Finite Group Databasebeta

← C2×#83292 #11664946

Subgroups — C2×#83292 #11664946

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×#532C211yesmaximal, characteristic, complemented
2C2×#5479C211yesmaximal, characteristic, complemented
3#83292C2116yesmaximal, direct product factor
4C2×#5081C211yesmaximal, characteristic, complemented
5C2×#5089C211yesmaximal, characteristic, complemented
6C2×#5074C211yesmaximal, characteristic, complemented
7C2×#5071C211yesmaximal, characteristic, complemented
8C2×#5039C212yesmaximal, complemented
9C2×#5042C212yesmaximal, complemented
10C2×C4×#214C212yesmaximal, complemented
11C2×C4×#213C212yesmaximal, complemented
12C2×C4×#208C211yesmaximal, characteristic, complemented
13C2×#3171—31nomaximal, Sylow
14C22×C3×#41—21no—
15C3×#532—24no—
16C3×#532C2218yescomplemented
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124—21no—
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2212yes—
24C22×C3×C42C2211yescharacteristic
25C22×#214C2212yes—
26C22×#213C2212yes—
27C2×#930C2211yescharacteristic
28C2×#929—21no—
29C2×C4×#67—21no—
30C2×#924C2211yescharacteristic
31C22×#208C2212yes—
32#5479—24no—
33#5479C2218yescomplemented
34C2×#876—21no—
35C2×#875C2211yescharacteristic
36C2×C4×D12C2211yescharacteristic
37C2×#872—21no—
38C2×#870C2211yescharacteristic
39C2×C4×Dic6C2211yescharacteristic
40C2×#832C2212yes—
41C2×#832C2212yes—
42#5089C2218yescomplemented
43#5081C2218yescomplemented
44#5074C2218yescomplemented
45#5071C2218yescomplemented
46#5042C22116yescomplemented
47#5039C22116yescomplemented
48C4×#214C22116yescomplemented
49C4×#213C22116yescomplemented
50C2×#810C2212yes—
51C2×#809C2212yes—
52C2×#808C2212yes—
53C2×#807C2212yes—
54C4×#208C2218yescomplemented
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2212yes—
58C2×#532D311yescharacteristic, complemented
59C2×#529—31no—
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—
70C2×C3×#41—28no—
71C2×C3×#41—48no—
72C22×C3×Q8—21no—
73C22×C3×D4—22no—
74C22×C3×D4—21no—
75C22×C3×D4—21no—
76C22×C3×D4C2312yes—
77C23×C3×C4—22no—
78C23×C3×C4D411yescharacteristic
79C23×C3×C4D411yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×Dic6C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8C2314yes—
87C3×#127—22no—
88C3×#127C2314yes—
89C3×#124—28no—
90C3×#121—28no—
91C3×#121—24no—
92C3×#121C2318yes—
93C3×C4×D4—28no—
94C3×C4×D4—24no—
95C3×C4×D4C2318yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42D414yes—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2314yes—
106C2×#217—21no—
107C2×#214—28no—
108C2×#214C2318yes—
109C2×#213—28no—
110C2×#213C2318yes—
111#930—22no—
112#930C2314yes—
113#929—28no—
114C4×#67—28no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—21no—
118C2×(C2×D3)⋊C4—21no—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121#924—22no—
122#924C2314yes—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×C4×Dic3—21no—
128C2×#208—28no—
129C2×#208C2318yes—
130C22×C3⋊C8C2312yes—
131C22×C3⋊C8C2312yes—
132#876—28no—
133#875—22no—
134#875C2314yes—
135C4×D12—22no—
136C4×D12C2314yes—
137#872—28no—
138#870—22no—
139#870C2314yes—
140C4×Dic6—22no—
141C4×Dic6C2314yes—
142#832C2318yes—
143#832C2318yes—
144#810C2318yes—
145#809C2318yes—
146#808C2318yes—
147#807C2318yes—
148#803C2314yes—
149#803C2314yes—
150C4×C3⋊C8C2318yes—
151C22×#41—21no—
152C22×SD16—32no—
153C22×D8—32no—
154C22×M16—32no—
155#532—24no—
156#532C2×D318yescomplemented
157C2×C4⋊Q8C2×D311yescharacteristic
158C2×#127C2×D311yescharacteristic
159#529—38no—
160#529—64no—
161C2×#126—61no—
162C2×#125—31no—
163C2×#124—21no—
164C2×#124—31no—
165C2×#123—61no—
166C2×C22⋊Q8—31no—
167C2×#121—21no—
168C2×#121—31no—
169C2×#121C2×D312yes—
170C2×C4×Q8—31no—
171C2×C4×D4—21no—
172C2×C4×D4—31no—
173C2×C4×D4—61no—
174C2×C4×D4C2×D312yes—
175C2×#117—61no—
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—22no—
196C24×C3C2×D411yescharacteristic
197C23×D3—21no—
198C3×#41—432no—
199C2×C3×Q8—24no—
200C2×C3×Q8—44no—
201C2×C3×D4—28no—
202C2×C3×D4—22no—
203C2×C3×D4—24no—
204C2×C3×D4—24no—
205C2×C3×D4—24no—
206C2×C3×D4—24no—
207C2×C3×D4—44no—
208C2×C3×D4—44no—
209C2×C3×D4—48no—
210C2×C3×D4C2414yes—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—22no—
214C22×C3×C4—28no—
215C22×C3×C4—42no—
216C22×C3×C4—48no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D414yes—
223C22×C3×C4C2×D414yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C2×#67—24no—
228C2×#67—44no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C2×D12—24no—
233C2×D12—21no—
234C2×D12C2412yes—
235C2×C4×D3—24no—
236C2×C4×D3—44no—
237C2×Dic6—24no—
238C2×Dic6—21no—
239C2×Dic6C2412yes—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4C2414yes—
243C3×C22⋊C4—28no—
244C3×C22⋊C4—28no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247C3×C42C2×D414yes—
248C3×C42C2×D414yes—
249C3×C42C2412yes—
250C3×C42C2412yes—
251#217—24no—
252#214—216no—
253#213—216no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2412yes—
261C4⋊Dic3C2412yes—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264Dic3⋊C4—24no—
265Dic3⋊C4—24no—
266C4×Dic3—24no—
267#208—216no—
268C2×C3⋊C8—24no—
269C2×C3⋊C8#4118yescomplemented
270C2×C3⋊C8C2414yes—
271C2×C3⋊C8C2414yes—
272C2×#41—28no—
273C2×#41—48no—
274C22×Q8—21no—
275C22×Q8—31no—
276C22×D4—22no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—31no—
280C22×D4—61no—
281C22×D4C22×D312yes—
282C23×C4—22no—
283C23×C4—61no—
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—22no—
296C4⋊Q8C22×D314yes—
297#127—22no—
298#127C22×D314yes—
299#126—68no—
300#125—34no—
301#125—62no—
302#124—28no—
303#124—34no—
304#124—62no—
305#123—68no—
306C22⋊Q8—34no—
307C22⋊Q8—62no—
308#121—28no—
309#121—24no—
310#121—34no—
311#121—62no—
312#121C22×D318yes—
313C4×Q8—34no—
314C4×Q8—62no—
315C4×D4—28no—
316C4×D4—24no—
317C4×D4—34no—
318C4×D4—68no—
319C4×D4—62no—
320C4×D4C22×D318yes—
321#117—68no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4C22×D312yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C42—61no—
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—24no—
357C23×C3—44no—
358C23×C3—44no—
359C23×C3—44no—
360C23×C3C22×D412yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—42no—
364C22×D3—42no—
365C22×D3—42no—
366C3×Q8—48no—
367C3×D4—24no—
368C3×D4—24no—
369C3×D4—48no—
370C3×D4—48no—
371C3×D4—416no—
372C3×D4—44no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—28no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—44no—
382C2×C3×C4—44no—
383C2×C3×C4—416no—
384C2×C3×C4AGL(1,ℤ/8)14yes—
385C2×C3×C4C22×D412yes—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4114yes—
392C2×C3×C4C2511yescharacteristic
393#67—48no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400D12—24no—
401D12—42no—
402C4×D3—48no—
403Dic6—24no—
404Dic6—42no—
405C3⋊C8—24no—
406C3⋊C8C2×#4118yescomplemented
407C24—22no—
408C24—61no—
409C24C2×#6711yescharacteristic
410#41—432no—
411C2×Q8—24no—
412C2×Q8—32no—
413C2×Q8—44no—
414C2×Q8—61no—
415C2×Q8—64no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—32no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—64no—
427C2×D4—64no—
428C2×D4—61no—
429C2×D4—124no—
430C2×D4C23×D314yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—64no—
441C22×C4—124no—
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—22no—
461C4⋊C4—32no—
462C4⋊C4—32no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4C23×D314yes—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—64no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C42—64no—
478C42C2×#6714yes—
479C42C2×#6712yes—
480C42C2×#6714yes—
481C42C2×#6712yes—
482C42C23×D312yes—
483C42C23×D312yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—88no—
492C22×C3#53212yes—
493C22×C3C2×AGL(1,ℤ/8)12yes—
494C22×C3#57512yes—
495C22×C3C23×D411yescharacteristic
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—84no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—44no—
505C3×C4#53212yes—
506C3×C4C2×AGL(1,ℤ/8)14yes—
507C3×C4C22×#4112yes—
508Dic3—42no—
509Dic3—42no—
510Dic3—42no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—62no—
519C23—122no—
520C23—122no—
521C23—122no—
522C23C22×#6712yes—
523C23C22×#6711yescharacteristic
524Q8—48no—
525Q8—64no—
526Q8—122no—
527D4—24no—
528D4—24no—
529D4—48no—
530D4—48no—
531D4—416no—
532D4—44no—
533D4—64no—
534D4—128no—
535D4—122no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—416no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—128no—
554C2×C4#93214yes—
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—84no—
567C2×C3#317114yescomplemented
568C2×C3C2×#53211yescharacteristic
569C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
570C2×C3C2×#57511yescharacteristic
571D3—82no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—88no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—244no—
584C22C2×#93212yes—
585C22#582212yes—
586C22#583212yes—
587C22C23×#6711yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—44no—
593C4—122no—
594C4—122no—
595C4—122no—
596C4C2×#93214yes—
597C4#583212yes—
598C4C2×D3×#4112yes—
599C3C2×#317111yescharacteristic, complemented, Sylow
600C2—42no—
601C2—84no—
602C2—242no—
603C2#8329214yesdirect product factor
604C2C22×#93211yescharacteristic
605C2C2×#582211yescharacteristic
606C2C2×#583211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3C2×#3171
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×#5479C2semidirect
3. #83292C2direct (N × H)
4. C2×#5081C2semidirect
5. C2×#5089C2semidirect
6. C2×#5074C2semidirect
7. C2×#5071C2semidirect
8. C2×#5039C2semidirect
9. C2×#5042C2semidirect
10. C2×C4×#214C2semidirect
11. C2×C4×#213C2semidirect
12. C2×C4×#208C2semidirect
16. C3×#532C22semidirect
33. #5479C22semidirect
42. #5089C22semidirect
43. #5081C22semidirect
44. #5074C22semidirect
45. #5071C22semidirect
46. #5042C22semidirect
47. #5039C22semidirect
48. C4×#214C22semidirect
49. C4×#213C22semidirect
54. C4×#208C22semidirect
58. C2×#532D3semidirect
156. #532C2×D3semidirect
269. C2×C3⋊C8#41semidirect
406. C3⋊C8C2×#41semidirect
567. C2×C3#3171semidirect
599. C3C2×#3171semidirect
603. C2#83292direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#54792
#832922
C2×#50812
C2×#50892
C2×#50742
C2×#50712
C2×#50392
C2×#50422
C2×C4×#2142
C2×C4×#2132
C2×C4×#2082
C2×#31713