Finite Group Databasebeta

← #10594843

Subgroups — #10594843

Order 576.

523 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64139C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#63978C212yesmaximal, complemented
5#63952C212yesmaximal, complemented
6#63909C211yesmaximal, characteristic, complemented
7#63894C211yesmaximal, characteristic, complemented
8#63869C211yesmaximal, characteristic, complemented
9#63854C212yesmaximal, complemented
10#63848C212yesmaximal, complemented
11#63841C211yesmaximal, characteristic, complemented
12#5565—31nomaximal
13#5514—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#67C2211yescharacteristic
18C22×C3×Dic3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×#626C2212yes—
21C2×#624C2212yes—
22#3752C2212yes—
23C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#217C2211yescharacteristic
25C3×(C2×D3)⋊C4C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2212yes—
28C3×C4⋊Dic3C2211yescharacteristic
29C3×Dic3⋊C4C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31#3724C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×C4×D4—31no—
38C2×C3×C4⋊C4—31no—
39#939—32no—
40#930—31no—
41#930—31no—
42#930—32no—
43#930—31no—
44#930—61no—
45C4×#67—32no—
46C4×#67D311yescharacteristic, complemented
47#926—31no—
48C2×C4⋊Dic3D311yescharacteristic, complemented
49#898—32no—
50#898—32no—
51#895—31no—
52#894—32no—
53#893—31no—
54#892—31no—
55#890—31no—
56#883—32no—
57#67⋊C4—32no—
58C4×D12—31no—
59#870—31no—
60C23×C32C2311yescharacteristic
61C22×C3⋊D3C2312yes—
62C22×C3×D3C2311yescharacteristic
63C2×C32×C4—22no—
64C2×C32×C4C2311yescharacteristic
65C2×C32×C4C2311yescharacteristic
66#638—24no—
67C2×C3⋊Dic3C2312yes—
68C4⋊(C3⋊D3)—22no—
69C3×#67—22no—
70C2×C3×Dic3—42no—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C3×C4×D3—22no—
77#626—42no—
78#624—42no—
79#553—91noSylow
80C2×C3×D4—31no—
81C22×C3×C4—21no—
82C22×C3×C4—32no—
83C22×C3×C4—31no—
84C22×C3×C4C2×D311yescharacteristic
85C22×C3×C4C2×D311yescharacteristic
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—62no—
90C2×#67C2×D311yescharacteristic
91C22×Dic3C2×D312yes—
92C2×D12—31no—
93C2×D12—31no—
94C2×D12—32no—
95C2×D12—61no—
96C2×C4×D3—32no—
97C2×C4×D3—32no—
98C2×C4×D3C2×D311yescharacteristic
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—31no—
103C3×C22⋊C4—31no—
104C3×C22⋊C4—31no—
105C3×C42—31no—
106#217—32no—
107#217C2×D311yescharacteristic
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—62no—
116(C2×D3)⋊C4C2×D311yescharacteristic
117C4⋊Dic3—31no—
118C4⋊Dic3—31no—
119C4⋊Dic3—32no—
120C4⋊Dic3—61no—
121C4⋊Dic3C2×D311yescharacteristic
122C4⋊Dic3C2×D311yescharacteristic
123C4⋊Dic3C2×D312yes—
124Dic3⋊C4—32no—
125Dic3⋊C4—32no—
126Dic3⋊C4—32no—
127Dic3⋊C4C2×D311yescharacteristic
128C4×Dic3—32no—
129C4×Dic3—32no—
130C4×Dic3C2×D311yescharacteristic
131C22×C32—21no—
132C22×C32—21no—
133C22×C32#4112yes—
134C22×C32C2411yescharacteristic
135C2×C3⋊D3—22no—
136C2×C3⋊D3—22no—
137C2×C3⋊D3—22no—
138C2×C3×D3—21no—
139C2×C3×D3—21no—
140C2×C3×D3—21no—
141C32×C4—21no—
142C32×C4#4112yes—
143C3⋊Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3—21no—
146C3×Dic3—21no—
147C3×Dic3—22no—
148C3×Dic3—22no—
149#126—92no—
150#125—91no—
151#123—92no—
152#121—91no—
153#121—92no—
154#121—91no—
155C4×D4—31no—
156C4×D4—91no—
157C4×D4—92no—
158#117—91no—
159C2×C4⋊C4—31no—
160C23×C3—21no—
161C23×C3—31no—
162C23×C3C22×D311yescharacteristic
163C23×C3C22×D311yescharacteristic
164C22×D3—32no—
165C22×D3—32no—
166C22×D3—62no—
167C22×D3C22×D311yescharacteristic
168C3×D4—62no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—31no—
174C2×C3×C4—32no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—32no—
178C2×C3×C4—42no—
179C2×C3×C4—62no—
180C2×C3×C4—122no—
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D311yescharacteristic
183C2×C3×C4C22×D311yescharacteristic
184C2×C3×C4C22×D311yescharacteristic
185#67—22no—
186#67—64no—
187#67—64no—
188#67—124no—
189#67—122no—
190C2×Dic3—32no—
191C2×Dic3—32no—
192C2×Dic3—42no—
193C2×Dic3—62no—
194C2×Dic3C22×D311yescharacteristic
195C2×Dic3C22×D312yes—
196C2×Dic3C22×D311yescharacteristic
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D312yes—
199D12—62no—
200D12—62no—
201D12—122no—
202D12—122no—
203C4×D3—22no—
204C4×D3—122no—
205C4×D3—122no—
206C2×C32—22no—
207C2×C32C2×#4111yescharacteristic
208C2×C32C2×#4111yescharacteristic
209C2×C32H(5,𝔽2)11yescharacteristic
210C3⋊D3—42no—
211C3×D3—41no—
212C2×D4—31no—
213C2×D4—92no—
214C2×D4—92no—
215C2×D4—91no—
216C22×C4—32no—
217C22×C4—31no—
218C22×C4—92no—
219C22×C4D3211yescharacteristic
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—32no—
223C4⋊C4—31no—
224C4⋊C4—92no—
225C4⋊C4—92no—
226C4⋊C4—91no—
227C22⋊C4—31no—
228C22⋊C4—31no—
229C22⋊C4—92no—
230C22⋊C4—92no—
231C22⋊C4—92no—
232C22⋊C4—92no—
233C42—31no—
234C42—92no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—61no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3#23512yes—
247C22×C3#23912yes—
248C22×C3C23×D311yescharacteristic
249C22×C3C23×D311yescharacteristic
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—62no—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—122no—
260C2×D3—122no—
261C2×D3—122no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—22no—
265C3×C4—41no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—62no—
270C3×C4—62no—
271C3×C4#23512yes—
272C3×C4#23712yes—
273Dic3—21no—
274Dic3—21no—
275Dic3—22no—
276Dic3—22no—
277Dic3—21no—
278Dic3—62no—
279Dic3—62no—
280Dic3—122no—
281C32#55311yescharacteristic, complemented, Sylow
282C23—31no—
283C23—92no—
284C23C2×D3211yescharacteristic
285D4—62no—
286D4—184no—
287D4—182no—
288D4—362no—
289C2×C4—22no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—62no—
296C2×C4—92no—
297C2×C4—122no—
298C2×C4—362no—
299C2×C4C2×D3211yescharacteristic
300C2×C4C2×D3211yescharacteristic
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—42no—
307C2×C3—121no—
308C2×C3C2×#23511yescharacteristic
309C2×C3C2×#23511yescharacteristic
310C2×C3C2×#23711yescharacteristic
311C2×C3#100411yescharacteristic
312C2×C3C2×#23911yescharacteristic
313C2×C3#100911yescharacteristic
314D3—41no—
315D3—122no—
316D3—122no—
317D3—242no—
318C22—21no—
319C22—21no—
320C22—61no—
321C22—61no—
322C22—61no—
323C22—182no—
324C22—182no—
325C22—182no—
326C22#379912yes—
327C22C22×D3211yescharacteristic
328C4—21no—
329C4—61no—
330C4—62no—
331C4—62no—
332C4—61no—
333C4—61no—
334C4—182no—
335C4#380412yes—
336C3—21no—
337C3#551411yescharacteristic, complemented
338C3#556511yescharacteristic, complemented
339C2—22no—
340C2—121no—
341C2—362no—
342C2C2×#379911yescharacteristic
343C2#6430711yescharacteristic
344C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64139C2semidirect
2. C3×C4×#67C2semidirect
3. C2×C3×C4⋊Dic3C2semidirect
4. #63978C2semidirect
5. #63952C2semidirect
6. #63909C2semidirect
7. #63894C2semidirect
8. #63869C2semidirect
9. #63854C2semidirect
10. #63848C2semidirect
11. #63841C2semidirect
46. C4×#67D3semidirect
48. C2×C4⋊Dic3D3semidirect
281. C32#553semidirect
337. C3#5514semidirect
338. C3#5565semidirect

Maximal subgroups

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

subgroupindex
#641392
C3×C4×#672
C2×C3×C4⋊Dic32
#639782
#639522
#639092
#638942
#638692
#638542
#638482
#638412
#55653
#55143