Finite Group Databasebeta

← #10594983

Subgroups — #10594983

Order 576.

469 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#64133C211yesmaximal, characteristic, complemented
2C3×#930C211yesmaximal, characteristic
3C3×#926C211yesmaximal, characteristic, complemented
4#63969C212yesmaximal, complemented
5#63959C212yesmaximal, complemented
6#63918C211yesmaximal, characteristic, complemented
7#63899C212yesmaximal, complemented
8Dic3×D12C211yesmaximal, characteristic, complemented
9#63860C211yesmaximal, characteristic, complemented
10#63858C212yesmaximal, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5603—31nomaximal
13#5534—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×#67C2212yes—
17C2×C3×D12C2211yescharacteristic
18C2×#627C2212yes—
19C2×D3×Dic3C2212yes—
20#3757C2212yes—
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22#3750C2212yes—
23C3×#217C2211yescharacteristic
24C3×#217C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic3C2211yescharacteristic
29C3×C4×Dic3C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31#3724C2212yes—
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×#121—31no—
38C3×#117—31no—
39#936—32no—
40#936—31no—
41#935—32no—
42D4×Dic3—31no—
43#930D311yescharacteristic
44#926D311yescharacteristic, complemented
45#924—31no—
46#924—31no—
47#924—61no—
48#898—32no—
49#896—32no—
50#888—31no—
51#887—31no—
52#885—32no—
53#883—32no—
54#879—32no—
55#878—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
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2312yes—
67#634—22no—
68C3×#67—42no—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C3×D12—22no—
75#627—42no—
76D3×Dic3—42no—
77#542—91noSylow
78C2×C3×D4—32no—
79C2×C3×D4—31no—
80C22×C3×C4—21no—
81C22×C3×C4C2×D311yescharacteristic
82C22×C3×C4C2×D311yescharacteristic
83C2×#67C2×D312yes—
84C22×Dic3—32no—
85C2×D12C2×D311yescharacteristic
86C2×C4×D3—32no—
87C2×Dic6—31no—
88C2×Dic6—31no—
89C2×Dic6—32no—
90C2×Dic6—32no—
91C2×Dic6—61no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—31no—
94C3×C4⋊C4—31no—
95C3×C22⋊C4—31no—
96C3×C22⋊C4—31no—
97C3×C22⋊C4—32no—
98C3×C42—31no—
99C3×C42—31no—
100#217—32no—
101#217—32no—
102#217—32no—
103#217—32no—
104#217—32no—
105#217—62no—
106#217C2×D311yescharacteristic
107#217C2×D311yescharacteristic
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4C2×D312yes—
112C4⋊Dic3—31no—
113C4⋊Dic3—31no—
114C4⋊Dic3—32no—
115C4⋊Dic3—61no—
116C4⋊Dic3C2×D311yescharacteristic
117C4⋊Dic3C2×D311yescharacteristic
118C4⋊Dic3C2×D311yescharacteristic
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124Dic3⋊C4—62no—
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—21no—
142C3×Dic3—22no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C4⋊Q8—91no—
146#126—92no—
147#124—91no—
148#124—92no—
149#123—92no—
150C22⋊Q8—92no—
151C22⋊Q8—91no—
152#121—31no—
153C4×Q8—91no—
154C4×D4—91no—
155#117—31no—
156C23×C3—21no—
157C23×C3—32no—
158C23×C3C22×D311yescharacteristic
159C23×C3C22×D311yescharacteristic
160C22×D3C22×D312yes—
161C3×D4—62no—
162C3×D4—122no—
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—31no—
171C2×C3×C4—31no—
172C2×C3×C4—42no—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177#67—42no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—122no—
185C2×Dic3C22×D311yescharacteristic
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D311yescharacteristic
190D12—22no—
191C4×D3—122no—
192Dic6—62no—
193Dic6—62no—
194Dic6—122no—
195Dic6—122no—
196Dic6—122no—
197C2×C32—41no—
198C2×C32C2×#4111yescharacteristic
199C2×C32H(5,𝔽2)11yescharacteristic
200C2×C322−1+411yescharacteristic
201C3×D3—42no—
202C2×Q8—92no—
203C2×Q8—91no—
204C2×D4—32no—
205C2×D4—31no—
206C22×C4—92no—
207C22×C4D3211yescharacteristic
208C4⋊C4—31no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—92no—
212C4⋊C4—92no—
213C4⋊C4—91no—
214C4⋊C4—92no—
215C22⋊C4—31no—
216C22⋊C4—31no—
217C22⋊C4—32no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C22⋊C4—92no—
222C42—31no—
223C42—31no—
224C42—92no—
225C22×C3—22no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3C23×D311yescharacteristic
239C22×C3C23×D311yescharacteristic
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—22no—
246C3×C4—41no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4#23512yes—
253C3×C4#23712yes—
254Dic3—22no—
255Dic3—21no—
256Dic3—21no—
257Dic3—21no—
258Dic3—21no—
259Dic3—62no—
260Dic3—62no—
261Dic3—62no—
262Dic3—62no—
263Dic3—122no—
264Dic3—122no—
265C32#54211yescharacteristic, complemented, Sylow
266C23—32no—
267C23C2×D3211yescharacteristic
268Q8—182no—
269Q8—362no—
270D4—62no—
271D4—122no—
272C2×C4—22no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—32no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—362no—
281C2×C4C2×D3211yescharacteristic
282C2×C4C2×D3211yescharacteristic
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—42no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—122no—
290C2×C3C2×#23511yescharacteristic
291C2×C3C2×#23711yescharacteristic
292C2×C3#100411yescharacteristic
293C2×C3#100911yescharacteristic
294C2×C3#101011yescharacteristic
295C2×C3#101011yescharacteristic
296D3—42no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—62no—
301C22—62no—
302C22—62no—
303C22C22×D3211yescharacteristic
304C4—21no—
305C4—61no—
306C4—61no—
307C4—61no—
308C4—62no—
309C4—61no—
310C4—182no—
311C4—182no—
312C4#380412yes—
313C3—21no—
314C3#553411yescharacteristic, complemented
315C3#560311yescharacteristic, complemented
316C2—41no—
317C2—122no—
318C2#6429811yescharacteristic
319C2#6430711yescharacteristic
320C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC3×#930C2
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
1. #64133C2semidirect
3. C3×#926C2semidirect
4. #63969C2semidirect
5. #63959C2semidirect
6. #63918C2semidirect
7. #63899C2semidirect
8. Dic3×D12C2semidirect
9. #63860C2semidirect
10. #63858C2semidirect
11. #63844C2semidirect
44. #926D3semidirect
265. C32#542semidirect
314. C3#5534semidirect
315. C3#5603semidirect

Maximal subgroups

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

subgroupindex
#641332
C3×#9302
C3×#9262
#639692
#639592
#639182
#638992
Dic3×D122
#638602
#638582
#638442
#56033
#55343