Finite Group Databasebeta

← C2×#83135 #11664953

Subgroups — C2×#83135 #11664953

Order 768.

2266 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic, complemented
2C2×#5477C211yesmaximal, characteristic, complemented
3#83135C2116yesmaximal, direct product factor
4C2×#5064C211yesmaximal, characteristic, complemented
5C2×#5067C211yesmaximal, characteristic, complemented
6C2×#5058C211yesmaximal, characteristic, complemented
7C2×#5055C211yesmaximal, characteristic, complemented
8C2×#5049C212yesmaximal, complemented
9C2×#5039C212yesmaximal, complemented
10C2×C4×#215C212yesmaximal, complemented
11C2×C4×#213C212yesmaximal, complemented
12C2×C4×#208C211yesmaximal, characteristic, complemented
13C2×#3171—31nomaximal, Sylow
14C22×#235—21no—
15C3×#529—24no—
16C3×#529C2218yescomplemented
17C2×C3×#126—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×#123—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117—21no—
27C22×C3×C42C2211yescharacteristic
28C22×#215C2212yes—
29C22×#213C2212yes—
30C2×#930—21no—
31C2×#930C2212yes—
32C22×#208C2212yes—
33#5477—24no—
34#5477C2218yescomplemented
35C2×#875—21no—
36C2×#874C2211yescharacteristic
37C2×C4×D12—21no—
38C2×C4×D12C2212yes—
39C2×#869C2211yescharacteristic
40C2×#835C2212yes—
41C2×#832C2212yes—
42#5067C2218yescomplemented
43#5064C2218yescomplemented
44#5058C2218yescomplemented
45#5055C2218yescomplemented
46#5049C22116yescomplemented
47#5039C22116yescomplemented
48C4×#215C22116yescomplemented
49C4×#213C22116yescomplemented
50C2×#809C2212yes—
51C2×#809C2212yes—
52C2×#808C2212yes—
53C2×#807C2212yes—
54C4×#208C2218yescomplemented
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2212yes—
58C2×#532—31no—
59C2×#529D311yescharacteristic, complemented
60#3171—316no—
61C2×#501—31no—
62C2×#496—31no—
63C2×#490—31no—
64C2×#489—31no—
65C2×#469—32no—
66C2×#468—32no—
67C2×C4×SD16—32no—
68C2×C4×D8—32no—
69C2×C4×M16—31no—
70C22×C3×Q8C2311yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C23×C3×C4D411yescharacteristic
75C23×C3×C4D411yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×#67—22no—
78C2×#235—28no—
79C2×#235—48no—
80C22×D12—21no—
81C22×D12—21no—
82C22×D12C2312yes—
83C22×C4×D3—22no—
84C22×Dic6—21no—
85C3×#126—28no—
86C3×#125—22no—
87C3×#125C2314yes—
88C3×#124—22no—
89C3×#124C2314yes—
90C3×#123—28no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8C2314yes—
93C3×#121—22no—
94C3×#121C2314yes—
95C3×C4×Q8—22no—
96C3×C4×Q8C2314yes—
97C3×C4×D4—28no—
98C3×C4×D4—22no—
99C3×C4×D4C2314yes—
100C3×#117—28no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113C2×C3×C42D411yescharacteristic
114C2×C3×C42D411yescharacteristic
115C2×C3×C42D414yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2314yes—
119C2×#215—28no—
120C2×#215C2318yes—
121C2×#213—28no—
122C2×#213C2318yes—
123#930—24no—
124#930—28no—
125#930C2318yes—
126C2×(C2×D3)⋊C4—22no—
127C2×(C2×D3)⋊C4—22no—
128C2×C4⋊Dic3—21no—
129C2×C4⋊Dic3C2312yes—
130C2×#208—28no—
131C2×#208C2318yes—
132C22×C3⋊C8C2312yes—
133C22×C3⋊C8C2312yes—
134#875—28no—
135#874—22no—
136#874C2314yes—
137C4×D12—28no—
138C4×D12—24no—
139C4×D12C2318yes—
140#869—22no—
141#869C2314yes—
142#835C2318yes—
143#832C2318yes—
144#809C2318yes—
145#809C2318yes—
146#808C2318yes—
147#807C2318yes—
148#803C2314yes—
149#803C2314yes—
150C4×C3⋊C8C2318yes—
151C22×#41—61no—
152C22×SD16—32no—
153C22×D8—32no—
154C22×M16—32no—
155#532—38no—
156#532—64no—
157C2×C4⋊Q8—31no—
158C2×#127—31no—
159#529—24no—
160#529C2×D318yescomplemented
161C2×#126—21no—
162C2×#125C2×D311yescharacteristic
163C2×#124—61no—
164C2×#124C2×D311yescharacteristic
165C2×#123—21no—
166C2×C22⋊Q8C2×D311yescharacteristic
167C2×#121—32no—
168C2×#121—61no—
169C2×#121C2×D311yescharacteristic
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×D4—21no—
172C2×C4×D4—32no—
173C2×C4×D4—61no—
174C2×C4×D4C2×D311yescharacteristic
175C2×#117—21no—
176C22×C42C2×D311yescharacteristic
177#501—38no—
178#496—38no—
179#490—38no—
180#489—38no—
181#469—316no—
182#468—316no—
183C4×SD16—316no—
184C4×D8—316no—
185C2×#107—32no—
186C2×#106—32no—
187C2×#103—32no—
188C2×#102—32no—
189C2×#102—32no—
190C2×#102—32no—
191C4×M16—38no—
192C2×#97—31no—
193C2×#97—31no—
194C2×C4×C8—32no—
195C24×C3—21no—
196C24×C3C2×D411yescharacteristic
197C23×D3—22no—
198C2×C3×Q8—21no—
199C2×C3×Q8—24no—
200C2×C3×Q8C2412yes—
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—24no—
204C2×C3×D4—44no—
205C2×C3×D4C2412yes—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—24no—
212C22×C3×C4—42no—
213C22×C3×C4—44no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D414yes—
220C22×C3×C4C2×D414yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2412yes—
223C22×C3×C4C2411yescharacteristic
224C2×#67—28no—
225C2×#67—48no—
226C22×Dic3—22no—
227#235—432no—
228C2×D12—24no—
229C2×D12—24no—
230C2×D12—22no—
231C2×D12—24no—
232C2×D12—24no—
233C2×D12—44no—
234C2×D12—44no—
235C2×D12C2414yes—
236C2×C4×D3—28no—
237C2×C4×D3—48no—
238C2×Dic6—24no—
239C2×Dic6—44no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2412yes—
247C3×C4⋊C4C2412yes—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C42—24no—
254C3×C42C2×D412yes—
255C3×C42C2×D414yes—
256C3×C42C2×D412yes—
257C3×C42C2×D414yes—
258C3×C42C2412yes—
259C3×C42C2412yes—
260#215—216no—
261#213—216no—
262(C2×D3)⋊C4—28no—
263(C2×D3)⋊C4—28no—
264C4⋊Dic3—22no—
265C4⋊Dic3—24no—
266C4⋊Dic3C2414yes—
267#208—216no—
268C2×C3⋊C8—24no—
269C2×C3⋊C8#4118yescomplemented
270C2×C3⋊C8C2414yes—
271C2×C3⋊C8C2414yes—
272C2×#41—68no—
273C2×#41—128no—
274C22×Q8—61no—
275C22×Q8C22×D311yescharacteristic
276C22×D4—21no—
277C22×D4—32no—
278C22×D4—62no—
279C22×D4—61no—
280C22×D4—61no—
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—62no—
284C23×C4#6711yescharacteristic
285C23×C4#6711yescharacteristic
286C23×C4C22×D311yescharacteristic
287C2×SD16—38no—
288C2×SD16—68no—
289C2×D8—38no—
290C2×D8—68no—
291C2×M16—38no—
292C2×M16—68no—
293C22×C8—32no—
294C22×C8—32no—
295C4⋊Q8—34no—
296C4⋊Q8—62no—
297#127—34no—
298#127—62no—
299#126—28no—
300#125—22no—
301#125C22×D314yes—
302#124—22no—
303#124—68no—
304#124C22×D314yes—
305#123—28no—
306C22⋊Q8—22no—
307C22⋊Q8C22×D314yes—
308#121—22no—
309#121—38no—
310#121—64no—
311#121—68no—
312#121C22×D314yes—
313C4×Q8—22no—
314C4×Q8C22×D314yes—
315C4×D4—28no—
316C4×D4—22no—
317C4×D4—38no—
318C4×D4—64no—
319C4×D4—68no—
320C4×D4C22×D314yes—
321#117—28no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—32no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—62no—
336C2×C22⋊C4—62no—
337C2×C42—21no—
338C2×C42#6711yescharacteristic
339C2×C42#6714yes—
340C2×C42#6711yescharacteristic
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D314yes—
343C2×C42C22×D311yescharacteristic
344#107—38no—
345#106—38no—
346#103—38no—
347#102—38no—
348#102—38no—
349#102—38no—
350#97—34no—
351#97—34no—
352C4×C8—38no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—42no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3C22×D412yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—24no—
363C22×D3—44no—
364C22×D3—44no—
365C22×D3—44no—
366C3×Q8—24no—
367C3×Q8—42no—
368C3×D4—24no—
369C3×D4—48no—
370C3×D4—42no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—28no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—44no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—48no—
386C2×C3×C4AGL(1,ℤ/8)14yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D412yes—
393C2×C3×C4C2×#4114yes—
394C2×C3×C4C2511yescharacteristic
395#67—416no—
396C2×Dic3—24no—
397C2×Dic3—44no—
398D12—24no—
399D12—24no—
400D12—48no—
401D12—48no—
402D12—44no—
403C4×D3—416no—
404Dic6—48no—
405C3⋊C8—24no—
406C3⋊C8C2×#4118yescomplemented
407C24—21no—
408C24—62no—
409C24C2×#6711yescharacteristic
410#41—1232no—
411C2×Q8—21no—
412C2×Q8—24no—
413C2×Q8—64no—
414C2×Q8—124no—
415C2×Q8C23×D312yes—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—34no—
420C2×D4—44no—
421C2×D4—68no—
422C2×D4—64no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—62no—
426C2×D4—64no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—128no—
430C2×D4C23×D312yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—62no—
440C22×C4—68no—
441C22×C4—128no—
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6712yes—
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6714yes—
448C22×C4C2×#6714yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C23×D312yes—
451C22×C4C23×D311yescharacteristic
452SD16—616no—
453D8—616no—
454M16—616no—
455C2×C8—38no—
456C2×C8—34no—
457C2×C8—34no—
458C2×C8—64no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—34no—
466C4⋊C4—62no—
467C4⋊C4—64no—
468C4⋊C4C23×D312yes—
469C4⋊C4C23×D312yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—68no—
476C22⋊C4—68no—
477C42—24no—
478C42C2×#6714yes—
479C42C2×#6712yes—
480C42C2×#6712yes—
481C42C2×#6714yes—
482C42C23×D312yes—
483C42C23×D312yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—44no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—84no—
492C22×C3#53212yes—
493C22×C3C2×AGL(1,ℤ/8)12yes—
494C22×C3#57512yes—
495C22×C3C23×D411yescharacteristic
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—88no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4#53212yes—
508C3×C4C2×AGL(1,ℤ/8)14yes—
509C3×C4C22×#4112yes—
510Dic3—44no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—64no—
519C23—124no—
520C23—124no—
521C23—124no—
522C23C22×#6712yes—
523C23C22×#6711yescharacteristic
524Q8—24no—
525Q8—42no—
526Q8—128no—
527D4—24no—
528D4—48no—
529D4—42no—
530D4—64no—
531D4—64no—
532D4—128no—
533D4—128no—
534D4—1216no—
535D4—124no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—64no—
552C2×C4—124no—
553C2×C4—1216no—
554C2×C4#94914yes—
555C2×C4D3×#4114yes—
556C2×C4C22×#6712yes—
557C2×C4C22×#6712yes—
558C2×C4C22×#6712yes—
559C2×C4C22×#6712yes—
560C2×C4C22×#6712yes—
561C2×C4C22×#6711yescharacteristic
562C2×C4C24×D311yescharacteristic
563C8—38no—
564C8—64no—
565C2×C3—42no—
566C2×C3—82no—
567C2×C3#317114yescomplemented
568C2×C3C2×#53211yescharacteristic
569C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
570C2×C3C2×#57511yescharacteristic
571D3—84no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—84no—
580C22—124no—
581C22—124no—
582C22—124no—
583C22—248no—
584C22C2×#94912yes—
585C22#582212yes—
586C22#583212yes—
587C22C23×#6711yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—124no—
596C4C2×#94914yes—
597C4#583212yes—
598C4C2×D3×#4112yes—
599C3C2×#317111yescharacteristic, complemented, Sylow
600C2—42no—
601C2—82no—
602C2—244no—
603C2#8313514yesdirect product factor
604C2C22×#94911yescharacteristic
605C2C2×#582211yescharacteristic
606C2C2×#583211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
hypercenterC2×#529D3
hypocenterC3C2×#3171
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×C3×#529C2semidirect
2. C2×#5477C2semidirect
3. #83135C2direct (N × H)
4. C2×#5064C2semidirect
5. C2×#5067C2semidirect
6. C2×#5058C2semidirect
7. C2×#5055C2semidirect
8. C2×#5049C2semidirect
9. C2×#5039C2semidirect
10. C2×C4×#215C2semidirect
11. C2×C4×#213C2semidirect
12. C2×C4×#208C2semidirect
16. C3×#529C22semidirect
34. #5477C22semidirect
42. #5067C22semidirect
43. #5064C22semidirect
44. #5058C22semidirect
45. #5055C22semidirect
46. #5049C22semidirect
47. #5039C22semidirect
48. C4×#215C22semidirect
49. C4×#213C22semidirect
54. C4×#208C22semidirect
59. C2×#529D3semidirect
160. #529C2×D3semidirect
269. C2×C3⋊C8#41semidirect
406. C3⋊C8C2×#41semidirect
567. C2×C3#3171semidirect
599. C3C2×#3171semidirect
603. C2#83135direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#54772
#831352
C2×#50642
C2×#50672
C2×#50582
C2×#50552
C2×#50492
C2×#50392
C2×C4×#2152
C2×C4×#2132
C2×C4×#2082
C2×#31713