Finite Group Databasebeta

← C2×#83471 #11664440

Subgroups — C2×#83471 #11664440

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×#544C211yesmaximal, characteristic, complemented
2C2×#5526C211yesmaximal, characteristic
3#83471C2116yesmaximal, direct product factor
4C2×#5089C212yesmaximal, complemented
5C2×#5093C212yesmaximal, complemented
6C2×#5069C212yesmaximal, complemented
7C2×#5068C212yesmaximal, complemented
8C2×#5004C211yesmaximal, characteristic, complemented
9C2×#5002C211yesmaximal, characteristic
10C2×#4997C211yesmaximal, characteristic, complemented
11C22×#810C211yesmaximal, characteristic, complemented
12C2×#4966C211yesmaximal, characteristic
13C2×#3137—31nomaximal, Sylow
14C22×#235C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C3×#544—24no—
17C3×#544C2218yescomplemented
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×#125—21no—
21C2×C3×#125C2212yes—
22C2×C3×C22⋊Q8—22no—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28#5526—24no—
29#5526C2218yes—
30C2×#897—21no—
31C2×#897—21no—
32C2×#886—22no—
33C2×#875C2212yes—
34C2×#869C2212yes—
35#5093C22116yescomplemented
36#5089C22116yescomplemented
37#5069C22116yescomplemented
38#5068C22116yescomplemented
39#5004C2218yescomplemented
40#5002C2218yes—
41#4997C2218yescomplemented
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2212yes—
45C2×#810C2212yes—
46C2×#810C2212yes—
47C2×#810C2212yes—
48C2×#810C2212yes—
49C2×#810C2218yescomplemented
50#4966C2218yes—
51C2×#809C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#809C2212yes—
54C2×#803C2212yes—
55C2×C4×C3⋊C8C2212yes—
56C2×#544D311yescharacteristic, complemented
57C2×#536—31no—
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—
68C22×C3×Q8—22no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C22×#67—21no—
72C23×Dic3—21no—
73C2×#235—22no—
74C2×#235—24no—
75C2×#235—24no—
76C2×#235C2314yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—24no—
80C22×Dic6—24no—
81C22×Dic6—21no—
82C22×Dic6—22no—
83C22×Dic6C2312yes—
84C22×Dic6C2314yes—
85C22×Dic6C2311yescharacteristic
86C3×C4⋊Q8—28no—
87C3×C4⋊Q8—24no—
88C3×C4⋊Q8C2318yes—
89C3×#125—28no—
90C3×#125—24no—
91C3×#125C2318yes—
92C3×C22⋊Q8—216no—
93C3×#117—22no—
94C3×#117D414yes—
95C3×#117C2314yes—
96C3×#117C2314yes—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42—21no—
111C2×C3×C42C2312yes—
112C2×(C2×D3)⋊C4—22no—
113C2×(C2×D3)⋊C4—22no—
114C2×C4⋊Dic3—22no—
115C2×C4⋊Dic3—22no—
116C2×#208—24no—
117C2×#208C2314yes—
118C22×C3⋊C8—24no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2312yes—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2314yes—
123#897—28no—
124#897—28no—
125#886—216no—
126#875—24no—
127#875C2318yes—
128#869—24no—
129#869C2318yes—
130#810C2318yes—
131#810C2318yes—
132#810C2318yes—
133#810C2314yes—
134#810C2314yes—
135#810C2318yes—
136#810C2318yes—
137#809C2318yes—
138#809C2314yes—
139#809C2314yes—
140#803C2318yes—
141C4×C3⋊C8C2318yes—
142C22×#41—31no—
143C23×Q8—31no—
144C22×M16—31no—
145C23×C8—31no—
146#544—24no—
147#544C2×D318yescomplemented
148#536—38no—
149#536—64no—
150C2×C4⋊Q8—21no—
151C2×C4⋊Q8—32no—
152C2×C4⋊Q8C2×D312yes—
153C2×#125—21no—
154C2×#125C2×D312yes—
155C2×#124—32no—
156C2×#123—62no—
157C2×C22⋊Q8—22no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×#117C2×D311yescharacteristic
161C2×#117C2×D311yescharacteristic
162C22×C4⋊C4C2×D311yescharacteristic
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×C3C2×D411yescharacteristic
185C23×D3—21no—
186C2×C3×Q8—28no—
187C2×C3×Q8—84no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—82no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4114yes—
199C22×C3×C4C2411yescharacteristic
200C22×C3×C4C2412yes—
201C2×#67—24no—
202C2×#67—44no—
203C22×Dic3—24no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—44no—
207#235—28no—
208#235—28no—
209#235—48no—
210C2×D12—22no—
211C2×D12—22no—
212C2×D12—21no—
213C2×D12C2412yes—
214C2×C4×D3—24no—
215C2×C4×D3—44no—
216C2×Dic6—24no—
217C2×Dic6—24no—
218C2×Dic6—28no—
219C2×Dic6—28no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6—22no—
224C2×Dic6—24no—
225C2×Dic6—44no—
226C2×Dic6—48no—
227C2×Dic6C2412yes—
228C2×Dic6C2414yes—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—28no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—28no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C2×D414yes—
237C3×C4⋊C4C2×D414yes—
238C3×C4⋊C4C2414yes—
239C3×C4⋊C4C2412yes—
240C3×C4⋊C4C2412yes—
241C3×C22⋊C4—24no—
242C3×C22⋊C4C2×D418yes—
243C3×C42—24no—
244C3×C42C2×D414yes—
245C3×C42C2414yes—
246(C2×D3)⋊C4—28no—
247(C2×D3)⋊C4—28no—
248C4⋊Dic3—28no—
249C4⋊Dic3—28no—
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—34no—
259C2×#41—62no—
260C2×#41—64no—
261C2×#41—64no—
262C22×Q8—22no—
263C22×Q8—32no—
264C22×Q8—31no—
265C22×Q8—34no—
266C22×Q8—61no—
267C22×Q8—62no—
268C22×Q8—64no—
269C22×Q8—64no—
270C22×D4—31no—
271C22×D4—61no—
272C23×C4—21no—
273C23×C4—61no—
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—28no—
284C4⋊Q8—24no—
285C4⋊Q8—38no—
286C4⋊Q8—64no—
287C4⋊Q8C22×D318yes—
288#125—28no—
289#125—24no—
290#125C22×D318yes—
291#124—38no—
292#124—64no—
293#123—616no—
294C22⋊Q8—216no—
295C22⋊Q8—68no—
296C22⋊Q8—68no—
297#117—22no—
298#117#6714yes—
299#117C22×D314yes—
300#117C22×D314yes—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—62no—
307C2×C4⋊C4—62no—
308C2×C4⋊C4#6711yescharacteristic
309C2×C4⋊C4#6711yescharacteristic
310C2×C4⋊C4C22×D312yes—
311C2×C4⋊C4C22×D314yes—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—62no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4#6712yes—
318C2×C42—21no—
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#12414yes—
335C23×C3C22×D412yes—
336C23×C3C22×D411yescharacteristic
337C22×D3—22no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C3×Q8—88no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—44no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—84no—
351C2×C3×C4#12414yes—
352C2×C3×C4#12418yes—
353C2×C3×C4C22×D414yes—
354C2×C3×C4C22×D414yes—
355C2×C3×C4C22×D412yes—
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4112yes—
358C2×C3×C4C2×#4112yes—
359C2×C3×C4C2×#4114yes—
360C2×C3×C4C2×#4114yes—
361C2×C3×C4C2511yescharacteristic
362#67—48no—
363C2×Dic3—22no—
364C2×Dic3—24no—
365C2×Dic3—48no—
366C2×Dic3—44no—
367C2×Dic3—42no—
368D12—22no—
369D12—22no—
370D12—42no—
371C4×D3—48no—
372Dic6—24no—
373Dic6—24no—
374Dic6—22no—
375Dic6—22no—
376Dic6—44no—
377Dic6—48no—
378Dic6—42no—
379C3⋊C8—22no—
380C3⋊C8—22no—
381C3⋊C8—24no—
382C24—61no—
383C24C2×#6711yescharacteristic
384#41—68no—
385#41—68no—
386#41—128no—
387C2×Q8—28no—
388C2×Q8—34no—
389C2×Q8—32no—
390C2×Q8—64no—
391C2×Q8—62no—
392C2×Q8—61no—
393C2×Q8—64no—
394C2×Q8—64no—
395C2×Q8—68no—
396C2×Q8—68no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8—84no—
400C2×Q8—124no—
401C2×Q8—128no—
402C2×D4—32no—
403C2×D4—64no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—61no—
407C2×D4—124no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—64no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—64no—
416C22×C4—82no—
417C22×C4—124no—
418C22×C4—124no—
419C22×C4#23914yes—
420C22×C4#23914yes—
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—24no—
436C4⋊C4—22no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4C2×#6714yes—
445C4⋊C4C2×#6714yes—
446C4⋊C4C23×D314yes—
447C4⋊C4C23×D312yes—
448C4⋊C4C23×D312yes—
449C22⋊C4—24no—
450C22⋊C4—68no—
451C22⋊C4—68no—
452C22⋊C4C2×#6718yes—
453C42—24no—
454C42C2×#6714yes—
455C42C23×D314yes—
456C22×C3—28no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3C2×#12414yes—
460C22×C3C2×#12412yes—
461C22×C3#57712yes—
462C22×C3#57812yes—
463C22×C3C23×D411yescharacteristic
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—42no—
467C2×D3—84no—
468C3×C4—24no—
469C3×C4—24no—
470C3×C4—44no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4C2×#12412yes—
474C3×C4C2×#12414yes—
475C3×C4C22×#4111yescharacteristic
476C3×C4C22×#4111yescharacteristic
477Dic3—44no—
478Dic3—42no—
479C23—22no—
480C23—22no—
481C23—62no—
482C23—122no—
483C23—122no—
484C23—122no—
485C23#94714yes—
486C23C22×#6712yes—
487C23C22×#6711yescharacteristic
488Q8—64no—
489Q8—64no—
490Q8—62no—
491Q8—62no—
492Q8—88no—
493Q8—128no—
494Q8—124no—
495Q8—122no—
496D4—62no—
497D4—62no—
498D4—128no—
499D4—122no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—84no—
511C2×C4—122no—
512C2×C4—128no—
513C2×C4—124no—
514C2×C4—128no—
515C2×C4#94714yes—
516C2×C4#94718yes—
517C2×C4C2×#23912yes—
518C2×C4C2×#23912yes—
519C2×C4C2×#23914yes—
520C2×C4C2×#23914yes—
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#313714yescomplemented
531C2×C3C22×#12411yescharacteristic
532C2×C3C2×#57711yescharacteristic
533C2×C3C2×#57811yescharacteristic
534D3—82no—
535C22—28no—
536C22—22no—
537C22—22no—
538C22—122no—
539C22—122no—
540C22—122no—
541C22—244no—
542C22C2×#94714yes—
543C22C2×#94712yes—
544C22#583612yes—
545C22#583812yes—
546C22C23×#6711yescharacteristic
547C4—24no—
548C4—24no—
549C4—44no—
550C4—42no—
551C4—42no—
552C4—122no—
553C4—124no—
554C4C2×#94714yes—
555C4C2×#94712yes—
556C4C22×#23911yescharacteristic
557C4C22×#23911yescharacteristic
558C3C2×#313711yescharacteristic, complemented, Sylow
559C2—24no—
560C2—242no—
561C2#8347114yesdirect product factor
562C2C22×#94711yescharacteristic
563C2C2×#583611yescharacteristic
564C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#544C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#544D3
hypocenterC3C2×#3137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×C3×#544C2semidirect
3. #83471C2direct (N × H)
4. C2×#5089C2semidirect
5. C2×#5093C2semidirect
6. C2×#5069C2semidirect
7. C2×#5068C2semidirect
8. C2×#5004C2semidirect
10. C2×#4997C2semidirect
11. C22×#810C2semidirect
17. C3×#544C22semidirect
35. #5093C22semidirect
36. #5089C22semidirect
37. #5069C22semidirect
38. #5068C22semidirect
39. #5004C22semidirect
41. #4997C22semidirect
49. C2×#810C22semidirect
56. C2×#544D3semidirect
147. #544C2×D3semidirect
530. C2×C3#3137semidirect
558. C3C2×#3137semidirect
561. C2#83471direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×#55262
#834712
C2×#50892
C2×#50932
C2×#50692
C2×#50682
C2×#50042
C2×#50022
C2×#49972
C22×#8102
C2×#49662
C2×#31373