Finite Group Databasebeta

← #10594476

Subgroups — #10594476

Order 576.

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

normal subgroups only

This group has 475 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#64102C211yesmaximal, characteristic
2C3×#889C212yesmaximal, complemented
3#63900C211yesmaximal, characteristic, complemented
4#63897C212yesmaximal
5#63874C211yesmaximal, characteristic, complemented
6#63870C211yesmaximal, characteristic, complemented
7#63865C211yesmaximal, characteristic, complemented
8#63865C211yesmaximal, characteristic, complemented
9#63848C212yesmaximal, complemented
10#63838C211yesmaximal, characteristic, complemented
11#63836C212yesmaximal, complemented
12#5693—32nomaximal
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×#627C2211yescharacteristic
17C2×#624C2212yes—
18C32×C4⋊C4C2211yescharacteristic
19#3752C2211yescharacteristic
20#3752C2211yescharacteristic
21C4×C3⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic3C2212yes—
23C3×Dic3⋊C4C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26C3×Dic3⋊C4C2212yes—
27C3×C4×Dic3C2212yes—
28#3724C2211yescharacteristic
29#3724C2211yescharacteristic
30#3724C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32#3722C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35Dic32C2212yes—
36C3×#125—32no—
37#898—32no—
38#898—32no—
39#896—32no—
40#896—32no—
41#894—32no—
42#894—32no—
43#893—32no—
44#893—61no—
45#892—32no—
46D3×C4⋊C4—32no—
47#889—32no—
48#889D312yescomplemented
49#887—32no—
50#872—32no—
51#872—32no—
52C22×C3⋊D3C2311yescharacteristic
53C22×C3⋊D3C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C4⋊(C3⋊D3)—22no—
60C4×C3⋊D3—22no—
61C4×C3⋊D3—22no—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64C2×C3×Dic3C2312yes—
65C2×C3×Dic3C2312yes—
66#627—22no—
67#624—24no—
68#552—91noSylow
69C2×D12—32no—
70C2×D12—61no—
71C2×C4×D3—32no—
72C2×C4×D3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—32no—
75C2×C4×D3—61no—
76C2×C4×D3—61no—
77C2×Dic6—32no—
78C3×C4⋊C4—21no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—32no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4—32no—
84C3×C4⋊C4C2×D312yes—
85C3×C42—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—32no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—61no—
95(C2×D3)⋊C4—61no—
96C4⋊Dic3—32no—
97C4⋊Dic3C2×D312yes—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4C2×D312yes—
103Dic3⋊C4C2×D312yes—
104Dic3⋊C4C2×D312yes—
105Dic3⋊C4C2×D312yes—
106C4×Dic3—32no—
107C4×Dic3—32no—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C4×Dic3C2×D312yes—
111C22×C32C2411yescharacteristic
112C2×C3⋊D3—21no—
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3—21no—
116C2×C3⋊D3—21no—
117C2×C3⋊D3#4112yes—
118C32×C4—21no—
119C32×C4—21no—
120C32×C4—21no—
121C3⋊Dic3—21no—
122C3⋊Dic3#4112yes—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C3×Dic3—22no—
126C3×Dic3—22no—
127#126—92no—
128#125—32no—
129#125—91no—
130#123—92no—
131C22⋊Q8—91no—
132C22⋊Q8—91no—
133C4×Q8—91no—
134C4×D4—91no—
135#117—91no—
136#117—92no—
137C2×C4⋊C4—91no—
138C22×D3—32no—
139C22×D3—32no—
140C22×D3—61no—
141C22×D3—61no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4C22×D312yes—
150C2×C3×C4C22×D312yes—
151C2×C3×C4C22×D312yes—
152C2×Dic3—32no—
153C2×Dic3—32no—
154C2×Dic3—61no—
155C2×Dic3—61no—
156C2×Dic3C22×D312yes—
157C2×Dic3C22×D312yes—
158C2×Dic3C22×D312yes—
159C2×Dic3C22×D312yes—
160D12—64no—
161D12—64no—
162C4×D3—64no—
163C4×D3—64no—
164C4×D3—64no—
165C4×D3—64no—
166C4×D3—122no—
167C4×D3—122no—
168Dic6—64no—
169C2×C32C2×#4111yescharacteristic
170C2×C32C2×#4111yescharacteristic
171C2×C322−1+411yescharacteristic
172C3⋊D3—22no—
173C3⋊D3—41no—
174C2×Q8—91no—
175C2×D4—91no—
176C22×C4—92no—
177C22×C4—91no—
178C22×C4—91no—
179C4⋊C4—32no—
180C4⋊C4—32no—
181C4⋊C4—32no—
182C4⋊C4—32no—
183C4⋊C4—32no—
184C4⋊C4—91no—
185C4⋊C4—91no—
186C4⋊C4—91no—
187C4⋊C4—92no—
188C4⋊C4D3211yescharacteristic
189C22⋊C4—92no—
190C22⋊C4—92no—
191C22⋊C4—92no—
192C22⋊C4—91no—
193C22⋊C4—91no—
194C42—32no—
195C42—92no—
196C42—91no—
197C22×C3—21no—
198C22×C3C23×D312yes—
199C2×D3—34no—
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—62no—
206C2×D3—62no—
207C2×D3—121no—
208C2×D3—121no—
209C2×D3—121no—
210C2×D3—121no—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—62no—
218C3×C4—62no—
219C3×C4—62no—
220C3×C4—62no—
221Dic3—22no—
222Dic3—22no—
223Dic3—22no—
224Dic3—22no—
225Dic3—34no—
226Dic3—62no—
227Dic3—62no—
228Dic3—121no—
229C32#55211yescharacteristic, complemented, Sylow
230C23—91no—
231C23—91no—
232Q8—182no—
233D4—182no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—182no—
241C2×C4—182no—
242C2×C4—184no—
243C2×C4C2×D3211yescharacteristic
244C2×C4C2×D3211yescharacteristic
245C2×C4C2×D3211yescharacteristic
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—21no—
249C2×C3#100712yes—
250C2×C3D3×#4112yes—
251C2×C3D3×#4112yes—
252D3—64no—
253D3—122no—
254D3—122no—
255D3—122no—
256C22—92no—
257C22—181no—
258C22—181no—
259C22—181no—
260C22—181no—
261C22—181no—
262C22C22×D3211yescharacteristic
263C4—21no—
264C4—21no—
265C4—21no—
266C4—62no—
267C4—62no—
268C4—62no—
269C4—62no—
270C4—92no—
271C4—181no—
272C3—21no—
273C3#569312yescomplemented
274C2—182no—
275C2—361no—
276C2#6430611yescharacteristic
277C2#6431211yescharacteristic
278C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×C4⋊(C3⋊D3)C22
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
2. C3×#889C2semidirect
3. #63900C2semidirect
5. #63874C2semidirect
6. #63870C2semidirect
7. #63865C2semidirect
8. #63865C2semidirect
9. #63848C2semidirect
10. #63838C2semidirect
11. #63836C2semidirect
48. #889D3semidirect
229. C32#552semidirect
273. C3#5693semidirect

Maximal subgroups

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

subgroupindex
#641022
C3×#8892
#639002
#638972
#638742
#638702
#638652
#638482
#638382
#638362
#56933