Finite Group Databasebeta

← #10594168

Subgroups — #10594168

Order 576.

503 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 138 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
2C3×#878C212yesmaximal, complemented
3#63982C211yesmaximal, characteristic, complemented
4C2×#3724C211yesmaximal, characteristic, complemented
5#63952C212yesmaximal
6#63900C211yesmaximal, characteristic, complemented
7#63888C212yesmaximal, complemented
8#63873C211yesmaximal, characteristic, complemented
9#63865C211yesmaximal, characteristic, complemented
10#63847C212yesmaximal, complemented
11#63838C211yesmaximal, characteristic, complemented
12#5636—32nomaximal
13C2×#638C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C22×C3×Dic3C2212yes—
17C2×C3×Dic6C2212yes—
18C2×#627C2211yescharacteristic
19C2×#624C2212yes—
20C32×C22⋊C4C2211yescharacteristic
21#3752C2211yescharacteristic
22#3750C2211yescharacteristic
23C4×C3⋊Dic3C2211yescharacteristic
24C3×#217C2212yes—
25C3×C4⋊Dic3C2212yes—
26C3×Dic3⋊C4C2212yes—
27C3×Dic3⋊C4C2212yes—
28#3724C2211yescharacteristic
29#3724C2212yes—
30#3724C2212yes—
31#3724C2211yescharacteristic
32#3724C2211yescharacteristic
33Dic3⋊Dic3C2212yes—
34#3722C2212yes—
35Dic32C2212yes—
36C3×C22⋊Q8—32no—
37#946—32no—
38#944—32no—
39C4×#67—32no—
40C2×Dic3⋊C4—32no—
41#896—32no—
42#896—32no—
43D3×C4⋊C4—32no—
44#889—32no—
45#889—32no—
46#887—32no—
47#67⋊C4—32no—
48#67⋊C4—32no—
49#67⋊C4—61no—
50#878—32no—
51#878D312yescomplemented
52C23×C32C2311yescharacteristic
53C22×C3⋊D3C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56#638Q814yescomplemented
57C2×C3⋊Dic3—22no—
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C4×C3⋊D3—22no—
62C2×C3×Dic3—42no—
63C2×C3×Dic3C2312yes—
64C2×C3×Dic3C2312yes—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C3×Dic6—42no—
68#627—22no—
69#624—42no—
70#554—91noSylow
71C2×C3×Q8—32no—
72C22×C3×C4—32no—
73C2×#67—32no—
74C2×#67—61no—
75C22×Dic3—32no—
76C22×Dic3—61no—
77C22×Dic3C2×D312yes—
78C2×C4×D3—32no—
79C2×C4×D3—32no—
80C2×C4×D3—32no—
81C2×C4×D3—61no—
82C2×Dic6—32no—
83C2×Dic6C2×D312yes—
84C3×C4⋊C4—32no—
85C3×C4⋊C4—32no—
86C3×C4⋊C4—32no—
87C3×C22⋊C4—21no—
88C3×C22⋊C4—32no—
89C3×C22⋊C4C2×D312yes—
90#217C2×D312yes—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—32no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—61no—
95C4⋊Dic3—32no—
96C4⋊Dic3C2×D312yes—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—61no—
107Dic3⋊C4C2×D312yes—
108Dic3⋊C4C2×D312yes—
109C4×Dic3—32no—
110C4×Dic3—32no—
111C4×Dic3—32no—
112C4×Dic3—61no—
113C22×C32—21no—
114C22×C32—21no—
115C22×C32C2×Q812yes—
116C22×C32C2411yescharacteristic
117C2×C3⋊D3—21no—
118C2×C3⋊D3—21no—
119C2×C3⋊D3C2×Q812yes—
120C32×C4—21no—
121C32×C4—21no—
122C3⋊Dic3—21no—
123C3⋊Dic3C2×Q812yes—
124C3⋊Dic3#4112yes—
125C3×Dic3—22no—
126C3×Dic3—22no—
127C3×Dic3—22no—
128C3×Dic3—22no—
129C4⋊Q8—91no—
130#125—92no—
131C22⋊Q8—32no—
132C22⋊Q8—91no—
133C22⋊Q8—91no—
134C22⋊Q8—92no—
135C4×Q8—91no—
136C4×D4—92no—
137C4×D4—91no—
138C2×C4⋊C4—91no—
139C2×C4⋊C4—91no—
140C23×C3—21no—
141C23×C3C22×D312yes—
142C22×D3—32no—
143C22×D3—61no—
144C3×Q8—122no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—32no—
151C2×C3×C4—122no—
152C2×C3×C4C22×D312yes—
153C2×C3×C4C22×D312yes—
154#67—38no—
155#67—64no—
156C2×Dic3—32no—
157C2×Dic3—32no—
158C2×Dic3—32no—
159C2×Dic3—42no—
160C2×Dic3—61no—
161C2×Dic3—61no—
162C2×Dic3—61no—
163C2×Dic3—64no—
164C2×Dic3—122no—
165C2×Dic3C22×D312yes—
166C2×Dic3C22×D312yes—
167C2×Dic3C22×D312yes—
168C2×Dic3C22×D312yes—
169C4×D3—64no—
170C4×D3—122no—
171C4×D3—122no—
172C4×D3—122no—
173Dic6—42no—
174Dic6—64no—
175C2×C32—22no—
176C2×C32C22×Q811yescharacteristic
177C2×C32C2×#4111yescharacteristic
178C2×C32H(5,𝔽2)11yescharacteristic
179C3⋊D3—22no—
180C2×Q8—32no—
181C2×Q8—91no—
182C2×D4—91no—
183C22×C4—32no—
184C22×C4—92no—
185C22×C4—91no—
186C22×C4—91no—
187C4⋊C4—32no—
188C4⋊C4—32no—
189C4⋊C4—32no—
190C4⋊C4—91no—
191C4⋊C4—92no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C4⋊C4—92no—
195C4⋊C4—92no—
196C4⋊C4—91no—
197C22⋊C4—32no—
198C22⋊C4—92no—
199C22⋊C4—91no—
200C22⋊C4D3211yescharacteristic
201C42—92no—
202C42—91no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—21no—
206C22×C3—22no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3D3×Q814yes—
210C22×C3C23×D312yes—
211C2×D3—34no—
212C2×D3—62no—
213C2×D3—62no—
214C2×D3—62no—
215C2×D3—121no—
216C2×D3—121no—
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—41no—
220C3×C4—41no—
221C3×C4—62no—
222C3×C4—62no—
223C3×C4—62no—
224C3×C4—62no—
225Dic3—22no—
226Dic3—22no—
227Dic3—22no—
228Dic3—22no—
229Dic3—34no—
230Dic3—34no—
231Dic3—62no—
232Dic3—62no—
233Dic3—62no—
234Dic3—121no—
235C32#55411yescharacteristic, complemented, Sylow
236C23—91no—
237C23C2×D3211yescharacteristic
238Q8—122no—
239Q8—182no—
240D4—94no—
241C2×C4—32no—
242C2×C4—32no—
243C2×C4—32no—
244C2×C4—32no—
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—122no—
249C2×C4—182no—
250C2×C4—182no—
251C2×C4—362no—
252C2×C4C2×D3211yescharacteristic
253C2×C4C2×D3211yescharacteristic
254C2×C3—24no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—42no—
259C2×C3#100412yes—
260C2×C3C2×D3×Q812yes—
261C2×C3D3×#4112yes—
262D3—64no—
263D3—122no—
264C22—21no—
265C22—21no—
266C22—92no—
267C22—181no—
268C22—181no—
269C22#379712yes—
270C22C22×D3211yescharacteristic
271C4—21no—
272C4—21no—
273C4—62no—
274C4—62no—
275C4—62no—
276C4—62no—
277C4—92no—
278C4—92no—
279C4—181no—
280C3—21no—
281C3#563612yescomplemented
282C2—22no—
283C2—182no—
284C2C2×#379711yescharacteristic
285C2#6431211yescharacteristic
286C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×#638C22
hypercenterC22⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#878C2semidirect
3. #63982C2semidirect
4. C2×#3724C2semidirect
6. #63900C2semidirect
7. #63888C2semidirect
8. #63873C2semidirect
9. #63865C2semidirect
10. #63847C2semidirect
11. #63838C2semidirect
51. #878D3semidirect
56. #638Q8semidirect
235. C32#554semidirect
281. C3#5636semidirect

Maximal subgroups

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

subgroupindex
#640902
C3×#8782
#639822
C2×#37242
#639522
#639002
#638882
#638732
#638652
#638472
#638382
#56363