Finite Group Databasebeta

← C2×#5490 #85696

Subgroups — C2×#5490 #85696

Order 384.

1900 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#67C211yesmaximal, characteristic, complemented
2C22×#235C211yesmaximal, characteristic, complemented
3C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×#953C211yesmaximal, characteristic, complemented
5C2×#930C211yesmaximal, characteristic, complemented
6C2×#929C211yesmaximal, characteristic, complemented
7C2×C4×#67C211yesmaximal, characteristic, complemented
8C2×#885C211yesmaximal, characteristic, complemented
9#5490C2116yesmaximal, direct product factor
10C2×#884C211yesmaximal, characteristic, complemented
11C2×#884C211yesmaximal, characteristic, complemented
12C2×#883C211yesmaximal, characteristic, complemented
13C2×#882C211yesmaximal, characteristic, complemented
14C2×#67⋊C4C211yesmaximal, characteristic, complemented
15C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
16C2×#878C211yesmaximal, characteristic, complemented
17C2×#546—31nomaximal, Sylow
18C25×C3C2211yescharacteristic
19C24×D3C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic, complemented
21C23×C3×C4C2211yescharacteristic, complemented
22C22×#67—28no—
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic
26C22×#67C2211yescharacteristic
27C22×#67C2211yescharacteristic
28C22×#67C2218yescomplemented
29C22×#67C2211yescharacteristic, complemented
30C22×#67C2211yescharacteristic, complemented
31C23×Dic3C2211yescharacteristic
32C2×#235—28no—
33C2×#235C2218yescomplemented
34C22×D12C2211yescharacteristic, complemented
35C22×D12C2211yescharacteristic, complemented
36C22×C4×D3C2211yescharacteristic, complemented
37C22×C4×D3C2211yescharacteristic, complemented
38C22×C4×D3C2211yescharacteristic, complemented
39C22×Dic6C2211yescharacteristic, complemented
40C2×C3×C22⋊C4C2211yescharacteristic
41C2×C3×C22⋊C4C2211yescharacteristic
42C2×C3×C22⋊C4C2218yescomplemented
43C2×C3×C22⋊C4C2211yescharacteristic, complemented
44C2×C3×C22⋊C4C2211yescharacteristic, complemented
45#953C2218yescomplemented
46C2×#217C2211yescharacteristic
47C2×#217C2211yescharacteristic, complemented
48C2×#217C2211yescharacteristic, complemented
49#930C2218yescomplemented
50#929C2218yescomplemented
51C4×#67C2218yescomplemented
52C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
53C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
54C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
55C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
56C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
57C2×C4⋊Dic3C2211yescharacteristic, complemented
58C2×Dic3⋊C4C2211yescharacteristic, complemented
59C2×Dic3⋊C4C2211yescharacteristic, complemented
60C2×Dic3⋊C4C2211yescharacteristic, complemented
61C2×C4×Dic3C2211yescharacteristic, complemented
62#885C2218yescomplemented
63#884C2218yescomplemented
64#884C2218yescomplemented
65#883C2218yescomplemented
66#882C2218yescomplemented
67#67⋊C4C2218yescomplemented
68D3×C22⋊C4C2218yescomplemented
69#878C2218yescomplemented
70C22×#41—31no—
71C23×D4—31no—
72#546—316no—
73C2×#124—31no—
74C2×#123—31no—
75C2×#123—31no—
76C2×C22⋊Q8—31no—
77C2×#121—31no—
78C2×#121—31no—
79C2×#121—31no—
80C2×#120—31no—
81C2×#120—31no—
82C2×C4×D4—31no—
83C2×C4×D4—31no—
84C22×C22⋊C4—31no—
85C22×C22⋊C4D311yescharacteristic, complemented
86C24×C3—24no—
87C24×C3—24no—
88C24×C3—42no—
89C24×C3C2311yescharacteristic
90C24×C3C2314yes—
91C24×C3C2311yescharacteristic
92C24×C3C2311yescharacteristic
93C23×D3—24no—
94C23×D3—24no—
95C23×D3—42no—
96C23×D3C2311yescharacteristic
97C23×D3C2311yescharacteristic
98C23×D3C2314yes—
99C23×D3C2311yescharacteristic
100C22×C3×C4—24no—
101C22×C3×C4—42no—
102C22×C3×C4C2311yescharacteristic
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2314yescomplemented
107C22×C3×C4C2314yescomplemented
108C2×#67—24no—
109C2×#67—24no—
110C2×#67—24no—
111C2×#67—24no—
112C2×#67—232no—
113C2×#67—24no—
114C2×#67—42no—
115C2×#67D418yescomplemented
116C2×#67C2314yes—
117C2×#67C2314yes—
118C2×#67C2314yes—
119C2×#67C2314yes—
120C2×#67C2314yes—
121C2×#67C2314yescomplemented
122C2×#67C2314yescomplemented
123C22×Dic3—24no—
124C22×Dic3C2314yes—
125C22×Dic3C2311yescharacteristic
126C22×Dic3C2311yescharacteristic
127C22×Dic3C2311yescharacteristic
128C22×Dic3C2311yescharacteristic
129C22×Dic3C2311yescharacteristic
130#235—232no—
131C2×D12—24no—
132C2×D12—42no—
133C2×D12C2314yescomplemented
134C2×D12C2314yescomplemented
135C2×C4×D3—24no—
136C2×C4×D3—24no—
137C2×C4×D3—42no—
138C2×C4×D3C2314yescomplemented
139C2×C4×D3C2314yescomplemented
140C2×C4×D3C2314yescomplemented
141C2×Dic6—24no—
142C2×Dic6C2314yescomplemented
143C3×C22⋊C4C2314yes—
144C3×C22⋊C4C2314yes—
145C3×C22⋊C4C2314yescomplemented
146C3×C22⋊C4C2314yescomplemented
147#217C2314yes—
148#217C2314yescomplemented
149#217C2314yescomplemented
150(C2×D3)⋊C4C2314yescomplemented
151(C2×D3)⋊C4C2314yescomplemented
152(C2×D3)⋊C4C2314yescomplemented
153(C2×D3)⋊C4C2314yescomplemented
154(C2×D3)⋊C4C2314yescomplemented
155C4⋊Dic3C2314yescomplemented
156Dic3⋊C4C2314yescomplemented
157Dic3⋊C4C2314yescomplemented
158Dic3⋊C4C2314yescomplemented
159C4×Dic3C2314yescomplemented
160C25—31no—
161C25C2×D311yescharacteristic
162C2×#41—38no—
163C2×#41—68no—
164C22×Q8—31no—
165C22×D4—31no—
166C22×D4—31no—
167C22×D4—31no—
168C22×D4—31no—
169C22×D4—31no—
170C22×D4—31no—
171C22×D4—38no—
172C22×D4—31no—
173C22×D4—31no—
174C22×D4—31no—
175C22×D4—68no—
176C23×C4—31no—
177C23×C4—31no—
178C23×C4—31no—
179C23×C4—31no—
180C23×C4C2×D311yescharacteristic, complemented
181C23×C4C2×D311yescharacteristic, complemented
182#124—38no—
183#123—38no—
184#123—38no—
185C22⋊Q8—38no—
186#121—38no—
187#121—38no—
188#121—38no—
189#120—38no—
190#120—38no—
191C4×D4—38no—
192C4×D4—38no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—31no—
197C2×C22⋊C4—31no—
198C2×C22⋊C4—31no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—31no—
203C2×C22⋊C4—31no—
204C2×C22⋊C4—31no—
205C2×C22⋊C4—38no—
206C2×C22⋊C4C2×D311yescharacteristic
207C2×C22⋊C4C2×D311yescharacteristic
208C2×C22⋊C4C2×D318yescomplemented
209C2×C22⋊C4C2×D311yescharacteristic, complemented
210C2×C22⋊C4C2×D311yescharacteristic, complemented
211C2×C42—31no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—28no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—28no—
220C23×C3—22no—
221C23×C3—44no—
222C23×C3—416no—
223C23×C3C2×D414yes—
224C23×C3#4114yescomplemented
225C23×C3C2412yes—
226C23×C3C2412yes—
227C23×C3C2412yes—
228C23×C3C2411yescharacteristic
229C22×D3—22no—
230C22×D3—22no—
231C22×D3—22no—
232C22×D3—22no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—28no—
236C22×D3—22no—
237C22×D3—22no—
238C22×D3—28no—
239C22×D3—44no—
240C22×D3—416no—
241C22×D3C2×D414yes—
242C22×D3C2412yes—
243C22×D3C2412yes—
244C22×D3C2412yes—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—28no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—44no—
251C2×C3×C4C2412yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2412yes—
255#67—28no—
256#67—28no—
257#67—28no—
258#67—28no—
259#67—28no—
260#67—44no—
261#67C2×D4116yescomplemented
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—28no—
267C2×Dic3C2×D414yes—
268C2×Dic3C2412yes—
269C2×Dic3C2412yes—
270C2×Dic3C2412yes—
271C2×Dic3C2412yes—
272C2×Dic3C2412yes—
273D12—28no—
274D12—44no—
275C4×D3—28no—
276C4×D3—28no—
277C4×D3—44no—
278Dic6—28no—
279C24—24no—
280C24—24no—
281C24—31no—
282C24—31no—
283C24—31no—
284C24—34no—
285C24—42no—
286C24—64no—
287C24—64no—
288C24—122no—
289C24C22×D311yescharacteristic
290C24C22×D314yes—
291C24C22×D311yescharacteristic
292C24C22×D311yescharacteristic
293#41—632no—
294C2×Q8—34no—
295C2×Q8—64no—
296C2×D4—38no—
297C2×D4—34no—
298C2×D4—34no—
299C2×D4—34no—
300C2×D4—34no—
301C2×D4—34no—
302C2×D4—34no—
303C2×D4—34no—
304C2×D4—34no—
305C2×D4—34no—
306C2×D4—64no—
307C2×D4—64no—
308C2×D4—632no—
309C2×D4—64no—
310C2×D4—64no—
311C2×D4—64no—
312C2×D4—64no—
313C2×D4—122no—
314C2×D4—122no—
315C22×C4—24no—
316C22×C4—31no—
317C22×C4—34no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—34no—
323C22×C4—34no—
324C22×C4—34no—
325C22×C4—42no—
326C22×C4—64no—
327C22×C4—64no—
328C22×C4—64no—
329C22×C4—122no—
330C22×C4C22×D311yescharacteristic
331C22×C4C22×D311yescharacteristic
332C22×C4C22×D311yescharacteristic
333C22×C4C22×D311yescharacteristic
334C22×C4C22×D314yescomplemented
335C22×C4C22×D314yescomplemented
336C4⋊C4—34no—
337C4⋊C4—34no—
338C4⋊C4—34no—
339C4⋊C4—34no—
340C22⋊C4—34no—
341C22⋊C4—34no—
342C22⋊C4—34no—
343C22⋊C4—34no—
344C22⋊C4—34no—
345C22⋊C4—34no—
346C22⋊C4—34no—
347C22⋊C4—34no—
348C22⋊C4C22×D314yes—
349C22⋊C4C22×D314yes—
350C22⋊C4C22×D314yescomplemented
351C22⋊C4C22×D314yescomplemented
352C42—34no—
353C22×C3—28no—
354C22×C3—28no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—416no—
363C22×C3—44no—
364C22×C3C22×D414yes—
365C22×C3C22×D412yes—
366C22×C3C2×#4112yes—
367C22×C3C2×#4114yescomplemented
368C22×C3H(5,𝔽2)12yes—
369C22×C3C2511yescharacteristic
370C2×D3—28no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—22no—
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—22no—
379C2×D3—416no—
380C2×D3—44no—
381C2×D3—44no—
382C2×D3C22×D414yes—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387Dic3—22no—
388Dic3—22no—
389Dic3—22no—
390Dic3—22no—
391Dic3C22×D414yes—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—28no—
398C23—28no—
399C23—22no—
400C23—22no—
401C23—34no—
402C23—32no—
403C23—32no—
404C23—32no—
405C23—44no—
406C23—416no—
407C23—68no—
408C23—62no—
409C23—62no—
410C23—62no—
411C23—62no—
412C23—68no—
413C23—62no—
414C23—62no—
415C23—62no—
416C23—62no—
417C23—124no—
418C23—1216no—
419C23#23514yescomplemented
420C23D3×D414yes—
421C23C23×D312yes—
422C23C23×D312yes—
423C23C23×D312yes—
424C23C23×D311yescharacteristic
425Q8—68no—
426D4—316no—
427D4—68no—
428D4—68no—
429D4—68no—
430D4—68no—
431D4—68no—
432D4—68no—
433D4—124no—
434D4—124no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—34no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—44no—
447C2×C4—68no—
448C2×C4—68no—
449C2×C4—68no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—124no—
455C2×C4C23×D312yes—
456C2×C4C23×D312yes—
457C2×C4C23×D312yes—
458C2×C4C23×D312yes—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—42no—
462C2×C3#54614yescomplemented
463C2×C3C23×D411yescharacteristic
464C2×C3C22×#4111yescharacteristic
465C2×C3C2×H(5,𝔽2)11yescharacteristic
466D3—24no—
467D3—42no—
468D3—42no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—28no—
474C22—28no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—34no—
479C22—416no—
480C22—44no—
481C22—62no—
482C22—62no—
483C22—68no—
484C22—62no—
485C22—62no—
486C22—62no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—1216no—
491C22—124no—
492C22—124no—
493C22C2×#23512yes—
494C22C2×#23514yescomplemented
495C22C2×D3×D414yes—
496C22C2×D3×D412yes—
497C22#100412yes—
498C22C24×D311yescharacteristic
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—34no—
504C4—62no—
505C4—62no—
506C4—62no—
507C4—62no—
508C3C2×#54611yescharacteristic, complemented, Sylow
509C2—24no—
510C2—24no—
511C2—42no—
512C2—64no—
513C2—122no—
514C2—122no—
515C2#549014yesdirect product factor
516C2C22×#23511yescharacteristic
517C2C22×D3×D411yescharacteristic
518C2C2×#100411yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC22×C3×C22⋊C4C2
socleC23×C3C24
hypercenterC22×C22⋊C4D3
hypocenterC3C2×#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×#67C2semidirect
2. C22×#235C2semidirect
3. C22×C3×C22⋊C4C2semidirect
4. C2×#953C2semidirect
5. C2×#930C2semidirect
6. C2×#929C2semidirect
7. C2×C4×#67C2semidirect
8. C2×#885C2semidirect
9. #5490C2direct (N × H)
10. C2×#884C2semidirect
11. C2×#884C2semidirect
12. C2×#883C2semidirect
13. C2×#882C2semidirect
14. C2×#67⋊C4C2semidirect
15. C2×D3×C22⋊C4C2semidirect
16. C2×#878C2semidirect
20. C23×C3×C4C22semidirect
21. C23×C3×C4C22semidirect
28. C22×#67C22semidirect
29. C22×#67C22semidirect
30. C22×#67C22semidirect
33. C2×#235C22semidirect
34. C22×D12C22semidirect
35. C22×D12C22semidirect
36. C22×C4×D3C22semidirect
37. C22×C4×D3C22semidirect
38. C22×C4×D3C22semidirect
39. C22×Dic6C22semidirect
42. C2×C3×C22⋊C4C22semidirect
43. C2×C3×C22⋊C4C22semidirect
44. C2×C3×C22⋊C4C22semidirect
45. #953C22semidirect
47. C2×#217C22semidirect
48. C2×#217C22semidirect
49. #930C22semidirect
50. #929C22semidirect
51. C4×#67C22semidirect
52. C2×(C2×D3)⋊C4C22semidirect
53. C2×(C2×D3)⋊C4C22semidirect
54. C2×(C2×D3)⋊C4C22semidirect
55. C2×(C2×D3)⋊C4C22semidirect
56. C2×(C2×D3)⋊C4C22semidirect
57. C2×C4⋊Dic3C22semidirect
58. C2×Dic3⋊C4C22semidirect
59. C2×Dic3⋊C4C22semidirect
60. C2×Dic3⋊C4C22semidirect
61. C2×C4×Dic3C22semidirect
62. #885C22semidirect
63. #884C22semidirect
64. #884C22semidirect
65. #883C22semidirect
66. #882C22semidirect
67. #67⋊C4C22semidirect
68. D3×C22⋊C4C22semidirect
69. #878C22semidirect
85. C22×C22⋊C4D3semidirect
106. C22×C3×C4C23semidirect
107. C22×C3×C4C23semidirect
115. C2×#67D4semidirect
121. C2×#67C23semidirect
122. C2×#67C23semidirect
133. C2×D12C23semidirect
134. C2×D12C23semidirect
138. C2×C4×D3C23semidirect
139. C2×C4×D3C23semidirect
140. C2×C4×D3C23semidirect
142. C2×Dic6C23semidirect
145. C3×C22⋊C4C23semidirect
146. C3×C22⋊C4C23semidirect
148. #217C23semidirect
149. #217C23semidirect
150. (C2×D3)⋊C4C23semidirect
151. (C2×D3)⋊C4C23semidirect
152. (C2×D3)⋊C4C23semidirect
153. (C2×D3)⋊C4C23semidirect
154. (C2×D3)⋊C4C23semidirect
155. C4⋊Dic3C23semidirect
156. Dic3⋊C4C23semidirect
157. Dic3⋊C4C23semidirect
158. Dic3⋊C4C23semidirect
159. C4×Dic3C23semidirect
180. C23×C4C2×D3semidirect
181. C23×C4C2×D3semidirect
208. C2×C22⋊C4C2×D3semidirect
209. C2×C22⋊C4C2×D3semidirect
210. C2×C22⋊C4C2×D3semidirect
224. C23×C3#41semidirect
261. #67C2×D4semidirect
334. C22×C4C22×D3semidirect
335. C22×C4C22×D3semidirect
350. C22⋊C4C22×D3semidirect
351. C22⋊C4C22×D3semidirect
367. C22×C3C2×#41semidirect
419. C23#235semidirect
462. C2×C3#546semidirect
494. C22C2×#235semidirect
508. C3C2×#546semidirect
515. C2#5490direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#672
C22×#2352
C22×C3×C22⋊C42
C2×#9532
C2×#9302
C2×#9292
C2×C4×#672
C2×#8852
#54902
C2×#8842
C2×#8832
C2×#8822
C2×#67⋊C42
C2×D3×C22⋊C42
C2×#8782
C2×#5463