Finite Group Databasebeta

← #10591499

Subgroups — #10591499

Order 576.

273 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C3×#938C211yesmaximal, characteristic
2C2×C3×#214C211yesmaximal, characteristic, complemented
3#63856C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊C8C211yesmaximal, characteristic, complemented
5#63659C211yesmaximal, characteristic, complemented
6#63573C211yesmaximal, characteristic, complemented
7#63565C211yesmaximal, characteristic, complemented
8#5171—31nomaximal
9#5041—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C3×#67—21no—
12C2×C3×C4×D3C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#217—21no—
17C3×#214—22no—
18C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20#3724—21no—
21(C2×D3)⋊Dic3—21no—
22D3×C3⋊C8—22no—
23C2×C3×SD16—31no—
24C3×#121—31no—
25#946—31no—
26#938D311yescharacteristic
27C2×#214D311yescharacteristic, complemented
28#924—31no—
29C22×C3⋊C8—31no—
30C2×D3×C8—31no—
31#835—31no—
32#832—31no—
33#832—31no—
34#832—61no—
35#817—31no—
36#810—31no—
37#808—31no—
38C23×C32—21no—
39C22×C3×D3D411yescharacteristic
40C32×D4—21no—
41C32×D4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44C3×#67—42no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3D411yescharacteristic
48C3×C4×D3—21no—
49C3×C4×D3—21no—
50C3×Dic6—21no—
51C3×Dic6—21no—
52C3⋊(C3⋊C8)—21no—
53C3×C3⋊C8D412yescomplemented
54#465—91noSylow
55C2×C3×Q8—31no—
56C2×C3×D4—21no—
57C2×C3×D4—61no—
58C2×C3×D4C2×D311yescharacteristic
59C2×C3×D4C2×D311yescharacteristic
60C22×C3×C4—31no—
61C2×#67—21no—
62C2×C4×D3C2×D311yescharacteristic
63C2×Dic6C2×D311yescharacteristic
64C3×SD16—62no—
65C2×C3×C8—31no—
66C3×C4⋊C4—31no—
67C3×C22⋊C4—61no—
68#217—21no—
69#217—61no—
70#214—22no—
71(C2×D3)⋊C4—61no—
72C4⋊Dic3—31no—
73C4⋊Dic3—31no—
74C4⋊Dic3—61no—
75C4⋊Dic3C2×D311yescharacteristic
76Dic3⋊C4—61no—
77Dic3⋊C4—61no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—31no—
80C2×C3⋊C8—61no—
81C2×C3⋊C8—62no—
82C2×C3⋊C8C2×D311yescharacteristic
83D3×C8—62no—
84C22×C32—41no—
85C22×C32—41no—
86C22×C32—41no—
87C22×C32C2×D411yescharacteristic
88C2×C3×D3—21no—
89C2×C3×D3—21no—
90C2×C3×D3SD1612yescomplemented
91C32×C4C2×D411yescharacteristic
92C32×C4#4111yescharacteristic
93C3⋊Dic3—41no—
94C3×Dic3—21no—
95C3×Dic3—41no—
96C3×Dic3—41no—
97C2×SD16—31no—
98C22×C8—91no—
99C22⋊Q8—91no—
100#121—31no—
101#106—91no—
102#103—91no—
103#102—91no—
104C23×C3—21no—
105C23×C3—21no—
106C23×C3—31no—
107C23×C3—41no—
108C22×D3#6711yescharacteristic
109C3×Q8—61no—
110C3×Q8—61no—
111C3×D4—22no—
112C3×D4—21no—
113C3×D4—21no—
114C3×D4—21no—
115C3×D4—21no—
116C3×D4—41no—
117C3×D4—122no—
118C2×C3×C4—21no—
119C2×C3×C4—31no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4C22×D311yescharacteristic
125C2×C3×C4C22×D311yescharacteristic
126#67—42no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131C2×Dic3—121no—
132C2×Dic3#6711yescharacteristic
133C4×D3—21no—
134C4×D3—21no—
135Dic6—21no—
136Dic6—21no—
137C3×C8—32no—
138C3⋊C8—61no—
139C3⋊C8—61no—
140C3⋊C8—121no—
141C3⋊C8#6712yescomplemented
142C2×C32—81no—
143C2×C32#12111yescharacteristic
144C2×C32C2×SD1611yescharacteristic
145C2×C32#13511yescharacteristic
146C3×D3—22no—
147C2×Q8—31no—
148C2×D4—61no—
149C2×D4D3211yescharacteristic
150C22×C4—31no—
151SD16—62no—
152C2×C8—31no—
153C2×C8—91no—
154C2×C8—182no—
155C4⋊C4—31no—
156C4⋊C4—91no—
157C4⋊C4—181no—
158C22⋊C4—61no—
159C22⋊C4—181no—
160C22×C3—21no—
161C22×C3—32no—
162C22×C3—42no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—61no—
170C22×C3—61no—
171C22×C3—81no—
172C22×C3—81no—
173C22×C3D3×D411yescharacteristic
174C22×C3C2×#6711yescharacteristic
175C2×D3—21no—
176C2×D3—21no—
177C2×D3#21412yescomplemented
178C3×C4—21no—
179C3×C4—21no—
180C3×C4—61no—
181C3×C4—121no—
182C3×C4—121no—
183C3×C4D3×D411yescharacteristic
184C3×C4#23711yescharacteristic
185C3×C4#23711yescharacteristic
186C3×C4C2×#6711yescharacteristic
187Dic3—21no—
188Dic3—41no—
189Dic3—41no—
190Dic3—121no—
191Dic3—121no—
192Dic3—241no—
193C32#46511yescharacteristic, complemented, Sylow
194C23—21no—
195C23—31no—
196Q8—61no—
197Q8—61no—
198D4—21no—
199D4—21no—
200D4—122no—
201C2×C4—31no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—181no—
207C2×C4C2×D3211yescharacteristic
208C8—32no—
209C8—181no—
210C2×C3—21no—
211C2×C3—21no—
212C2×C3—21no—
213C2×C3—62no—
214C2×C3—82no—
215C2×C3—81no—
216C2×C3—81no—
217C2×C3D3×SD1611yescharacteristic
218C2×C3#91611yescharacteristic
219C2×C3C2×#21411yescharacteristic
220C2×C3#93811yescharacteristic
221C2×C3#93911yescharacteristic
222C2×C3#95011yescharacteristic
223D3—22no—
224C22—32no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—61no—
229C22—61no—
230C22D3×#6711yescharacteristic
231C4—61no—
232C4—121no—
233C4—121no—
234C4—361no—
235C4#380511yescharacteristic
236C4D3×#6711yescharacteristic
237C3—21no—
238C3#504111yescharacteristic, complemented
239C3#517111yescharacteristic, complemented
240C2—62no—
241C2—81no—
242C2D3×#21411yescharacteristic
243C2#6393511yescharacteristic
244C2#6397011yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC3×#938C2
hypercenterC2×D4D32
hypocenterC32#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#214C2semidirect
3. #63856C2semidirect
4. C2×D3×C3⋊C8C2semidirect
5. #63659C2semidirect
6. #63573C2semidirect
7. #63565C2semidirect
27. C2×#214D3semidirect
53. C3×C3⋊C8D4semidirect
90. C2×C3×D3SD16semidirect
141. C3⋊C8#67semidirect
177. C2×D3#214semidirect
193. C32#465semidirect
238. C3#5041semidirect
239. C3#5171semidirect

Maximal subgroups

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

subgroupindex
C3×#9382
C2×C3×#2142
#638562
C2×D3×C3⋊C82
#636592
#635732
#635652
#51713
#50413