Finite Group Databasebeta

← #10594814

Subgroups — #10594814

Order 576.

479 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64139C211yesmaximal, characteristic
2C3×#926C211yesmaximal, characteristic, complemented
3C3×#924C211yesmaximal, characteristic, complemented
4#63975C212yesmaximal, complemented
5#63953C212yesmaximal, complemented
6#63918C211yesmaximal, characteristic, complemented
7#63895C212yesmaximal, complemented
8#63894C211yesmaximal, characteristic, complemented
9#63864C211yesmaximal, characteristic, complemented
10#63848C212yesmaximal, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5643—31nomaximal
13#5534—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×#627C2212yes—
19C2×#624C2212yes—
20#3752C2212yes—
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#217C2212yes—
23C3×#217C2211yescharacteristic
24C3×#217C2211yescharacteristic
25C3×C4⋊Dic3C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×C4×Dic3C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35#3722C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C3×#117—31no—
39#947—31no—
40Q8×Dic3—31no—
41#930—31no—
42#930—31no—
43#930—61no—
44#926D311yescharacteristic, complemented
45#924D311yescharacteristic, complemented
46#898—32no—
47#898—32no—
48#897—32no—
49#896—32no—
50#893—31no—
51#888—31no—
52#885—32no—
53#885—32no—
54#883—32no—
55#883—32no—
56#875—31no—
57C4×D12—31no—
58#869—31no—
59C4×Dic6—31no—
60C23×C32C2311yescharacteristic
61C22×C3⋊D3C2312yes—
62C2×C32×C4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65#638—42no—
66C2×C3⋊Dic3C2312yes—
67C4⋊(C3⋊D3)—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C3×Dic6—22no—
75#627—42no—
76#624—42no—
77#542—91noSylow
78C2×C3×Q8—31no—
79C22×C3×C4—21no—
80C22×C3×C4C2×D311yescharacteristic
81C22×C3×C4C2×D311yescharacteristic
82C2×#67—32no—
83C2×#67—32no—
84C2×#67—62no—
85C2×D12—31no—
86C2×D12—31no—
87C2×D12—61no—
88C2×C4×D3—32no—
89C2×C4×D3—32no—
90C2×Dic6—32no—
91C2×Dic6—32no—
92C2×Dic6C2×D311yescharacteristic
93C3×C4⋊C4—31no—
94C3×C4⋊C4—31no—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—31no—
97C3×C22⋊C4—31no—
98C3×C22⋊C4—32no—
99C3×C22⋊C4—31no—
100C3×C42—31no—
101C3×C42—31no—
102#217C2×D311yescharacteristic
103#217C2×D312yes—
104#217C2×D311yescharacteristic
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
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—62no—
114C4⋊Dic3—31no—
115C4⋊Dic3—31no—
116C4⋊Dic3—32no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119C4⋊Dic3C2×D311yescharacteristic
120C4⋊Dic3C2×D311yescharacteristic
121C4⋊Dic3C2×D311yescharacteristic
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124Dic3⋊C4C2×D312yes—
125C4×Dic3—32no—
126C4×Dic3—32no—
127C4×Dic3C2×D311yescharacteristic
128C4×Dic3C2×D311yescharacteristic
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32C2411yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3—22no—
135C2×C3⋊D3—22no—
136C32×C4—21no—
137C32×C4#4112yes—
138C3⋊Dic3—22no—
139C3×Dic3—22no—
140C3×Dic3—21no—
141C3×Dic3—22no—
142C3×Dic3—21no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C4⋊Q8—91no—
146#126—92no—
147#124—91no—
148#124—92no—
149#123—92no—
150C22⋊Q8—31no—
151C22⋊Q8—92no—
152#121—91no—
153C4×Q8—91no—
154C4×D4—91no—
155#117—31no—
156C23×C3—21no—
157C23×C3C22×D311yescharacteristic
158C23×C3C22×D311yescharacteristic
159C22×D3—32no—
160C22×D3—32no—
161C22×D3—62no—
162C3×Q8—62no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—31no—
168C2×C3×C4—32no—
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—42no—
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177C2×C3×C4C22×D311yescharacteristic
178#67—122no—
179#67—122no—
180#67—124no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—62no—
184C2×Dic3C22×D311yescharacteristic
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D311yescharacteristic
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D312yes—
190D12—62no—
191D12—62no—
192D12—122no—
193C4×D3—122no—
194C4×D3—122no—
195Dic6—22no—
196Dic6—122no—
197Dic6—122no—
198C2×C32—41no—
199C2×C32C2×#4111yescharacteristic
200C2×C32H(5,𝔽2)11yescharacteristic
201C2×C322−1+411yescharacteristic
202C3⋊D3—42no—
203C2×Q8—31no—
204C2×Q8—92no—
205C2×D4—92no—
206C2×D4—91no—
207C22×C4—92no—
208C22×C4D3211yescharacteristic
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—92no—
214C4⋊C4—92no—
215C4⋊C4—91no—
216C22⋊C4—31no—
217C22⋊C4—31no—
218C22⋊C4—32no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C22⋊C4—92no—
222C22⋊C4—92no—
223C42—31no—
224C42—31no—
225C42—92no—
226C22×C3—22no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3C23×D311yescharacteristic
237C22×C3C23×D311yescharacteristic
238C2×D3—62no—
239C2×D3—62no—
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—122no—
245C2×D3—122no—
246C2×D3—122no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—22no—
250C3×C4—41no—
251C3×C4—62no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—62no—
256C3×C4—61no—
257C3×C4#23512yes—
258C3×C4#23712yes—
259Dic3—22no—
260Dic3—21no—
261Dic3—22no—
262Dic3—21no—
263Dic3—21no—
264Dic3—21no—
265Dic3—62no—
266Dic3—62no—
267Dic3—122no—
268C32#54211yescharacteristic, complemented, Sylow
269C23—92no—
270C23C2×D3211yescharacteristic
271Q8—62no—
272Q8—362no—
273D4—182no—
274D4—362no—
275C2×C4—22no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—92no—
283C2×C4—362no—
284C2×C4C2×D3211yescharacteristic
285C2×C4C2×D3211yescharacteristic
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—42no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3C2×#23511yescharacteristic
293C2×C3C2×#23711yescharacteristic
294C2×C3#100711yescharacteristic
295C2×C3#100911yescharacteristic
296C2×C3#100911yescharacteristic
297C2×C3#101011yescharacteristic
298D3—122no—
299D3—122no—
300D3—242no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—182no—
305C22—182no—
306C22—182no—
307C22C22×D3211yescharacteristic
308C4—21no—
309C4—61no—
310C4—61no—
311C4—61no—
312C4—62no—
313C4—61no—
314C4—62no—
315C4—182no—
316C4#380412yes—
317C3—21no—
318C3#553411yescharacteristic, complemented
319C3#564311yescharacteristic, complemented
320C2—41no—
321C2—362no—
322C2#6429711yescharacteristic
323C2#6430811yescharacteristic
324C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#64139C2
hypercenterC22×C4D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#926C2semidirect
3. C3×#924C2semidirect
4. #63975C2semidirect
5. #63953C2semidirect
6. #63918C2semidirect
7. #63895C2semidirect
8. #63894C2semidirect
9. #63864C2semidirect
10. #63848C2semidirect
11. Dic3×Dic6C2semidirect
44. #926D3semidirect
45. #924D3semidirect
268. C32#542semidirect
318. C3#5534semidirect
319. C3#5643semidirect

Maximal subgroups

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

subgroupindex
#641392
C3×#9262
C3×#9242
#639752
#639532
#639182
#638952
#638942
#638642
#638482
Dic3×Dic62
#56433
#55343