Finite Group Databasebeta

← #10591459

Subgroups — #10591459

Order 576.

410 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×C3×D3×D4C211yesmaximal, characteristic
2C2×#3754C211yesmaximal, characteristic, complemented
3#63857C211yesmaximal, characteristic, complemented
4C2×#3716C211yesmaximal, characteristic, complemented
5C3×#832C211yesmaximal, characteristic, complemented
6#63562C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#5218—31nomaximal
9#4776—31nomaximal
10C23×C3×D3—21no—
11C2×C32×D4C2211yescharacteristic
12C2×#634C2211yescharacteristic, complemented
13C2×C3×#67—21no—
14C3×D3×D4—22no—
15C3×D3×D4—22no—
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18#3754—22no—
19C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C4⋊Dic3C2211yescharacteristic, complemented
21C2×C3×C3⋊C8C2211yescharacteristic, complemented
22Dic3⋊Dic3—21no—
23(C2×D3)⋊Dic3—21no—
24#3716—22no—
25C22×C3×D4—31no—
26C2×D3×D4D311yescharacteristic
27C2×#214—31no—
28C2×#214—31no—
29C2×#214—31no—
30C2×#214—61no—
31#924—31no—
32C2×#204—31no—
33#897—31no—
34C3×#102—31no—
35#832—31no—
36#832D311yescharacteristic, complemented
37#830—31no—
38(C2×D3)⋊C8—31no—
39#809—31no—
40C23×C32—21no—
41C22×C3×D3—24no—
42C22×C3×D3—21no—
43C22×C3×D3—42no—
44C22×C3×D3—42no—
45C22×C3×D3D411yescharacteristic
46C32×D4—21no—
47C32×D4D412yes—
48C2×C32×C4C2311yescharacteristic
49C2×C3⋊Dic3—21no—
50#634—21no—
51#634—21no—
52C3×#67—24no—
53C2×C3×Dic3—21no—
54C2×C3×Dic3D411yescharacteristic
55C3×D12—21no—
56C3×D12D412yes—
57C3×C4×D3—21no—
58C3×C4×D3—21no—
59C3⋊(C3⋊C8)—21no—
60C3×C3⋊C8—21no—
61#450—91noSylow
62C24×C3—61no—
63C23×D3—21no—
64C2×C3×D4—21no—
65C2×C3×D4—31no—
66C2×C3×D4—61no—
67C2×C3×D4—62no—
68C2×C3×D4—62no—
69C2×C3×D4C2×D311yescharacteristic
70C2×C3×D4C2×D311yescharacteristic
71C22×C3×C4—31no—
72C2×#67—21no—
73D3×D4—22no—
74D3×D4—22no—
75C2×D12C2×D311yescharacteristic
76C2×C4×D3C2×D311yescharacteristic
77C2×Dic6—31no—
78C2×Dic6—31no—
79C2×Dic6—61no—
80C2×C3×C8—31no—
81C3×C4⋊C4—31no—
82#217—61no—
83#214—62no—
84#214—64no—
85#214—62no—
86#214—62no—
87(C2×D3)⋊C4—61no—
88C4⋊Dic3—61no—
89C4⋊Dic3C2×D311yescharacteristic, complemented
90Dic3⋊C4—61no—
91C2×C3⋊C8—31no—
92C2×C3⋊C8—31no—
93C2×C3⋊C8—61no—
94C2×C3⋊C8C2×D311yescharacteristic, complemented
95#204—62no—
96C22×C32—22no—
97C22×C32—41no—
98C22×C32—41no—
99C22×C32C2×D411yescharacteristic
100C2×C3×D3—21no—
101C2×C3×D3—21no—
102C2×C3×D3—22no—
103C2×C3×D3—48no—
104C2×C3×D3—41no—
105C2×C3×D3—41no—
106C2×C3×D3SD1612yescomplemented
107C32×C4C2×D411yescharacteristic
108C32×C4C2×D411yescharacteristic
109C3⋊Dic3—41no—
110C3×Dic3—21no—
111C3×Dic3—41no—
112C22×D4—31no—
113C2×SD16—91no—
114C2×SD16—91no—
115C22⋊Q8—91no—
116#102—31no—
117#102—91no—
118C22⋊C8—91no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—31no—
122C23×C3—41no—
123C23×C3—64no—
124C23×C3—61no—
125C23×C3—122no—
126C23×C3—122no—
127C22×D3—24no—
128C22×D3—21no—
129C22×D3—42no—
130C22×D3—42no—
131C22×D3#6711yescharacteristic
132C3×D4—21no—
133C3×D4—21no—
134C3×D4—22no—
135C3×D4—32no—
136C3×D4—41no—
137C3×D4—64no—
138C3×D4—61no—
139C3×D4D1212yes—
140C3×D4#6712yes—
141C2×C3×C4—21no—
142C2×C3×C4—31no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4C22×D311yescharacteristic
147C2×C3×C4C22×D311yescharacteristic
148#67—24no—
149C2×Dic3—21no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3—121no—
153C2×Dic3#6711yescharacteristic
154D12—21no—
155D12#6712yes—
156C4×D3—21no—
157C4×D3—21no—
158Dic6—62no—
159Dic6—61no—
160Dic6—61no—
161Dic6—61no—
162Dic6—61no—
163Dic6—121no—
164C3×C8—61no—
165C3⋊C8—21no—
166C3⋊C8—62no—
167C3⋊C8—61no—
168C3⋊C8—61no—
169C2×C32—42no—
170C2×C32#12011yescharacteristic
171C2×C32C2×SD1611yescharacteristic
172C2×C32AGL(1,ℤ/8)11yescharacteristic
173C3×D3—22no—
174C3×D3—42no—
175C24—61no—
176C2×Q8—91no—
177C2×D4—31no—
178C2×D4—61no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4D3211yescharacteristic
182C22×C4—31no—
183SD16—182no—
184SD16—182no—
185C2×C8—31no—
186C2×C8—91no—
187C4⋊C4—31no—
188C4⋊C4—181no—
189C22⋊C4—181no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—21no—
193C22×C3—32no—
194C22×C3—42no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—61no—
200C22×C3—61no—
201C22×C3—62no—
202C22×C3—81no—
203C22×C3—81no—
204C22×C3—128no—
205C22×C3—121no—
206C22×C3—121no—
207C22×C3D3×D411yescharacteristic
208C22×C3C2×#6711yescharacteristic
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—22no—
212C2×D3—48no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3#21412yescomplemented
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—61no—
219C3×C4—121no—
220C3×C4C2×D1211yescharacteristic
221C3×C4D3×D411yescharacteristic
222C3×C4C2×#6711yescharacteristic
223C3×C4C2×#6711yescharacteristic
224Dic3—21no—
225Dic3—41no—
226Dic3—122no—
227Dic3—121no—
228Dic3—121no—
229C32#45011yescharacteristic, complemented, Sylow
230C23—21no—
231C23—31no—
232C23—64no—
233C23—61no—
234C23—122no—
235C23—122no—
236Q8—181no—
237Q8—181no—
238D4—21no—
239D4—32no—
240D4—64no—
241D4—61no—
242D4#62612yes—
243C2×C4—31no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—181no—
248C2×C4C2×D3211yescharacteristic
249C8—61no—
250C8—181no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—42no—
255C2×C3—42no—
256C2×C3—62no—
257C2×C3—82no—
258C2×C3—122no—
259C2×C3#88211yescharacteristic
260C2×C3#91111yescharacteristic
261C2×C3D3×SD1611yescharacteristic
262C2×C3#93211yescharacteristic
263C2×C3C2×#21411yescharacteristic
264C2×C3#95311yescharacteristic
265D3—22no—
266D3—42no—
267C22—22no—
268C22—32no—
269C22—41no—
270C22—41no—
271C22—61no—
272C22—61no—
273C22—62no—
274C22—128no—
275C22—121no—
276C22—121no—
277C22D3×#6711yescharacteristic
278C4—61no—
279C4—121no—
280C4—361no—
281C4C2×#62611yescharacteristic
282C4D3×#6711yescharacteristic
283C3—21no—
284C3#477611yescharacteristic, complemented
285C3#521811yescharacteristic, complemented
286C2—42no—
287C2—62no—
288C2—122no—
289C2D3×#21411yescharacteristic
290C2#6393211yescharacteristic
291C2#6397711yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC2×C3×D3×D4C2
hypercenterC2×D4D32
hypocenterC32#450
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×#3754C2semidirect
3. #63857C2semidirect
4. C2×#3716C2semidirect
5. C3×#832C2semidirect
6. #63562C2semidirect
7. (C2×D3)⋊(C3⋊C8)C2semidirect
12. C2×#634C22semidirect
19. C2×C3⋊(C3⋊C8)C22semidirect
20. C3×C4⋊Dic3C22semidirect
21. C2×C3×C3⋊C8C22semidirect
36. #832D3semidirect
89. C4⋊Dic3C2×D3semidirect
94. C2×C3⋊C8C2×D3semidirect
106. C2×C3×D3SD16semidirect
215. C2×D3#214semidirect
229. C32#450semidirect
284. C3#4776semidirect
285. C3#5218semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×D42
C2×#37542
#638572
C2×#37162
C3×#8322
#635622
(C2×D3)⋊(C3⋊C8)2
#52183
#47763