Finite Group Databasebeta

← #10595142

Subgroups — #10595142

Order 576.

627 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3805C211yesmaximal, characteristic, complemented
3D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#939C212yesmaximal, complemented
5#63982C211yesmaximal, characteristic, complemented
6#63980C211yesmaximal, characteristic, complemented
7Dic3×#67C212yesmaximal, complemented
8#63959C212yesmaximal, complemented
9#63915C211yesmaximal, characteristic, complemented
10#63893C212yesmaximal, complemented
11#63838C211yesmaximal, characteristic, complemented
12#5598—32nomaximal
13C2×C32×D4C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C22×C3⋊Dic3C2211yescharacteristic
16C2×C3×#67C2212yes—
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2212yes—
19C2×#627—24no—
20C2×#627C2212yes—
21C2×#627C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×#625C2211yescharacteristic
24#3805—24no—
25C2×D3×Dic3C2212yes—
26#3757C2211yescharacteristic
27#3757C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4×C3⋊Dic3C2211yescharacteristic
30C3×#217C2212yes—
31C3×(C2×D3)⋊C4C2212yes—
32C3×Dic3⋊C4C2212yes—
33#3724C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37Dic32C2212yes—
38C2×#237—32no—
39C22×Dic6—32no—
40C3×#121—32no—
41#939—32no—
42#939D312yescomplemented
43#936—32no—
44D4×Dic3—32no—
45D4×Dic3—32no—
46D4×Dic3—61no—
47C4×#67—32no—
48#897—32no—
49#887—32no—
50#885—32no—
51#885—32no—
52#878—32no—
53#878—32no—
54C23×C32C2311yescharacteristic
55C23×C32C2311yescharacteristic
56C22×C3×D3C2312yes—
57C32×D4—22no—
58C2×C32×C4C2311yescharacteristic
59C2×C3⋊Dic3—22no—
60C2×C3⋊Dic3—22no—
61C2×C3⋊Dic3C2311yescharacteristic
62C2×C3⋊Dic3C2311yescharacteristic
63C2×C3⋊Dic3C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C3×#67—24no—
66C3×#67—24no—
67C2×C3×Dic3—24no—
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2312yes—
71#627—22no—
72#627—24no—
73#627D414yescomplemented
74#625—22no—
75D3×Dic3—24no—
76#548—91noSylow
77C2×C3×D4—21no—
78C2×C3×D4—32no—
79C2×C3×D4—32no—
80C2×C3×D4C2×D312yes—
81C22×C3×C4—32no—
82C2×#67—32no—
83C2×#67C2×D312yes—
84C2×#67C2×D312yes—
85C22×Dic3—32no—
86C22×Dic3—32no—
87C22×Dic3—32no—
88C22×Dic3—61no—
89C22×Dic3—61no—
90C22×Dic3C2×D312yes—
91#237—68no—
92C2×C4×D3—32no—
93C2×Dic6—32no—
94C2×Dic6—32no—
95C2×Dic6—32no—
96C2×Dic6—32no—
97C2×Dic6—68no—
98C3×C4⋊C4—32no—
99C3×C22⋊C4—32no—
100C3×C22⋊C4—32no—
101#217—32no—
102#217—32no—
103#217—32no—
104#217—32no—
105#217—61no—
106#217—61no—
107#217C2×D312yes—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4C2×D312yes—
111C4⋊Dic3—32no—
112C4⋊Dic3—32no—
113C4⋊Dic3—61no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4C2×D312yes—
117C4×Dic3—32no—
118C4×Dic3—32no—
119C4×Dic3—32no—
120C4×Dic3—61no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32#4112yescomplemented
127C22×C32C2411yescharacteristic
128C2×C3×D3—22no—
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C32×C4—21no—
132C3⋊Dic3—21no—
133C3⋊Dic3—21no—
134C3⋊Dic3—21no—
135C3⋊Dic3C2×D412yes—
136C3×Dic3—22no—
137C3×Dic3—22no—
138C3×Dic3C2×D414yes—
139C2×#41—91no—
140C22×Q8—91no—
141#124—91no—
142#124—92no—
143C22⋊Q8—91no—
144C22⋊Q8—92no—
145#121—32no—
146#121—91no—
147C4×Q8—91no—
148C4×D4—92no—
149C4×D4—91no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—32no—
153C23×C3C22×D312yes—
154C23×C3C22×D312yes—
155C22×D3C22×D312yes—
156C3×D4—24no—
157C3×D4—24no—
158C3×D4—64no—
159C3×D4—64no—
160C2×C3×C4—21no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—32no—
164C2×C3×C4—64no—
165C2×C3×C4C22×D312yes—
166#67—24no—
167#67—24no—
168#67—64no—
169C2×Dic3—24no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—32no—
173C2×Dic3—32no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—64no—
179C2×Dic3—64no—
180C2×Dic3—64no—
181C2×Dic3—122no—
182C2×Dic3—122no—
183C2×Dic3C22×D312yes—
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D312yes—
186C4×D3—64no—
187Dic6—38no—
188Dic6—64no—
189Dic6—64no—
190Dic6—64no—
191C2×C32—22no—
192C2×C32—41no—
193C2×C32C22×D411yescharacteristic
194C2×C32C2×#4111yescharacteristic
195C2×C322−1+411yescharacteristic
196C3×D3—42no—
197#41—184no—
198C2×Q8—91no—
199C2×Q8—92no—
200C2×Q8—91no—
201C2×Q8—184no—
202C2×D4—32no—
203C2×D4—32no—
204C2×D4—91no—
205C2×D4D3211yescharacteristic
206C22×C4—32no—
207C22×C4—92no—
208C22×C4—91no—
209C22×C4—91no—
210C4⋊C4—32no—
211C4⋊C4—92no—
212C4⋊C4—91no—
213C4⋊C4—91no—
214C22⋊C4—32no—
215C22⋊C4—32no—
216C22⋊C4—92no—
217C22⋊C4—92no—
218C22⋊C4—91no—
219C22⋊C4—91no—
220C42—92no—
221C42—91no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—22no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—62no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3#23714yescomplemented
238C22×C3C23×D312yes—
239C2×D3—22no—
240C2×D3—22no—
241C2×D3—22no—
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—34no—
245C3×C4—62no—
246C3×C4—62no—
247Dic3—22no—
248Dic3—22no—
249Dic3—34no—
250Dic3—62no—
251Dic3—62no—
252Dic3—62no—
253Dic3—62no—
254Dic3—121no—
255Dic3—121no—
256Dic3—121no—
257Dic3D3×D414yes—
258C32#54811yescharacteristic, complemented, Sylow
259C23—32no—
260C23C2×D3211yescharacteristic
261C23C2×D3211yescharacteristic
262Q8—94no—
263Q8—184no—
264Q8—182no—
265D4—22no—
266D4—64no—
267D4—64no—
268D4—182no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—64no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4—184no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4C2×D3211yescharacteristic
281C2×C3—24no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—42no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—122no—
289C2×C3C2×D3×D412yes—
290C2×C3C2×#23712yes—
291C2×C3#101012yes—
292D3—42no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—62no—
299C22—62no—
300C22—62no—
301C22#380512yescomplemented
302C22C22×D3211yescharacteristic
303C4—21no—
304C4—34no—
305C4—62no—
306C4—62no—
307C4—92no—
308C4—181no—
309C4—181no—
310C4—181no—
311C3—21no—
312C3#559812yescomplemented
313C2—22no—
314C2—41no—
315C2—122no—
316C2#6430911yescharacteristic
317C2D32×D411yescharacteristic
318C2C2×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#627C2semidirect
2. C2×#3805C2semidirect
3. D4×C3⋊Dic3C2semidirect
4. C3×#939C2semidirect
5. #63982C2semidirect
6. #63980C2semidirect
7. Dic3×#67C2semidirect
8. #63959C2semidirect
9. #63915C2semidirect
10. #63893C2semidirect
11. #63838C2semidirect
42. #939D3semidirect
73. #627D4semidirect
126. C22×C32#41semidirect
237. C22×C3#237semidirect
258. C32#548semidirect
301. C22#3805semidirect
312. C3#5598semidirect

Maximal subgroups

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

subgroupindex
C22×#6272
C2×#38052
D4×C3⋊Dic32
C3×#9392
#639822
#639802
Dic3×#672
#639592
#639152
#638932
#638382
#55983