Finite Group Databasebeta

← #11805222

Subgroups — #11805222

Order 1152.

1644 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4⋊C4C211yesmaximal, characteristic
2#10592911C211yesmaximal, characteristic
3#10592911C212yesmaximal
4C4×#3757C211yesmaximal, characteristic
5C4×#3757C212yesmaximal
6#10592683C212yesmaximal, complemented
7#10592679C214yesmaximal, complemented
8C4⋊C4×C3⋊Dic3C212yesmaximal, complemented
9#82160—34nomaximal
10C23×C32×C4C2211yescharacteristic
11C23×C32×C4C2212yes—
12C2×C32×C4⋊C4C418yescomplemented
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×#3757C2211yescharacteristic
17C2×#3757C2212yes—
18C2×#3757C2211yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2212yes—
20C2×C4⋊(C3⋊Dic3)C2214yes—
21C2×C4×C3⋊Dic3C2212yes—
22C2×C4×C3⋊Dic3C2214yes—
23#63658C2212yes—
24#63658C2214yes—
25#63658C2214yes—
26#63658C2212yes—
27C22×C3×C4⋊C4D314yes—
28#5212—38no—
29#5212—34no—
30C4×#217—34no—
31C4×#217—38no—
32#4984—38no—
33#4980—316no—
34Dic3×C4⋊C4—38no—
35C24×C32C2311yescharacteristic
36C22×C32×C4—24no—
37C22×C32×C4—44no—
38C22×C32×C4C2×C414yes—
39C22×C32×C4C2×C418yes—
40C22×C32×C4C2312yes—
41C22×C32×C4C2312yes—
42C22×C32×C4C2311yescharacteristic
43C22×C32×C4C2311yescharacteristic
44C22×C3⋊Dic3C2312yes—
45C22×C3⋊Dic3C2314yes—
46C22×C3⋊Dic3C2312yes—
47C32×C4⋊C4C2×C418yes—
48C32×C4⋊C4C2×C414yes—
49C32×C4⋊C4C2×C414yes—
50#3757—22no—
51#3757—24no—
52#3757Q814yes—
53C4⋊(C3⋊Dic3)—24no—
54C4⋊(C3⋊Dic3)—44no—
55C4×C3⋊Dic3—24no—
56C4×C3⋊Dic3—44no—
57#2345—91noSylow
58C23×C3×C4C2×D314yes—
59C23×C3×C4C2×D318yes—
60C2×C3×C4⋊C4Dic3132yescomplemented
61C2×C3×C4⋊C4C2×D314yes—
62C2×C3×C4⋊C4C2×D318yes—
63C2×C3×C4⋊C4C2×D314yes—
64C2×#217—34no—
65C2×#217—34no—
66C2×#217—38no—
67C2×C4⋊Dic3—38no—
68C2×C4⋊Dic3—316no—
69C2×C4×Dic3—38no—
70C2×C4×Dic3—316no—
71C3⋊#95—38no—
72C3⋊#95—316no—
73C3⋊#95—316no—
74C3⋊#95—38no—
75C23×C32—22no—
76C23×C32—22no—
77C23×C32C22×C412yes—
78C23×C32C2×Q812yes—
79C23×C32#4112yes—
80C23×C32C2411yescharacteristic
81C2×C32×C4—22no—
82C2×C32×C4—24no—
83C2×C32×C4—24no—
84C2×C32×C4—28no—
85C2×C32×C4—48no—
86C2×C32×C4C22×C412yes—
87C2×C32×C4C22×C414yes—
88C2×C32×C4C22×C414yes—
89C2×C32×C4C22×C412yes—
90C2×C32×C4#4114yes—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—24no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—24no—
97C2×C3⋊Dic3—24no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3C2×Q814yes—
100C22×C4⋊C4C3⋊D311yescharacteristic
101#389—92no—
102#387—94no—
103#380—92no—
104#380—91no—
105C4×C4⋊C4—92no—
106C4×C22⋊C4—91no—
107C4×C22⋊C4—92no—
108C24×C3C22×D314yes—
109C22×C3×C4—216no—
110C22×C3×C4—416no—
111C22×C3×C4C2×Dic3116yes—
112C22×C3×C4C2×Dic3132yes—
113C22×C3×C4C22×D314yes—
114C22×C3×C4C22×D318yes—
115C22×C3×C4C22×D318yes—
116C22×C3×C4C22×D314yes—
117C22×Dic3—38no—
118C22×Dic3—38no—
119C22×Dic3—316no—
120C3×C4⋊C4C2×Dic3132yes—
121C3×C4⋊C4C2×Dic3116yes—
122C3×C4⋊C4C2×Dic3116yes—
123#217—316no—
124#217—68no—
125#217—616no—
126C4⋊Dic3—616no—
127C4⋊Dic3—1216no—
128C4×Dic3—616no—
129C4×Dic3—1216no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—24no—
133C22×C32—24no—
134C22×C32C4×Q814yes—
135C22×C32C23×C411yescharacteristic
136C22×C32C22×Q811yescharacteristic
137C22×C32C2×#4112yes—
138C22×C32C2×#4111yescharacteristic
139C22×C32H(5,𝔽2)12yes—
140C32×C4—22no—
141C32×C4—24no—
142C32×C4—24no—
143C32×C4#11714yes—
144C3⋊Dic3—24no—
145C3⋊Dic3—42no—
146C3⋊Dic3—44no—
147C23×C4C2×C3⋊D311yescharacteristic
148C23×C4C2×C3⋊D312yes—
149C2×C4⋊C4—92no—
150C2×C4⋊C4—94no—
151C2×C4⋊C4C3⋊Dic318yescomplemented
152C2×C4⋊C4C2×C3⋊D311yescharacteristic
153C2×C4⋊C4C2×C3⋊D312yes—
154C2×C4⋊C4C2×C3⋊D311yescharacteristic
155C2×C22⋊C4—91no—
156C2×C22⋊C4—91no—
157C2×C22⋊C4—92no—
158C2×C42—92no—
159C2×C42—94no—
160#95—92no—
161#95—94no—
162#95—94no—
163#95—92no—
164C23×C3—28no—
165C23×C3—28no—
166C23×C3D3×Q818yes—
167C23×C3#23918yes—
168C23×C3C22×Dic318yes—
169C23×C3C23×D314yes—
170C2×C3×C4—28no—
171C2×C3×C4—216no—
172C2×C3×C4—216no—
173C2×C3×C4—232no—
174C2×C3×C4—432no—
175C2×C3×C4#235116yes—
176C2×C3×C4C22×Dic318yes—
177C2×C3×C4C22×Dic3116yes—
178C2×C3×C4C22×Dic3116yes—
179C2×C3×C4C22×Dic318yes—
180C2×Dic3—316no—
181C2×Dic3—68no—
182C2×Dic3—68no—
183C2×Dic3—68no—
184C2×Dic3—616no—
185C2×Dic3—616no—
186C2×Dic3—68no—
187C2×Dic3—68no—
188C2×Dic3—616no—
189C2×C32—24no—
190C2×C32C2×#11711yescharacteristic
191C2×C32C2×C4×Q811yescharacteristic
192C2×C32#51811yescharacteristic
193C2×C32#55312yes—
194C2×C32#55412yes—
195C24C22×C3⋊D311yescharacteristic
196C22×C4—24no—
197C22×C4—44no—
198C22×C4—92no—
199C22×C4—92no—
200C22×C4—94no—
201C22×C4C2×C3⋊Dic314yes—
202C22×C4C2×C3⋊Dic318yes—
203C22×C4C22×C3⋊D311yescharacteristic
204C22×C4C22×C3⋊D312yes—
205C22×C4C22×C3⋊D312yes—
206C22×C4C22×C3⋊D311yescharacteristic
207C4⋊C4—184no—
208C4⋊C4—364no—
209C4⋊C4C2×C3⋊Dic318yes—
210C4⋊C4C2×C3⋊Dic314yes—
211C4⋊C4C2×C3⋊Dic314yes—
212C22⋊C4—94no—
213C22⋊C4—182no—
214C22⋊C4—184no—
215C42—184no—
216C42—364no—
217C22×C3—216no—
218C22×C3—216no—
219C22×C3—28no—
220C22×C3—28no—
221C22×C3Q8×Dic3116yes—
222C22×C3C2×#23518yes—
223C22×C3#100418yes—
224C22×C3C2×D3×Q814yes—
225C22×C3C2×#23914yes—
226C22×C3C23×Dic314yes—
227C3×C4—28no—
228C3×C4—216no—
229C3×C4—216no—
230C3×C4#926116yes—
231Dic3—616no—
232Dic3—128no—
233Dic3—1216no—
234C32#234511yescharacteristic, complemented, Sylow
235C23—22no—
236C23—22no—
237C23Q8×C3⋊D312yes—
238C23#383212yes—
239C23C22×C3⋊Dic312yes—
240C23C23×C3⋊D311yescharacteristic
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—28no—
245C2×C4—48no—
246C2×C4—94no—
247C2×C4—182no—
248C2×C4—182no—
249C2×C4—182no—
250C2×C4—184no—
251C2×C4—184no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—184no—
255C2×C4#382814yes—
256C2×C4C22×C3⋊Dic312yes—
257C2×C4C22×C3⋊Dic314yes—
258C2×C4C22×C3⋊Dic314yes—
259C2×C4C22×C3⋊Dic312yes—
260C2×C3—216no—
261C2×C3#551118yes—
262C2×C3#551418yes—
263C2×C3C2×#92614yes—
264C2×C3#579814yes—
265C2×C3C2×Q8×Dic314yes—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22Q8×C3⋊Dic314yes—
271C22C2×#382812yes—
272C22#6436112yes—
273C22C2×Q8×C3⋊D311yescharacteristic
274C22C2×#383211yescharacteristic
275C22C23×C3⋊Dic311yescharacteristic
276C4—22no—
277C4—24no—
278C4—24no—
279C4—184no—
280C4—362no—
281C4—364no—
282C4#6413514yes—
283C3#8216014yescomplemented
284C2—24no—
285C2#1059598212yes—
286C2#1059598512yes—
287C2C2×#6413511yescharacteristic
288C2#1059626911yescharacteristic
289C2C2×Q8×C3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC22×C32×C4⋊C4C2
socle, cosocleC23×C32C24
prime radicalC24×C32C23
hypercenterC22×C4⋊C4C3⋊D3
hypocenterC32#2345
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10592683C2semidirect
7. #10592679C2semidirect
8. C4⋊C4×C3⋊Dic3C2semidirect
12. C2×C32×C4⋊C4C4semidirect
60. C2×C3×C4⋊C4Dic3semidirect
151. C2×C4⋊C4C3⋊Dic3semidirect
234. C32#2345semidirect
283. C3#82160semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C4⋊C42
#105929112
C4×#37572
#105926832
#105926792
C4⋊C4×C3⋊Dic32
#821603