Finite Group Databasebeta

← #10593959

Subgroups — #10593959

Order 576.

446 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64078C211yesmaximal, characteristic, complemented
2C3×#872C211yesmaximal, characteristic
3C3×#870C211yesmaximal, characteristic, complemented
4#63918C211yesmaximal, characteristic, complemented
5#63896C212yesmaximal
6#63893C212yesmaximal
7D3×C4⋊Dic3C211yesmaximal, characteristic
8#63869C211yesmaximal, characteristic, complemented
9#63845C212yesmaximal, complemented
10#63841C211yesmaximal, characteristic
11#63840C212yesmaximal, complemented
12#5689—31nomaximal
13#5519—31nomaximal
14C2×#634C2212yes—
15C2×C3×C4×D3C2211yescharacteristic
16C2×#627C2212yes—
17C2×D3×Dic3C2212yes—
18C32×C42C2211yescharacteristic
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20C4⋊(C3⋊Dic3)C2212yes—
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×(C2×D3)⋊C4C2211yescharacteristic
23C3×(C2×D3)⋊C4C2211yescharacteristic
24C3×C4⋊Dic3C2211yescharacteristic
25C3×C4⋊Dic3C2211yescharacteristic
26C3×Dic3⋊C4C2211yescharacteristic
27C3×Dic3⋊C4C2212yes—
28C3×Dic3⋊C4C2211yescharacteristic
29C3×Dic3⋊C4C2212yes—
30C3×C4×Dic3C2211yescharacteristic
31#3724C2212yes—
32Dic3⋊Dic3C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×#125—31no—
38C3×#117—31no—
39#926—31no—
40C2×C4⋊Dic3—31no—
41#897—32no—
42#896—32no—
43#892—31no—
44D3×C4⋊C4—31no—
45#890—32no—
46#890—31no—
47#889—32no—
48#888—32no—
49#888—32no—
50#888—31no—
51#878—32no—
52#878—32no—
53#872D311yescharacteristic
54#870—31no—
55#870D311yescharacteristic, complemented
56#869—31no—
57#869—31no—
58#869—31no—
59#869—61no—
60C22×C3×D3C2311yescharacteristic
61C2×C32×C4C2311yescharacteristic
62C2×C32×C4C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2312yes—
66#634—42no—
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C3×C4×D3Q814yescomplemented
73#627—42no—
74D3×Dic3—42no—
75#544—91noSylow
76C22×C3×C4—31no—
77C22×Dic3—32no—
78C2×C4×D3—32no—
79C2×C4×D3C2×D311yescharacteristic
80C2×Dic6—32no—
81C2×Dic6—32no—
82C2×Dic6—32no—
83C2×Dic6—32no—
84C2×Dic6—62no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—31no—
87C3×C4⋊C4—32no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—31no—
91C3×C22⋊C4—31no—
92C3×C22⋊C4—31no—
93C3×C42—21no—
94C3×C42—31no—
95C3×C42C2×D311yescharacteristic
96C3×C42C2×D311yescharacteristic
97#217—32no—
98(C2×D3)⋊C4—32no—
99(C2×D3)⋊C4C2×D311yescharacteristic
100(C2×D3)⋊C4C2×D311yescharacteristic
101C4⋊Dic3—31no—
102C4⋊Dic3—31no—
103C4⋊Dic3—32no—
104C4⋊Dic3—31no—
105C4⋊Dic3—32no—
106C4⋊Dic3—31no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3—32no—
110C4⋊Dic3—61no—
111C4⋊Dic3—61no—
112C4⋊Dic3—62no—
113C4⋊Dic3C2×D311yescharacteristic
114C4⋊Dic3C2×D311yescharacteristic
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4C2×D312yes—
121Dic3⋊C4C2×D312yes—
122Dic3⋊C4C2×D311yescharacteristic
123Dic3⋊C4C2×D311yescharacteristic
124C4×Dic3—32no—
125C4×Dic3—32no—
126C4×Dic3C2×D311yescharacteristic
127C22×C32C2411yescharacteristic
128C2×C3×D3—21no—
129C2×C3×D3—21no—
130C2×C3×D3C2×Q812yes—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4C2×Q812yes—
134C3⋊Dic3—22no—
135C3⋊Dic3—22no—
136C3×Dic3—22no—
137C3×Dic3—21no—
138C3×Dic3—22no—
139C3×Dic3—21no—
140C3×Dic3C2×Q812yes—
141C4⋊Q8—91no—
142C4⋊Q8—92no—
143C4⋊Q8—91no—
144#125—31no—
145#125—91no—
146#125—92no—
147C22⋊Q8—92no—
148C22⋊Q8—92no—
149#117—31no—
150#117—91no—
151C2×C4⋊C4—91no—
152C23×C3—31no—
153C22×D3C22×D311yescharacteristic
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—31no—
158C2×C3×C4—34no—
159C2×C3×C4—32no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—32no—
163C2×C3×C4C22×D311yescharacteristic
164C2×C3×C4C22×D311yescharacteristic
165C2×C3×C4C22×D311yescharacteristic
166C2×C3×C4C22×D311yescharacteristic
167C2×C3×C4C22×D311yescharacteristic
168C2×C3×C4C22×D311yescharacteristic
169C2×Dic3—32no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—32no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—122no—
176C2×Dic3C22×D312yes—
177C2×Dic3C22×D311yescharacteristic
178C2×Dic3C22×D311yescharacteristic
179C2×Dic3C22×D312yes—
180C2×Dic3C22×D311yescharacteristic
181C4×D3—122no—
182C4×D3Dic614yescomplemented
183Dic6—122no—
184Dic6—122no—
185Dic6—122no—
186Dic6—122no—
187Dic6—124no—
188C2×C32C22×Q811yescharacteristic
189C2×C32H(5,𝔽2)11yescharacteristic
190C2×C322−1+411yescharacteristic
191C3×D3—22no—
192C2×Q8—92no—
193C2×Q8—92no—
194C22×C4—31no—
195C22×C4—92no—
196C4⋊C4—31no—
197C4⋊C4—31no—
198C4⋊C4—31no—
199C4⋊C4—31no—
200C4⋊C4—32no—
201C4⋊C4—32no—
202C4⋊C4—92no—
203C4⋊C4—92no—
204C4⋊C4—92no—
205C4⋊C4—92no—
206C4⋊C4—91no—
207C4⋊C4—91no—
208C4⋊C4—92no—
209C22⋊C4—31no—
210C22⋊C4—31no—
211C22⋊C4—92no—
212C42—31no—
213C42—92no—
214C42D3211yescharacteristic
215C22×C3—21no—
216C22×C3—32no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3C23×D311yescharacteristic
220C22×C3C23×D311yescharacteristic
221C2×D3—21no—
222C2×D3—21no—
223C2×D3C2×Dic612yes—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—22no—
231C3×C4—32no—
232C3×C4—61no—
233C3×C4—62no—
234C3×C4—61no—
235C3×C4—62no—
236C3×C4C2×Dic612yes—
237C3×C4D3×Q812yes—
238Dic3—21no—
239Dic3—22no—
240Dic3—22no—
241Dic3—21no—
242Dic3—62no—
243Dic3—62no—
244Dic3—62no—
245Dic3—62no—
246Dic3—122no—
247Dic3—122no—
248Dic3C2×Dic612yes—
249C32#54411yescharacteristic, complemented, Sylow
250C23—31no—
251Q8—362no—
252Q8—362no—
253C2×C4—31no—
254C2×C4—34no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—92no—
260C2×C4—92no—
261C2×C4—362no—
262C2×C4C2×D3211yescharacteristic
263C2×C4C2×D3211yescharacteristic
264C2×C4C2×D3211yescharacteristic
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—62no—
269C2×C3C22×Dic611yescharacteristic
270C2×C3C2×D3×Q811yescharacteristic
271C2×C3#100911yescharacteristic
272C2×C3#100911yescharacteristic
273C2×C3#101011yescharacteristic
274C2×C3#101011yescharacteristic
275D3—22no—
276C22—32no—
277C22—61no—
278C22—61no—
279C22C22×D3211yescharacteristic
280C4—21no—
281C4—21no—
282C4—32no—
283C4—61no—
284C4—62no—
285C4—62no—
286C4—61no—
287C4—182no—
288C4—182no—
289C4D3×Dic612yes—
290C3—21no—
291C3#551911yescharacteristic, complemented
292C3#568911yescharacteristic, complemented
293C2—62no—
294C2C2×D3×Dic611yescharacteristic
295C2#6429811yescharacteristic
296C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radicalC22×C3×D3C23
hypercenterC42D32
hypocenterC32#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64078C2semidirect
3. C3×#870C2semidirect
4. #63918C2semidirect
8. #63869C2semidirect
9. #63845C2semidirect
11. #63840C2semidirect
55. #870D3semidirect
72. C3×C4×D3Q8semidirect
182. C4×D3Dic6semidirect
249. C32#544semidirect
291. C3#5519semidirect
292. C3#5689semidirect

Maximal subgroups

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

subgroupindex
#640782
C3×#8722
C3×#8702
#639182
#638962
#638932
D3×C4⋊Dic32
#638692
#638452
#638412
#638402
#56893
#55193