Finite Group Databasebeta

← #10594971

Subgroups — #10594971

Order 576.

521 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 144 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#64133C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C3×#924C211yesmaximal, characteristic, complemented
4D3×#217C212yesmaximal
5#63964C212yesmaximal, complemented
6#63899C212yesmaximal, complemented
7#63896C212yesmaximal, complemented
8#63893C212yesmaximal, complemented
9#63887C212yesmaximal, complemented
10#5634—31nomaximal
11#5483—31nomaximal
12C23×C3×D3C2211yescharacteristic
13C22×C32×C4C2211yescharacteristic
14C2×#634C2211yescharacteristic, complemented
15C22×C3×Dic3C2211yescharacteristic
16C2×C3×Dic6C2211yescharacteristic, complemented
17C2×#627C2212yescomplemented
18C2×D3×Dic3C2212yes—
19C2×D3×Dic3C2212yes—
20#3757C2212yes—
21C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
22#3750C2212yescomplemented
23C3×#217C2212yes—
24C3×(C2×D3)⋊C4C2212yes—
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic, complemented
27C3×Dic3⋊C4C2212yescomplemented
28#3724C2212yescomplemented
29Dic3⋊Dic3C2212yescomplemented
30Dic3⋊Dic3C2212yescomplemented
31(C2×D3)⋊Dic3C2212yes—
32(C2×D3)⋊Dic3C2212yes—
33C3×C22⋊Q8—31no—
34C2×C3×C22⋊C4—31no—
35C2×#217—32no—
36#946—32no—
37C2×(C2×D3)⋊C4D311yescharacteristic
38#924—31no—
39#924—31no—
40#924—32no—
41#924—61no—
42#924D311yescharacteristic, complemented
43#897—32no—
44#896—32no—
45#896—32no—
46D3×C22⋊C4—32no—
47#878—32no—
48#878—32no—
49#878—32no—
50#878—32no—
51#878—32no—
52C23×C32C2311yescharacteristic
53C22×C3×D3—41no—
54C22×C3×D3—41no—
55C22×C3×D3Q814yescomplemented
56C22×C3×D3C2312yes—
57C2×C32×C4—41no—
58C2×C32×C4C2311yescharacteristic
59C2×C32×C4C2311yescharacteristic
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2312yes—
62#634—41no—
63C2×C3×Dic3—41no—
64C2×C3×Dic3C2312yes—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C3×Dic6—41no—
68#627—42no—
69D3×Dic3—42no—
70D3×Dic3—42no—
71#543—91noSylow
72C24×C3—31no—
73C23×D3C2×D311yescharacteristic
74C2×C3×Q8—31no—
75C22×C3×C4—21no—
76C22×C3×C4—31no—
77C22×C3×C4C2×D311yescharacteristic
78C22×C3×C4C2×D311yescharacteristic
79C22×Dic3—32no—
80C22×Dic3—32no—
81C22×Dic3C2×D311yescharacteristic
82C2×C4×D3—32no—
83C2×C4×D3—32no—
84C2×Dic6—31no—
85C2×Dic6—31no—
86C2×Dic6—32no—
87C2×Dic6—32no—
88C2×Dic6—61no—
89C2×Dic6C2×D311yescharacteristic, complemented
90C3×C4⋊C4—31no—
91C3×C4⋊C4—32no—
92C3×C22⋊C4—32no—
93C3×C22⋊C4—32no—
94C3×C22⋊C4—32no—
95#217—32no—
96#217—32no—
97#217—32no—
98#217—32no—
99#217—62no—
100#217C2×D312yes—
101(C2×D3)⋊C4—32no—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4C2×D312yes—
104(C2×D3)⋊C4C2×D312yes—
105C4⋊Dic3—31no—
106C4⋊Dic3—31no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3—61no—
110C4⋊Dic3C2×D311yescharacteristic, complemented
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—62no—
118Dic3⋊C4C2×D312yescomplemented
119C22×C32—21no—
120C22×C32—21no—
121C22×C32C2×Q812yes—
122C22×C32C2411yescharacteristic
123C2×C3×D3—22no—
124C2×C3×D3—22no—
125C2×C3×D3—44no—
126C2×C3×D3C2×Q814yes—
127C32×C4—21no—
128C32×C4—21no—
129C3⋊Dic3—22no—
130C3⋊Dic3—22no—
131C3×Dic3—22no—
132C3×Dic3—22no—
133C3×Dic3—22no—
134C22⋊Q8—31no—
135C22⋊Q8—92no—
136C22⋊Q8—92no—
137C22⋊Q8—92no—
138C22⋊Q8—92no—
139C22⋊Q8—92no—
140C22⋊Q8—91no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—92no—
143C23×C3—21no—
144C23×C3—32no—
145C23×C3—34no—
146C23×C3—121no—
147C23×C3—121no—
148C23×C3C22×D311yescharacteristic
149C23×C3C22×D311yescharacteristic
150C22×D3—41no—
151C22×D3—41no—
152C22×D3Dic614yescomplemented
153C22×D3C22×D312yes—
154C3×Q8—121no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—32no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—121no—
164C2×C3×C4C22×D311yescharacteristic
165C2×C3×C4C22×D311yescharacteristic
166C2×C3×C4C22×D311yescharacteristic
167C2×C3×C4C22×D311yescharacteristic
168C2×Dic3—32no—
169C2×Dic3—32no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—41no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—122no—
176C2×Dic3—122no—
177C2×Dic3C22×D312yes—
178C2×Dic3C22×D312yes—
179C2×Dic3C22×D312yes—
180C4×D3—122no—
181C4×D3—122no—
182Dic6—41no—
183Dic6—122no—
184Dic6—122no—
185Dic6—121no—
186Dic6—121no—
187Dic6—122no—
188C2×C32—22no—
189C2×C32C22×Q811yescharacteristic
190C2×C32H(5,𝔽2)11yescharacteristic
191C2×C32H(5,𝔽2)11yescharacteristic
192C3×D3—24no—
193C24—31no—
194C2×Q8—31no—
195C2×Q8—92no—
196C2×Q8—91no—
197C22×C4—31no—
198C22×C4—92no—
199C22×C4—92no—
200C22×C4D3211yescharacteristic
201C4⋊C4—32no—
202C4⋊C4—31no—
203C4⋊C4—92no—
204C4⋊C4—92no—
205C4⋊C4—92no—
206C4⋊C4—92no—
207C4⋊C4—91no—
208C22⋊C4—32no—
209C22⋊C4—32no—
210C22⋊C4—32no—
211C22⋊C4—92no—
212C22⋊C4—92no—
213C22⋊C4—92no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—22no—
220C22×C3—34no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—62no—
224C22×C3—62no—
225C22×C3—124no—
226C22×C3C2×Dic612yes—
227C22×C3D3×Q812yes—
228C22×C3C23×D311yescharacteristic
229C22×C3C23×D311yescharacteristic
230C2×D3—22no—
231C2×D3—22no—
232C2×D3—44no—
233C2×D3C2×Dic614yes—
234C3×C4—22no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—41no—
240C3×C4—62no—
241C3×C4—62no—
242C3×C4—62no—
243Dic3—22no—
244Dic3—22no—
245Dic3—22no—
246Dic3—62no—
247Dic3—62no—
248Dic3—62no—
249Dic3—62no—
250Dic3—122no—
251Dic3—122no—
252C32#54311yescharacteristic, complemented, Sylow
253C23—32no—
254C23—34no—
255C23—121no—
256C23—121no—
257C23C2×D3211yescharacteristic
258Q8—121no—
259Q8—361no—
260Q8—362no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—41no—
265C2×C4—92no—
266C2×C4—92no—
267C2×C4—121no—
268C2×C4—362no—
269C2×C4—362no—
270C2×C4C2×D3211yescharacteristic
271C2×C4C2×D3211yescharacteristic
272C2×C3—22no—
273C2×C3—22no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—42no—
278C2×C3—64no—
279C2×C3C22×Dic611yescharacteristic
280C2×C3#100411yescharacteristic
281C2×C3#100411yescharacteristic
282C2×C3#100411yescharacteristic
283C2×C3C2×D3×Q811yescharacteristic
284C2×C3#100911yescharacteristic
285D3—24no—
286C22—21no—
287C22—21no—
288C22—34no—
289C22—62no—
290C22—62no—
291C22—124no—
292C22D3×Dic612yes—
293C22C22×D3211yescharacteristic
294C4—21no—
295C4—21no—
296C4—62no—
297C4—62no—
298C4—62no—
299C4—182no—
300C4—182no—
301C3—21no—
302C3#548311yescharacteristic, complemented
303C3#563411yescharacteristic, complemented
304C2—22no—
305C2—64no—
306C2C2×D3×Dic611yescharacteristic
307C2#6430711yescharacteristic
308C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64133C2semidirect
3. C3×#924C2semidirect
5. #63964C2semidirect
6. #63899C2semidirect
7. #63896C2semidirect
8. #63893C2semidirect
9. #63887C2semidirect
14. C2×#634C22semidirect
16. C2×C3×Dic6C22semidirect
17. C2×#627C22semidirect
21. C4⋊(C3⋊Dic3)C22semidirect
22. #3750C22semidirect
26. C3×C4⋊Dic3C22semidirect
27. C3×Dic3⋊C4C22semidirect
28. #3724C22semidirect
29. Dic3⋊Dic3C22semidirect
30. Dic3⋊Dic3C22semidirect
42. #924D3semidirect
55. C22×C3×D3Q8semidirect
89. C2×Dic6C2×D3semidirect
110. C4⋊Dic3C2×D3semidirect
118. Dic3⋊C4C2×D3semidirect
152. C22×D3Dic6semidirect
252. C32#543semidirect
302. C3#5483semidirect
303. C3#5634semidirect

Maximal subgroups

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

subgroupindex
#641332
C2×C3×(C2×D3)⋊C42
C3×#9242
D3×#2172
#639642
#638992
#638962
#638932
#638872
#56343
#54833