Finite Group Databasebeta

← #10594427

Subgroups — #10594427

Order 576.

503 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 134 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#64096C211yesmaximal, characteristic, complemented
2C3×#895C211yesmaximal, characteristic
3C3×#890C211yesmaximal, characteristic, complemented
4#63898C212yesmaximal, complemented
5#63898C211yesmaximal, characteristic, complemented
6#63884C212yesmaximal, complemented
7#63878C212yesmaximal, complemented
8C4×D3×Dic3C211yesmaximal, characteristic, complemented
9#63860C211yesmaximal, characteristic, complemented
10#63858C212yesmaximal, complemented
11#63857C211yesmaximal, characteristic, complemented
12#5691—31nomaximal
13#5600—31nomaximal
14C2×#634C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×D12C2212yes—
17C2×C3×C4×D3C2211yescharacteristic
18C2×D3×Dic3C2211yescharacteristic
19C2×D3×Dic3C2211yescharacteristic
20C2×D3×Dic3C2212yes—
21C32×C4⋊C4C2211yescharacteristic
22#3750C2212yes—
23C4×C3⋊Dic3C2211yescharacteristic
24C4×C3⋊Dic3C2212yes—
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2212yes—
28C3×Dic3⋊C4C2212yes—
29C3×C4×Dic3C2211yescharacteristic
30Dic3⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32(C2×D3)⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35Dic32C2211yescharacteristic
36Dic32C2211yescharacteristic
37C3×#125—31no—
38C3×#121—31no—
39#936—31no—
40#935—32no—
41D4×Dic3—32no—
42D4×Dic3—31no—
43#924—31no—
44C2×C4×Dic3—31no—
45#898—32no—
46#897—31no—
47#895D311yescharacteristic
48#894—32no—
49#893—31no—
50#893—32no—
51#892—32no—
52#890D311yescharacteristic, complemented
53#887—31no—
54#887—31no—
55#887—61no—
56#879—32no—
57#877—32no—
58#875—31no—
59C42×D3—31no—
60C22×C3×D3C2312yes—
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2311yescharacteristic
67#634—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C3×D12—24no—
73C3×D12—22no—
74C3×C4×D3—22no—
75D3×Dic3—22no—
76D3×Dic3—22no—
77D3×Dic3—24no—
78#529—91noSylow
79C2×C3×D4—32no—
80C2×C3×D4—31no—
81C22×C3×C4—31no—
82C22×Dic3—32no—
83C22×Dic3—31no—
84C22×Dic3—31no—
85C2×D12C2×D311yescharacteristic
86C2×D12C2×D312yes—
87C2×C4×D3—31no—
88C2×C4×D3—31no—
89C2×C4×D3—32no—
90C2×C4×D3C2×D311yescharacteristic
91C2×Dic6—31no—
92C2×Dic6—31no—
93C2×Dic6—61no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—31no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4C2×D311yescharacteristic
99C3×C4⋊C4C2×D311yescharacteristic
100C3×C22⋊C4—32no—
101C3×C42—31no—
102#217—32no—
103#217—32no—
104#217—32no—
105#217—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4C2×D312yes—
111C4⋊Dic3—32no—
112C4⋊Dic3C2×D312yes—
113C4⋊Dic3C2×D311yescharacteristic
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—62no—
118Dic3⋊C4C2×D312yes—
119C4×Dic3—31no—
120C4×Dic3—32no—
121C4×Dic3—32no—
122C4×Dic3—31no—
123C4×Dic3—31no—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—31no—
127C4×Dic3—61no—
128C4×Dic3—62no—
129C4×Dic3C2×D311yescharacteristic
130C22×C32C2411yescharacteristic
131C2×C3×D3—21no—
132C2×C3×D3—21no—
133C2×C3×D3—22no—
134C2×C3×D3—22no—
135C2×C3×D3—22no—
136C2×C3×D3#4112yes—
137C32×C4—22no—
138C32×C4#4112yes—
139C3⋊Dic3—22no—
140C3⋊Dic3—21no—
141C3⋊Dic3#4112yes—
142C3×Dic3—21no—
143C3×Dic3—21no—
144C3×Dic3—22no—
145C3×Dic3—21no—
146#126—92no—
147#125—31no—
148#124—91no—
149#123—92no—
150C22⋊Q8—91no—
151#121—31no—
152C4×Q8—91no—
153C4×D4—91no—
154C4×D4—92no—
155#117—92no—
156C2×C42—91no—
157C23×C3—31no—
158C23×C3—32no—
159C22×D3C22×D312yes—
160C22×D3C22×D311yescharacteristic
161C3×D4—64no—
162C3×D4—62no—
163C2×C3×C4—21no—
164C2×C3×C4—22no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—32no—
169C2×C3×C4—62no—
170C2×C3×C4C22×D312yes—
171C2×C3×C4C22×D312yes—
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—61no—
181C2×Dic3—62no—
182C2×Dic3—61no—
183C2×Dic3—62no—
184C2×Dic3—64no—
185C2×Dic3—62no—
186C2×Dic3C22×D311yescharacteristic
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D312yes—
190D12—24no—
191D12—22no—
192C4×D3—22no—
193C4×D3—62no—
194C4×D3—64no—
195C4×D3—62no—
196Dic6—64no—
197Dic6—62no—
198Dic6—62no—
199C2×C32C2×#4111yescharacteristic
200C2×C32C2×#4111yescharacteristic
201C2×C32C2×#4111yescharacteristic
202C3×D3—22no—
203C3×D3—42no—
204C2×Q8—91no—
205C2×D4—32no—
206C2×D4—31no—
207C22×C4—31no—
208C22×C4—91no—
209C22×C4—91no—
210C22×C4—92no—
211C4⋊C4—32no—
212C4⋊C4—31no—
213C4⋊C4—32no—
214C4⋊C4—92no—
215C4⋊C4—92no—
216C4⋊C4D3211yescharacteristic
217C22⋊C4—32no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C22⋊C4—92no—
222C42—31no—
223C42—91no—
224C42—91no—
225C42—91no—
226C42—92no—
227C22×C3—21no—
228C22×C3—32no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—62no—
232C22×C3—62no—
233C22×C3—62no—
234C22×C3C23×D311yescharacteristic
235C22×C3C23×D311yescharacteristic
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—22no—
239C2×D3—22no—
240C2×D3—22no—
241C2×D3#23712yes—
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—42no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4#23712yes—
251C3×C4#23912yes—
252Dic3—21no—
253Dic3—21no—
254Dic3—21no—
255Dic3—22no—
256Dic3—32no—
257Dic3—32no—
258Dic3—62no—
259Dic3—64no—
260Dic3—62no—
261Dic3—61no—
262Dic3—62no—
263Dic3—61no—
264Dic3—121no—
265C32#52911yescharacteristic, complemented, Sylow
266C23—31no—
267C23—32no—
268Q8—182no—
269D4—64no—
270D4—62no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—62no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—92no—
279C2×C4—182no—
280C2×C4—184no—
281C2×C4—182no—
282C2×C4C2×D3212yes—
283C2×C4C2×D3211yescharacteristic
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—62no—
288C2×C3—122no—
289C2×C3C2×#23711yescharacteristic
290C2×C3C2×#23711yescharacteristic
291C2×C3C2×#23911yescharacteristic
292C2×C3D3×#4111yescharacteristic
293C2×C3D3×#4111yescharacteristic
294C2×C3D3×#4111yescharacteristic
295D3—22no—
296D3—42no—
297C22—32no—
298C22—61no—
299C22—61no—
300C22—62no—
301C22—62no—
302C22—62no—
303C22C22×D3211yescharacteristic
304C4—22no—
305C4—61no—
306C4—61no—
307C4—61no—
308C4—62no—
309C4—92no—
310C4—182no—
311C4—181no—
312C4#379612yes—
313C3—21no—
314C3#560011yescharacteristic, complemented
315C3#569111yescharacteristic, complemented
316C2—62no—
317C2—122no—
318C2C2×#379611yescharacteristic
319C2D3×#23711yescharacteristic
320C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC3×#895C2
hypercenterC4⋊C4D32
hypocenterC32#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096C2semidirect
3. C3×#890C2semidirect
4. #63898C2semidirect
5. #63898C2semidirect
6. #63884C2semidirect
7. #63878C2semidirect
8. C4×D3×Dic3C2semidirect
9. #63860C2semidirect
10. #63858C2semidirect
11. #63857C2semidirect
52. #890D3semidirect
265. C32#529semidirect
314. C3#5600semidirect
315. C3#5691semidirect

Maximal subgroups

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

subgroupindex
#640962
C3×#8952
C3×#8902
#638982
#638842
#638782
C4×D3×Dic32
#638602
#638582
#638572
#56913
#56003