Finite Group Databasebeta

← C2×#5705 #86114

Subgroups — C2×#5705 #86114

Order 384.

1328 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic, complemented
2#5705C2116yesmaximal, direct product factor
3C2×#897C211yesmaximal, characteristic, complemented
4C2×#896C211yesmaximal, characteristic, complemented
5C2×#894C211yesmaximal, characteristic, complemented
6C2×#894C211yesmaximal, characteristic, complemented
7C2×#893C211yesmaximal, characteristic, complemented
8C2×#892C211yesmaximal, characteristic, complemented
9C2×#885C211yesmaximal, characteristic, complemented
10C2×#883C211yesmaximal, characteristic, complemented
11C2×#882C211yesmaximal, characteristic
12C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
13C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
14C2×#879C211yesmaximal, characteristic, complemented
15C2×C4×D12C211yesmaximal, characteristic, complemented
16C2×#872C211yesmaximal, characteristic, complemented
17C2×#551—31nomaximal, Sylow
18C24×D3C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic, complemented
27C22×Dic6C2211yescharacteristic, complemented
28C3×#126C2218yescomplemented
29C2×C3×C4⋊C4C2211yescharacteristic, complemented
30C2×C3×C4⋊C4C2211yescharacteristic, complemented
31C2×C3×C4⋊C4C2211yescharacteristic, complemented
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×C3×C42C2211yescharacteristic, complemented
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C2×C4⋊Dic3C2211yescharacteristic, complemented
48C2×C4⋊Dic3C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51C2×Dic3⋊C4C2211yescharacteristic, complemented
52C2×C4×Dic3C2211yescharacteristic, complemented
53C2×C4×Dic3C2211yescharacteristic, complemented
54#897C2218yescomplemented
55#896C2218yescomplemented
56#894C2218yescomplemented
57#894C2218yescomplemented
58#893C2218yescomplemented
59#892C2218yescomplemented
60#885C2218yescomplemented
61#883C2218yescomplemented
62#882C2218yes—
63D3×C22⋊C4C2218yescomplemented
64D3×C22⋊C4C2218yescomplemented
65#879C2218yescomplemented
66C4×D12C2218yescomplemented
67#872C2218yescomplemented
68#551—316no—
69C2×#126—31no—
70C2×#126D311yescharacteristic, complemented
71C2×#124—31no—
72C2×#123—31no—
73C2×#123—31no—
74C2×#123—31no—
75C2×C22⋊Q8—31no—
76C2×C22⋊Q8—31no—
77C2×#120—31no—
78C2×C4×D4—31no—
79C2×C4×D4—31no—
80C2×#117—31no—
81C2×#117—31no—
82C22×C22⋊C4—31no—
83C22×C22⋊C4—31no—
84C24×C3C2311yescharacteristic
85C23×D3—24no—
86C23×D3—24no—
87C23×D3—42no—
88C23×D3C2311yescharacteristic
89C23×D3C2314yes—
90C23×D3C2311yescharacteristic
91C23×D3C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C2×#67—42no—
99C2×#67C2314yes—
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C2×D12—24no—
106C2×D12—24no—
107C2×D12C2314yes—
108C2×D12C2314yes—
109C2×C4×D3—24no—
110C2×C4×D3—24no—
111C2×C4×D3—42no—
112C2×C4×D3—42no—
113C2×C4×D3—42no—
114C2×C4×D3C2314yes—
115C2×C4×D3C2314yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yescomplemented
119C2×Dic6—42no—
120C2×Dic6C2314yescomplemented
121C3×C4⋊C4C2314yescomplemented
122C3×C4⋊C4C2314yescomplemented
123C3×C4⋊C4C2314yescomplemented
124C3×C22⋊C4C2314yes—
125C3×C22⋊C4C2314yes—
126C3×C22⋊C4C2314yes—
127C3×C42C2314yescomplemented
128#217C2314yes—
129#217C2314yes—
130(C2×D3)⋊C4C2314yes—
131(C2×D3)⋊C4C2314yes—
132(C2×D3)⋊C4C2314yes—
133(C2×D3)⋊C4C2314yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2314yescomplemented
137(C2×D3)⋊C4C2314yescomplemented
138(C2×D3)⋊C4C2314yescomplemented
139C4⋊Dic3C2314yescomplemented
140C4⋊Dic3C2314yescomplemented
141Dic3⋊C4C2314yescomplemented
142Dic3⋊C4C2314yescomplemented
143Dic3⋊C4C2314yescomplemented
144C4×Dic3C2314yescomplemented
145C4×Dic3C2314yescomplemented
146C25—31no—
147C22×Q8—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—31no—
151C23×C4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156#126—38no—
157#126C2×D318yescomplemented
158#124—38no—
159#123—38no—
160#123—38no—
161#123—38no—
162C22⋊Q8—38no—
163C22⋊Q8—38no—
164#120—38no—
165C4×D4—38no—
166C4×D4—38no—
167#117—38no—
168#117—38no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4C2×D311yescharacteristic, complemented
175C2×C4⋊C4C2×D311yescharacteristic, complemented
176C2×C4⋊C4C2×D311yescharacteristic, complemented
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊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—38no—
189C2×C22⋊C4—38no—
190C2×C22⋊C4C2×D311yescharacteristic
191C2×C22⋊C4C2×D311yescharacteristic
192C2×C22⋊C4C2×D311yescharacteristic
193C2×C42—31no—
194C2×C42—31no—
195C2×C42C2×D311yescharacteristic, complemented
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3C2412yes—
200C23×C3C2411yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3—22no—
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—28no—
209C22×D3—28no—
210C22×D3—44no—
211C22×D3—416no—
212C22×D3#4114yes—
213C22×D3#4114yes—
214C22×D3C2412yes—
215C22×D3C2412yes—
216C22×D3C2412yes—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4C2412yes—
224C2×C3×C4C2412yes—
225C2×C3×C4C2412yes—
226C2×C3×C4C2412yes—
227C2×C3×C4C2412yes—
228C2×C3×C4C2412yes—
229#67—44no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3C2412yes—
236C2×Dic3C2412yes—
237C2×Dic3C2412yes—
238C2×Dic3C2412yes—
239C2×Dic3C2412yes—
240D12—28no—
241D12—28no—
242C4×D3—28no—
243C4×D3—28no—
244C4×D3—44no—
245C4×D3—44no—
246C4×D3—44no—
247Dic6—44no—
248C24—31no—
249C24—31no—
250C24—31no—
251C24—34no—
252C24—64no—
253C24—64no—
254C24—122no—
255C24C22×D311yescharacteristic
256C2×Q8—34no—
257C2×Q8—122no—
258C2×D4—34no—
259C2×D4—34no—
260C2×D4—34no—
261C2×D4—64no—
262C2×D4—64no—
263C2×D4—122no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—34no—
270C22×C4—34no—
271C22×C4—34no—
272C22×C4—34no—
273C22×C4—34no—
274C22×C4—64no—
275C22×C4—64no—
276C22×C4—122no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C4⋊C4—34no—
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4C22×D314yescomplemented
291C4⋊C4C22×D314yescomplemented
292C4⋊C4C22×D314yescomplemented
293C22⋊C4—34no—
294C22⋊C4—34no—
295C22⋊C4—34no—
296C22⋊C4—34no—
297C22⋊C4—34no—
298C22⋊C4—34no—
299C22⋊C4—34no—
300C22⋊C4—34no—
301C22⋊C4—34no—
302C22⋊C4—34no—
303C22⋊C4—34no—
304C22⋊C4C22×D314yes—
305C22⋊C4C22×D314yes—
306C22⋊C4C22×D314yes—
307C42—34no—
308C42—34no—
309C42C22×D314yescomplemented
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—44no—
314C22×C3C2×#4112yes—
315C22×C3C2×#4112yes—
316C22×C3H(5,𝔽2)12yes—
317C22×C3C2511yescharacteristic
318C2×D3—28no—
319C2×D3—28no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—416no—
328C2×D3—44no—
329C2×D3C2×#4114yes—
330C2×D3C2×#4114yes—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—22no—
337Dic3—22no—
338Dic3—22no—
339Dic3—22no—
340Dic3—22no—
341Dic3—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—34no—
346C23—34no—
347C23—32no—
348C23—32no—
349C23—32no—
350C23—68no—
351C23—68no—
352C23—62no—
353C23—62no—
354C23—62no—
355C23—62no—
356C23—62no—
357C23—62no—
358C23—62no—
359C23—124no—
360C23—1216no—
361C23C23×D312yes—
362C23C23×D311yescharacteristic
363Q8—124no—
364D4—68no—
365D4—68no—
366D4—124no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—68no—
379C2×C4—68no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4C23×D312yes—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C4C23×D312yes—
392C2×C4C23×D312yes—
393C2×C4C23×D312yes—
394C2×C3—42no—
395C2×C3#55114yescomplemented
396C2×C3C22×#4111yescharacteristic
397C2×C3C22×#4111yescharacteristic
398C2×C3C2×H(5,𝔽2)11yescharacteristic
399D3—24no—
400D3—24no—
401D3—42no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—34no—
406C22—34no—
407C22—44no—
408C22—68no—
409C22—68no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22—62no—
414C22—62no—
415C22—62no—
416C22—62no—
417C22—1216no—
418C22—124no—
419C22D3×#4112yes—
420C22D3×#4112yes—
421C22#100912yes—
422C22C24×D311yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—62no—
430C4—62no—
431C4—62no—
432C4—62no—
433C4—62no—
434C3C2×#55111yescharacteristic, complemented, Sylow
435C2—42no—
436C2—64no—
437C2—64no—
438C2—122no—
439C2#570514yesdirect product factor
440C2C2×D3×#4111yescharacteristic
441C2C2×D3×#4111yescharacteristic
442C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
prime radicalC2×#882C2
hypercenterC2×#126D3
hypocenterC3C2×#551
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. #5705C2direct (N × H)
3. C2×#897C2semidirect
4. C2×#896C2semidirect
5. C2×#894C2semidirect
6. C2×#894C2semidirect
7. C2×#893C2semidirect
8. C2×#892C2semidirect
9. C2×#885C2semidirect
10. C2×#883C2semidirect
12. C2×D3×C22⋊C4C2semidirect
13. C2×D3×C22⋊C4C2semidirect
14. C2×#879C2semidirect
15. C2×C4×D12C2semidirect
16. C2×#872C2semidirect
26. C22×C4×D3C22semidirect
27. C22×Dic6C22semidirect
28. C3×#126C22semidirect
29. C2×C3×C4⋊C4C22semidirect
30. C2×C3×C4⋊C4C22semidirect
31. C2×C3×C4⋊C4C22semidirect
35. C2×C3×C42C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×(C2×D3)⋊C4C22semidirect
47. C2×C4⋊Dic3C22semidirect
48. C2×C4⋊Dic3C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×Dic3⋊C4C22semidirect
51. C2×Dic3⋊C4C22semidirect
52. C2×C4×Dic3C22semidirect
53. C2×C4×Dic3C22semidirect
54. #897C22semidirect
55. #896C22semidirect
56. #894C22semidirect
57. #894C22semidirect
58. #893C22semidirect
59. #892C22semidirect
60. #885C22semidirect
61. #883C22semidirect
63. D3×C22⋊C4C22semidirect
64. D3×C22⋊C4C22semidirect
65. #879C22semidirect
66. C4×D12C22semidirect
67. #872C22semidirect
70. C2×#126D3semidirect
118. C2×C4×D3C23semidirect
120. C2×Dic6C23semidirect
121. C3×C4⋊C4C23semidirect
122. C3×C4⋊C4C23semidirect
123. C3×C4⋊C4C23semidirect
127. C3×C42C23semidirect
136. (C2×D3)⋊C4C23semidirect
137. (C2×D3)⋊C4C23semidirect
138. (C2×D3)⋊C4C23semidirect
139. C4⋊Dic3C23semidirect
140. C4⋊Dic3C23semidirect
141. Dic3⋊C4C23semidirect
142. Dic3⋊C4C23semidirect
143. Dic3⋊C4C23semidirect
144. C4×Dic3C23semidirect
145. C4×Dic3C23semidirect
157. #126C2×D3semidirect
174. C2×C4⋊C4C2×D3semidirect
175. C2×C4⋊C4C2×D3semidirect
176. C2×C4⋊C4C2×D3semidirect
195. C2×C42C2×D3semidirect
290. C4⋊C4C22×D3semidirect
291. C4⋊C4C22×D3semidirect
292. C4⋊C4C22×D3semidirect
309. C42C22×D3semidirect
395. C2×C3#551semidirect
434. C3C2×#551semidirect
439. C2#5705direct (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
#57052
C2×#8972
C2×#8962
C2×#8942
C2×#8932
C2×#8922
C2×#8852
C2×#8832
C2×#8822
C2×D3×C22⋊C42
C2×#8792
C2×C4×D122
C2×#8722
C2×#5513