Finite Group Databasebeta

← #10594584

Subgroups — #10594584

Order 576.

465 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64106C211yesmaximal, characteristic, complemented
2C3×#892C211yesmaximal, characteristic, complemented
3C3×#890C211yesmaximal, characteristic, complemented
4C4×#626C211yesmaximal, characteristic
5#63900C211yesmaximal, characteristic, complemented
6#63869C212yesmaximal, complemented
7#63861C212yesmaximal, complemented
8#63857C211yesmaximal, characteristic, complemented
9#63854C212yesmaximal
10#63844C211yesmaximal, characteristic, complemented
11#63836C212yesmaximal, complemented
12#5694—31nomaximal
13#5529—31nomaximal
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×#626C2211yescharacteristic
18C2×#624C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20#3752C2212yes—
21C4⋊(C3⋊Dic3)C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2212yes—
24C3×C4⋊Dic3C2211yescharacteristic
25C3×C4⋊Dic3C2211yescharacteristic
26C3×Dic3⋊C4C2212yes—
27C3×C4×Dic3C2211yescharacteristic
28C3×C4×Dic3C2212yes—
29#3724C2212yes—
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2211yescharacteristic
33#3722C2211yescharacteristic
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2211yescharacteristic
36Dic32C2212yes—
37C3×#125—31no—
38C3×#117—31no—
39C4×#67—31no—
40#924—31no—
41#898—32no—
42#897—31no—
43#897—31no—
44#897—31no—
45#897—61no—
46#894—32no—
47#892—32no—
48#892D311yescharacteristic, complemented
49D3×C4⋊C4—31no—
50D3×C4⋊C4—31no—
51#890—32no—
52#890D311yescharacteristic, complemented
53#887—31no—
54#883—32no—
55#879—32no—
56C4×D12—31no—
57#872—32no—
58#870—32no—
59C4×Dic6—31no—
60C22×C3⋊D3C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2311yescharacteristic
66C4×C3⋊D3—22no—
67#634—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×C4×D3—22no—
74#626—22no—
75#624—24no—
76#552—91noSylow
77C22×C3×C4—31no—
78C2×#67—31no—
79C2×D12—31no—
80C2×C4×D3—31no—
81C2×C4×D3—31no—
82C2×C4×D3—32no—
83C2×C4×D3—32no—
84C2×C4×D3—61no—
85C2×C4×D3C2×D311yescharacteristic
86C2×Dic6—31no—
87C2×Dic6—31no—
88C2×Dic6—61no—
89C3×C4⋊C4—21no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—32no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—32no—
94C3×C4⋊C4C2×D311yescharacteristic
95C3×C4⋊C4C2×D311yescharacteristic
96C3×C22⋊C4—32no—
97C3×C42—31no—
98C3×C42—32no—
99#217—31no—
100#217—32no—
101(C2×D3)⋊C4—31no—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—31no—
106(C2×D3)⋊C4—31no—
107(C2×D3)⋊C4—62no—
108(C2×D3)⋊C4C2×D312yes—
109C4⋊Dic3—32no—
110C4⋊Dic3—32no—
111C4⋊Dic3—31no—
112C4⋊Dic3—32no—
113C4⋊Dic3—62no—
114C4⋊Dic3C2×D311yescharacteristic
115C4⋊Dic3C2×D312yes—
116C4⋊Dic3C2×D311yescharacteristic
117Dic3⋊C4—32no—
118Dic3⋊C4—31no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123Dic3⋊C4C2×D312yes—
124C4×Dic3—32no—
125C4×Dic3—32no—
126C4×Dic3C2×D311yescharacteristic
127C4×Dic3C2×D312yes—
128C22×C32C2411yescharacteristic
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C2×C3⋊D3#4112yes—
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C32×C4—22no—
136C32×C4#4112yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—21no—
139C3×Dic3—22no—
140C3×Dic3—22no—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3—21no—
144#126—92no—
145#125—31no—
146#125—92no—
147#123—92no—
148C22⋊Q8—91no—
149C22⋊Q8—91no—
150C4×Q8—91no—
151C4×D4—91no—
152#117—31no—
153#117—92no—
154C2×C4⋊C4—91no—
155C23×C3—31no—
156C22×D3—31no—
157C22×D3—31no—
158C22×D3—61no—
159C22×D3C22×D311yescharacteristic
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—62no—
168C2×C3×C4C22×D312yes—
169C2×C3×C4C22×D312yes—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172#67—62no—
173C2×Dic3—32no—
174C2×Dic3—32no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—61no—
178C2×Dic3—62no—
179C2×Dic3C22×D311yescharacteristic
180C2×Dic3C22×D312yes—
181C2×Dic3C22×D311yescharacteristic
182C2×Dic3C22×D311yescharacteristic
183C2×Dic3C22×D312yes—
184D12—62no—
185C4×D3—22no—
186C4×D3—64no—
187C4×D3—64no—
188C4×D3—62no—
189C4×D3—62no—
190C4×D3—122no—
191Dic6—62no—
192Dic6—62no—
193Dic6—122no—
194C2×C32C2×#4111yescharacteristic
195C2×C32C2×#4111yescharacteristic
196C2×C322−1+411yescharacteristic
197C3⋊D3—22no—
198C3×D3—41no—
199C2×Q8—91no—
200C2×D4—91no—
201C22×C4—31no—
202C22×C4—92no—
203C22×C4—91no—
204C4⋊C4—31no—
205C4⋊C4—31no—
206C4⋊C4—32no—
207C4⋊C4—32no—
208C4⋊C4—92no—
209C4⋊C4—92no—
210C4⋊C4—91no—
211C4⋊C4—92no—
212C4⋊C4—92no—
213C4⋊C4D3211yescharacteristic
214C22⋊C4—32no—
215C22⋊C4—91no—
216C22⋊C4—91no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C42—32no—
220C42—31no—
221C42—92no—
222C22×C3—21no—
223C22×C3—61no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3C23×D311yescharacteristic
227C22×C3C23×D311yescharacteristic
228C2×D3—21no—
229C2×D3—21no—
230C2×D3—21no—
231C2×D3—32no—
232C2×D3—32no—
233C2×D3—62no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—121no—
239C2×D3—121no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—42no—
244C3×C4—61no—
245C3×C4—62no—
246C3×C4—62no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4#23512yes—
250C3×C4#23912yes—
251Dic3—21no—
252Dic3—22no—
253Dic3—22no—
254Dic3—21no—
255Dic3—21no—
256Dic3—62no—
257Dic3—61no—
258Dic3—62no—
259Dic3—61no—
260Dic3—122no—
261Dic3—121no—
262C32#55211yescharacteristic, complemented, Sylow
263C23—31no—
264C23—91no—
265Q8—182no—
266D4—182no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—31no—
271C2×C4—32no—
272C2×C4—62no—
273C2×C4—92no—
274C2×C4—91no—
275C2×C4—184no—
276C2×C4—182no—
277C2×C4C2×D3212yes—
278C2×C4C2×D3211yescharacteristic
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—121no—
283C2×C3C2×#23511yescharacteristic
284C2×C3C2×#23911yescharacteristic
285C2×C3D3×#4111yescharacteristic
286C2×C3D3×#4111yescharacteristic
287C2×C3#101011yescharacteristic
288C2×C3#101011yescharacteristic
289D3—41no—
290D3—62no—
291D3—62no—
292D3—122no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22—92no—
297C22—181no—
298C22—181no—
299C22C22×D3211yescharacteristic
300C4—22no—
301C4—61no—
302C4—61no—
303C4—62no—
304C4—62no—
305C4—61no—
306C4—181no—
307C4—182no—
308C4#379912yes—
309C3—21no—
310C3#552911yescharacteristic, complemented
311C3#569411yescharacteristic, complemented
312C2—121no—
313C2—182no—
314C2C2×#379911yescharacteristic
315C2#6430911yescharacteristic
316C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#626C22
hypercenterC4⋊C4D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64106C2semidirect
2. C3×#892C2semidirect
3. C3×#890C2semidirect
5. #63900C2semidirect
6. #63869C2semidirect
7. #63861C2semidirect
8. #63857C2semidirect
10. #63844C2semidirect
11. #63836C2semidirect
48. #892D3semidirect
52. #890D3semidirect
262. C32#552semidirect
310. C3#5529semidirect
311. C3#5694semidirect

Maximal subgroups

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

subgroupindex
#641062
C3×#8922
C3×#8902
C4×#6262
#639002
#638692
#638612
#638572
#638542
#638442
#638362
#56943
#55293