Finite Group Databasebeta

← #10594841

Subgroups — #10594841

Order 576.

462 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64135C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic
3C3×#924C211yesmaximal, characteristic, complemented
4#63959C212yesmaximal
5#63956C212yesmaximal, complemented
6#63918C211yesmaximal, characteristic, complemented
7#63862C212yesmaximal, complemented
8#63856C211yesmaximal, characteristic, complemented
9Dic3×Dic6C211yesmaximal, characteristic, complemented
10#63841C211yesmaximal, characteristic, complemented
11#63838C212yesmaximal, complemented
12#5641—31nomaximal
13#5542—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×C3×#67C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×#627C2212yes—
19#3757C2212yes—
20C4⋊(C3⋊Dic3)C2211yescharacteristic
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22C4×C3⋊Dic3C2212yes—
23C3×#217C2212yes—
24C3×#217C2211yescharacteristic
25C3×(C2×D3)⋊C4C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×Dic3⋊C4C2212yes—
28C3×Dic3⋊C4C2211yescharacteristic
29C3×C4×Dic3C2211yescharacteristic
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35Dic32C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C3×C4×D4—31no—
39#946—31no—
40Q8×Dic3—31no—
41#936—32no—
42C4×#67D311yescharacteristic
43#926—31no—
44#926—31no—
45#926—31no—
46#926—61no—
47#924—31no—
48#924D311yescharacteristic, complemented
49#898—32no—
50#892—31no—
51#888—31no—
52#887—32no—
53#887—32no—
54#885—32no—
55#879—32no—
56#879—32no—
57#879—32no—
58#869—31no—
59C4×Dic6—31no—
60C23×C32C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2312yes—
67C3×#67—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×C4×D3—22no—
74C3×Dic6—22no—
75#627—24no—
76#556—91noSylow
77C2×C3×Q8—31no—
78C2×C3×D4—31no—
79C22×C3×C4—21no—
80C22×C3×C4—31no—
81C22×C3×C4C2×D311yescharacteristic
82C22×C3×C4C2×D311yescharacteristic
83C2×#67C2×D311yescharacteristic
84C2×C4×D3C2×D311yescharacteristic
85C2×Dic6—32no—
86C2×Dic6—32no—
87C2×Dic6C2×D311yescharacteristic
88C3×C4⋊C4—31no—
89C3×C4⋊C4—32no—
90C3×C4⋊C4—31no—
91C3×C22⋊C4—31no—
92C3×C22⋊C4—32no—
93C3×C22⋊C4—31no—
94C3×C42—31no—
95#217—32no—
96#217—32no—
97#217—32no—
98#217—32no—
99#217—62no—
100#217C2×D311yescharacteristic
101#217C2×D312yes—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4C2×D311yescharacteristic
105C4⋊Dic3—31no—
106C4⋊Dic3—31no—
107C4⋊Dic3—31no—
108C4⋊Dic3—31no—
109C4⋊Dic3—32no—
110C4⋊Dic3—32no—
111C4⋊Dic3—61no—
112C4⋊Dic3—61no—
113C4⋊Dic3C2×D311yescharacteristic
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4C2×D312yes—
119Dic3⋊C4C2×D311yescharacteristic
120C4×Dic3—32no—
121C4×Dic3—32no—
122C4×Dic3—32no—
123C4×Dic3—32no—
124C4×Dic3—32no—
125C4×Dic3—32no—
126C4×Dic3—62no—
127C4×Dic3C2×D311yescharacteristic
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32C2411yescharacteristic
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C32×C4—21no—
136C32×C4#4112yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—22no—
139C3×Dic3—22no—
140C3×Dic3—22no—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3#4112yes—
144C4⋊Q8—91no—
145#126—92no—
146#126—92no—
147#124—92no—
148C22⋊Q8—31no—
149C22⋊Q8—91no—
150C4×Q8—91no—
151C4×Q8—92no—
152C4×D4—31no—
153#117—91no—
154#117—91no—
155C23×C3—21no—
156C23×C3—31no—
157C23×C3C22×D311yescharacteristic
158C23×C3C22×D311yescharacteristic
159C22×D3C22×D311yescharacteristic
160C3×Q8—62no—
161C3×D4—62no—
162C2×C3×C4—24no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—62no—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177#67—22no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3C22×D311yescharacteristic
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C4×D3—22no—
190Dic6—22no—
191Dic6—64no—
192Dic6—64no—
193C2×C32—41no—
194C2×C32C2×#4111yescharacteristic
195C2×C32C2×#4111yescharacteristic
196C2×C322−1+411yescharacteristic
197C3×D3—41no—
198C2×Q8—31no—
199C2×Q8—92no—
200C2×D4—31no—
201C22×C4—31no—
202C22×C4D3211yescharacteristic
203C4⋊C4—31no—
204C4⋊C4—31no—
205C4⋊C4—32no—
206C4⋊C4—92no—
207C4⋊C4—92no—
208C4⋊C4—92no—
209C4⋊C4—91no—
210C4⋊C4—91no—
211C22⋊C4—31no—
212C22⋊C4—32no—
213C22⋊C4—31no—
214C22⋊C4—92no—
215C22⋊C4—92no—
216C22⋊C4—92no—
217C42—31no—
218C42—92no—
219C42—92no—
220C42—92no—
221C22×C3—22no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3C23×D311yescharacteristic
235C22×C3C23×D311yescharacteristic
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—21no—
239C3×C4—22no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—22no—
243C3×C4—32no—
244C3×C4—62no—
245C3×C4—61no—
246C3×C4—62no—
247C3×C4—61no—
248C3×C4#23712yes—
249C3×C4#23712yes—
250Dic3—22no—
251Dic3—21no—
252Dic3—22no—
253Dic3—21no—
254Dic3—62no—
255Dic3—62no—
256Dic3—62no—
257Dic3—62no—
258Dic3—122no—
259Dic3—122no—
260Dic3#23512yes—
261C32#55611yescharacteristic, complemented, Sylow
262C23—31no—
263C23C2×D3211yescharacteristic
264Q8—62no—
265Q8—184no—
266D4—62no—
267C2×C4—22no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—62no—
274C2×C4—92no—
275C2×C4—92no—
276C2×C4C2×D3211yescharacteristic
277C2×C4C2×D3211yescharacteristic
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—121no—
285C2×C3C2×#23511yescharacteristic
286C2×C3C2×#23711yescharacteristic
287C2×C3C2×#23711yescharacteristic
288C2×C3#100711yescharacteristic
289C2×C3D3×#4111yescharacteristic
290C2×C3#101011yescharacteristic
291D3—41no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—61no—
296C22—61no—
297C22—61no—
298C22C22×D3211yescharacteristic
299C4—21no—
300C4—32no—
301C4—61no—
302C4—62no—
303C4—61no—
304C4—62no—
305C4—182no—
306C4—182no—
307C4#380512yes—
308C3—21no—
309C3#554211yescharacteristic, complemented
310C3#564111yescharacteristic, complemented
311C2—41no—
312C2—121no—
313C2#6429711yescharacteristic
314C2D3×#23511yescharacteristic
315C2C2×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×C3×#67C22
hypercenterC22×C4D32
hypocenterC32#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64135C2semidirect
3. C3×#924C2semidirect
5. #63956C2semidirect
6. #63918C2semidirect
7. #63862C2semidirect
8. #63856C2semidirect
9. Dic3×Dic6C2semidirect
10. #63841C2semidirect
11. #63838C2semidirect
48. #924D3semidirect
261. C32#556semidirect
309. C3#5542semidirect
310. C3#5641semidirect

Maximal subgroups

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

subgroupindex
#641352
C3×C4×#672
C3×#9242
#639592
#639562
#639182
#638622
#638562
Dic3×Dic62
#638412
#638382
#56413
#55423