Finite Group Databasebeta

← #86018

Subgroups — #86018

Order 384.

1212 conjugacy classes of subgroups.

The subgroup lattice has 418 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
5C22×C4×Dic3C211yesmaximal, characteristic, complemented
6#5649C214yesmaximal, complemented
7#5624C212yesmaximal, complemented
8#5622C212yesmaximal, complemented
9#5600C212yesmaximal, complemented
10#5596C212yesmaximal, complemented
11#5586C214yesmaximal, complemented
12#5538C214yesmaximal, complemented
13C4×#237C214yesmaximal, complemented
14#5522C211yesmaximal, characteristic, complemented
15#5517C211yesmaximal, characteristic, complemented
16#3444—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×Dic3C2212yes—
20C2×#237C2214yes—
21C2×#235C2211yescharacteristic
22C3×#123C2212yes—
23C3×C22⋊Q8C2212yes—
24C3×#121C2212yes—
25C3×#120C2212yes—
26C3×C4×D4C2214yes—
27C3×#117C2211yescharacteristic
28#953C2212yes—
29#947C2211yescharacteristic, complemented
30Q8×Dic3C2212yescomplemented
31#944C2211yescharacteristic, complemented
32#940C2211yescharacteristic, complemented
33#939C2214yes—
34#939C2214yes—
35#936C2211yescharacteristic, complemented
36#935C2214yes—
37D4×Dic3C2214yes—
38D4×Dic3C2214yes—
39D4×Dic3C2212yescomplemented
40#930C2212yes—
41#929C2212yes—
42C4×#67C2214yes—
43C4×#67C2214yescomplemented
44#926C2211yescharacteristic
45#926C2212yescomplemented
46#924C2212yes—
47C2×C4×Dic3C2212yes—
48C2×C4×Dic3C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50C2×C4×Dic3C2212yes—
51C2×C4×Dic3C2212yes—
52C2×C4×Dic3C2212yes—
53C2×C4×Dic3C2212yes—
54C2×C4×Dic3C2212yescomplemented
55#898C2214yescomplemented
56#893C2212yescomplemented
57#892C2214yescomplemented
58#890C2212yescomplemented
59#889C2214yescomplemented
60#888C2212yescomplemented
61#887C2214yescomplemented
62#887C2212yescomplemented
63#886C2214yes—
64#885C2214yescomplemented
65#67⋊C4C2214yes—
66#67⋊C4C2214yes—
67#67⋊C4C2214yescomplemented
68#879C2214yescomplemented
69#878C2214yes—
70#878C2214yes—
71#877C2214yes—
72#877C2214yescomplemented
73#872C2212yescomplemented
74C42×D3C2212yescomplemented
75C4×Dic6C2214yescomplemented
76#533—32no—
77#533—31no—
78#532—31no—
79#532—32no—
80#529—34no—
81#529—32no—
82#529—32no—
83#529—34no—
84#525—31no—
85#525—34no—
86#525D311yescharacteristic, complemented
87C4×#41—31no—
88C4×#41—34no—
89C4×#41—31no—
90C22×C42—31no—
91C24×C3C2311yescharacteristic
92C3×#41—24no—
93C2×C3×Q8C2311yescharacteristic
94C2×C3×D4C2312yes—
95C2×C3×D4C2314yes—
96C2×C3×D4C2311yescharacteristic
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×Dic3—28no—
107C22×Dic3C2314yes—
108C22×Dic3C2312yes—
109C22×Dic3C2312yes—
110C22×Dic3C2312yes—
111C22×Dic3C2312yes—
112C22×Dic3C2312yes—
113C22×Dic3C2312yes—
114#237—216no—
115#235—24no—
116C2×D12C2311yescharacteristic
117C2×C4×D3C2312yes—
118C2×C4×D3C2314yes—
119C2×Dic6C2314yes—
120C2×Dic6C2311yescharacteristic
121C3×C4⋊C4C2312yes—
122C3×C4⋊C4C2314yes—
123C3×C22⋊C4C2314yes—
124C3×C22⋊C4C2314yes—
125C3×C22⋊C4C2312yes—
126C3×C42C2312yes—
127#217C2314yes—
128#217C2314yes—
129#217C2312yes—
130#217C2314yes—
131(C2×D3)⋊C4C2314yes—
132(C2×D3)⋊C4C2314yes—
133(C2×D3)⋊C4C2314yes—
134C4⋊Dic3C2314yes—
135C4⋊Dic3C2314yes—
136C4⋊Dic3C2312yes—
137Dic3⋊C4C2314yes—
138Dic3⋊C4C2314yes—
139Dic3⋊C4C2314yes—
140Dic3⋊C4C2314yes—
141Dic3⋊C4C2314yes—
142C4×Dic3C2312yes—
143C4×Dic3C2312yes—
144C4×Dic3C2311yescharacteristic
145C4×Dic3C2314yes—
146C4×Dic3C2311yescharacteristic
147C4×Dic3C2314yes—
148C4×Dic3C2312yes—
149C4×Dic3C2312yes—
150C4×Dic3C2312yes—
151C4×Dic3C2312yes—
152C4×Dic3C2312yes—
153C4×Dic3C2311yescharacteristic
154C4×Dic3C2311yescharacteristic
155C2×#41—34no—
156C2×#41—31no—
157C2×#41C2×D311yescharacteristic
158C23×C4—32no—
159C23×C4C2×D311yescharacteristic
160C4⋊Q8—31no—
161C4⋊Q8—32no—
162#127—31no—
163#126—34no—
164#126—34no—
165#125—34no—
166#125—32no—
167#124—31no—
168#124—31no—
169#124—34no—
170#123—34no—
171#123—32no—
172#123—34no—
173#123C2×D312yes—
174C22⋊Q8—32no—
175C22⋊Q8—34no—
176C22⋊Q8—34no—
177C22⋊Q8C2×D312yes—
178#121—34no—
179#121—34no—
180#121—32no—
181#121C2×D312yes—
182#120—32no—
183#120C2×D312yes—
184C4×Q8—32no—
185C4×Q8—34no—
186C4×Q8—32no—
187C4×Q8—34no—
188C4×D4—34no—
189C4×D4—34no—
190C4×D4—32no—
191C4×D4—34no—
192C4×D4—34no—
193C4×D4—32no—
194C4×D4—34no—
195C4×D4—34no—
196C4×D4—34no—
197C4×D4C2×D314yes—
198#117—31no—
199#117—32no—
200#117—34no—
201#117—34no—
202#117—34no—
203#117—32no—
204#117C2×D311yescharacteristic
205C2×C42—32no—
206C2×C42—32no—
207C2×C42—31no—
208C2×C42—32no—
209C2×C42—32no—
210C2×C42—32no—
211C2×C42—32no—
212C2×C42—31no—
213C2×C42—32no—
214C23×C3—24no—
215C23×C3—22no—
216C23×C3C2412yes—
217C23×C3C2411yescharacteristic
218C23×C3C2412yes—
219C22×D3C2412yes—
220C3×Q8—22no—
221C3×D4—28no—
222C3×D4—24no—
223C3×D4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4#4114yescomplemented
229C2×C3×C4C2412yes—
230C2×C3×C4C2412yes—
231C2×C3×C4C2414yes—
232C2×C3×C4C2411yescharacteristic
233C2×C3×C4C2411yescharacteristic
234#67—24no—
235#67—28no—
236#67—28no—
237C2×Dic3—24no—
238C2×Dic3—28no—
239C2×Dic3—24no—
240C2×Dic3—24no—
241C2×Dic3—24no—
242C2×Dic3—24no—
243C2×Dic3#4118yescomplemented
244C2×Dic3C2412yes—
245C2×Dic3C2412yes—
246C2×Dic3C2412yes—
247C2×Dic3C2414yes—
248C2×Dic3C2412yes—
249C2×Dic3C2412yes—
250D12—22no—
251C4×D3—24no—
252C4×D3—28no—
253Dic6—22no—
254Dic6—28no—
255C24C22×D311yescharacteristic
256#41—24no—
257#41—616no—
258#41—64no—
259C2×Q8—34no—
260C2×Q8—31no—
261C2×Q8C22×D311yescharacteristic
262C2×D4—32no—
263C2×D4—34no—
264C2×D4—34no—
265C2×D4—31no—
266C2×D4C22×D312yes—
267C2×D4C22×D311yescharacteristic
268C2×D4C22×D314yes—
269C22×C4—24no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—32no—
275C22×C4—32no—
276C22×C4—34no—
277C22×C4—34no—
278C22×C4—32no—
279C22×C4—68no—
280C22×C4C22×D312yes—
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D312yes—
283C22×C4C22×D312yes—
284C22×C4C22×D311yescharacteristic
285C4⋊C4—34no—
286C4⋊C4—34no—
287C4⋊C4—32no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4C22×D314yes—
294C4⋊C4C22×D312yes—
295C22⋊C4—32no—
296C22⋊C4—34no—
297C22⋊C4—34no—
298C22⋊C4—34no—
299C22⋊C4—34no—
300C22⋊C4—34no—
301C22⋊C4—34no—
302C22⋊C4C22×D314yes—
303C22⋊C4C22×D314yes—
304C22⋊C4C22×D312yes—
305C42—32no—
306C42—32no—
307C42—32no—
308C42—32no—
309C42—31no—
310C42—32no—
311C42—31no—
312C42—34no—
313C42—34no—
314C42—31no—
315C42—31no—
316C42—32no—
317C42—32no—
318C42C22×D312yes—
319C22×C3—28no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—24no—
324C22×C3—22no—
325C22×C3C2×#4112yes—
326C22×C3C2×#4114yes—
327C22×C3C2511yescharacteristic
328C2×D3—22no—
329C2×D3—24no—
330C3×C4—24no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4C2×#4112yes—
334C3×C4C2×#4112yes—
335Dic3—22no—
336Dic3—24no—
337Dic3—22no—
338Dic3—22no—
339Dic3C2×#4114yes—
340Dic3C2×#4114yes—
341C23—22no—
342C23—24no—
343C23—32no—
344C23C23×D312yes—
345C23C23×D311yescharacteristic
346C23C23×D312yes—
347Q8—22no—
348Q8—68no—
349Q8—62no—
350D4—24no—
351D4—28no—
352D4—22no—
353D4—68no—
354D4—64no—
355D4—68no—
356D4—62no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—38no—
362C2×C4—32no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—32no—
367C2×C4—34no—
368C2×C4—68no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—64no—
372C2×C4—64no—
373C2×C4—68no—
374C2×C4—64no—
375C2×C4—64no—
376C2×C4#23714yescomplemented
377C2×C4C23×D312yes—
378C2×C4C23×D312yes—
379C2×C4C23×D314yes—
380C2×C4C23×D311yescharacteristic
381C2×C4C23×D311yescharacteristic
382C2×C3—22no—
383C2×C3—24no—
384C2×C3—42no—
385C2×C3C22×#4111yescharacteristic
386C2×C3C22×#4112yes—
387D3—42no—
388C22—28no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—24no—
394C22—62no—
395C22—64no—
396C22C2×#23712yes—
397C22D3×#4114yes—
398C22C24×D311yescharacteristic
399C4—22no—
400C4—22no—
401C4—24no—
402C4—34no—
403C4—34no—
404C4—62no—
405C4—62no—
406C4—62no—
407C4—64no—
408C4C2×#23712yes—
409C4C2×#23712yes—
410C3#344411yescharacteristic, complemented, Sylow
411C2—22no—
412C2—24no—
413C2—42no—
414C2—122no—
415C2C22×#23711yescharacteristic
416C2C2×D3×#4112yes—
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3444
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. C22×C4×Dic3C2semidirect
6. #5649C2semidirect
7. #5624C2semidirect
8. #5622C2semidirect
9. #5600C2semidirect
10. #5596C2semidirect
11. #5586C2semidirect
12. #5538C2semidirect
13. C4×#237C2semidirect
14. #5522C2semidirect
15. #5517C2semidirect
29. #947C22semidirect
30. Q8×Dic3C22semidirect
31. #944C22semidirect
32. #940C22semidirect
35. #936C22semidirect
39. D4×Dic3C22semidirect
43. C4×#67C22semidirect
45. #926C22semidirect
54. C2×C4×Dic3C22semidirect
55. #898C22semidirect
56. #893C22semidirect
57. #892C22semidirect
58. #890C22semidirect
59. #889C22semidirect
60. #888C22semidirect
61. #887C22semidirect
62. #887C22semidirect
64. #885C22semidirect
67. #67⋊C4C22semidirect
68. #879C22semidirect
72. #877C22semidirect
73. #872C22semidirect
74. C42×D3C22semidirect
75. C4×Dic6C22semidirect
86. #525D3semidirect
228. C2×C3×C4#41semidirect
243. C2×Dic3#41semidirect
376. C2×C4#237semidirect
410. C3#3444semidirect

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
C22×C4×Dic32
#56492
#56242
#56222
#56002
#55962
#55862
#55382
C4×#2372
#55222
#55172
#34443