Finite Group Databasebeta

← #10594229

Subgroups — #10594229

Order 576.

523 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 124 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#64090C211yesmaximal, characteristic, complemented
2C3×#883C212yesmaximal, complemented
3#63980C211yesmaximal, characteristic, complemented
4#63958C212yesmaximal, complemented
5#63957C211yesmaximal, characteristic, complemented
6#63912C211yesmaximal, characteristic, complemented
7#63904C212yesmaximal, complemented
8C4×#625C211yesmaximal, characteristic, complemented
9#63900C211yesmaximal, characteristic, complemented
10#63870C211yesmaximal, characteristic, complemented
11#63858C212yesmaximal, complemented
12#5661—32nomaximal
13C2×#638C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#67C2212yes—
17C2×C3×C4×D3C2212yes—
18C2×#626C2212yes—
19C2×#625C2211yescharacteristic
20C2×#624C2212yes—
21C32×C22⋊C4C2211yescharacteristic
22#3752C2211yescharacteristic
23#3750C2211yescharacteristic
24C4×C3⋊Dic3C2211yescharacteristic
25C3×#217C2212yes—
26C3×(C2×D3)⋊C4C2212yes—
27C3×C4⋊Dic3C2212yes—
28C3×Dic3⋊C4C2212yes—
29#3724C2211yescharacteristic
30#3724C2211yescharacteristic
31#3724C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35Dic32C2212yes—
36C3×#123—32no—
37#939—32no—
38#938—32no—
39C4×#67—32no—
40C4×#67—32no—
41#898—32no—
42#893—32no—
43D3×C4⋊C4—32no—
44#889—32no—
45#884—32no—
46#884—32no—
47#883D312yescomplemented
48#67⋊C4—32no—
49#67⋊C4—61no—
50#879—32no—
51#877—32no—
52C23×C32C2311yescharacteristic
53C22×C3⋊D3C2311yescharacteristic
54C22×C3×D3C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57#638—22no—
58C2×C3⋊Dic3—22no—
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2311yescharacteristic
62C4×C3⋊D3—22no—
63C3×#67—42no—
64C2×C3×Dic3C2312yes—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C3×C4×D3—42no—
68#626—24no—
69#625—22no—
70#624—42no—
71#553—91noSylow
72C2×C3×D4—32no—
73C22×C3×C4—32no—
74C2×#67—32no—
75C2×#67—32no—
76C2×#67—32no—
77C2×#67—61no—
78C2×#67C2×D312yes—
79C22×Dic3—32no—
80C22×Dic3—61no—
81C2×D12—32no—
82C2×C4×D3—32no—
83C2×C4×D3—32no—
84C2×C4×D3—32no—
85C2×C4×D3—61no—
86C2×C4×D3C2×D312yes—
87C3×C4⋊C4—32no—
88C3×C4⋊C4—32no—
89C3×C22⋊C4—21no—
90C3×C22⋊C4—32no—
91C3×C22⋊C4—32no—
92C3×C22⋊C4C2×D312yes—
93#217—32no—
94#217—32no—
95#217C2×D312yes—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4—32no—
99(C2×D3)⋊C4—61no—
100(C2×D3)⋊C4C2×D312yes—
101C4⋊Dic3—32no—
102C4⋊Dic3C2×D312yes—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—61no—
109Dic3⋊C4C2×D312yes—
110C4×Dic3—32no—
111C4×Dic3—32no—
112C4×Dic3—32no—
113C4×Dic3—61no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32C2411yescharacteristic
118C2×C3⋊D3—21no—
119C2×C3⋊D3—21no—
120C2×C3⋊D3#4112yes—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C2×C3×D3—22no—
124C32×C4—21no—
125C32×C4—21no—
126C3⋊Dic3—21no—
127C3⋊Dic3—21no—
128C3⋊Dic3#4112yes—
129C3×Dic3—22no—
130C3×Dic3—22no—
131C3×Dic3—22no—
132#126—92no—
133#125—91no—
134#123—32no—
135#121—91no—
136#121—92no—
137#121—91no—
138C4×D4—92no—
139C4×D4—91no—
140C4×D4—91no—
141#117—91no—
142C2×C4⋊C4—91no—
143C23×C3—21no—
144C23×C3—32no—
145C23×C3C22×D312yes—
146C22×D3—32no—
147C22×D3—61no—
148C22×D3C22×D312yes—
149C3×D4—122no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—122no—
156C2×C3×C4C22×D312yes—
157C2×C3×C4C22×D312yes—
158#67—42no—
159#67—64no—
160#67—64no—
161#67—64no—
162#67—64no—
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—61no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—64no—
170C2×Dic3—122no—
171C2×Dic3C22×D312yes—
172C2×Dic3C22×D312yes—
173C2×Dic3C22×D312yes—
174D12—64no—
175C4×D3—42no—
176C4×D3—64no—
177C4×D3—122no—
178C4×D3—122no—
179C4×D3—122no—
180C2×C32—41no—
181C2×C32C2×#4111yescharacteristic
182C2×C32C2×#4111yescharacteristic
183C2×C32H(5,𝔽2)11yescharacteristic
184C3⋊D3—22no—
185C3×D3—42no—
186C2×D4—32no—
187C2×D4—92no—
188C2×D4—91no—
189C2×D4—91no—
190C22×C4—32no—
191C22×C4—92no—
192C22×C4—91no—
193C22×C4—91no—
194C4⋊C4—32no—
195C4⋊C4—32no—
196C4⋊C4—91no—
197C4⋊C4—91no—
198C4⋊C4—91no—
199C4⋊C4—92no—
200C4⋊C4—91no—
201C22⋊C4—32no—
202C22⋊C4—32no—
203C22⋊C4—92no—
204C22⋊C4—92no—
205C22⋊C4—91no—
206C22⋊C4D3211yescharacteristic
207C42—92no—
208C42—91no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—62no—
217C22×C3—62no—
218C22×C3—62no—
219C22×C3C23×D312yes—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—34no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—121no—
228C2×D3—121no—
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4—62no—
234C3×C4—62no—
235C3×C4—62no—
236Dic3—22no—
237Dic3—22no—
238Dic3—22no—
239Dic3—34no—
240Dic3—62no—
241Dic3—62no—
242Dic3—62no—
243Dic3—62no—
244Dic3—121no—
245C32#55311yescharacteristic, complemented, Sylow
246C23—32no—
247C23—91no—
248C23C2×D3211yescharacteristic
249D4—122no—
250D4—184no—
251D4—182no—
252D4—182no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—122no—
260C2×C4—182no—
261C2×C4—182no—
262C2×C4—362no—
263C2×C4C2×D3211yescharacteristic
264C2×C4C2×D3211yescharacteristic
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3—122no—
271C2×C3#100412yes—
272C2×C3D3×#4112yes—
273C2×C3D3×#4112yes—
274D3—42no—
275D3—64no—
276D3—122no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22—92no—
284C22—181no—
285C22—181no—
286C22C22×D3211yescharacteristic
287C4—21no—
288C4—21no—
289C4—62no—
290C4—62no—
291C4—62no—
292C4—92no—
293C4—181no—
294C4—181no—
295C3—21no—
296C3#566112yescomplemented
297C2—41no—
298C2—122no—
299C2—182no—
300C2#6430611yescharacteristic
301C2#6431211yescharacteristic
302C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64090C2semidirect
2. C3×#883C2semidirect
3. #63980C2semidirect
4. #63958C2semidirect
5. #63957C2semidirect
6. #63912C2semidirect
7. #63904C2semidirect
8. C4×#625C2semidirect
9. #63900C2semidirect
10. #63870C2semidirect
11. #63858C2semidirect
47. #883D3semidirect
245. C32#553semidirect
296. C3#5661semidirect

Maximal subgroups

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

subgroupindex
#640902
C3×#8832
#639802
#639582
#639572
#639122
#639042
C4×#6252
#639002
#638702
#638582
#56613