Finite Group Databasebeta

← C2×#5556 #85783

Subgroups — C2×#5556 #85783

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×C4×D4C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic
3C2×#935C211yesmaximal, characteristic, complemented
4C2×#929C211yesmaximal, characteristic, complemented
5C2×C4×#67C211yesmaximal, characteristic, complemented
6C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7C2×#924C211yesmaximal, characteristic, complemented
8#5556C2116yesmaximal, direct product factor
9C2×#898C211yesmaximal, characteristic, complemented
10C2×#896C211yesmaximal, characteristic, complemented
11C2×#885C211yesmaximal, characteristic, complemented
12C2×#883C211yesmaximal, characteristic, complemented
13C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
14C2×#877C211yesmaximal, characteristic, complemented
15C2×#876C211yesmaximal, characteristic, complemented
16C2×#872C211yesmaximal, characteristic, complemented
17C2×#551—31nomaximal, Sylow
18C24×D3C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C23×C3×C4C2211yescharacteristic, complemented
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×Dic6C2211yescharacteristic, complemented
28C3×C4×D4C2218yescomplemented
29C2×C3×C4⋊C4C2211yescharacteristic, complemented
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic, complemented
32C2×C3×C42C2211yescharacteristic, complemented
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic, complemented
36C2×#217C2211yescharacteristic, complemented
37#937C2218yes—
38#935C2218yescomplemented
39#929C2218yescomplemented
40C4×#67C2218yescomplemented
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
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)⋊C4C2218yescomplemented
50C2×C4⋊Dic3C2211yescharacteristic, complemented
51#924C2218yescomplemented
52C2×Dic3⋊C4C2211yescharacteristic, complemented
53C2×Dic3⋊C4C2211yescharacteristic, complemented
54C2×Dic3⋊C4C2211yescharacteristic, complemented
55C2×Dic3⋊C4C2211yescharacteristic, complemented
56C2×Dic3⋊C4C2211yescharacteristic, complemented
57C2×Dic3⋊C4C2211yescharacteristic, complemented
58C2×C4×Dic3C2211yescharacteristic, complemented
59C2×C4×Dic3C2211yescharacteristic, complemented
60#898C2218yescomplemented
61#896C2218yescomplemented
62#885C2218yescomplemented
63#883C2218yescomplemented
64D3×C22⋊C4C2218yescomplemented
65#877C2218yescomplemented
66#876C2218yescomplemented
67#872C2218yescomplemented
68#551—316no—
69C2×#126—31no—
70C2×#126—31no—
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×D4D311yescharacteristic, complemented
80C2×#117—31no—
81C2×#117—31no—
82C22×C22⋊C4—31no—
83C22×C22⋊C4—31no—
84C24×C3C2311yescharacteristic
85C24×C3C2311yescharacteristic
86C23×D3—24no—
87C23×D3—24no—
88C23×D3—42no—
89C23×D3C2311yescharacteristic
90C23×D3C2314yes—
91C23×D3C2311yescharacteristic
92C2×C3×D4—24no—
93C2×C3×D4C2314yes—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2314yes—
102C22×C3×C4C2314yescomplemented
103C2×#67—24no—
104C2×#67—42no—
105C2×#67C2314yes—
106C2×#67C2314yes—
107C22×Dic3—24no—
108C22×Dic3C2314yes—
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C2×C4×D3—24no—
116C2×C4×D3—42no—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yes—
119C2×Dic6—42no—
120C2×Dic6C2314yescomplemented
121C3×C4⋊C4C2314yescomplemented
122C3×C22⋊C4C2314yes—
123C3×C22⋊C4C2314yescomplemented
124C3×C42C2314yescomplemented
125#217C2314yes—
126#217C2314yes—
127#217C2314yescomplemented
128#217C2314yescomplemented
129(C2×D3)⋊C4C2314yes—
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)⋊C4C2314yes—
137C4⋊Dic3C2314yescomplemented
138Dic3⋊C4C2314yescomplemented
139Dic3⋊C4C2314yescomplemented
140Dic3⋊C4C2314yescomplemented
141Dic3⋊C4C2314yescomplemented
142Dic3⋊C4C2314yescomplemented
143Dic3⋊C4C2314yescomplemented
144C4×Dic3C2314yescomplemented
145C4×Dic3C2314yescomplemented
146C25—31no—
147C22×Q8—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4C2×D311yescharacteristic
151C23×C4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4C2×D311yescharacteristic
155C23×C4C2×D311yescharacteristic, complemented
156#126—38no—
157#126—38no—
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×D4C2×D318yescomplemented
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⋊C4—31no—
175C2×C4⋊C4—31no—
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—38no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—38no—
191C2×C22⋊C4C2×D311yescharacteristic
192C2×C22⋊C4C2×D311yescharacteristic, complemented
193C2×C42—31no—
194C2×C42—31no—
195C2×C42C2×D311yescharacteristic, complemented
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3#4114yes—
202C23×C3C2412yes—
203C23×C3C2412yes—
204C23×C3C2411yescharacteristic
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—28no—
208C22×D3—22no—
209C22×D3—22no—
210C22×D3—22no—
211C22×D3—28no—
212C22×D3—44no—
213C22×D3—416no—
214C22×D3#4114yes—
215C22×D3C2412yes—
216C22×D3C2412yes—
217C3×D4—28no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4C2412yes—
226C2×C3×C4C2412yes—
227C2×C3×C4C2412yes—
228C2×C3×C4C2412yes—
229C2×C3×C4C2412yes—
230#67—28no—
231#67—44no—
232C2×Dic3—22no—
233C2×Dic3—28no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3C2412yes—
240C2×Dic3C2412yes—
241C2×Dic3C2412yes—
242C2×Dic3C2412yes—
243C2×Dic3C2412yes—
244C2×Dic3C2412yes—
245C4×D3—28no—
246C4×D3—44no—
247Dic6—44no—
248C24—31no—
249C24—31no—
250C24—34no—
251C24—64no—
252C24—64no—
253C24—122no—
254C24C22×D311yescharacteristic
255C24C22×D311yescharacteristic
256C2×Q8—34no—
257C2×Q8—122no—
258C2×D4—24no—
259C2×D4—34no—
260C2×D4—34no—
261C2×D4—64no—
262C2×D4—122no—
263C2×D4C22×D314yes—
264C22×C4—34no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—34no—
272C22×C4—34no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—64no—
276C22×C4—64no—
277C22×C4—122no—
278C22×C4C22×D311yescharacteristic
279C22×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D314yes—
284C22×C4C22×D314yescomplemented
285C4⋊C4—34no—
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—34no—
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⋊C4—34no—
305C22⋊C4C22×D314yes—
306C22⋊C4C22×D314yescomplemented
307C42—34no—
308C42—34no—
309C42C22×D314yescomplemented
310C22×C3—28no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—44no—
317C22×C3C2×#4114yes—
318C22×C3C2×#4112yes—
319C22×C3C2×#4112yes—
320C22×C3H(5,𝔽2)12yes—
321C22×C3C2511yescharacteristic
322C2×D3—28no—
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—416no—
329C2×D3—44no—
330C2×D3C2×#4114yes—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—22no—
336Dic3—22no—
337Dic3—22no—
338Dic3—22no—
339Dic3—22no—
340Dic3—22no—
341Dic3—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—34no—
348C23—32no—
349C23—32no—
350C23—68no—
351C23—62no—
352C23—62no—
353C23—68no—
354C23—62no—
355C23—62no—
356C23—62no—
357C23—124no—
358C23—1216no—
359C23#23514yes—
360C23C23×D312yes—
361C23C23×D312yes—
362C23C23×D311yescharacteristic
363Q8—124no—
364D4—28no—
365D4—68no—
366D4—124no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—68no—
381C2×C4—68no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—124no—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C4C23×D312yes—
392C2×C4C23×D312yes—
393C2×C4C23×D312yes—
394C2×C3—24no—
395C2×C3—42no—
396C2×C3#55114yescomplemented
397C2×C3C22×#4111yescharacteristic
398C2×C3C22×#4111yescharacteristic
399C2×C3C2×H(5,𝔽2)11yescharacteristic
400D3—24no—
401D3—42no—
402C22—22no—
403C22—22no—
404C22—28no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—34no—
409C22—44no—
410C22—62no—
411C22—62no—
412C22—68no—
413C22—62no—
414C22—62no—
415C22—62no—
416C22—1216no—
417C22—124no—
418C22C2×#23514yes—
419C22C2×#23512yes—
420C22#100412yes—
421C22D3×#4112yes—
422C22C24×D311yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—62no—
429C4—62no—
430C4—62no—
431C4—62no—
432C4—62no—
433C4—62no—
434C3C2×#55111yescharacteristic, complemented, Sylow
435C2—24no—
436C2—42no—
437C2—64no—
438C2—122no—
439C2#555614yesdirect product factor
440C2C22×#23511yescharacteristic
441C2C2×#100411yescharacteristic
442C2C2×D3×#4111yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
socleC23×C3C24
prime radicalC2×#937C2
hypercenterC2×C4×D4D3
hypocenterC3C2×#551
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
3. C2×#935C2semidirect
4. C2×#929C2semidirect
5. C2×C4×#67C2semidirect
6. C22×(C2×D3)⋊C4C2semidirect
7. C2×#924C2semidirect
8. #5556C2direct (N × H)
9. C2×#898C2semidirect
10. C2×#896C2semidirect
11. C2×#885C2semidirect
12. C2×#883C2semidirect
13. C2×D3×C22⋊C4C2semidirect
14. C2×#877C2semidirect
15. C2×#876C2semidirect
16. C2×#872C2semidirect
21. C23×C3×C4C22semidirect
27. C22×Dic6C22semidirect
28. C3×C4×D4C22semidirect
29. C2×C3×C4⋊C4C22semidirect
31. C2×C3×C22⋊C4C22semidirect
32. C2×C3×C42C22semidirect
35. C2×#217C22semidirect
36. C2×#217C22semidirect
38. #935C22semidirect
39. #929C22semidirect
40. C4×#67C22semidirect
49. C2×(C2×D3)⋊C4C22semidirect
50. C2×C4⋊Dic3C22semidirect
51. #924C22semidirect
52. C2×Dic3⋊C4C22semidirect
53. C2×Dic3⋊C4C22semidirect
54. C2×Dic3⋊C4C22semidirect
55. C2×Dic3⋊C4C22semidirect
56. C2×Dic3⋊C4C22semidirect
57. C2×Dic3⋊C4C22semidirect
58. C2×C4×Dic3C22semidirect
59. C2×C4×Dic3C22semidirect
60. #898C22semidirect
61. #896C22semidirect
62. #885C22semidirect
63. #883C22semidirect
64. D3×C22⋊C4C22semidirect
65. #877C22semidirect
66. #876C22semidirect
67. #872C22semidirect
79. C2×C4×D4D3semidirect
102. C22×C3×C4C23semidirect
120. C2×Dic6C23semidirect
121. C3×C4⋊C4C23semidirect
123. C3×C22⋊C4C23semidirect
124. C3×C42C23semidirect
127. #217C23semidirect
128. #217C23semidirect
137. C4⋊Dic3C23semidirect
138. Dic3⋊C4C23semidirect
139. Dic3⋊C4C23semidirect
140. Dic3⋊C4C23semidirect
141. Dic3⋊C4C23semidirect
142. Dic3⋊C4C23semidirect
143. Dic3⋊C4C23semidirect
144. C4×Dic3C23semidirect
145. C4×Dic3C23semidirect
155. C23×C4C2×D3semidirect
166. C4×D4C2×D3semidirect
176. C2×C4⋊C4C2×D3semidirect
192. C2×C22⋊C4C2×D3semidirect
195. C2×C42C2×D3semidirect
284. C22×C4C22×D3semidirect
292. C4⋊C4C22×D3semidirect
306. C22⋊C4C22×D3semidirect
309. C42C22×D3semidirect
396. C2×C3#551semidirect
434. C3C2×#551semidirect
439. C2#5556direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9372
C2×#9352
C2×#9292
C2×C4×#672
C22×(C2×D3)⋊C42
C2×#9242
#55562
C2×#8982
C2×#8962
C2×#8852
C2×#8832
C2×D3×C22⋊C42
C2×#8772
C2×#8762
C2×#8722
C2×#5513