Finite Group Databasebeta

← C2×#83401 #11664435

Subgroups — C2×#83401 #11664435

Order 768.

2070 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3C2×#5232C211yesmaximal, characteristic
4C2×#5233C211yesmaximal, characteristic
5C2×#5225C211yesmaximal, characteristic, complemented
6C22×#835C211yesmaximal, characteristic, complemented
7#83401C2116yesmaximal, direct product factor
8C2×#5082C212yesmaximal, complemented
9C2×#5079C212yesmaximal, complemented
10C2×#5055C212yesmaximal, complemented
11C2×#5056C212yesmaximal, complemented
12C2×#4997C211yesmaximal, characteristic, complemented
13C2×#3137—31nomaximal, Sylow
14C22×C3×#41C2211yescharacteristic
15C23×C3×Q8C2211yescharacteristic
16C3×#536—24no—
17C3×#536C2218yes—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×#124C2212yes—
20C2×C3×#123—22no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8—21no—
23C2×C3×#117C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5519—24no—
29#5519C2218yescomplemented
30C2×#890—21no—
31C2×#888—21no—
32C2×#878—22no—
33C2×#870C2212yes—
34C2×#869C2212yes—
35#5233C2218yes—
36#5232C2218yes—
37#5225C2218yescomplemented
38C2×#835C2212yes—
39C2×#835C2212yes—
40C2×#835C2212yes—
41C2×#835C2212yes—
42C2×#835C2211yescharacteristic
43C2×#835C2211yescharacteristic
44C2×#835C2212yes—
45C2×#835C2218yescomplemented
46C2×#832C2211yescharacteristic
47C2×#832C2211yescharacteristic
48C2×#832C2212yes—
49#5082C22116yescomplemented
50#5079C22116yescomplemented
51#5056C22116yescomplemented
52#5055C22116yescomplemented
53#4997C2218yescomplemented
54C2×#803C2212yes—
55C2×C4×C3⋊C8C2212yes—
56C2×#544—31no—
57C2×#536D311yescharacteristic
58#3137—316no—
59C2×#491—32no—
60C2×#489—32no—
61C2×#488—32no—
62C2×#487—32no—
63C2×#417—31no—
64C2×#419—31no—
65C2×#416—31no—
66C22×#103—31no—
67C2×#405—31no—
68C2×C3×#41—22no—
69C2×C3×#41—24no—
70C2×C3×#41—24no—
71C2×C3×#41C2314yes—
72C22×C3×Q8—21no—
73C22×C3×Q8—22no—
74C22×C3×Q8—24no—
75C22×C3×Q8—24no—
76C22×C3×Q8C2312yes—
77C22×C3×Q8C2314yes—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C23×C3×C4C2311yescharacteristic
84C23×Dic3—21no—
85C22×Dic6—22no—
86C3×C4⋊Q8—24no—
87C3×C4⋊Q8C2318yes—
88C3×#124—24no—
89C3×#124C2318yes—
90C3×#123—216no—
91C3×C22⋊Q8—28no—
92C3×C22⋊Q8—28no—
93C3×#117D414yes—
94C3×#117C2314yes—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4D412yes—
102C2×C3×C42C2312yes—
103C2×#217—21no—
104#926—22no—
105#926C2314yes—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2314yes—
113C2×Dic3⋊C4—22no—
114C2×Dic3⋊C4—22no—
115C2×C4×Dic3—21no—
116C2×#208—24no—
117C2×#208C2314yes—
118C22×C3⋊C8—24no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2314yes—
123#890—28no—
124#888—28no—
125#878—216no—
126#870—24no—
127#870C2318yes—
128#869—24no—
129#869C2318yes—
130#835C2318yes—
131#835C2318yes—
132#835C2318yes—
133#835C2318yes—
134#835C2318yes—
135#835C2314yes—
136#835C2314yes—
137#832C2318yes—
138#832C2314yes—
139#832C2314yes—
140#803C2318yes—
141C4×C3⋊C8C2318yes—
142C22×#41C2×D311yescharacteristic
143C23×Q8C2×D311yescharacteristic
144C22×M16—31no—
145C23×C8—31no—
146#544—38no—
147#544—64no—
148#536—24no—
149#536C2×D318yes—
150C2×C4⋊Q8—32no—
151C2×C4⋊Q8—61no—
152C2×C4⋊Q8C2×D312yes—
153C2×#125—32no—
154C2×#125—61no—
155C2×#124C2×D312yes—
156C2×#123—22no—
157C2×C22⋊Q8—21no—
158C2×C22⋊Q8—21no—
159C2×C22⋊Q8—62no—
160C2×#117—31no—
161C2×#117C2×D311yescharacteristic
162C22×C4⋊C4—31no—
163#491—316no—
164#489—316no—
165#488—316no—
166#487—316no—
167#419—38no—
168#417—38no—
169#416—38no—
170C2×#103—31no—
171C2×#103—32no—
172C2×#103—32no—
173C2×#103—32no—
174C2×#103—32no—
175C2×#103—32no—
176C2×#103—31no—
177C2×#103—38no—
178C2×#102—31no—
179C2×#102—32no—
180C2×#102—31no—
181#405—38no—
182C2×#97—32no—
183C2×C4×C8—32no—
184C24×C3—21no—
185C24×C3C2×D411yescharacteristic
186C3×#41—28no—
187C3×#41—28no—
188C3×#41—48no—
189C2×C3×Q8—24no—
190C2×C3×Q8—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—24no—
193C2×C3×Q8—24no—
194C2×C3×Q8—28no—
195C2×C3×Q8—28no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—44no—
199C2×C3×Q8—48no—
200C2×C3×Q8C2414yes—
201C2×C3×Q8C2412yes—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—22no—
206C2×C3×D4—44no—
207C2×C3×D4C2412yes—
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—44no—
213C22×C3×C4—44no—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4#4114yes—
219C22×C3×C4#4114yes—
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2412yes—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—22no—
225C22×Dic3—24no—
226C22×Dic3—82no—
227C2×Dic6—28no—
228C2×Dic6—84no—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4C2×D414yes—
232C3×C4⋊C4C2×D414yes—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—28no—
235C3×C22⋊C4C2×D418yes—
236C3×C42C2×D414yes—
237C3×C42C2414yes—
238#217—24no—
239C4⋊Dic3—21no—
240C4⋊Dic3—22no—
241C4⋊Dic3—28no—
242C4⋊Dic3—21no—
243C4⋊Dic3—24no—
244C4⋊Dic3C2412yes—
245C4⋊Dic3C2414yes—
246C4⋊Dic3C2412yes—
247Dic3⋊C4—28no—
248Dic3⋊C4—28no—
249C4×Dic3—24no—
250#208—28no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—28no—
254C2×C3⋊C8—24no—
255C2×C3⋊C8C2412yes—
256C2×C3⋊C8C2414yes—
257C2×C3⋊C8C2412yes—
258C2×#41—22no—
259C2×#41—24no—
260C2×#41—24no—
261C2×#41C22×D314yes—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—24no—
265C22×Q8—24no—
266C22×Q8—62no—
267C22×Q8C22×D312yes—
268C22×Q8C22×D311yescharacteristic
269C22×Q8C22×D314yes—
270C22×D4—21no—
271C22×D4C22×D311yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276C2×M16—34no—
277C2×M16—64no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8—34no—
281C22×C8—32no—
282C22×C8—64no—
283C4⋊Q8—24no—
284C4⋊Q8—38no—
285C4⋊Q8—68no—
286C4⋊Q8—64no—
287C4⋊Q8C22×D318yes—
288#125—38no—
289#125—68no—
290#125—64no—
291#124—24no—
292#124C22×D318yes—
293#123—216no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—616no—
297#117—34no—
298#117—62no—
299#117#6714yes—
300#117C22×D314yes—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—62no—
311C2×C4⋊C4—62no—
312C2×C4⋊C4#6711yescharacteristic
313C2×C4⋊C4#6711yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4#6712yes—
318C2×C42—61no—
319C2×C42C22×D312yes—
320#103—38no—
321#103—38no—
322#103—34no—
323#103—34no—
324#103—38no—
325#103—38no—
326#103—38no—
327#102—34no—
328#102—34no—
329#102—38no—
330#97—38no—
331C4×C8—38no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3#12414yes—
339C23×C3C22×D412yes—
340C23×C3C22×D411yescharacteristic
341C3×Q8—24no—
342C3×Q8—24no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—48no—
346C3×Q8—44no—
347C3×Q8—42no—
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—48no—
351C3×D4—42no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—48no—
358C2×C3×C4—44no—
359C2×C3×C4—48no—
360C2×C3×C4#12414yes—
361C2×C3×C4#12418yes—
362C2×C3×C4C22×D414yes—
363C2×C3×C4C22×D414yes—
364C2×C3×C4C22×D412yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4112yes—
368C2×C3×C4C2×#4114yes—
369C2×C3×C4C2×#4114yes—
370C2×C3×C4C2511yescharacteristic
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—24no—
374C2×Dic3—42no—
375C2×Dic3—44no—
376C2×Dic3—42no—
377C2×Dic3—84no—
378Dic6—88no—
379C3⋊C8—22no—
380C3⋊C8—22no—
381C3⋊C8—24no—
382C24—21no—
383C24C2×#6711yescharacteristic
384#41—28no—
385#41—28no—
386#41—48no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—68no—
399C2×Q8—244no—
400C2×Q8C23×D314yes—
401C2×Q8C23×D312yes—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—44no—
407C2×D4C23×D312yes—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—64no—
418C22×C4—242no—
419C22×C4#23714yes—
420C22×C4#23714yes—
421C22×C4C2×#6712yes—
422C22×C4C2×#6712yes—
423C22×C4C2×#6712yes—
424C22×C4C2×#6711yescharacteristic
425C22×C4C23×D312yes—
426C22×C4C23×D311yescharacteristic
427M16—68no—
428C2×C8—32no—
429C2×C8—34no—
430C2×C8—32no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—68no—
434C2×C8—64no—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—34no—
438C4⋊C4—32no—
439C4⋊C4—32no—
440C4⋊C4—64no—
441C4⋊C4—62no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—68no—
447C4⋊C4C2×#6714yes—
448C4⋊C4C2×#6714yes—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—64no—
452C22⋊C4C2×#6718yes—
453C42—64no—
454C42C2×#6714yes—
455C42C23×D314yes—
456C22×C3—28no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—84no—
463C22×C3C2×#12414yes—
464C22×C3C2×#12412yes—
465C22×C3#57712yes—
466C22×C3#57812yes—
467C22×C3C23×D411yescharacteristic
468C3×C4—24no—
469C3×C4—24no—
470C3×C4—42no—
471C3×C4—44no—
472C3×C4C2×#12414yes—
473C3×C4C2×#12412yes—
474C3×C4C22×#4111yescharacteristic
475C3×C4C22×#4111yescharacteristic
476Dic3—42no—
477Dic3—44no—
478Dic3—42no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23#93614yes—
486C23C22×#6712yes—
487C23C22×#6711yescharacteristic
488Q8—24no—
489Q8—24no—
490Q8—22no—
491Q8—22no—
492Q8—48no—
493Q8—44no—
494Q8—42no—
495Q8—248no—
496D4—22no—
497D4—22no—
498D4—48no—
499D4—42no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—124no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—244no—
515C2×C4#93614yes—
516C2×C4#93618yes—
517C2×C4C2×#23712yes—
518C2×C4C2×#23712yes—
519C2×C4C2×#23714yes—
520C2×C4C2×#23714yes—
521C2×C4C22×#6714yes—
522C2×C4C22×#6714yes—
523C2×C4C22×#6712yes—
524C2×C4C22×#6711yescharacteristic
525C2×C4C24×D311yescharacteristic
526C8—62no—
527C8—62no—
528C8—64no—
529C2×C3—24no—
530C2×C3—82no—
531C2×C3#313714yescomplemented
532C2×C3C22×#12411yescharacteristic
533C2×C3C2×#57711yescharacteristic
534C2×C3C2×#57811yescharacteristic
535C22—28no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—84no—
542C22C2×#93614yes—
543C22C2×#93612yes—
544C22#583712yes—
545C22#583812yes—
546C22C23×#6711yescharacteristic
547C4—24no—
548C4—24no—
549C4—42no—
550C4—44no—
551C4—124no—
552C4—122no—
553C4—122no—
554C4C2×#93614yes—
555C4C2×#93612yes—
556C4C22×#23711yescharacteristic
557C4C22×#23711yescharacteristic
558C3C2×#313711yescharacteristic, complemented, Sylow
559C2—24no—
560C2—82no—
561C2#8340114yesdirect product factor
562C2C22×#93611yescharacteristic
563C2C2×#583711yescharacteristic
564C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#536D3
hypocenterC3C2×#3137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
5. C2×#5225C2semidirect
6. C22×#835C2semidirect
7. #83401C2direct (N × H)
8. C2×#5082C2semidirect
9. C2×#5079C2semidirect
10. C2×#5055C2semidirect
11. C2×#5056C2semidirect
12. C2×#4997C2semidirect
29. #5519C22semidirect
37. #5225C22semidirect
45. C2×#835C22semidirect
49. #5082C22semidirect
50. #5079C22semidirect
51. #5056C22semidirect
52. #5055C22semidirect
53. #4997C22semidirect
531. C2×C3#3137semidirect
558. C3C2×#3137semidirect
561. C2#83401direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×#55192
C2×#52322
C2×#52332
C2×#52252
C22×#8352
#834012
C2×#50822
C2×#50792
C2×#50552
C2×#50562
C2×#49972
C2×#31373