Finite Group Databasebeta

← #10591458

Subgroups — #10591458

Order 576.

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

normal subgroups only

This group has 401 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×C3×#214C211yesmaximal, characteristic, complemented
3#63856C211yesmaximal, characteristic, complemented
4C2×#3715C211yesmaximal, characteristic, complemented
5#63659C211yesmaximal, characteristic, complemented
6#63563C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#5218—31nomaximal
9#5169—31nomaximal
10C23×C3×D3—21no—
11C2×C32×D4C2211yescharacteristic
12C2×C3×#67—21no—
13C3×D3×D4—22no—
14C3×D3×D4—22no—
15C2×C3×D12C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic, complemented
18C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×#214—22no—
21C2×C3×C3⋊C8C2211yescharacteristic, complemented
22#3724—21no—
23(C2×D3)⋊Dic3—21no—
24#3715—22no—
25C22×C3×D4—31no—
26C2×D3×D4D311yescharacteristic
27C2×C3×SD16—31no—
28#946—31no—
29C2×#215—31no—
30C2×#214—31no—
31C2×#214D311yescharacteristic, complemented
32#924—31no—
33#832—31no—
34#832—31no—
35#832—31no—
36#832—61no—
37#830—31no—
38#821—31no—
39(C2×D3)⋊C8—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—
50C3×#67—24no—
51C2×C3×Dic3—21no—
52C2×C3×Dic3D411yescharacteristic
53C3×D12—21no—
54C3×D12D412yes—
55C3×C4×D3—21no—
56C3×C4×D3—21no—
57C3×Dic6—21no—
58C3×Dic6—21no—
59C3⋊(C3⋊C8)—21no—
60C3×C3⋊C8—21no—
61#450—91noSylow
62C24×C3—61no—
63C23×D3—21no—
64C2×C3×Q8—31no—
65C2×C3×D4—21no—
66C2×C3×D4—31no—
67C2×C3×D4—61no—
68C2×C3×D4—62no—
69C2×C3×D4—62no—
70C2×C3×D4C2×D311yescharacteristic
71C2×C3×D4C2×D311yescharacteristic
72C22×C3×C4—31no—
73C2×#67—21no—
74D3×D4—22no—
75D3×D4—22no—
76C2×D12C2×D311yescharacteristic
77C2×C4×D3C2×D311yescharacteristic
78C2×Dic6C2×D311yescharacteristic, complemented
79C3×SD16—62no—
80C2×C3×C8—31no—
81#217—61no—
82#215—62no—
83#214—22no—
84#214—62no—
85(C2×D3)⋊C4—61no—
86C4⋊Dic3—31no—
87C4⋊Dic3—31no—
88C4⋊Dic3—61no—
89Dic3⋊C4—61no—
90Dic3⋊C4—61no—
91C2×C3⋊C8—31no—
92C2×C3⋊C8—31no—
93C2×C3⋊C8—61no—
94C2×C3⋊C8C2×D311yescharacteristic, complemented
95C22×C32—22no—
96C22×C32—41no—
97C22×C32—41no—
98C22×C32C2×D411yescharacteristic
99C2×C3×D3—21no—
100C2×C3×D3—21no—
101C2×C3×D3—22no—
102C2×C3×D3—48no—
103C2×C3×D3—41no—
104C2×C3×D3—41no—
105C2×C3×D3SD1612yescomplemented
106C32×C4C2×D411yescharacteristic
107C32×C4C2×D411yescharacteristic
108C3⋊Dic3—41no—
109C3×Dic3—21no—
110C3×Dic3—41no—
111C22×D4—31no—
112C2×SD16—31no—
113C2×SD16—91no—
114C22⋊Q8—91no—
115#102—91no—
116#102—91no—
117C22⋊C8—91no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—31no—
121C23×C3—41no—
122C23×C3—64no—
123C23×C3—61no—
124C23×C3—122no—
125C23×C3—122no—
126C22×D3—24no—
127C22×D3—21no—
128C22×D3—42no—
129C22×D3—42no—
130C22×D3#6711yescharacteristic
131C3×Q8—61no—
132C3×Q8—61no—
133C3×D4—21no—
134C3×D4—21no—
135C3×D4—22no—
136C3×D4—32no—
137C3×D4—41no—
138C3×D4—64no—
139C3×D4—61no—
140C3×D4#6712yes—
141C3×D4#6712yes—
142C2×C3×C4—21no—
143C2×C3×C4—31no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—61no—
147C2×C3×C4C22×D311yescharacteristic
148C2×C3×C4C22×D311yescharacteristic
149#67—24no—
150C2×Dic3—21no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—121no—
154C2×Dic3#6711yescharacteristic
155D12—21no—
156D12#6712yes—
157C4×D3—21no—
158C4×D3—21no—
159Dic6—21no—
160Dic6—21no—
161C3×C8—61no—
162C3⋊C8—21no—
163C3⋊C8—61no—
164C3⋊C8—61no—
165C3⋊C8—121no—
166C2×C32—42no—
167C2×C32#12011yescharacteristic
168C2×C32C2×SD1611yescharacteristic
169C2×C32AGL(1,ℤ/8)11yescharacteristic
170C3×D3—22no—
171C3×D3—42no—
172C24—61no—
173C2×Q8—31no—
174C2×D4—31no—
175C2×D4—61no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4D3211yescharacteristic
179C22×C4—31no—
180SD16—62no—
181SD16—182no—
182C2×C8—31no—
183C2×C8—91no—
184C4⋊C4—91no—
185C4⋊C4—181no—
186C22⋊C4—181no—
187C22×C3—22no—
188C22×C3—22no—
189C22×C3—21no—
190C22×C3—32no—
191C22×C3—42no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—61no—
197C22×C3—61no—
198C22×C3—62no—
199C22×C3—81no—
200C22×C3—81no—
201C22×C3—128no—
202C22×C3—121no—
203C22×C3—121no—
204C22×C3D3×D411yescharacteristic
205C22×C3C2×#6711yescharacteristic
206C2×D3—21no—
207C2×D3—21no—
208C2×D3—22no—
209C2×D3—48no—
210C2×D3—41no—
211C2×D3—41no—
212C2×D3#21412yescomplemented
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—61no—
216C3×C4—121no—
217C3×C4D3×D411yescharacteristic
218C3×C4C2×#6711yescharacteristic
219C3×C4C2×#6711yescharacteristic
220C3×C4C2×#6711yescharacteristic
221Dic3—21no—
222Dic3—41no—
223Dic3—121no—
224Dic3—121no—
225Dic3—241no—
226C32#45011yescharacteristic, complemented, Sylow
227C23—21no—
228C23—31no—
229C23—64no—
230C23—61no—
231C23—122no—
232C23—122no—
233Q8—61no—
234Q8—61no—
235D4—21no—
236D4—32no—
237D4—64no—
238D4—61no—
239D4#62512yes—
240C2×C4—31no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—181no—
245C2×C4C2×D3211yescharacteristic
246C8—61no—
247C8—181no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—42no—
252C2×C3—42no—
253C2×C3—62no—
254C2×C3—82no—
255C2×C3—122no—
256C2×C3D3×SD1611yescharacteristic
257C2×C3#91411yescharacteristic
258C2×C3#93211yescharacteristic
259C2×C3C2×#21411yescharacteristic
260C2×C3#93711yescharacteristic
261C2×C3#95311yescharacteristic
262D3—22no—
263D3—42no—
264C22—22no—
265C22—32no—
266C22—41no—
267C22—41no—
268C22—61no—
269C22—61no—
270C22—62no—
271C22—128no—
272C22—121no—
273C22—121no—
274C22D3×#6711yescharacteristic
275C4—61no—
276C4—121no—
277C4—361no—
278C4C2×#62511yescharacteristic
279C4D3×#6711yescharacteristic
280C3—21no—
281C3#516911yescharacteristic, complemented
282C3#521811yescharacteristic, complemented
283C2—42no—
284C2—62no—
285C2—122no—
286C2D3×#21411yescharacteristic
287C2#6393411yescharacteristic
288C2#6397611yescharacteristic
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×C3×#214C2semidirect
3. #63856C2semidirect
4. C2×#3715C2semidirect
5. #63659C2semidirect
6. #63563C2semidirect
7. (C2×D3)⋊(C3⋊C8)C2semidirect
17. C2×C3×Dic6C22semidirect
18. C4⋊(C3⋊Dic3)C22semidirect
19. C2×C3⋊(C3⋊C8)C22semidirect
21. C2×C3×C3⋊C8C22semidirect
31. C2×#214D3semidirect
78. C2×Dic6C2×D3semidirect
94. C2×C3⋊C8C2×D3semidirect
105. C2×C3×D3SD16semidirect
212. C2×D3#214semidirect
226. C32#450semidirect
281. C3#5169semidirect
282. 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×C3×#2142
#638562
C2×#37152
#636592
#635632
(C2×D3)⋊(C3⋊C8)2
#52183
#51693