no—
27C3×#379D314yescomplemented
28#4958—312no—
29#4946—312no—
30Dic3×C22⋊C4—312no—
31C24×C32—26no—
32C24×C32—26no—
33C24×C32C2×C414yescomplemented
34C24×C32C2313yes—
35C23×C3⋊D3C2311yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C22×C32×C4C2313yes—
38C2×#638—26no—
39C2×#638—26no—
40C2×#638C2×C4112yescomplemented
41C22×C3⋊Dic3—232no—
42C22×C3⋊Dic3—26no—
43C22×C3⋊Dic3—212no—
44C22×C3⋊Dic3—212no—
45C22×C3⋊Dic3D4112yescomplemented
46C22×C3⋊Dic3C2313yes—
47C22×C3⋊Dic3C2313yes—
48C22×C3⋊Dic3C2311yescharacteristic
49C2×C4×C3⋊D3—26no—
50C32×C22⋊C4—26no—
51C32×C22⋊C4—26no—
52#3757C2×C4112yescomplemented
53#3752—26no—
54#3750—26no—
55C4×C3⋊Dic3—26no—
56C4×#120—91noSylow
57C25×C3C2×D314yes—
58C22×#67—312no—
59C23×Dic3—312no—
60C23×Dic3—312no—
61C23×Dic3—648no—
62C22×C4×D3—34no—
63C2×C3×C22⋊C4C2×D3112yes—
64C2×C3×C22⋊C4C2×D3112yes—
65#953—332no—
66C2×#217—312no—
67C2×(C2×D3)⋊C4—312no—
68C2×Dic3⋊C4—312no—
69C2×C4×Dic3—312no—
70#67⋊C4—648no—
71C3⋊#95—324no—
72C23×C32—224no—
73C23×C32—28no—
74C23×C32—224no—
75C23×C32—26no—
76C23×C32—26no—
77C23×C32C22×C416yes—
78C23×C32C2×D416yes—
79C23×C32#4116yes—
80C23×C32C2411yescharacteristic
81C22×C3⋊D3—23no—
82C22×C3⋊D3—23no—
83C22×C3⋊D3C22×C412yescomplemented
84C2×C32×C4—23no—
85C2×C32×C4—23no—
86C2×C32×C4—26no—
87#638—224no—
88C2×C3⋊Dic3—224no—
89C2×C3⋊Dic3—23no—
90C2×C3⋊Dic3—26no—
91C2×C3⋊Dic3—23no—
92C2×C3⋊Dic3—224no—
93C2×C3⋊Dic3C22×C416yescomplemented
94C2×C3⋊Dic3C2×D416yes—
95C2×C3⋊Dic3C2×D416yes—
96C4×C3⋊D3—44no—
97C24×C4—91no—
98C2×#120—91no—
99C2×C4×D4—93no—
100#386—93no—
101#385—93no—
102#379C3⋊D311yescharacteristic, complemented
103C4×C22⋊C4—93no—
104C24×C3—224no—
105C24×C3—224no—
106C24×C3C4×D3116yescomplemented
107C24×C3C22×D3112yes—
108C23×D3—34no—
109C22×C3×C4C22×D3112yes—
110C22×C3×C4C22×D314yes—
111C2×#67—348no—
112C2×#67—624no—
113C2×#67—624no—
114C22×Dic3—34no—
115C22×Dic3—348no—
116C22×Dic3—312no—
117C22×Dic3—312no—
118C22×Dic3—648no—
119C22×Dic3—648no—
120C22×Dic3—6128no—
121C22×Dic3—624no—
122C2×C4×D3—624no—
123C3×C22⋊C4—224no—
124C3×C22⋊C4—224no—
125#217—348no—
126(C2×D3)⋊C4—624no—
127Dic3⋊C4—624no—
128C4×Dic3—624no—
129C22×C32—232no—
130C22×C32—212no—
131C22×C32—26no—
132C22×C32—26no—
133C22×C32—212no—
134C22×C32C4×D4112yescomplemented
135C22×C32C23×C411yescharacteristic
136C22×C32C22×D413yes—
137C22×C32C2×#4113yes—
138C2×C3⋊D3—26no—
139C2×C3⋊D3—41no—
140C2×C3⋊D3—43no—
141C32×C4—41no—
142C32×C4—43no—
143C3⋊Dic3—26no—
144C3⋊Dic3—26no—
145C3⋊Dic3#12014yescomplemented
146C25C2×C3⋊D311yescharacteristic
147C22×D4—93no—
148C23×C4—93no—
149C23×C4—93no—
150C23×C4—91no—
151C23×C4—1812no—
152#120—98no—
153C4×D4—1812no—
154C2×C4⋊C4—93no—
155C2×C22⋊C4—93no—
156C2×C22⋊C4—93no—
157C2×C22⋊C4C2×C3⋊D313yes—
158C2×C22⋊C4C2×C3⋊D313yes—
159C2×C42—93no—
160#95—96no—
161C23×C3—232no—
162C23×C3—296no—
163C23×C3—296no—
164C23×C3—224no—
165C23×C3—224no—
166C23×C3C2×C4×D3124yes—
167C23×C3D3×D4124yes—
168C23×C3#237124yes—
169C23×C3C23×D314yes—
170C22×D3—38no—
171C22×D3—612no—
172C22×D3—612no—
173C2×C3×C4—212no—
174C2×C3×C4—212no—
175C2×C3×C4—224no—
176#67—696no—
177C2×Dic3—324no—
178C2×Dic3—324no—
179C2×Dic3—324no—
180C2×Dic3—696no—
181C2×Dic3—696no—
182C2×Dic3—612no—
183C2×Dic3—624no—
184C2×Dic3—612no—
185C4×D3—1216no—
186C2×C32—212no—
187C2×C32C2×C4×D413yes—
188C2×C32C2×#12011yescharacteristic
189C2×C32#52513yes—
190C3⋊D3—42no—
191C24—26no—
192C24—26no—
193C24—91no—
194C24C4×C3⋊D314yescomplemented
195C24C22×C3⋊D313yes—
196C2×D4—912no—
197C2×D4—186no—
198C2×D4—186no—
199C22×C4—91no—
200C22×C4—93no—
201C22×C4—93no—
202C22×C4—912no—
203C22×C4—186no—
204C22×C4—1812no—
205C22×C4—1812no—
206C22×C4—1832no—
207C22×C4—186no—
208C22×C4C22×C3⋊D313yes—
209C22×C4C22×C3⋊D311yescharacteristic
210C4⋊C4—186no—
211C22⋊C4—26no—
212C22⋊C4—26no—
213C22⋊C4—912no—
214C22⋊C4—186no—
215C42—186no—
216C22×C3—2128no—
217C22×C3—248no—
218C22×C3—224no—
219C22×C3—248no—
220C22×C3—224no—
221C22×C3#67⋊C4148yescomplemented
222C22×C3C22×C4×D314yes—
223C22×C3C2×D3×D4112yes—
224C22×C3C2×#237112yes—
225C2×D3—624no—
226C2×D3—1212no—
227C2×D3—124no—
228C3×C4—412no—
229C3×C4—44no—
230Dic3—316no—
231Dic3—624no—
232Dic3—624no—
233C32C4×#12011yescharacteristic, complemented, Sylow
234C23—28no—
235C23—224no—
236C23—224no—
237C23—26no—
238C23—26no—
239C23—92no—
240C23—183no—
241C23—183no—
242C23C2×C4×C3⋊D316yes—
243C23D4×C3⋊D316yes—
244C23#383016yes—
245C23C23×C3⋊D311yescharacteristic
246D4—1824no—
247C2×C4—23no—
248C2×C4—23no—
249C2×C4—26no—
250C2×C4—96no—
251C2×C4—96no—
252C2×C4—96no—
253C2×C4—1824no—
254C2×C4—1824no—
255C2×C4—183no—
256C2×C4—186no—
257C2×C4—183no—
258C2×C4—364no—
259C2×C3—248no—
260C2×C3C2×#67⋊C4112yes—
261C2×C3#5586112yes—
262C2×C3D3×#12014yes—
263D3—128no—
264C22—232no—
265C22—212no—
266C22—26no—
267C22—212no—
268C22—26no—
269C22—186no—
270C22—363no—
271C22—361no—
272C22#64090112yescomplemented
273C22C22×C4×C3⋊D311yescharacteristic
274C22C2×D4×C3⋊D313yes—
275C22C2×#383013yes—
276C4—43no—
277C4—41no—
278C4—94no—
279C4—186no—
280C4—186no—
281C3#7381814yescomplemented
282C2—212no—
283C2—362no—
284C2C2×#6409013yes—
285C2#1059605713yes—
286C2C3⋊D3×#12011yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×#379C2
socle, cosocleC23×C32C24
prime radicalC2×#64162C2
hypercenter#379C3⋊D3
hypocenterC32C4×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊Dic3C2semidirect
3. C2×#64090C2semidirect
4. C32×#379C2semidirect
5. #10592657C2semidirect
6. #10592645C2semidirect
7. C22⋊C4×C3⋊Dic3C2semidirect
14. C22×C4×C3⋊D3C22semidirect
17. #64162C4semidirect
19. C2×#3752C22semidirect
20. C2×#3750C22semidirect
21. C2×C4×C3⋊Dic3C22semidirect
23. #63658C22semidirect
27. C3×#379D3semidirect
33. C24×C32C2×C4semidirect
40. C2×#638C2×C4semidirect
45. C22×C3⋊Dic3D4semidirect
52. #3757C2×C4semidirect
83. C22×C3⋊D3C22×C4semidirect
93. C2×C3⋊Dic3C22×C4semidirect
102. #379C3⋊D3semidirect
106. C24×C3C4×D3semidirect
134. C22×C32C4×D4semidirect
145. C3⋊Dic3#120semidirect
194. C24C4×C3⋊D3semidirect
221. C22×C3#67⋊C4semidirect
233. C32C4×#120semidirect
272. C22#64090semidirect
281. C3#73818semidirect

Maximal subgroups

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

subgroupindex
C24×C3⋊Dic32
C2×#641622
C2×#640902
C32×#3792
#105926572
#105926452
C22⋊C4×C3⋊Dic32
#738183