Finite Group Databasebeta

← #10595501

Subgroups — #10595501

Order 576.

778 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C23×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×#3805C211yesmaximal, characteristic, complemented
3C3×#953C212yesmaximal, complemented
4#63982C212yesmaximal, complemented
5#63980C212yesmaximal, complemented
6#67⋊Dic3C214yesmaximal, complemented
7#63962C212yesmaximal, complemented
8#63957C211yesmaximal, characteristic, complemented
9#5586—32nomaximal
10C24×C32C2211yescharacteristic
11C22×C3⋊Dic3—24no—
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2212yes—
14C22×C3⋊Dic3C2212yes—
15C22×C3⋊Dic3C2211yescharacteristic
16C2×C3×#67C2212yes—
17C2×C3×#67C2214yes—
18C2×#627C2211yescharacteristic, complemented
19C2×#625C2211yescharacteristic, complemented
20#3805—24no—
21C2×D3×Dic3C2212yescomplemented
22C3×#217C2212yes—
23C3×#217C2214yes—
24#3724C2212yescomplemented
25Dic3⋊Dic3C2214yescomplemented
26(C2×D3)⋊Dic3C2214yescomplemented
27Dic32C2212yescomplemented
28C23×Dic3—32no—
29C23×Dic3—61no—
30C2×#237—32no—
31C3×#120—32no—
32#953D312yescomplemented
33#939—34no—
34#935—32no—
35D4×Dic3—34no—
36#886—32no—
37#67⋊C4—34no—
38#878—34no—
39#877—32no—
40C23×C32—22no—
41C23×C32—24no—
42C23×C32C2311yescharacteristic
43C23×C32C2312yes—
44C22×C3×D3C2312yes—
45C2×C3⋊Dic3—22no—
46C2×C3⋊Dic3—24no—
47C2×C3⋊Dic3—24no—
48C2×C3⋊Dic3D414yescomplemented
49C2×C3⋊Dic3C2312yes—
50C2×C3⋊Dic3C2311yescharacteristic
51C2×C3⋊Dic3C2311yescharacteristic
52C3×#67—24no—
53C3×#67—28no—
54C2×C3×Dic3C2314yes—
55C2×C3×Dic3C2312yes—
56#627—22no—
57#625—22no—
58D3×Dic3—24no—
59#525—91noSylow
60C24×C3—21no—
61C24×C3C2×D312yes—
62C2×C3×D4—34no—
63C2×C3×D4—32no—
64C2×#67—32no—
65C2×#67C2×D314yes—
66C2×#67C2×D312yes—
67C22×Dic3—32no—
68C22×Dic3—34no—
69C22×Dic3—34no—
70C22×Dic3—32no—
71C22×Dic3—32no—
72C22×Dic3—61no—
73C22×Dic3—62no—
74C22×Dic3—62no—
75C22×Dic3—61no—
76C22×Dic3—68no—
77C22×Dic3—68no—
78#237—68no—
79C2×C4×D3—32no—
80C2×Dic6—32no—
81C3×C22⋊C4—32no—
82C3×C22⋊C4—34no—
83#217—34no—
84#217C2×D312yes—
85#217C2×D314yes—
86(C2×D3)⋊C4—34no—
87C4⋊Dic3—34no—
88Dic3⋊C4—34no—
89Dic3⋊C4—34no—
90C4×Dic3—34no—
91C22×C32—28no—
92C22×C32—22no—
93C22×C32—22no—
94C22×C32—22no—
95C22×C32C2×D412yes—
96C22×C32#4114yescomplemented
97C22×C32C2411yescharacteristic
98C2×C3×D3—22no—
99C2×C3×D3—24no—
100C3⋊Dic3—22no—
101C3⋊Dic3C2×D412yes—
102C3⋊Dic3C2×D412yes—
103C3×Dic3—22no—
104C3×Dic3—24no—
105C2×#41—91no—
106C23×C4—91no—
107#123—92no—
108C22⋊Q8—92no—
109#121—92no—
110#120—32no—
111C4×D4—94no—
112#117—91no—
113C23×C3—24no—
114C23×C3—28no—
115C23×C3—24no—
116C23×C3—28no—
117C23×C3—22no—
118C23×C3—21no—
119C23×C3—32no—
120C23×C3C22×D312yes—
121C23×C3C22×D314yes—
122C22×D3C22×D312yes—
123C3×D4—64no—
124C3×D4—68no—
125C2×C3×C4—34no—
126C2×C3×C4—32no—
127#67—24no—
128#67—28no—
129#67—64no—
130C2×Dic3—38no—
131C2×Dic3—32no—
132C2×Dic3—32no—
133C2×Dic3—34no—
134C2×Dic3—64no—
135C2×Dic3—61no—
136C2×Dic3—61no—
137C2×Dic3—62no—
138C2×Dic3—64no—
139C2×Dic3—68no—
140C2×Dic3—68no—
141C2×Dic3—68no—
142C2×Dic3—68no—
143C2×Dic3—64no—
144C2×Dic3—64no—
145C2×Dic3C22×D314yes—
146C2×Dic3C22×D312yes—
147C4×D3—64no—
148Dic6—64no—
149C2×C32—22no—
150C2×C32—24no—
151C2×C32C22×D411yescharacteristic
152C2×C32C2×#4112yes—
153C3×D3—42no—
154C24D3211yescharacteristic
155#41—184no—
156C2×Q8—91no—
157C2×D4—34no—
158C2×D4—32no—
159C2×D4—91no—
160C22×C4—92no—
161C22×C4—91no—
162C22×C4—92no—
163C22×C4—92no—
164C22×C4—91no—
165C22×C4—184no—
166C4⋊C4—92no—
167C4⋊C4—94no—
168C22⋊C4—32no—
169C22⋊C4—34no—
170C22⋊C4—94no—
171C42—92no—
172C22×C3—24no—
173C22×C3—216no—
174C22×C3—216no—
175C22×C3—24no—
176C22×C3—24no—
177C22×C3—24no—
178C22×C3—24no—
179C22×C3—24no—
180C22×C3—21no—
181C22×C3—24no—
182C22×C3—22no—
183C22×C3—62no—
184C22×C3—64no—
185C22×C3D3×D414yes—
186C22×C3#23718yescomplemented
187C22×C3C23×D312yes—
188C2×D3—22no—
189C2×D3—24no—
190C3×C4—62no—
191C3×C4—64no—
192Dic3—22no—
193Dic3—24no—
194Dic3—34no—
195Dic3—34no—
196Dic3—62no—
197Dic3—62no—
198Dic3—64no—
199Dic3—64no—
200C32#52511yescharacteristic, complemented, Sylow
201C23—24no—
202C23—22no—
203C23—32no—
204C23C2×D3212yes—
205C23C2×D3211yescharacteristic
206Q8—182no—
207D4—68no—
208D4—64no—
209D4—182no—
210C2×C4—34no—
211C2×C4—32no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—94no—
215C2×C4—92no—
216C2×C4—184no—
217C2×C4—184no—
218C2×C4—184no—
219C2×C4—182no—
220C2×C3—28no—
221C2×C3—24no—
222C2×C3—24no—
223C2×C3—28no—
224C2×C3—22no—
225C2×C3—21no—
226C2×C3—122no—
227C2×C3C2×D3×D412yes—
228C2×C3C2×#23714yes—
229D3—42no—
230C22—28no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—64no—
235C22—62no—
236C22#380514yescomplemented
237C22#381112yes—
238C22C22×D3211yescharacteristic
239C4—64no—
240C4—62no—
241C4—92no—
242C4—92no—
243C4—182no—
244C3—21no—
245C3#558612yescomplemented
246C2—24no—
247C2—22no—
248C2—122no—
249C2C2×#380512yes—
250C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC24×C32C22
hypercenterC24D32
hypocenterC32#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊Dic3C2semidirect
2. C2×#3805C2semidirect
3. C3×#953C2semidirect
4. #63982C2semidirect
5. #63980C2semidirect
6. #67⋊Dic3C2semidirect
7. #63962C2semidirect
8. #63957C2semidirect
18. C2×#627C22semidirect
19. C2×#625C22semidirect
21. C2×D3×Dic3C22semidirect
24. #3724C22semidirect
25. Dic3⋊Dic3C22semidirect
26. (C2×D3)⋊Dic3C22semidirect
27. Dic32C22semidirect
32. #953D3semidirect
48. C2×C3⋊Dic3D4semidirect
96. C22×C32#41semidirect
186. C22×C3#237semidirect
200. C32#525semidirect
236. C22#3805semidirect
245. C3#5586semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊Dic32
C2×#38052
C3×#9532
#639822
#639802
#67⋊Dic32
#639622
#639572
#55863