Finite Group Databasebeta

← C2×#5704 #86112

Subgroups — C2×#5704 #86112

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×#126C211yesmaximal, characteristic, complemented
2#5704C2116yesmaximal, direct product factor
3C2×#898C211yesmaximal, characteristic, complemented
4C2×#896C211yesmaximal, characteristic, complemented
5C2×#895C211yesmaximal, characteristic, complemented
6C2×#895C211yesmaximal, characteristic, complemented
7C2×#894C211yesmaximal, characteristic, complemented
8C2×#893C211yesmaximal, characteristic, complemented
9C2×#885C211yesmaximal, characteristic, complemented
10C2×#884C211yesmaximal, characteristic, complemented
11C2×#883C211yesmaximal, characteristic, complemented
12C2×#882C211yesmaximal, characteristic, complemented
13C2×#882C211yesmaximal, characteristic, complemented
14C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
15C2×#875C211yesmaximal, characteristic, complemented
16C2×C4×D12C211yesmaximal, characteristic, complemented
17C2×#538—31nomaximal, Sylow
18C24×D3C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C22×D12C2211yescharacteristic, complemented
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic, complemented
29C22×C4×D3C2211yescharacteristic, complemented
30C22×Dic6C2211yescharacteristic, complemented
31C3×#126C2218yescomplemented
32C2×C3×C4⋊C4C2211yescharacteristic, complemented
33C2×C3×C4⋊C4C2211yescharacteristic, complemented
34C2×C3×C4⋊C4C2211yescharacteristic, complemented
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×C3×C22⋊C4C2211yescharacteristic
38C2×C3×C42C2211yescharacteristic, complemented
39C2×#217C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
48C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
49C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
50C2×C4⋊Dic3C2211yescharacteristic, complemented
51C2×Dic3⋊C4C2211yescharacteristic, complemented
52C2×Dic3⋊C4C2211yescharacteristic, complemented
53C2×C4×Dic3C2211yescharacteristic, complemented
54#898C2218yescomplemented
55#896C2218yescomplemented
56#895C2218yescomplemented
57#895C2218yescomplemented
58#894C2218yescomplemented
59#893C2218yescomplemented
60#885C2218yescomplemented
61#884C2218yescomplemented
62#883C2218yescomplemented
63#882C2218yescomplemented
64#882C2218yescomplemented
65D3×C22⋊C4C2218yescomplemented
66#875C2218yescomplemented
67C4×D12C2218yescomplemented
68#538—316no—
69C2×#126—31no—
70C2×#126D311yescharacteristic, complemented
71C2×#124—31no—
72C2×#124—31no—
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
85C23×D3—24no—
86C23×D3—42no—
87C23×D3—42no—
88C23×D3C2311yescharacteristic
89C23×D3C2311yescharacteristic
90C23×D3C2311yescharacteristic
91C23×D3C2311yescharacteristic
92C23×D3C2314yes—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C2×#67—42no—
100C2×#67—42no—
101C2×#67C2314yes—
102C2×#67C2314yes—
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C2×D12—24no—
108C2×D12—24no—
109C2×D12—42no—
110C2×D12—42no—
111C2×D12—42no—
112C2×D12C2314yes—
113C2×D12C2314yes—
114C2×D12C2314yes—
115C2×D12C2314yes—
116C2×D12C2314yescomplemented
117C2×C4×D3—42no—
118C2×C4×D3—42no—
119C2×C4×D3—42no—
120C2×C4×D3—42no—
121C2×C4×D3C2314yes—
122C2×C4×D3C2314yes—
123C2×C4×D3C2314yescomplemented
124C2×C4×D3C2314yescomplemented
125C2×Dic6—42no—
126C2×Dic6C2314yescomplemented
127C3×C4⋊C4C2314yescomplemented
128C3×C4⋊C4C2314yescomplemented
129C3×C4⋊C4C2314yescomplemented
130C3×C22⋊C4C2314yes—
131C3×C22⋊C4C2314yes—
132C3×C22⋊C4C2314yes—
133C3×C42C2314yescomplemented
134#217C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yescomplemented
142(C2×D3)⋊C4C2314yescomplemented
143(C2×D3)⋊C4C2314yescomplemented
144(C2×D3)⋊C4C2314yescomplemented
145C4⋊Dic3C2314yescomplemented
146Dic3⋊C4C2314yescomplemented
147Dic3⋊C4C2314yescomplemented
148C4×Dic3C2314yescomplemented
149C25—31no—
150C22×Q8—31no—
151C22×D4—31no—
152C22×D4—31no—
153C22×D4—31no—
154C22×D4—31no—
155C22×D4—31no—
156C22×D4—31no—
157C22×D4—31no—
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—31no—
162#126—38no—
163#126C2×D318yescomplemented
164#124—38no—
165#124—38no—
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⋊C4C2×D311yescharacteristic, complemented
180C2×C4⋊C4C2×D311yescharacteristic, complemented
181C2×C4⋊C4C2×D311yescharacteristic, complemented
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—31no—
193C2×C22⋊C4—38no—
194C2×C22⋊C4C2×D311yescharacteristic
195C2×C22⋊C4C2×D311yescharacteristic
196C2×C22⋊C4C2×D311yescharacteristic
197C2×C42—31no—
198C2×C42C2×D311yescharacteristic, complemented
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3C2412yes—
203C23×C3C2411yescharacteristic
204C22×D3—22no—
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—22no—
210C22×D3—22no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—22no—
214C22×D3—22no—
215C22×D3—28no—
216C22×D3—44no—
217C22×D3—44no—
218C22×D3—416no—
219C22×D3#4114yes—
220C22×D3C2412yes—
221C22×D3C2412yes—
222C22×D3C2412yes—
223C22×D3C2412yes—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4C2412yes—
231C2×C3×C4C2412yes—
232C2×C3×C4C2412yes—
233C2×C3×C4C2412yes—
234C2×C3×C4C2412yes—
235C2×C3×C4C2412yes—
236#67—44no—
237#67—44no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3C2412yes—
243C2×Dic3C2412yes—
244C2×Dic3C2412yes—
245C2×Dic3C2412yes—
246D12—28no—
247D12—28no—
248D12—44no—
249D12—44no—
250D12—44no—
251C4×D3—44no—
252C4×D3—44no—
253C4×D3—44no—
254C4×D3—44no—
255Dic6—44no—
256C24—31no—
257C24—31no—
258C24—31no—
259C24—31no—
260C24—34no—
261C24—64no—
262C24—122no—
263C24—122no—
264C24C22×D311yescharacteristic
265C2×Q8—34no—
266C2×Q8—122no—
267C2×D4—34no—
268C2×D4—34no—
269C2×D4—34no—
270C2×D4—34no—
271C2×D4—34no—
272C2×D4—34no—
273C2×D4—34no—
274C2×D4—64no—
275C2×D4—64no—
276C2×D4—122no—
277C2×D4—122no—
278C2×D4—122no—
279C2×D4—122no—
280C2×D4—122no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—34no—
286C22×C4—34no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—122no—
290C22×C4—122no—
291C22×C4—122no—
292C22×C4—122no—
293C22×C4C22×D311yescharacteristic
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⋊C4C22×D314yescomplemented
303C4⋊C4C22×D314yescomplemented
304C4⋊C4C22×D314yescomplemented
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⋊C4—34no—
316C22⋊C4C22×D314yes—
317C22⋊C4C22×D314yes—
318C22⋊C4C22×D314yes—
319C42—34no—
320C42C22×D314yescomplemented
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—44no—
325C22×C3C2×#4112yes—
326C22×C3H(5,𝔽2)12yes—
327C22×C3H(5,𝔽2)12yes—
328C22×C3C2511yescharacteristic
329C2×D3—28no—
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—416no—
342C2×D3—44no—
343C2×D3—44no—
344C2×D3—44no—
345C2×D3C2×#4114yes—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—22no—
352Dic3—22no—
353Dic3—22no—
354Dic3—22no—
355Dic3—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—34no—
360C23—32no—
361C23—32no—
362C23—32no—
363C23—32no—
364C23—68no—
365C23—62no—
366C23—62no—
367C23—62no—
368C23—62no—
369C23—62no—
370C23—62no—
371C23—62no—
372C23—62no—
373C23—62no—
374C23—62no—
375C23—62no—
376C23—124no—
377C23—124no—
378C23—1216no—
379C23C23×D312yes—
380C23C23×D311yescharacteristic
381Q8—124no—
382D4—68no—
383D4—68no—
384D4—124no—
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—22no—
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—124no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4C23×D312yes—
408C2×C4C23×D312yes—
409C2×C4C23×D312yes—
410C2×C4C23×D312yes—
411C2×C4C23×D312yes—
412C2×C4C23×D312yes—
413C2×C3—42no—
414C2×C3#53814yescomplemented
415C2×C3C22×#4111yescharacteristic
416C2×C3C2×H(5,𝔽2)11yescharacteristic
417C2×C3C2×H(5,𝔽2)11yescharacteristic
418D3—24no—
419D3—42no—
420D3—42no—
421D3—42no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—34no—
426C22—44no—
427C22—68no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—1216no—
440C22—124no—
441C22—124no—
442C22—124no—
443C22D3×#4112yes—
444C22#100912yes—
445C22#100912yes—
446C22C24×D311yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—62no—
454C4—62no—
455C4—62no—
456C4—62no—
457C3C2×#53811yescharacteristic, complemented, Sylow
458C2—42no—
459C2—64no—
460C2—122no—
461C2—122no—
462C2—122no—
463C2#570414yesdirect product factor
464C2C2×D3×#4111yescharacteristic
465C2C2×#100911yescharacteristic
466C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
hypercenterC2×#126D3
hypocenterC3C2×#538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
2. #5704C2direct (N × H)
3. C2×#898C2semidirect
4. C2×#896C2semidirect
5. C2×#895C2semidirect
6. C2×#895C2semidirect
7. C2×#894C2semidirect
8. C2×#893C2semidirect
9. C2×#885C2semidirect
10. C2×#884C2semidirect
11. C2×#883C2semidirect
12. C2×#882C2semidirect
13. C2×#882C2semidirect
14. C2×D3×C22⋊C4C2semidirect
15. C2×#875C2semidirect
16. C2×C4×D12C2semidirect
25. C22×D12C22semidirect
28. C22×C4×D3C22semidirect
29. C22×C4×D3C22semidirect
30. C22×Dic6C22semidirect
31. C3×#126C22semidirect
32. C2×C3×C4⋊C4C22semidirect
33. C2×C3×C4⋊C4C22semidirect
34. C2×C3×C4⋊C4C22semidirect
38. C2×C3×C42C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×(C2×D3)⋊C4C22semidirect
48. C2×(C2×D3)⋊C4C22semidirect
49. C2×(C2×D3)⋊C4C22semidirect
50. C2×C4⋊Dic3C22semidirect
51. C2×Dic3⋊C4C22semidirect
52. C2×Dic3⋊C4C22semidirect
53. C2×C4×Dic3C22semidirect
54. #898C22semidirect
55. #896C22semidirect
56. #895C22semidirect
57. #895C22semidirect
58. #894C22semidirect
59. #893C22semidirect
60. #885C22semidirect
61. #884C22semidirect
62. #883C22semidirect
63. #882C22semidirect
64. #882C22semidirect
65. D3×C22⋊C4C22semidirect
66. #875C22semidirect
67. C4×D12C22semidirect
70. C2×#126D3semidirect
116. C2×D12C23semidirect
123. C2×C4×D3C23semidirect
124. C2×C4×D3C23semidirect
126. C2×Dic6C23semidirect
127. C3×C4⋊C4C23semidirect
128. C3×C4⋊C4C23semidirect
129. C3×C4⋊C4C23semidirect
133. C3×C42C23semidirect
141. (C2×D3)⋊C4C23semidirect
142. (C2×D3)⋊C4C23semidirect
143. (C2×D3)⋊C4C23semidirect
144. (C2×D3)⋊C4C23semidirect
145. C4⋊Dic3C23semidirect
146. Dic3⋊C4C23semidirect
147. Dic3⋊C4C23semidirect
148. C4×Dic3C23semidirect
163. #126C2×D3semidirect
179. C2×C4⋊C4C2×D3semidirect
180. C2×C4⋊C4C2×D3semidirect
181. C2×C4⋊C4C2×D3semidirect
198. C2×C42C2×D3semidirect
302. C4⋊C4C22×D3semidirect
303. C4⋊C4C22×D3semidirect
304. C4⋊C4C22×D3semidirect
320. C42C22×D3semidirect
414. C2×C3#538semidirect
457. C3C2×#538semidirect
463. C2#5704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1262
#57042
C2×#8982
C2×#8962
C2×#8952
C2×#8942
C2×#8932
C2×#8852
C2×#8842
C2×#8832
C2×#8822
C2×D3×C22⋊C42
C2×#8752
C2×C4×D122
C2×#5383