Finite Group Databasebeta

← #10594183

Subgroups — #10594183

Order 576.

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

normal subgroups only

This group has 715 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×#625C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3#64090C211yesmaximal, characteristic, complemented
4C3×D3×C22⋊C4C212yesmaximal, complemented
5#63982C211yesmaximal, characteristic, complemented
6#63977C212yesmaximal, complemented
7#63915C211yesmaximal, characteristic, complemented
8#63912C211yesmaximal, characteristic, complemented
9#63906C212yesmaximal, complemented
10#63905C212yesmaximal, complemented
11C4×#625C211yesmaximal, characteristic, complemented
12#5490—32nomaximal
13C23×C3×D3C2212yes—
14C2×#638C2211yescharacteristic
15C22×C3⋊Dic3C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic, complemented
17C2×C3×C4×D3C2212yescomplemented
18C2×C3×C4×D3C2212yescomplemented
19C2×#627C2211yescharacteristic, complemented
20C2×#626C2212yescomplemented
21C2×#626C2212yescomplemented
22C2×#625—24no—
23C2×#625C2212yes—
24C2×#625C2211yescharacteristic
25C2×#625C2211yescharacteristic
26#3798—24no—
27C32×C22⋊C4C2211yescharacteristic
28#3752C2211yescharacteristic, complemented
29#3750C2211yescharacteristic, complemented
30C4×C3⋊Dic3C2211yescharacteristic, complemented
31C3×#217C2212yes—
32C3×(C2×D3)⋊C4C2212yescomplemented
33C3×(C2×D3)⋊C4C2212yescomplemented
34#3724C2211yescharacteristic, complemented
35Dic3⋊Dic3C2212yescomplemented
36(C2×D3)⋊Dic3C2212yescomplemented
37(C2×D3)⋊Dic3C2212yescomplemented
38C22×#67—32no—
39C2×#235—32no—
40C2×C3×C22⋊C4—32no—
41#953—32no—
42#930—32no—
43#929—32no—
44C4×#67—32no—
45#885—32no—
46#884—32no—
47#884—32no—
48#883—32no—
49#882—32no—
50#67⋊C4—32no—
51#67⋊C4—61no—
52D3×C22⋊C4D312yescomplemented
53#878—32no—
54C23×C32C2311yescharacteristic
55C22×C3⋊D3C2311yescharacteristic
56C22×C3×D3—24no—
57C22×C3×D3—24no—
58C22×C3×D3—42no—
59C22×C3×D3C2312yes—
60C22×C3×D3C2312yes—
61C2×C32×C4C2311yescharacteristic
62C2×C32×C4C2311yescharacteristic
63#638—22no—
64C2×C3⋊Dic3—22no—
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C4×C3⋊D3—22no—
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2312yes—
71C3×C4×D3—24no—
72C3×C4×D3—42no—
73#627—22no—
74#626—24no—
75#626—42no—
76#625—24no—
77#625—22no—
78#625D414yescomplemented
79#546—91noSylow
80C24×C3—32no—
81C23×D3C2×D312yes—
82C22×C3×C4—32no—
83C22×C3×C4—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—32no—
90C2×#67—32no—
91C2×#67—61no—
92C2×#67—68no—
93C22×Dic3—32no—
94C22×Dic3—61no—
95#235—68no—
96C2×D12—32no—
97C2×D12—32no—
98C2×C4×D3—32no—
99C2×C4×D3—61no—
100C2×C4×D3C2×D312yescomplemented
101C2×C4×D3C2×D312yescomplemented
102C2×Dic6—32no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—32no—
105C3×C22⋊C4—32no—
106C3×C22⋊C4—32no—
107C3×C22⋊C4C2×D312yes—
108#217—32no—
109#217—32no—
110#217C2×D312yes—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—61no—
115(C2×D3)⋊C4C2×D312yescomplemented
116(C2×D3)⋊C4C2×D312yescomplemented
117C4⋊Dic3—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—61no—
122C4×Dic3—32no—
123C4×Dic3—61no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32#4112yescomplemented
127C22×C32C2411yescharacteristic
128C2×C3⋊D3—21no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C2×C3×D3—22no—
132C2×C3×D3—22no—
133C2×C3×D3—22no—
134C2×C3×D3—22no—
135C2×C3×D3—22no—
136C2×C3×D3—48no—
137C2×C3×D3C2×D414yes—
138C32×C4—21no—
139C32×C4—21no—
140C3⋊Dic3—21no—
141C3⋊Dic3—21no—
142C3⋊Dic3C2×D412yes—
143C3×Dic3—22no—
144C3×Dic3—22no—
145C2×#41—91no—
146C22×D4—91no—
147#124—91no—
148#123—92no—
149C22⋊Q8—91no—
150#121—91no—
151#121—92no—
152#120—92no—
153C4×D4—91no—
154C4×D4—91no—
155C2×C22⋊C4—32no—
156C23×C3—21no—
157C23×C3—32no—
158C23×C3—32no—
159C23×C3—64no—
160C23×C3—64no—
161C23×C3—122no—
162C23×C3C22×D312yes—
163C22×D3—24no—
164C22×D3—24no—
165C22×D3—32no—
166C22×D3—42no—
167C22×D3—61no—
168C22×D3C22×D312yes—
169C22×D3C22×D312yes—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—64no—
175C2×C3×C4—122no—
176C2×C3×C4C22×D312yes—
177C2×C3×C4C22×D312yes—
178#67—38no—
179#67—64no—
180#67—64no—
181#67—64no—
182#67—64no—
183#67—64no—
184#67—64no—
185#67—122no—
186C2×Dic3—32no—
187C2×Dic3—32no—
188C2×Dic3—32no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3—64no—
193C2×Dic3—122no—
194C2×Dic3C22×D312yes—
195C2×Dic3C22×D312yes—
196D12—64no—
197D12—122no—
198C4×D3—24no—
199C4×D3—42no—
200C4×D3—64no—
201C4×D3—122no—
202Dic6—64no—
203C2×C32—22no—
204C2×C32C22×D411yescharacteristic
205C2×C32C2×#4111yescharacteristic
206C2×C32H(5,𝔽2)11yescharacteristic
207C3⋊D3—41no—
208C3×D3—24no—
209C3×D3—42no—
210C24—32no—
211#41—184no—
212C2×Q8—91no—
213C2×D4—92no—
214C2×D4—92no—
215C2×D4—91no—
216C2×D4—92no—
217C2×D4—91no—
218C2×D4—91no—
219C2×D4—184no—
220C22×C4—32no—
221C22×C4—32no—
222C22×C4—91no—
223C22×C4—91no—
224C4⋊C4—91no—
225C4⋊C4—92no—
226C4⋊C4—91no—
227C22⋊C4—32no—
228C22⋊C4—32no—
229C22⋊C4—32no—
230C22⋊C4—92no—
231C22⋊C4—92no—
232C22⋊C4—91no—
233C22⋊C4D3211yescharacteristic
234C42—91no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—21no—
238C22×C3—22no—
239C22×C3—34no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—62no—
243C22×C3—62no—
244C22×C3—62no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—128no—
248C22×C3#23514yescomplemented
249C22×C3C23×D312yes—
250C2×D3—22no—
251C2×D3—22no—
252C2×D3—22no—
253C2×D3—22no—
254C2×D3—22no—
255C2×D3—48no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—121no—
261C2×D3—121no—
262C2×D3D3×D414yes—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—62no—
268C3×C4—62no—
269Dic3—22no—
270Dic3—22no—
271Dic3—34no—
272Dic3—62no—
273Dic3—62no—
274Dic3—62no—
275Dic3—62no—
276Dic3—121no—
277C32#54611yescharacteristic, complemented, Sylow
278C23—32no—
279C23—32no—
280C23—64no—
281C23—64no—
282C23—91no—
283C23—122no—
284C23C2×D3211yescharacteristic
285Q8—182no—
286D4—94no—
287D4—184no—
288D4—182no—
289D4—184no—
290D4—182no—
291D4—362no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—64no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—122no—
299C2×C4—182no—
300C2×C4—182no—
301C2×C4C2×D3211yescharacteristic
302C2×C4C2×D3211yescharacteristic
303C2×C3—24no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—64no—
309C2×C3—122no—
310C2×C3C2×#23512yes—
311C2×C3C2×D3×D412yes—
312C2×C3#100412yes—
313D3—24no—
314D3—42no—
315D3—122no—
316D3—122no—
317C22—21no—
318C22—21no—
319C22—34no—
320C22—62no—
321C22—62no—
322C22—62no—
323C22—62no—
324C22—62no—
325C22—128no—
326C22—181no—
327C22—181no—
328C22—181no—
329C22#379812yescomplemented
330C22C22×D3211yescharacteristic
331C4—21no—
332C4—21no—
333C4—62no—
334C4—62no—
335C4—92no—
336C4—181no—
337C4—181no—
338C3—21no—
339C3#549012yescomplemented
340C2—22no—
341C2—64no—
342C2—122no—
343C2—361no—
344C2C2×#379811yescharacteristic
345C2D32×D411yescharacteristic
346C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#546
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#625C2semidirect
2. C2×#3798C2semidirect
3. #64090C2semidirect
4. C3×D3×C22⋊C4C2semidirect
5. #63982C2semidirect
6. #63977C2semidirect
7. #63915C2semidirect
8. #63912C2semidirect
9. #63906C2semidirect
10. #63905C2semidirect
11. C4×#625C2semidirect
16. C2×C4×C3⋊D3C22semidirect
17. C2×C3×C4×D3C22semidirect
18. C2×C3×C4×D3C22semidirect
19. C2×#627C22semidirect
20. C2×#626C22semidirect
21. C2×#626C22semidirect
28. #3752C22semidirect
29. #3750C22semidirect
30. C4×C3⋊Dic3C22semidirect
32. C3×(C2×D3)⋊C4C22semidirect
33. C3×(C2×D3)⋊C4C22semidirect
34. #3724C22semidirect
35. Dic3⋊Dic3C22semidirect
36. (C2×D3)⋊Dic3C22semidirect
37. (C2×D3)⋊Dic3C22semidirect
52. D3×C22⋊C4D3semidirect
78. #625D4semidirect
100. C2×C4×D3C2×D3semidirect
101. C2×C4×D3C2×D3semidirect
115. (C2×D3)⋊C4C2×D3semidirect
116. (C2×D3)⋊C4C2×D3semidirect
126. C22×C32#41semidirect
248. C22×C3#235semidirect
277. C32#546semidirect
329. C22#3798semidirect
339. C3#5490semidirect

Maximal subgroups

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

subgroupindex
C22×#6252
C2×#37982
#640902
C3×D3×C22⋊C42
#639822
#639772
#639152
#639122
#639062
#639052
C4×#6252
#54903