Finite Group Databasebeta

← #10594072

Subgroups — #10594072

Order 576.

607 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C22×#627C211yesmaximal, characteristic, complemented
2C2×#3797C211yesmaximal, characteristic, complemented
3#64090C211yesmaximal, characteristic
4C3×#878C212yesmaximal, complemented
5#63982C211yesmaximal, characteristic, complemented
6#63952C212yesmaximal
7#63919C211yesmaximal, characteristic, complemented
8#63895C212yesmaximal, complemented
9#63865C211yesmaximal, characteristic, complemented
10#63839C212yesmaximal, complemented
11#63838C211yesmaximal, characteristic, complemented
12#5632—32nomaximal
13C2×#638C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C22×C3×Dic3C2212yes—
17C2×C3×Dic6C2212yes—
18C2×#627—24no—
19C2×#627C2211yescharacteristic
20C2×#627C2212yes—
21C2×#627C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×#624C2212yes—
24#3797—24no—
25C32×C22⋊C4C2211yescharacteristic
26#3752C2211yescharacteristic
27#3750C2211yescharacteristic
28C4×C3⋊Dic3C2211yescharacteristic
29C3×#217C2212yes—
30C3×C4⋊Dic3C2212yes—
31C3×Dic3⋊C4C2212yes—
32C3×Dic3⋊C4C2212yes—
33#3724C2211yescharacteristic
34#3724C2212yes—
35Dic3⋊Dic3C2212yes—
36#3722C2212yes—
37Dic32C2212yes—
38C2×D3×Q8—32no—
39C22×Dic6—32no—
40C3×C22⋊Q8—32no—
41#944—32no—
42C4×#67—32no—
43#897—32no—
44#896—32no—
45#896—32no—
46#888—32no—
47#888—32no—
48#887—32no—
49#67⋊C4—32no—
50#67⋊C4—32no—
51#67⋊C4—61no—
52#878—32no—
53#878D312yescomplemented
54C23×C32C2311yescharacteristic
55C22×C3⋊D3C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C32×C4C2311yescharacteristic
58#638Q814yescomplemented
59C2×C3⋊Dic3—22no—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2311yescharacteristic
62C2×C3⋊Dic3C2311yescharacteristic
63C4×C3⋊D3—22no—
64C2×C3×Dic3—24no—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2312yes—
69C3×Dic6—24no—
70#627—22no—
71#627—22no—
72#627—24no—
73#627D414yescomplemented
74#624—24no—
75D4×Q8—91noSylow
76C2×C3×Q8—32no—
77C22×C3×C4—32no—
78C2×#67—32no—
79C2×#67—61no—
80C22×Dic3—32no—
81C22×Dic3—61no—
82C22×Dic3C2×D312yes—
83D3×Q8—68no—
84C2×C4×D3—32no—
85C2×C4×D3—32no—
86C2×C4×D3—32no—
87C2×C4×D3—61no—
88C2×Dic6—32no—
89C2×Dic6—32no—
90C2×Dic6—32no—
91C2×Dic6—32no—
92C2×Dic6—32no—
93C2×Dic6—68no—
94C2×Dic6C2×D312yes—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—32no—
98C3×C22⋊C4—21no—
99C3×C22⋊C4—32no—
100C3×C22⋊C4C2×D312yes—
101#217C2×D312yes—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—61no—
106C4⋊Dic3—32no—
107C4⋊Dic3C2×D312yes—
108Dic3⋊C4—32no—
109Dic3⋊C4—32no—
110Dic3⋊C4—32no—
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—61no—
114Dic3⋊C4C2×D312yes—
115Dic3⋊C4C2×D312yes—
116C4×Dic3—32no—
117C4×Dic3—32no—
118C4×Dic3—32no—
119C4×Dic3—61no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32C2×Q812yescomplemented
123C22×C32C2411yescharacteristic
124C2×C3⋊D3—21no—
125C2×C3⋊D3—21no—
126C2×C3⋊D3C2×Q812yescomplemented
127C32×C4—21no—
128C32×C4—21no—
129C3⋊Dic3—21no—
130C3⋊Dic3C2×D412yes—
131C3⋊Dic3C2×Q812yescomplemented
132C3×Dic3—22no—
133C3×Dic3—22no—
134C3×Dic3—22no—
135C3×Dic3C2×D414yes—
136C22×Q8—91no—
137C22×Q8—91no—
138C4⋊Q8—91no—
139C4⋊Q8—92no—
140C22⋊Q8—32no—
141C22⋊Q8—91no—
142C22⋊Q8—91no—
143C22⋊Q8—92no—
144C4×Q8—91no—
145C4×D4—92no—
146C4×D4—91no—
147C23×C3—21no—
148C23×C3C22×D312yes—
149C22×D3—32no—
150C22×D3—61no—
151C3×Q8—64no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—32no—
157C2×C3×C4—32no—
158C2×C3×C4—64no—
159C2×C3×C4C22×D312yes—
160C2×C3×C4C22×D312yes—
161#67—38no—
162#67—64no—
163C2×Dic3—24no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—32no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—64no—
171C2×Dic3—122no—
172C2×Dic3C22×D312yes—
173C2×Dic3C22×D312yes—
174C2×Dic3C22×D312yes—
175C2×Dic3C22×D312yes—
176C4×D3—64no—
177C4×D3—64no—
178C4×D3—64no—
179C4×D3—122no—
180Dic6—24no—
181Dic6—38no—
182Dic6—64no—
183Dic6—64no—
184Dic6—64no—
185Dic6—64no—
186C2×C32—22no—
187C2×C32C22×D411yescharacteristic
188C2×C32C22×Q811yescharacteristic
189C2×C322−1+411yescharacteristic
190C3⋊D3—22no—
191C2×Q8—32no—
192C2×Q8—91no—
193C2×Q8—92no—
194C2×Q8—91no—
195C2×Q8—91no—
196C2×Q8—184no—
197C2×Q8—184no—
198C2×D4—91no—
199C22×C4—32no—
200C22×C4—92no—
201C22×C4—91no—
202C22×C4—91no—
203C4⋊C4—32no—
204C4⋊C4—32no—
205C4⋊C4—32no—
206C4⋊C4—92no—
207C4⋊C4—91no—
208C4⋊C4—92no—
209C4⋊C4—91no—
210C22⋊C4—32no—
211C22⋊C4—92no—
212C22⋊C4—91no—
213C22⋊C4D3211yescharacteristic
214C42—92no—
215C42—91no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—21no—
219C22×C3—22no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3D3×Q814yescomplemented
223C22×C3C23×D312yes—
224C2×D3—34no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—62no—
228C2×D3—121no—
229C2×D3—121no—
230C3×C4—22no—
231C3×C4—22no—
232C3×C4—34no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—62no—
236C3×C4—62no—
237C3×C4—62no—
238Dic3—22no—
239Dic3—22no—
240Dic3—22no—
241Dic3—34no—
242Dic3—34no—
243Dic3—62no—
244Dic3—62no—
245Dic3—62no—
246Dic3—121no—
247Dic3D3×D414yes—
248C32D4×Q811yescharacteristic, complemented, Sylow
249C23—91no—
250C23C2×D3211yescharacteristic
251Q8—64no—
252Q8—94no—
253Q8—184no—
254Q8—182no—
255Q8—182no—
256D4—94no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—64no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—182no—
266C2×C4—184no—
267C2×C4—182no—
268C2×C4C2×D3211yescharacteristic
269C2×C4C2×D3211yescharacteristic
270C2×C3—24no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—42no—
275C2×C3C2×D3×D412yes—
276C2×C3C2×D3×Q812yes—
277C2×C3#101012yes—
278D3—64no—
279D3—122no—
280C22—21no—
281C22—21no—
282C22—92no—
283C22—181no—
284C22—181no—
285C22#379712yescomplemented
286C22C22×D3211yescharacteristic
287C4—21no—
288C4—21no—
289C4—34no—
290C4—62no—
291C4—62no—
292C4—62no—
293C4—92no—
294C4—92no—
295C4—181no—
296C3—21no—
297C3#563212yescomplemented
298C2—22no—
299C2—182no—
300C2C2×#379711yescharacteristic
301C2#6430911yescharacteristic
302C2D32×D411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×#638C22
hypercenterC22⋊C4D32
hypocenterC32D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#627C2semidirect
2. C2×#3797C2semidirect
4. C3×#878C2semidirect
5. #63982C2semidirect
7. #63919C2semidirect
8. #63895C2semidirect
9. #63865C2semidirect
10. #63839C2semidirect
11. #63838C2semidirect
53. #878D3semidirect
58. #638Q8semidirect
73. #627D4semidirect
122. C22×C32C2×Q8semidirect
126. C2×C3⋊D3C2×Q8semidirect
131. C3⋊Dic3C2×Q8semidirect
222. C22×C3D3×Q8semidirect
248. C32D4×Q8semidirect
285. C22#3797semidirect
297. C3#5632semidirect

Maximal subgroups

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

subgroupindex
C22×#6272
C2×#37972
#640902
C3×#8782
#639822
#639522
#639192
#638952
#638652
#638392
#638382
#56323