Finite Group Databasebeta

← #672 #10595525

Subgroups — #672 #10595525

Order 576.

921 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 921 conjugacy classes of subgroups — too many to draw. Only its 168 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×C3×#67C212yesmaximal, complemented
2C2×D3×#67C212yesmaximal, complemented
3#64162C211yesmaximal, characteristic, complemented
4#63983C211yesmaximal, characteristic, complemented
5#63979C211yesmaximal, characteristic, complemented
6#63978C212yesmaximal, complemented
7#63976C212yesmaximal, complemented
8Dic3×#67C212yesmaximal, complemented
9#63970C212yesmaximal, complemented
10D4×#67—32nomaximal
11C24×C32C2211yescharacteristic, complemented
12C23×C3×D3C2212yescomplemented
13C22×D32C2211yescharacteristic, complemented
14C2×#638C2212yescomplemented
15C2×#638C2211yescharacteristic, complemented
16C2×C3×#67—28no—
17C2×C3×#67C2212yes—
18C2×C3×#67C2212yescomplemented
19C2×C3×#67C2212yescomplemented
20C2×C3×#67C2212yescomplemented
21C22×C3×Dic3C2212yescomplemented
22D3×#67—28no—
23C2×#626C2212yescomplemented
24C2×#625C2212yescomplemented
25C2×#625C2211yescharacteristic, complemented
26C2×D3×Dic3C2212yescomplemented
27#3757C2211yescharacteristic, complemented
28#3757C2212yescomplemented
29Dic3⋊Dic3C2212yescomplemented
30#3722C2211yescharacteristic, complemented
31(C2×D3)⋊Dic3C2212yescomplemented
32(C2×D3)⋊Dic3C2212yescomplemented
33Dic32C2211yescharacteristic, complemented
34C22×C3×D4—32no—
35C22×#67—32no—
36C22×#67D312yescomplemented
37C2×D3×D4—32no—
38#953—32no—
39#953—32no—
40#953—61no—
41#940—32no—
42#939—32no—
43#939—32no—
44#938—32no—
45#937—32no—
46#937—32no—
47D4×Dic3—32no—
48#930—32no—
49C4×#67—32no—
50C23×C32—24no—
51C23×C32—41no—
52C23×C32C2311yescharacteristic
53C23×C32C2312yes—
54C22×C3⋊D3C2311yescharacteristic
55C22×C3×D3—24no—
56C22×C3×D3—24no—
57C22×C3×D3—42no—
58C22×C3×D3C2312yes—
59C22×C3×D3C2312yes—
60C2×D32—24no—
61C2×D32—41no—
62#638—24no—
63#638—41no—
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C3×#67—24no—
67C3×#67—24no—
68C3×#67—24no—
69C3×#67D418yescomplemented
70C2×C3×Dic3—24no—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73#626—24no—
74#625—24no—
75#625—41no—
76D3×Dic3—24no—
77D42—91noSylow
78C24×C3—21no—
79C24×C3—32no—
80C24×C3C2×D312yescomplemented
81C23×D3—32no—
82C23×D3C2×D312yescomplemented
83C2×C3×D4—32no—
84C2×C3×D4—32no—
85C2×C3×D4—32no—
86C2×C3×D4—32no—
87C2×C3×D4—68no—
88C22×C3×C4—32no—
89C2×#67—28no—
90C2×#67—32no—
91C2×#67—32no—
92C2×#67—32no—
93C2×#67—32no—
94C2×#67—32no—
95C2×#67—32no—
96C2×#67—32no—
97C2×#67—62no—
98C2×#67—61no—
99C2×#67—68no—
100C2×#67C2×D312yes—
101C2×#67C2×D312yescomplemented
102C2×#67C2×D312yescomplemented
103C2×#67C2×D312yescomplemented
104C22×Dic3—32no—
105C22×Dic3C2×D312yescomplemented
106D3×D4—68no—
107C2×D12—32no—
108C2×C4×D3—32no—
109#217—32no—
110#217—32no—
111#217—32no—
112#217—32no—
113#217—32no—
114#217—62no—
115#217—61no—
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—32no—
118(C2×D3)⋊C4—32no—
119C4⋊Dic3—32no—
120Dic3⋊C4—32no—
121C4×Dic3—32no—
122C22×C32—21no—
123C22×C32—22no—
124C22×C32—22no—
125C22×C32—22no—
126C22×C32—44no—
127C22×C32C2×D414yescomplemented
128C22×C32C2411yescharacteristic
129C2×C3⋊D3—22no—
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×D414yescomplemented
138D32—44no—
139C3⋊Dic3—22no—
140C3⋊Dic3—21no—
141C3×Dic3—22no—
142C3×Dic3C2×D414yescomplemented
143C22×D4—32no—
144C22×D4—92no—
145#127—91no—
146#121—92no—
147#121—92no—
148#120—91no—
149#120—92no—
150#120—91no—
151C4×D4—92no—
152C23×C3—24no—
153C23×C3—24no—
154C23×C3—21no—
155C23×C3—22no—
156C23×C3—32no—
157C23×C3—32no—
158C23×C3—44no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3—64no—
162C23×C3—64no—
163C23×C3—122no—
164C23×C3C22×D312yes—
165C23×C3C22×D312yes—
166C23×C3C22×D312yes—
167C22×D3—24no—
168C22×D3—24no—
169C22×D3—32no—
170C22×D3—42no—
171C22×D3—61no—
172C22×D3—64no—
173C22×D3—64no—
174C22×D3—122no—
175C22×D3C22×D312yes—
176C22×D3C22×D312yes—
177C3×D4—38no—
178C3×D4—64no—
179C3×D4—64no—
180C3×D4—64no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—64no—
184#67—24no—
185#67—24no—
186#67—24no—
187#67—64no—
188#67—64no—
189#67—64no—
190#67—64no—
191#67—64no—
192#67—124no—
193#67—122no—
194#67—122no—
195#67—122no—
196#67#6718yesdirect product factor
197C2×Dic3—24no—
198C2×Dic3—32no—
199C2×Dic3—32no—
200C2×Dic3—32no—
201C2×Dic3—61no—
202C2×Dic3—62no—
203C2×Dic3—64no—
204C2×Dic3C22×D312yes—
205C2×Dic3C22×D312yes—
206D12—64no—
207C4×D3—64no—
208C2×C32—24no—
209C2×C32—41no—
210C2×C32C22×D412yes—
211C2×C32H(5,𝔽2)11yescharacteristic
212C3⋊D3—41no—
213C3×D3—24no—
214C3×D3—42no—
215C24—32no—
216C24—91no—
217C24D3211yescharacteristic, complemented
218C2×D4—32no—
219C2×D4—32no—
220C2×D4—32no—
221C2×D4—32no—
222C2×D4—68no—
223C2×D4—92no—
224C2×D4—92no—
225C2×D4—92no—
226C2×D4—91no—
227C2×D4—91no—
228C2×D4—188no—
229C22×C4—32no—
230C22×C4—92no—
231C4⋊C4—92no—
232C22⋊C4—91no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C22⋊C4—92no—
236C22⋊C4—91no—
237C42—91no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—24no—
248C22×C3—34no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—48no—
253C22×C3—48no—
254C22×C3—62no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—128no—
260C22×C3D3×D414yescomplemented
261C22×C3C2×#6714yescomplemented
262C22×C3C23×D312yes—
263C2×D3—22no—
264C2×D3—22no—
265C2×D3—22no—
266C2×D3—22no—
267C2×D3—22no—
268C2×D3—48no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—122no—
273C2×D3—121no—
274C2×D3—128no—
275C2×D3C2×#6714yescomplemented
276C3×C4—34no—
277C3×C4—62no—
278Dic3—22no—
279Dic3—62no—
280Dic3—62no—
281Dic3—62no—
282Dic3—122no—
283Dic3—121no—
284Dic3C2×#6714yescomplemented
285C32D4211yescharacteristic, complemented, Sylow
286C23—24no—
287C23—32no—
288C23—32no—
289C23—41no—
290C23—64no—
291C23—64no—
292C23—91no—
293C23—122no—
294C23—184no—
295C23—361no—
296C23C2×D3211yescharacteristic
297C23C2×D3212yes—
298D4—38no—
299D4—64no—
300D4—64no—
301D4—64no—
302D4—184no—
303D4—184no—
304D4—184no—
305D4—361no—
306D4—361no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—64no—
310C2×C4—91no—
311C2×C4—92no—
312C2×C4—184no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—22no—
316C2×C3—21no—
317C2×C3—44no—
318C2×C3—42no—
319C2×C3—42no—
320C2×C3—64no—
321C2×C3—122no—
322C2×C3C2×D3×D412yes—
323C2×C3#100412yes—
324C2×C3C22×#6712yes—
325D3—24no—
326D3—42no—
327D3—122no—
328D3—241no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—21no—
333C22—34no—
334C22—44no—
335C22—62no—
336C22—62no—
337C22—62no—
338C22—62no—
339C22—62no—
340C22—128no—
341C22—182no—
342C22—181no—
343C22—364no—
344C22D3×#6714yescomplemented
345C22C22×D3211yescharacteristic
346C4—34no—
347C4—62no—
348C4—182no—
349C4—181no—
350C3—21no—
351C3D4×#6712yescomplemented
352C2—24no—
353C2—41no—
354C2—64no—
355C2—122no—
356C2—361no—
357C2C2×D3×#6712yes—
358C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC24×C32C22
hypercenterC24D32
hypocenterC32D42
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#67C2semidirect
2. C2×D3×#67C2semidirect
3. #64162C2semidirect
4. #63983C2semidirect
5. #63979C2semidirect
6. #63978C2semidirect
7. #63976C2semidirect
8. Dic3×#67C2semidirect
9. #63970C2semidirect
11. C24×C32C22semidirect
12. C23×C3×D3C22semidirect
13. C22×D32C22semidirect
14. C2×#638C22semidirect
15. C2×#638C22semidirect
18. C2×C3×#67C22semidirect
19. C2×C3×#67C22semidirect
20. C2×C3×#67C22semidirect
21. C22×C3×Dic3C22semidirect
23. C2×#626C22semidirect
24. C2×#625C22semidirect
25. C2×#625C22semidirect
26. C2×D3×Dic3C22semidirect
27. #3757C22semidirect
28. #3757C22semidirect
29. Dic3⋊Dic3C22semidirect
30. #3722C22semidirect
31. (C2×D3)⋊Dic3C22semidirect
32. (C2×D3)⋊Dic3C22semidirect
33. Dic32C22semidirect
36. C22×#67D3semidirect
69. C3×#67D4semidirect
80. C24×C3C2×D3semidirect
82. C23×D3C2×D3semidirect
101. C2×#67C2×D3semidirect
102. C2×#67C2×D3semidirect
103. C2×#67C2×D3semidirect
105. C22×Dic3C2×D3semidirect
127. C22×C32C2×D4semidirect
137. C2×C3×D3C2×D4semidirect
142. C3×Dic3C2×D4semidirect
196. #67#67direct (N × H)
217. C24D32semidirect
260. C22×C3D3×D4semidirect
261. C22×C3C2×#67semidirect
275. C2×D3C2×#67semidirect
284. Dic3C2×#67semidirect
285. C32D42semidirect
344. C22D3×#67semidirect
351. C3D4×#67semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#672
C2×D3×#672
#641622
#639832
#639792
#639782
#639762
Dic3×#672
#639702
D4×#673