Finite Group Databasebeta

← #86019

Subgroups — #86019

Order 384.

1040 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#525C211yesmaximal, characteristic, complemented
2#5832C211yesmaximal, characteristic, complemented
3Dic3×#41C211yesmaximal, characteristic, complemented
4#5791C211yesmaximal, characteristic, complemented
5C2×#926C211yesmaximal, characteristic, complemented
6#5665C214yesmaximal, complemented
7#5643C212yesmaximal, complemented
8#5625C212yesmaximal, complemented
9#5619C212yesmaximal, complemented
10#5603C212yesmaximal, complemented
11#5593C214yesmaximal, complemented
12#5557C214yesmaximal, complemented
13#5538C214yesmaximal, complemented
14#5522C211yesmaximal, characteristic, complemented
15#5517C211yesmaximal, characteristic, complemented
16#3455—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C3×#123C2212yes—
21C3×C22⋊Q8C2212yes—
22C3×#121C2212yes—
23C3×#120C2212yes—
24C3×C4×D4C2214yes—
25C3×#117C2211yescharacteristic
26#953C2212yes—
27C2×#217C2212yes—
28#947C2211yescharacteristic, complemented
29Q8×Dic3C2212yescomplemented
30#944C2211yescharacteristic, complemented
31#940C2211yescharacteristic, complemented
32#939C2214yes—
33#939C2214yes—
34#938C2214yes—
35#936C2214yes—
36#936C2211yescharacteristic, complemented
37#935C2214yes—
38D4×Dic3C2214yes—
39D4×Dic3C2212yescomplemented
40#930C2212yes—
41#929C2212yes—
42C4×#67C2214yes—
43C4×#67C2214yescomplemented
44#926C2212yes—
45#926C2211yescharacteristic
46#926C2212yes—
47#926C2212yes—
48#926C2212yes—
49#926C2212yescomplemented
50C2×C4⋊Dic3C2212yes—
51#924C2212yes—
52C2×C4×Dic3C2211yescharacteristic
53C2×C4×Dic3C2211yescharacteristic
54C2×C4×Dic3C2212yescomplemented
55#898C2214yescomplemented
56#897C2214yescomplemented
57#893C2212yescomplemented
58#890C2214yescomplemented
59#890C2212yescomplemented
60#889C2214yescomplemented
61#888C2212yescomplemented
62#887C2212yescomplemented
63#886C2214yescomplemented
64#885C2214yes—
65#883C2214yes—
66#883C2214yescomplemented
67#67⋊C4C2214yes—
68#879C2214yes—
69#879C2214yes—
70#879C2214yescomplemented
71#878C2214yes—
72#878C2214yescomplemented
73#872C2212yescomplemented
74C42×D3C2212yescomplemented
75C4×Dic6C2214yescomplemented
76#542—32no—
77#542—32no—
78#541—32no—
79#540—32no—
80#539—34no—
81#538—34no—
82#533—31no—
83#532—31no—
84#529—34no—
85#529—34no—
86#525—31no—
87#525D311yescharacteristic, complemented
88C4×#41—31no—
89C4×#41—31no—
90C2×#117—31no—
91C24×C3C2311yescharacteristic
92C3×#41—24no—
93C2×C3×Q8C2311yescharacteristic
94C2×C3×D4C2312yes—
95C2×C3×D4C2311yescharacteristic
96C2×C3×D4C2314yes—
97C22×C3×C4—24no—
98C22×C3×C4C2312yes—
99C22×C3×C4C2312yes—
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2312yes—
102C22×C3×C4C2311yescharacteristic
103C2×#67C2312yes—
104C2×#67C2314yes—
105C2×#67C2314yes—
106C22×Dic3C2312yes—
107C22×Dic3C2314yes—
108C22×Dic3C2312yes—
109#235—24no—
110C2×D12C2311yescharacteristic
111C2×C4×D3C2312yes—
112C2×C4×D3C2314yes—
113C2×Dic6C2311yescharacteristic
114C2×Dic6C2314yes—
115C3×C4⋊C4C2312yes—
116C3×C4⋊C4C2314yes—
117C3×C22⋊C4C2312yes—
118C3×C22⋊C4C2314yes—
119C3×C22⋊C4C2314yes—
120C3×C42C2312yes—
121#217C2312yes—
122#217C2314yes—
123#217C2314yes—
124#217C2312yes—
125#217C2312yes—
126#217C2314yes—
127#217C2312yes—
128#217C2312yes—
129(C2×D3)⋊C4C2314yes—
130(C2×D3)⋊C4C2314yes—
131(C2×D3)⋊C4C2314yes—
132C4⋊Dic3C2314yes—
133C4⋊Dic3C2312yes—
134C4⋊Dic3C2314yes—
135C4⋊Dic3C2312yes—
136C4⋊Dic3C2312yes—
137C4⋊Dic3C2312yes—
138C4⋊Dic3C2312yes—
139Dic3⋊C4C2314yes—
140Dic3⋊C4C2314yes—
141Dic3⋊C4C2314yes—
142Dic3⋊C4C2314yes—
143Dic3⋊C4C2314yes—
144C4×Dic3C2312yes—
145C4×Dic3C2312yes—
146C4×Dic3C2311yescharacteristic
147C4×Dic3C2314yes—
148C4×Dic3C2311yescharacteristic
149C4×Dic3C2314yes—
150C4×Dic3C2312yes—
151C4×Dic3C2311yescharacteristic
152C4×Dic3C2311yescharacteristic
153C2×#41—31no—
154C2×#41C2×D311yescharacteristic
155C23×C4C2×D311yescharacteristic
156C4⋊Q8—31no—
157C4⋊Q8—32no—
158#127—31no—
159#126—34no—
160#126—34no—
161#126—34no—
162#126—34no—
163#125—34no—
164#125—34no—
165#125—32no—
166#124—31no—
167#124—34no—
168#124—31no—
169#124—34no—
170#123—34no—
171#123—32no—
172#123—34no—
173#123—34no—
174#123—34no—
175#123C2×D312yes—
176C22⋊Q8—32no—
177C22⋊Q8—34no—
178C22⋊Q8—34no—
179C22⋊Q8—34no—
180C22⋊Q8C2×D312yes—
181#121—34no—
182#121—34no—
183#121—34no—
184#121—32no—
185#121C2×D312yes—
186#120—32no—
187#120C2×D312yes—
188C4×Q8—32no—
189C4×Q8—32no—
190C4×Q8—34no—
191C4×D4—34no—
192C4×D4—32no—
193C4×D4—34no—
194C4×D4—34no—
195C4×D4—32no—
196C4×D4—34no—
197C4×D4C2×D314yes—
198#117—32no—
199#117—32no—
200#117—32no—
201#117—32no—
202#117—31no—
203#117—32no—
204#117—32no—
205#117C2×D311yescharacteristic
206C2×C4⋊C4—32no—
207C2×C22⋊C4—32no—
208C2×C42—31no—
209C2×C42—31no—
210C2×C42—32no—
211C2×C42—32no—
212C23×C3—22no—
213C23×C3—42no—
214C23×C3C2412yes—
215C23×C3C2412yes—
216C23×C3C2411yescharacteristic
217C22×D3C2412yes—
218C3×Q8—22no—
219C3×D4—24no—
220C3×D4—22no—
221C3×D4—44no—
222C2×C3×C4—22no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4#4114yes—
227C2×C3×C4C2412yes—
228C2×C3×C4C2414yes—
229C2×C3×C4C2412yes—
230C2×C3×C4C2411yescharacteristic
231C2×C3×C4C2411yescharacteristic
232#67—24no—
233#67—44no—
234#67—44no—
235C2×Dic3—42no—
236C2×Dic3—44no—
237C2×Dic3—42no—
238C2×Dic3C2412yes—
239C2×Dic3C2412yes—
240C2×Dic3C2412yes—
241C2×Dic3C2412yes—
242C2×Dic3C2414yes—
243C2×Dic3C2412yes—
244D12—22no—
245C4×D3—24no—
246C4×D3—44no—
247Dic6—22no—
248Dic6—44no—
249C24C22×D311yescharacteristic
250#41—24no—
251#41—64no—
252C2×Q8—31no—
253C2×Q8—34no—
254C2×Q8C22×D311yescharacteristic
255C2×D4—32no—
256C2×D4—34no—
257C2×D4—34no—
258C2×D4—31no—
259C2×D4C22×D312yes—
260C2×D4C22×D311yescharacteristic
261C2×D4C22×D314yes—
262C22×C4—24no—
263C22×C4—32no—
264C22×C4—32no—
265C22×C4—34no—
266C22×C4—32no—
267C22×C4—34no—
268C22×C4C22×D311yescharacteristic
269C22×C4C22×D312yes—
270C22×C4C22×D312yes—
271C22×C4C22×D312yes—
272C22×C4C22×D311yescharacteristic
273C4⋊C4—32no—
274C4⋊C4—32no—
275C4⋊C4—32no—
276C4⋊C4—32no—
277C4⋊C4—32no—
278C4⋊C4—34no—
279C4⋊C4—34no—
280C4⋊C4—34no—
281C4⋊C4—34no—
282C4⋊C4—34no—
283C4⋊C4—34no—
284C4⋊C4—34no—
285C4⋊C4C22×D312yes—
286C4⋊C4C22×D314yes—
287C22⋊C4—32no—
288C22⋊C4—32no—
289C22⋊C4—32no—
290C22⋊C4—32no—
291C22⋊C4—32no—
292C22⋊C4—34no—
293C22⋊C4—34no—
294C22⋊C4—34no—
295C22⋊C4—34no—
296C22⋊C4—34no—
297C22⋊C4—34no—
298C22⋊C4C22×D312yes—
299C22⋊C4C22×D314yes—
300C22⋊C4C22×D314yes—
301C42—32no—
302C42—31no—
303C42—31no—
304C42—34no—
305C42—34no—
306C42—31no—
307C42—31no—
308C42—32no—
309C42—32no—
310C42C22×D312yes—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—24no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—44no—
318C22×C3C2×#4112yes—
319C22×C3C2511yescharacteristic
320C2×D3—22no—
321C2×D3—24no—
322C3×C4—22no—
323C3×C4—22no—
324C3×C4—24no—
325C3×C4C2×#4112yes—
326C3×C4C2×#4112yes—
327Dic3—22no—
328Dic3—22no—
329Dic3—24no—
330Dic3—22no—
331Dic3—22no—
332Dic3—22no—
333C23—22no—
334C23—32no—
335C23—42no—
336C23C23×D312yes—
337C23C23×D312yes—
338C23C23×D311yescharacteristic
339Q8—22no—
340Q8—62no—
341Q8—124no—
342D4—24no—
343D4—22no—
344D4—44no—
345D4—64no—
346D4—62no—
347D4—124no—
348D4—124no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—34no—
358C2×C4—32no—
359C2×C4—64no—
360C2×C4—124no—
361C2×C4—122no—
362C2×C4—124no—
363C2×C4—122no—
364C2×C4#23714yes—
365C2×C4C23×D312yes—
366C2×C4C23×D314yes—
367C2×C4C23×D312yes—
368C2×C4C23×D311yescharacteristic
369C2×C4C23×D311yescharacteristic
370C2×C3—22no—
371C2×C3—42no—
372C2×C3—42no—
373C2×C3C22×#4111yescharacteristic
374C2×C3#58512yes—
375D3—42no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—24no—
380C22—22no—
381C22—22no—
382C22—44no—
383C22—62no—
384C22—64no—
385C22C2×#23712yes—
386C22C24×D311yescharacteristic
387C4—22no—
388C4—22no—
389C4—24no—
390C4—62no—
391C4—62no—
392C4—62no—
393C4—62no—
394C4—64no—
395C4—62no—
396C4C2×#23712yes—
397C4C2×#23712yes—
398C3#345511yescharacteristic, complemented, Sylow
399C2—22no—
400C2—42no—
401C2—42no—
402C2—122no—
403C2C22×#23711yescharacteristic
404C2#596412yes—
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3455
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C3×#525C2semidirect
2. #5832C2semidirect
3. Dic3×#41C2semidirect
4. #5791C2semidirect
5. C2×#926C2semidirect
6. #5665C2semidirect
7. #5643C2semidirect
8. #5625C2semidirect
9. #5619C2semidirect
10. #5603C2semidirect
11. #5593C2semidirect
12. #5557C2semidirect
13. #5538C2semidirect
14. #5522C2semidirect
15. #5517C2semidirect
28. #947C22semidirect
29. Q8×Dic3C22semidirect
30. #944C22semidirect
31. #940C22semidirect
36. #936C22semidirect
39. D4×Dic3C22semidirect
43. C4×#67C22semidirect
49. #926C22semidirect
54. C2×C4×Dic3C22semidirect
55. #898C22semidirect
56. #897C22semidirect
57. #893C22semidirect
58. #890C22semidirect
59. #890C22semidirect
60. #889C22semidirect
61. #888C22semidirect
62. #887C22semidirect
63. #886C22semidirect
66. #883C22semidirect
70. #879C22semidirect
72. #878C22semidirect
73. #872C22semidirect
74. C42×D3C22semidirect
75. C4×Dic6C22semidirect
87. #525D3semidirect
398. C3#3455semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
#58322
Dic3×#412
#57912
C2×#9262
#56652
#56432
#56252
#56192
#56032
#55932
#55572
#55382
#55222
#55172
#34553