Finite Group Databasebeta

← C2×#83432 #11665015

Subgroups — C2×#83432 #11665015

Order 768.

2098 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic, complemented
2C2×#5535C211yesmaximal, characteristic, complemented
3#83432C2116yesmaximal, direct product factor
4C2×#5073C211yesmaximal, characteristic, complemented
5C2×#5071C211yesmaximal, characteristic, complemented
6C2×#5065C211yesmaximal, characteristic, complemented
7C2×#5067C211yesmaximal, characteristic, complemented
8C2×#5039C212yesmaximal, complemented
9C2×#5038C212yesmaximal, complemented
10C2×#5014C212yesmaximal, complemented
11C2×C4×#213C212yesmaximal, complemented
12C2×#4997C211yesmaximal, characteristic, complemented
13C2×#3176—31nomaximal, Sylow
14C3×#540—24no—
15C3×#540C2218yescomplemented
16C2×C3×#127C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—22no—
19C2×C3×#121—22no—
20C2×C3×#121C2212yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117C2211yescharacteristic
23C22×#213C2212yes—
24C22×#213C2212yes—
25C2×#930C2212yes—
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5535—24no—
29#5535C2218yescomplemented
30C2×#894—22no—
31C2×#884—22no—
32C2×#874C2211yescharacteristic
33C2×C4×D12C2212yes—
34C2×#870C2211yescharacteristic
35C2×#832C2212yes—
36C2×#832C2212yes—
37#5073C2218yescomplemented
38#5071C2218yescomplemented
39#5067C2218yescomplemented
40#5065C2218yescomplemented
41#5039C22116yescomplemented
42#5038C22116yescomplemented
43#5014C22116yescomplemented
44C4×#213C22116yescomplemented
45#4997C2218yescomplemented
46C2×#809C2212yes—
47C2×#809C2212yes—
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#803C2211yescharacteristic
51C2×#803C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54C2×#540—31no—
55C2×#540D311yescharacteristic, complemented
56#3176—316no—
57C2×#499—31no—
58C2×#493—31no—
59C2×#490—31no—
60C2×#490—31no—
61C2×#469—32no—
62C2×#466—32no—
63C2×#442—32no—
64C2×C4×D8—32no—
65C2×#405—31no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4C2312yes—
69C23×C3×C4—22no—
70C23×C3×C4C2311yescharacteristic
71C22×#67—22no—
72C22×D12—22no—
73C22×D12C2312yes—
74C22×C4×D3—22no—
75C3×#127—22no—
76C3×#127C2314yes—
77C3×#125—22no—
78C3×#125C2314yes—
79C3×#123—216no—
80C3×#121—24no—
81C3×#121—216no—
82C3×#121C2318yes—
83C3×C4×D4—24no—
84C3×C4×D4C2318yes—
85C3×#117D414yes—
86C3×#117C2314yes—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4D412yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4D412yes—
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#213—28no—
96C2×#213—28no—
97C2×#213C2318yes—
98C2×#213C2318yes—
99#930—24no—
100#930C2318yes—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4—22no—
103C2×C4⋊Dic3C2312yes—
104C2×Dic3⋊C4—22no—
105C2×#208—24no—
106C2×#208C2314yes—
107C22×C3⋊C8—24no—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2314yes—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#894—216no—
114#884—216no—
115#874—22no—
116#874C2314yes—
117C4×D12—24no—
118C4×D12C2318yes—
119#870—22no—
120#870C2314yes—
121#832C2318yes—
122#832C2318yes—
123#809C2318yes—
124#809C2318yes—
125#808C2318yes—
126#807C2318yes—
127#803C2314yes—
128#803C2314yes—
129C4×C3⋊C8C2314yes—
130C4×C3⋊C8C2314yes—
131C22×D8—32no—
132C22×D8—32no—
133C22×M16—31no—
134C23×C8—31no—
135#540—24no—
136#540—38no—
137#540—64no—
138#540C2×D318yescomplemented
139C2×#127—31no—
140C2×#127C2×D311yescharacteristic
141C2×#125—31no—
142C2×#125C2×D311yescharacteristic
143C2×#123—22no—
144C2×#123—62no—
145C2×#121—22no—
146C2×#121—32no—
147C2×#121—62no—
148C2×#121C2×D312yes—
149C2×C4×D4—32no—
150C2×C4×D4C2×D312yes—
151C2×#117C2×D311yescharacteristic
152#499—38no—
153#493—38no—
154#490—38no—
155#490—38no—
156#469—316no—
157#466—316no—
158#442—316no—
159C4×D8—316no—
160C2×#107—32no—
161C2×#106—32no—
162C2×#102—32no—
163C2×#102—32no—
164C2×#102—32no—
165C2×#102—32no—
166#405—38no—
167C2×#97—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171C24×C3—22no—
172C24×C3C2×D411yescharacteristic
173C23×D3—22no—
174C2×C3×D4—28no—
175C2×C3×D4—22no—
176C2×C3×D4—28no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—84no—
180C2×C3×D4—84no—
181C2×C3×D4C2414yes—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—28no—
186C22×C3×C4—84no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2412yes—
192C22×C3×C4C2411yescharacteristic
193C2×#67—28no—
194C2×#67—84no—
195C22×Dic3—22no—
196C2×D12—24no—
197C2×D12—24no—
198C2×D12—22no—
199C2×D12—28no—
200C2×D12—84no—
201C2×D12C2414yes—
202C2×C4×D3—28no—
203C2×C4×D3—84no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4C2×D418yes—
207C3×C4⋊C4C2414yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4C2×D418yes—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C3×C42C2412yes—
214C3×C42C2412yes—
215#213—216no—
216#213—216no—
217(C2×D3)⋊C4—28no—
218(C2×D3)⋊C4—28no—
219C4⋊Dic3—22no—
220C4⋊Dic3C2414yes—
221Dic3⋊C4—28no—
222#208—28no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—28no—
226C2×C3⋊C8#4114yes—
227C2×C3⋊C8#4114yes—
228C2×C3⋊C8C2412yes—
229C2×C3⋊C8C2412yes—
230C2×C3⋊C8C2412yes—
231C2×C3⋊C8C2412yes—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—32no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4C22×D312yes—
238C23×C4—22no—
239C23×C4—62no—
240C23×C4C22×D311yescharacteristic
241C2×D8—38no—
242C2×D8—38no—
243C2×D8—68no—
244C2×D8—68no—
245C2×M16—34no—
246C2×M16—64no—
247C22×C8—31no—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8—34no—
251C22×C8—31no—
252C22×C8—64no—
253#127—22no—
254#127—34no—
255#127—62no—
256#127C22×D314yes—
257#125—22no—
258#125—34no—
259#125—62no—
260#125C22×D314yes—
261#123—216no—
262#123—616no—
263#121—24no—
264#121—216no—
265#121—38no—
266#121—64no—
267#121—616no—
268#121C22×D318yes—
269C4×D4—24no—
270C4×D4—38no—
271C4×D4—64no—
272C4×D4C22×D318yes—
273#117#6714yes—
274#117C22×D314yes—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4#6712yes—
279C2×C4⋊C4C22×D312yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4#6712yes—
285C2×C42C22×D311yescharacteristic
286C2×C42C22×D311yescharacteristic
287#107—38no—
288#106—38no—
289#102—38no—
290#102—38no—
291#102—38no—
292#102—38no—
293#97—34no—
294#97—34no—
295C4×C8—34no—
296C4×C8—34no—
297C23×C3—22no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—44no—
303C23×C3C22×D412yes—
304C23×C3C22×D411yescharacteristic
305C22×D3—24no—
306C22×D3—44no—
307C22×D3—44no—
308C22×D3—44no—
309C3×D4—24no—
310C3×D4—24no—
311C3×D4—44no—
312C3×D4—88no—
313C3×D4—88no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—44no—
322C2×C3×C4—88no—
323C2×C3×C4C22×D414yes—
324C2×C3×C4C22×D412yes—
325C2×C3×C4C22×D414yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4112yes—
328C2×C3×C4C2×#4112yes—
329C2×C3×C4C2511yescharacteristic
330#67—88no—
331C2×Dic3—24no—
332C2×Dic3—44no—
333D12—24no—
334D12—24no—
335D12—44no—
336D12—88no—
337C4×D3—88no—
338C3⋊C8—22no—
339C3⋊C8—22no—
340C3⋊C8C2×#4114yes—
341C3⋊C8C2×#4114yes—
342C24—22no—
343C24—62no—
344C24C2×#6711yescharacteristic
345C2×D4—28no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—28no—
350C2×D4—34no—
351C2×D4—68no—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—62no—
355C2×D4—68no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—244no—
359C2×D4—244no—
360C2×D4C23×D314yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—62no—
366C22×C4—68no—
367C22×C4—84no—
368C22×C4—244no—
369C22×C4C2×#6712yes—
370C22×C4C2×#6712yes—
371C22×C4C2×#6712yes—
372C22×C4C2×#6711yescharacteristic
373C22×C4C23×D312yes—
374C22×C4C23×D311yescharacteristic
375D8—616no—
376D8—616no—
377M16—68no—
378C2×C8—34no—
379C2×C8—34no—
380C2×C8—32no—
381C2×C8—32no—
382C2×C8—32no—
383C2×C8—32no—
384C2×C8—68no—
385C2×C8—62no—
386C2×C8—62no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—34no—
390C4⋊C4—62no—
391C4⋊C4—68no—
392C4⋊C4C2×#6718yes—
393C4⋊C4C23×D314yes—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—68no—
397C22⋊C4—68no—
398C22⋊C4C2×#6718yes—
399C42C2×#6712yes—
400C42C2×#6712yes—
401C42C23×D312yes—
402C42C23×D312yes—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—88no—
410C22×C3#53212yes—
411C22×C3#57614yes—
412C22×C3C23×D411yescharacteristic
413C2×D3—44no—
414C2×D3—44no—
415C2×D3—44no—
416C2×D3—88no—
417C3×C4—22no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—44no—
421C3×C4#53212yes—
422C3×C4C22×#4111yescharacteristic
423C3×C4C22×#4111yescharacteristic
424Dic3—44no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—64no—
432C23—124no—
433C23—124no—
434C23—124no—
435C23C22×#6712yes—
436C23C22×#6711yescharacteristic
437D4—24no—
438D4—24no—
439D4—44no—
440D4—64no—
441D4—64no—
442D4—88no—
443D4—88no—
444D4—124no—
445D4—248no—
446D4—248no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—64no—
456C2×C4—88no—
457C2×C4—124no—
458C2×C4—248no—
459C2×C4D3×#4112yes—
460C2×C4D3×#4112yes—
461C2×C4C22×#6714yes—
462C2×C4C22×#6714yes—
463C2×C4C22×#6712yes—
464C2×C4C22×#6711yescharacteristic
465C2×C4C24×D311yescharacteristic
466C8—34no—
467C8—34no—
468C8—62no—
469C8—62no—
470C2×C3—42no—
471C2×C3—84no—
472C2×C3#317614yescomplemented
473C2×C3C2×#53211yescharacteristic
474C2×C3C2×#57612yes—
475D3—84no—
476C22—22no—
477C22—24no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—88no—
483C22—124no—
484C22—124no—
485C22—124no—
486C22—248no—
487C22#583212yes—
488C22#583514yes—
489C22C23×#6711yescharacteristic
490C4—22no—
491C4—24no—
492C4—24no—
493C4—44no—
494C4—124no—
495C4#583212yes—
496C4C2×D3×#4111yescharacteristic
497C4C2×D3×#4111yescharacteristic
498C3C2×#317611yescharacteristic, complemented, Sylow
499C2—42no—
500C2—84no—
501C2—244no—
502C2#8343214yesdirect product factor
503C2C2×#583211yescharacteristic
504C2C2×#583512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#540C2
socleC23×C3C22×D4
hypercenterC2×#540D3
hypocenterC3C2×#3176
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×C3×#540C2semidirect
2. C2×#5535C2semidirect
3. #83432C2direct (N × H)
4. C2×#5073C2semidirect
5. C2×#5071C2semidirect
6. C2×#5065C2semidirect
7. C2×#5067C2semidirect
8. C2×#5039C2semidirect
9. C2×#5038C2semidirect
10. C2×#5014C2semidirect
11. C2×C4×#213C2semidirect
12. C2×#4997C2semidirect
15. C3×#540C22semidirect
29. #5535C22semidirect
37. #5073C22semidirect
38. #5071C22semidirect
39. #5067C22semidirect
40. #5065C22semidirect
41. #5039C22semidirect
42. #5038C22semidirect
43. #5014C22semidirect
44. C4×#213C22semidirect
45. #4997C22semidirect
55. C2×#540D3semidirect
138. #540C2×D3semidirect
472. C2×C3#3176semidirect
498. C3C2×#3176semidirect
502. C2#83432direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×#55352
#834322
C2×#50732
C2×#50712
C2×#50652
C2×#50672
C2×#50392
C2×#50382
C2×#50142
C2×C4×#2132
C2×#49972
C2×#31763