Finite Group Databasebeta

← C2×#5678 #86071

Subgroups — C2×#5678 #86071

Order 384.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#124C211yesmaximal, characteristic, complemented
2C2×#947C211yesmaximal, characteristic, complemented
3C2×#946C211yesmaximal, characteristic, complemented
4C2×#939C211yesmaximal, characteristic, complemented
5C2×#937C211yesmaximal, characteristic, complemented
6#5678C2116yesmaximal, direct product factor
7C2×#886C211yesmaximal, characteristic, complemented
8C2×#884C211yesmaximal, characteristic, complemented
9C2×#884C211yesmaximal, characteristic, complemented
10C2×#883C211yesmaximal, characteristic, complemented
11C2×#882C211yesmaximal, characteristic, complemented
12C2×#67⋊C4C211yesmaximal, characteristic, complemented
13C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
14C2×#879C211yesmaximal, characteristic, complemented
15C2×#876C211yesmaximal, characteristic, complemented
16C2×C4×D12C211yesmaximal, characteristic, complemented
17C2×#538—31nomaximal, Sylow
18C24×D3C2211yescharacteristic
19C22×C3×Q8C2211yescharacteristic, complemented
20C22×C3×D4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic, complemented
25C23×Dic3C2211yescharacteristic, complemented
26C22×D12C2211yescharacteristic
27C22×D12C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C22×C4×D3C2211yescharacteristic, complemented
31C3×#124C2218yescomplemented
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic, complemented
35C2×C3×C22⋊C4C2211yescharacteristic, complemented
36C2×C3×C42C2211yescharacteristic, complemented
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39#947C2218yescomplemented
40#946C2218yescomplemented
41#939C2218yescomplemented
42#937C2218yescomplemented
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×(C2×D3)⋊C4C2211yescharacteristic
48C2×(C2×D3)⋊C4C2211yescharacteristic
49C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
50C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
51C2×C4⋊Dic3C2211yescharacteristic, complemented
52C2×C4⋊Dic3C2211yescharacteristic, complemented
53C2×Dic3⋊C4C2211yescharacteristic, complemented
54C2×Dic3⋊C4C2211yescharacteristic, complemented
55C2×Dic3⋊C4C2211yescharacteristic, complemented
56C2×Dic3⋊C4C2211yescharacteristic, complemented
57C2×C4×Dic3C2211yescharacteristic, complemented
58#886C2218yescomplemented
59#884C2218yescomplemented
60#884C2218yescomplemented
61#883C2218yescomplemented
62#882C2218yescomplemented
63#67⋊C4C2218yescomplemented
64D3×C22⋊C4C2218yescomplemented
65#879C2218yescomplemented
66#876C2218yescomplemented
67C4×D12C2218yescomplemented
68#538—316no—
69C2×#126—31no—
70C2×#126—31no—
71C2×#124—31no—
72C2×#124D311yescharacteristic, complemented
73C2×#123—31no—
74C2×#123—31no—
75C2×C22⋊Q8—31no—
76C2×#121—31no—
77C2×#121—31no—
78C2×#121—31no—
79C2×#120—31no—
80C2×#120—31no—
81C2×C4×D4—31no—
82C2×C4×D4—31no—
83C22×C22⋊C4—31no—
84C24×C3C2311yescharacteristic
85C24×C3C2311yescharacteristic
86C23×D3—24no—
87C23×D3—42no—
88C23×D3—42no—
89C23×D3C2311yescharacteristic
90C23×D3C2311yescharacteristic
91C23×D3C2311yescharacteristic
92C23×D3C2314yes—
93C2×C3×Q8—42no—
94C2×C3×Q8C2314yescomplemented
95C2×C3×D4—42no—
96C2×C3×D4C2314yes—
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C2×#67—24no—
103C2×#67—42no—
104C2×#67—42no—
105C2×#67—42no—
106C2×#67C2314yes—
107C2×#67C2314yes—
108C2×#67C2314yes—
109C2×#67C2314yescomplemented
110C22×Dic3—42no—
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2314yescomplemented
117C2×D12—24no—
118C2×D12—42no—
119C2×D12C2314yes—
120C2×D12C2314yes—
121C2×C4×D3—42no—
122C2×C4×D3—42no—
123C2×C4×D3—42no—
124C2×C4×D3C2314yes—
125C2×C4×D3C2314yes—
126C2×C4×D3C2314yescomplemented
127C3×C22⋊C4C2314yes—
128C3×C22⋊C4C2314yes—
129C3×C22⋊C4C2314yescomplemented
130C3×C22⋊C4C2314yescomplemented
131C3×C42C2314yescomplemented
132#217C2314yes—
133#217C2314yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yescomplemented
141(C2×D3)⋊C4C2314yescomplemented
142C4⋊Dic3C2314yescomplemented
143C4⋊Dic3C2314yescomplemented
144Dic3⋊C4C2314yescomplemented
145Dic3⋊C4C2314yescomplemented
146Dic3⋊C4C2314yescomplemented
147Dic3⋊C4C2314yescomplemented
148C4×Dic3C2314yescomplemented
149C25—31no—
150C22×Q8C2×D311yescharacteristic, complemented
151C22×D4—31no—
152C22×D4—31no—
153C22×D4—31no—
154C22×D4—31no—
155C22×D4—31no—
156C22×D4—31no—
157C22×D4C2×D311yescharacteristic
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—31no—
162#126—38no—
163#126—38no—
164#124—38no—
165#124C2×D318yescomplemented
166#123—38no—
167#123—38no—
168C22⋊Q8—38no—
169#121—38no—
170#121—38no—
171#121—38no—
172#120—38no—
173#120—38no—
174C4×D4—38no—
175C4×D4—38no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—38no—
193C2×C22⋊C4C2×D311yescharacteristic
194C2×C22⋊C4C2×D311yescharacteristic
195C2×C22⋊C4C2×D311yescharacteristic, complemented
196C2×C22⋊C4C2×D311yescharacteristic, complemented
197C2×C42—31no—
198C2×C42C2×D311yescharacteristic, complemented
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—22no—
205C23×C3C2412yes—
206C23×C3C2412yes—
207C23×C3C2411yescharacteristic
208C22×D3—22no—
209C22×D3—22no—
210C22×D3—22no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—22no—
214C22×D3—22no—
215C22×D3—22no—
216C22×D3—28no—
217C22×D3—44no—
218C22×D3—44no—
219C22×D3—416no—
220C22×D3#4114yes—
221C22×D3C2412yes—
222C22×D3C2412yes—
223C22×D3C2412yes—
224C3×Q8—44no—
225C3×D4—44no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4C2412yes—
232C2×C3×C4C2412yes—
233C2×C3×C4C2412yes—
234C2×C3×C4C2412yes—
235C2×C3×C4C2412yes—
236#67—28no—
237#67—44no—
238#67—44no—
239#67—44no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—44no—
246C2×Dic3C2412yes—
247C2×Dic3C2412yes—
248C2×Dic3C2412yes—
249C2×Dic3C2412yes—
250C2×Dic3C2412yes—
251D12—28no—
252D12—44no—
253C4×D3—44no—
254C4×D3—44no—
255C4×D3—44no—
256C24—31no—
257C24—31no—
258C24—31no—
259C24—34no—
260C24—64no—
261C24—122no—
262C24—122no—
263C24C22×D311yescharacteristic
264C24C22×D311yescharacteristic
265C2×Q8—42no—
266C2×Q8C22×D314yescomplemented
267C2×D4—34no—
268C2×D4—34no—
269C2×D4—34no—
270C2×D4—34no—
271C2×D4—34no—
272C2×D4—34no—
273C2×D4—42no—
274C2×D4—64no—
275C2×D4—64no—
276C2×D4—122no—
277C2×D4—122no—
278C2×D4—122no—
279C2×D4—122no—
280C2×D4C22×D314yes—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—34no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—34no—
290C22×C4—122no—
291C22×C4—122no—
292C22×C4—122no—
293C22×C4—122no—
294C22×C4C22×D311yescharacteristic
295C22×C4C22×D311yescharacteristic
296C22×C4C22×D311yescharacteristic
297C22×C4C22×D311yescharacteristic
298C22×C4C22×D311yescharacteristic
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—34no—
304C4⋊C4—34no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—34no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—34no—
311C22⋊C4—34no—
312C22⋊C4—34no—
313C22⋊C4—34no—
314C22⋊C4—34no—
315C22⋊C4C22×D314yes—
316C22⋊C4C22×D314yes—
317C22⋊C4C22×D314yescomplemented
318C22⋊C4C22×D314yescomplemented
319C42—34no—
320C42C22×D314yescomplemented
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—44no—
328C22×C3—44no—
329C22×C3C2×#4112yes—
330C22×C3H(5,𝔽2)12yes—
331C22×C3H(5,𝔽2)12yes—
332C22×C3C2511yescharacteristic
333C2×D3—28no—
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—22no—
342C2×D3—416no—
343C2×D3—44no—
344C2×D3—44no—
345C2×D3C2×#4114yes—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351Dic3—22no—
352Dic3—22no—
353Dic3—22no—
354Dic3—22no—
355Dic3—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—34no—
363C23—32no—
364C23—32no—
365C23—32no—
366C23—68no—
367C23—62no—
368C23—62no—
369C23—62no—
370C23—62no—
371C23—62no—
372C23—62no—
373C23—62no—
374C23—62no—
375C23—124no—
376C23—124no—
377C23—1216no—
378C23C23×D312yes—
379C23C23×D312yes—
380C23C23×D311yescharacteristic
381Q8—44no—
382D4—44no—
383D4—68no—
384D4—68no—
385D4—124no—
386D4—124no—
387D4—124no—
388D4—124no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4—124no—
408C2×C4C23×D312yes—
409C2×C4C23×D312yes—
410C2×C4C23×D312yes—
411C2×C4C23×D312yes—
412C2×C4C23×D312yes—
413C2×C3—42no—
414C2×C3—42no—
415C2×C3#53814yescomplemented
416C2×C3C22×#4111yescharacteristic
417C2×C3C2×H(5,𝔽2)11yescharacteristic
418C2×C3C2×H(5,𝔽2)11yescharacteristic
419D3—24no—
420D3—42no—
421D3—42no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—34no—
429C22—44no—
430C22—44no—
431C22—68no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—1216no—
441C22—124no—
442C22—124no—
443C22#100412yes—
444C22D3×#4112yes—
445C22#100912yes—
446C22C24×D311yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—62no—
453C4—62no—
454C4—62no—
455C4—62no—
456C4—62no—
457C3C2×#53811yescharacteristic, complemented, Sylow
458C2—42no—
459C2—42no—
460C2—64no—
461C2—122no—
462C2—122no—
463C2#567814yesdirect product factor
464C2C2×#100411yescharacteristic
465C2C2×D3×#4111yescharacteristic
466C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#124C2
socleC23×C3C24
hypercenterC2×#124D3
hypocenterC3C2×#538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
2. C2×#947C2semidirect
3. C2×#946C2semidirect
4. C2×#939C2semidirect
5. C2×#937C2semidirect
6. #5678C2direct (N × H)
7. C2×#886C2semidirect
8. C2×#884C2semidirect
9. C2×#884C2semidirect
10. C2×#883C2semidirect
11. C2×#882C2semidirect
12. C2×#67⋊C4C2semidirect
13. C2×D3×C22⋊C4C2semidirect
14. C2×#879C2semidirect
15. C2×#876C2semidirect
16. C2×C4×D12C2semidirect
19. C22×C3×Q8C22semidirect
24. C22×#67C22semidirect
25. C23×Dic3C22semidirect
30. C22×C4×D3C22semidirect
31. C3×#124C22semidirect
34. C2×C3×C22⋊C4C22semidirect
35. C2×C3×C22⋊C4C22semidirect
36. C2×C3×C42C22semidirect
39. #947C22semidirect
40. #946C22semidirect
41. #939C22semidirect
42. #937C22semidirect
49. C2×(C2×D3)⋊C4C22semidirect
50. C2×(C2×D3)⋊C4C22semidirect
51. C2×C4⋊Dic3C22semidirect
52. C2×C4⋊Dic3C22semidirect
53. C2×Dic3⋊C4C22semidirect
54. C2×Dic3⋊C4C22semidirect
55. C2×Dic3⋊C4C22semidirect
56. C2×Dic3⋊C4C22semidirect
57. C2×C4×Dic3C22semidirect
58. #886C22semidirect
59. #884C22semidirect
60. #884C22semidirect
61. #883C22semidirect
62. #882C22semidirect
63. #67⋊C4C22semidirect
64. D3×C22⋊C4C22semidirect
65. #879C22semidirect
66. #876C22semidirect
67. C4×D12C22semidirect
72. C2×#124D3semidirect
94. C2×C3×Q8C23semidirect
109. C2×#67C23semidirect
116. C22×Dic3C23semidirect
126. C2×C4×D3C23semidirect
129. C3×C22⋊C4C23semidirect
130. C3×C22⋊C4C23semidirect
131. C3×C42C23semidirect
140. (C2×D3)⋊C4C23semidirect
141. (C2×D3)⋊C4C23semidirect
142. C4⋊Dic3C23semidirect
143. C4⋊Dic3C23semidirect
144. Dic3⋊C4C23semidirect
145. Dic3⋊C4C23semidirect
146. Dic3⋊C4C23semidirect
147. Dic3⋊C4C23semidirect
148. C4×Dic3C23semidirect
150. C22×Q8C2×D3semidirect
165. #124C2×D3semidirect
195. C2×C22⋊C4C2×D3semidirect
196. C2×C22⋊C4C2×D3semidirect
198. C2×C42C2×D3semidirect
266. C2×Q8C22×D3semidirect
317. C22⋊C4C22×D3semidirect
318. C22⋊C4C22×D3semidirect
320. C42C22×D3semidirect
415. C2×C3#538semidirect
457. C3C2×#538semidirect
463. C2#5678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#9472
C2×#9462
C2×#9392
C2×#9372
#56782
C2×#8862
C2×#8842
C2×#8832
C2×#8822
C2×#67⋊C42
C2×D3×C22⋊C42
C2×#8792
C2×#8762
C2×C4×D122
C2×#5383