Finite Group Databasebeta

← #10594521

Subgroups — #10594521

Order 576.

627 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 136 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×#3798C211yesmaximal, characteristic, complemented
2C2×#3797C211yesmaximal, characteristic, complemented
3#64102C211yesmaximal, characteristic, complemented
4C3×#896C212yesmaximal, complemented
5#63915C211yesmaximal, characteristic, complemented
6#63912C211yesmaximal, characteristic, complemented
7#63910C212yesmaximal, complemented
8#63907C212yesmaximal, complemented
9#63904C212yesmaximal, complemented
10#63865C211yesmaximal, characteristic, complemented
11#63838C211yesmaximal, characteristic, complemented
12#5633—32nomaximal
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×D3C2212yes—
17C2×C3×Dic6C2212yes—
18C2×#627C2211yescharacteristic
19C2×#627C2211yescharacteristic
20C2×#626C2212yes—
21C2×#626C2212yes—
22C2×#625C2211yescharacteristic
23C2×#624C2212yes—
24#3798—24no—
25#3797—24no—
26C32×C4⋊C4C2211yescharacteristic
27#3752C2211yescharacteristic
28#3752C2211yescharacteristic
29C4×C3⋊Dic3C2211yescharacteristic
30C3×(C2×D3)⋊C4C2212yes—
31C3×(C2×D3)⋊C4C2212yes—
32C3×Dic3⋊C4C2212yes—
33C3×Dic3⋊C4C2212yes—
34#3724C2211yescharacteristic
35#3722C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37Dic32C2212yes—
38C2×D3×Q8—32no—
39C2×#235—32no—
40C3×C22⋊Q8—32no—
41#947—32no—
42C4×#67—32no—
43#896—32no—
44#896D312yescomplemented
45#895—32no—
46#893—32no—
47#893—32no—
48#893—61no—
49#887—32no—
50#885—32no—
51#885—32no—
52#884—32no—
53#884—32no—
54C22×C3⋊D3C2311yescharacteristic
55C22×C3⋊D3C2311yescharacteristic
56C22×C3×D3C2312yes—
57C2×C32×C4C2311yescharacteristic
58C2×C32×C4C2311yescharacteristic
59C2×C32×C4C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2311yescharacteristic
62C4⋊(C3⋊D3)—22no—
63C4×C3⋊D3—22no—
64C4×C3⋊D3—22no—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×C4×D3—24no—
69C3×Dic6—24no—
70#627—22no—
71#627D414yescomplemented
72#626—24no—
73#626—24no—
74#625—22no—
75#624—24no—
76#548—91noSylow
77C2×C3×Q8—32no—
78C22×C3×C4—32no—
79C2×#67—32no—
80C2×#67—32no—
81C2×#67—32no—
82D3×Q8—68no—
83#235—68no—
84C2×D12—32no—
85C2×D12—32no—
86C2×D12—32no—
87C2×D12—61no—
88C2×C4×D3—32no—
89C2×C4×D3—32no—
90C2×C4×D3—32no—
91C2×C4×D3—32no—
92C2×C4×D3—61no—
93C2×C4×D3—61no—
94C2×C4×D3C2×D312yes—
95C2×Dic6—32no—
96C2×Dic6—32no—
97C2×Dic6C2×D312yes—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—32no—
100C3×C4⋊C4—32no—
101C3×C4⋊C4C2×D312yes—
102C3×C22⋊C4—32no—
103C3×C22⋊C4—32no—
104#217—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—61no—
111(C2×D3)⋊C4—61no—
112(C2×D3)⋊C4C2×D312yes—
113(C2×D3)⋊C4C2×D312yes—
114Dic3⋊C4—32no—
115Dic3⋊C4C2×D312yes—
116Dic3⋊C4C2×D312yes—
117C4×Dic3—32no—
118C4×Dic3—32no—
119C4×Dic3—32no—
120C4×Dic3—61no—
121C22×C32C2411yescharacteristic
122C2×C3⋊D3—21no—
123C2×C3⋊D3—21no—
124C2×C3⋊D3—21no—
125C2×C3⋊D3—21no—
126C2×C3⋊D3—21no—
127C2×C3⋊D3#4112yescomplemented
128C2×C3×D3—22no—
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4—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—32no—
144C22⋊Q8—91no—
145#121—91no—
146#121—92no—
147C4×Q8—91no—
148C4×D4—92no—
149C4×D4—91no—
150C23×C3—32no—
151C22×D3—32no—
152C22×D3—32no—
153C22×D3—61no—
154C22×D3—61no—
155C22×D3C22×D312yes—
156C3×Q8—64no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—64no—
164C2×C3×C4C22×D312yes—
165C2×C3×C4C22×D312yes—
166C2×C3×C4C22×D312yes—
167#67—64no—
168#67—64no—
169#67—64no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3C22×D312yes—
175C2×Dic3C22×D312yes—
176C2×Dic3C22×D312yes—
177D12—64no—
178D12—64no—
179D12—64no—
180D12—64no—
181C4×D3—24no—
182C4×D3—64no—
183C4×D3—64no—
184C4×D3—64no—
185C4×D3—64no—
186C4×D3—122no—
187C4×D3—122no—
188Dic6—24no—
189Dic6—38no—
190Dic6—64no—
191C2×C32C22×D411yescharacteristic
192C2×C32C2×#4111yescharacteristic
193C2×C322−1+411yescharacteristic
194C3⋊D3—22no—
195C3⋊D3—41no—
196C3×D3—42no—
197#41—184no—
198C2×Q8—32no—
199C2×Q8—91no—
200C2×Q8—91no—
201C2×Q8—184no—
202C2×D4—92no—
203C2×D4—92no—
204C2×D4—91no—
205C2×D4—91no—
206C22×C4—32no—
207C22×C4—92no—
208C22×C4—91no—
209C22×C4—91no—
210C4⋊C4—32no—
211C4⋊C4—32no—
212C4⋊C4—91no—
213C4⋊C4D3211yescharacteristic
214C22⋊C4—32no—
215C22⋊C4—32no—
216C22⋊C4—92no—
217C22⋊C4—92no—
218C22⋊C4—91no—
219C22⋊C4—91no—
220C42—92no—
221C42—91no—
222C22×C3—21no—
223C22×C3—62no—
224C22×C3—62no—
225C22×C3—62no—
226C22×C3C23×D312yes—
227C2×D3—22no—
228C2×D3—22no—
229C2×D3—22no—
230C2×D3—34no—
231C2×D3—62no—
232C2×D3—62no—
233C2×D3—62no—
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—62no—
238C2×D3—121no—
239C2×D3—121no—
240C2×D3—121no—
241C2×D3—121no—
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—34no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4—62no—
250C3×C4—62no—
251Dic3—22no—
252Dic3—22no—
253Dic3—34no—
254Dic3—62no—
255Dic3—62no—
256Dic3—121no—
257Dic3D3×D414yes—
258C32#54811yescharacteristic, complemented, Sylow
259C23—32no—
260C23—91no—
261C23—91no—
262Q8—64no—
263Q8—94no—
264Q8—182no—
265D4—182no—
266D4—184no—
267D4—184no—
268D4—182no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—64no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—182no—
276C2×C4—182no—
277C2×C4—184no—
278C2×C4C2×D3211yescharacteristic
279C2×C4C2×D3211yescharacteristic
280C2×C4C2×D3211yescharacteristic
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—122no—
285C2×C3C2×D3×D412yes—
286C2×C3#100712yes—
287C2×C3D3×#4112yes—
288D3—42no—
289D3—64no—
290D3—122no—
291D3—122no—
292D3—122no—
293C22—62no—
294C22—62no—
295C22—62no—
296C22—92no—
297C22—181no—
298C22—181no—
299C22—181no—
300C22—181no—
301C22—181no—
302C22C22×D3211yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—34no—
307C4—62no—
308C4—62no—
309C4—92no—
310C4—181no—
311C3—21no—
312C3#563312yescomplemented
313C2—122no—
314C2—182no—
315C2—361no—
316C2#6430611yescharacteristic
317C2D32×D411yescharacteristic
318C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3798C2semidirect
2. C2×#3797C2semidirect
3. #64102C2semidirect
4. C3×#896C2semidirect
5. #63915C2semidirect
6. #63912C2semidirect
7. #63910C2semidirect
8. #63907C2semidirect
9. #63904C2semidirect
10. #63865C2semidirect
11. #63838C2semidirect
44. #896D3semidirect
71. #627D4semidirect
127. C2×C3⋊D3#41semidirect
258. C32#548semidirect
312. C3#5633semidirect

Maximal subgroups

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

subgroupindex
C2×#37982
C2×#37972
#641022
C3×#8962
#639152
#639122
#639102
#639072
#639042
#638652
#638382
#56333