Finite Group Databasebeta

← #10594532

Subgroups — #10594532

Order 576.

490 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64097C211yesmaximal, characteristic, complemented
2C3×#897C211yesmaximal, characteristic, complemented
3C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C4×#625C211yesmaximal, characteristic
5#63896C212yesmaximal, complemented
6D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#63879C212yesmaximal
8#63857C212yesmaximal, complemented
9#63856C211yesmaximal, characteristic, complemented
10#63847C212yesmaximal, complemented
11#63843C211yesmaximal, characteristic, complemented
12#5635—31nomaximal
13#5511—31nomaximal
14C2×#634C2212yes—
15C2×C3×C4×D3C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×C4×D3C2212yes—
18C2×C3×Dic6C2211yescharacteristic
19C2×#625C2211yescharacteristic
20C2×D3×Dic3C2212yes—
21C32×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23#3750C2212yes—
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2212yes—
28C3×Dic3⋊C4C2212yes—
29#3724C2211yescharacteristic
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2211yescharacteristic
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C2×C3×C4⋊C4—31no—
39#946—31no—
40Q8×Dic3—31no—
41C4×#67—31no—
42C4×#67—31no—
43C4×#67—32no—
44C2×C4⋊Dic3—31no—
45#924—31no—
46#924—32no—
47#897—32no—
48#897D311yescharacteristic, complemented
49#896—32no—
50D3×C4⋊C4—31no—
51D3×C4⋊C4D311yescharacteristic, complemented
52#889—32no—
53#889—32no—
54#888—31no—
55#888—31no—
56#888—61no—
57#887—31no—
58#67⋊C4—32no—
59#878—32no—
60C22×C3×D3C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2311yescharacteristic
67#634—42no—
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C3×C4×D3—22no—
73C3×C4×D3—22no—
74C3×C4×D3—42no—
75C3×Dic6—22no—
76#625Q814yescomplemented
77D3×Dic3—42no—
78#554—91noSylow
79C2×C3×Q8—31no—
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4—32no—
83C2×#67—31no—
84C2×#67—31no—
85C22×Dic3—32no—
86C2×C4×D3—32no—
87C2×C4×D3C2×D311yescharacteristic
88C2×C4×D3C2×D311yescharacteristic
89C2×C4×D3C2×D312yes—
90C2×Dic6—32no—
91C2×Dic6—32no—
92C2×Dic6—62no—
93C2×Dic6C2×D311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4—31no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4C2×D311yescharacteristic
99C3×C4⋊C4C2×D311yescharacteristic
100C3×C22⋊C4—32no—
101#217—31no—
102#217—31no—
103#217—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—31no—
106(C2×D3)⋊C4—31no—
107(C2×D3)⋊C4C2×D312yes—
108C4⋊Dic3—31no—
109C4⋊Dic3—31no—
110C4⋊Dic3—32no—
111C4⋊Dic3—32no—
112C4⋊Dic3—61no—
113C4⋊Dic3C2×D312yes—
114C4⋊Dic3C2×D311yescharacteristic
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123Dic3⋊C4—62no—
124Dic3⋊C4C2×D312yes—
125C4×Dic3—31no—
126C4×Dic3—31no—
127C4×Dic3—32no—
128C4×Dic3—32no—
129C4×Dic3—61no—
130C22×C32C2411yescharacteristic
131C2×C3×D3—21no—
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3C2×Q812yes—
136C2×C3×D3C2×Q812yes—
137C32×C4—22no—
138C32×C4#4112yes—
139C3⋊Dic3—21no—
140C3⋊Dic3—22no—
141C3⋊Dic3C2×Q812yes—
142C3×Dic3—22no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C3×Dic3—22no—
146C4⋊Q8—91no—
147#125—92no—
148C22⋊Q8—31no—
149C22⋊Q8—92no—
150C22⋊Q8—92no—
151C22⋊Q8—91no—
152C4×Q8—91no—
153C4×D4—92no—
154C4×D4—91no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—91no—
157C23×C3—31no—
158C23×C3—31no—
159C22×D3C22×D311yescharacteristic
160C22×D3C22×D311yescharacteristic
161C3×Q8—62no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—32no—
165C2×C3×C4—32no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—122no—
171C2×C3×C4C22×D312yes—
172C2×C3×C4C22×D312yes—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175#67—34no—
176#67—34no—
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—31no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—62no—
186C2×Dic3—122no—
187C2×Dic3C22×D312yes—
188C2×Dic3C22×D312yes—
189C2×Dic3C22×D311yescharacteristic
190C2×Dic3C22×D311yescharacteristic
191C4×D3—22no—
192C4×D3—22no—
193C4×D3—42no—
194C4×D3—122no—
195Dic6—22no—
196Dic6—122no—
197Dic6—122no—
198Dic6—124no—
199C2×C32C22×Q811yescharacteristic
200C2×C32C2×#4111yescharacteristic
201C2×C32H(5,𝔽2)11yescharacteristic
202C3×D3—22no—
203C3×D3—22no—
204C2×Q8—31no—
205C2×Q8—92no—
206C2×D4—91no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—32no—
210C22×C4—92no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—32no—
214C4⋊C4—92no—
215C4⋊C4—91no—
216C4⋊C4—92no—
217C4⋊C4—92no—
218C4⋊C4—91no—
219C4⋊C4—92no—
220C4⋊C4D3211yescharacteristic
221C22⋊C4—32no—
222C22⋊C4—92no—
223C22⋊C4—91no—
224C22⋊C4—91no—
225C42—92no—
226C42—91no—
227C22×C3—21no—
228C22×C3—32no—
229C22×C3—32no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3C23×D311yescharacteristic
235C22×C3C23×D311yescharacteristic
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—21no—
239C2×D3—21no—
240C2×D3D3×Q812yes—
241C2×D3D3×Q812yes—
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—42no—
246C3×C4—61no—
247C3×C4—62no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4#23512yes—
251C3×C4#23712yes—
252Dic3—21no—
253Dic3—22no—
254Dic3—21no—
255Dic3—22no—
256Dic3—32no—
257Dic3—32no—
258Dic3—62no—
259Dic3—62no—
260Dic3—61no—
261Dic3—62no—
262Dic3—61no—
263Dic3—62no—
264Dic3—122no—
265C32#55411yescharacteristic, complemented, Sylow
266C23—31no—
267C23—31no—
268Q8—62no—
269Q8—362no—
270D4—94no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—92no—
280C2×C4—122no—
281C2×C4—362no—
282C2×C4C2×D3212yes—
283C2×C4C2×D3211yescharacteristic
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—62no—
288C2×C3—62no—
289C2×C3C2×#23511yescharacteristic
290C2×C3C2×#23711yescharacteristic
291C2×C3#100411yescharacteristic
292C2×C3C2×D3×Q811yescharacteristic
293C2×C3C2×D3×Q811yescharacteristic
294C2×C3#100911yescharacteristic
295D3—22no—
296D3—22no—
297C22—32no—
298C22—32no—
299C22—61no—
300C22—61no—
301C22—61no—
302C22—61no—
303C22C22×D3211yescharacteristic
304C4—22no—
305C4—61no—
306C4—61no—
307C4—62no—
308C4—62no—
309C4—92no—
310C4—182no—
311C4—181no—
312C4#379512yes—
313C3—21no—
314C3#551111yescharacteristic, complemented
315C3#563511yescharacteristic, complemented
316C2—62no—
317C2—62no—
318C2C2×#379511yescharacteristic
319C2#6431411yescharacteristic
320C2D32×Q811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64097C2semidirect
2. C3×#897C2semidirect
3. C3×D3×C4⋊C4C2semidirect
5. #63896C2semidirect
6. D3×C4⋊Dic3C2semidirect
8. #63857C2semidirect
9. #63856C2semidirect
10. #63847C2semidirect
11. #63843C2semidirect
48. #897D3semidirect
51. D3×C4⋊C4D3semidirect
76. #625Q8semidirect
265. C32#554semidirect
314. C3#5511semidirect
315. C3#5635semidirect

Maximal subgroups

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

subgroupindex
#640972
C3×#8972
C3×D3×C4⋊C42
C4×#6252
#638962
D3×C4⋊Dic32
#638792
#638572
#638562
#638472
#638432
#56353
#55113