Finite Group Databasebeta

← C2×#73837 #11429764

Subgroups — C2×#73837 #11429764

Order 768.

5006 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#953C212yesmaximal, complemented
2C22×#885C212yesmaximal, complemented
3C2×C3×#379C211yesmaximal, characteristic, complemented
4C2×#4959C212yesmaximal, complemented
5C2×#4960C211yesmaximal, characteristic, complemented
6C2×#4960C212yesmaximal, complemented
7C2×#4947C211yesmaximal, characteristic, complemented
8C2×#4957C212yesmaximal, complemented
9C2×#4945C212yesmaximal, complemented
10#73837C2116yesmaximal, direct product factor
11C2×#2670—31nomaximal, Sylow
12C26×C3C2211yescharacteristic
13C23×#67C2212yes—
14C23×#67C2212yes—
15C23×#67C2212yes—
16C23×D12C2211yescharacteristic, complemented
17C23×Dic6C2211yescharacteristic, complemented
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C2×#953—216no—
23C2×#953C22116yescomplemented
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×(C2×D3)⋊C4C2212yescomplemented
28C22×(C2×D3)⋊C4C2212yescomplemented
29C22×(C2×D3)⋊C4C2212yescomplemented
30C22×Dic3⋊C4C2212yescomplemented
31C22×C4×Dic3C2212yescomplemented
32C2×#885—216no—
33C2×#885C22116yescomplemented
34C3×#379C2218yescomplemented
35C2×C3⋊#95C2211yescharacteristic, complemented
36C2×C3⋊#95C2212yescomplemented
37C2×C3⋊#95C2211yescharacteristic, complemented
38#4960C2218yescomplemented
39#4960C22116yescomplemented
40#4959C22116yescomplemented
41#4957C22116yescomplemented
42#4947C2218yescomplemented
43#4945C22116yescomplemented
44C22×#124—32no—
45C22×#120—32no—
46#2670—316no—
47C2×#394—32no—
48C2×#393—31no—
49C2×#392—31no—
50C2×#392—32no—
51C2×#390—32no—
52C2×#388—32no—
53C2×#379D311yescharacteristic, complemented
54C25×C3—28no—
55C25×C3—24no—
56C25×C3—42no—
57C25×C3—44no—
58C25×C3—42no—
59C25×C3C2312yes—
60C25×C3C2311yescharacteristic
61C25×C3C2314yes—
62C24×D3C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C22×#67—28no—
67C22×#67—28no—
68C22×#67—28no—
69C22×#67—28no—
70C22×#67—28no—
71C22×#67—44no—
72C22×#67—44no—
73C22×#67—44no—
74C22×#67—44no—
75C22×#67C2318yes—
76C22×#67C2318yes—
77C22×#67C2318yes—
78C23×Dic3C2312yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C22×D12—44no—
82C22×D12—42no—
83C22×D12C2314yescomplemented
84C22×Dic6—28no—
85C22×Dic6—42no—
86C22×Dic6C2314yescomplemented
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4—44no—
89C2×C3×C22⋊C4—42no—
90C2×C3×C22⋊C4—42no—
91C2×C3×C22⋊C4C2318yes—
92C2×C3×C22⋊C4C2318yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2314yes—
95#953—264no—
96C2×#217—28no—
97C2×#217—28no—
98C2×#217D4116yescomplemented
99C2×#217C2318yes—
100C2×#217C2318yes—
101C2×#217C2318yes—
102C2×(C2×D3)⋊C4—28no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4—44no—
105C2×(C2×D3)⋊C4C2318yescomplemented
106C2×(C2×D3)⋊C4C2318yescomplemented
107C2×(C2×D3)⋊C4C2318yescomplemented
108C2×Dic3⋊C4—44no—
109C2×Dic3⋊C4C2318yescomplemented
110C2×C4×Dic3—28no—
111C2×C4×Dic3C2318yescomplemented
112#885—264no—
113C3⋊#95C2314yescomplemented
114C3⋊#95C2318yescomplemented
115C3⋊#95C2314yescomplemented
116C26C2×D311yescharacteristic
117C23×Q8—31no—
118C23×D4—32no—
119C23×D4—32no—
120C23×D4—32no—
121C23×D4—31no—
122C2×#124—316no—
123C2×#124—616no—
124C2×#120—316no—
125C2×#120—616no—
126C22×C4⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C22⋊C4C2×D312yes—
134C22×C22⋊C4C2×D311yescharacteristic
135C22×C22⋊C4C2×D312yes—
136C22×C22⋊C4C2×D311yescharacteristic
137C22×C42—32no—
138#394—316no—
139#393—38no—
140#392—38no—
141#392—316no—
142#390—316no—
143#388—316no—
144#379C2×D318yescomplemented
145C2×#95—32no—
146C2×#95—31no—
147C2×#95—31no—
148C24×C3—24no—
149C24×C3—24no—
150C24×C3—24no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—24no—
157C24×C3—24no—
158C24×C3—216no—
159C24×C3—28no—
160C24×C3—44no—
161C24×C3—48no—
162C24×C3—44no—
163C24×C3—416no—
164C24×C3—432no—
165C24×C3—432no—
166C24×C3—416no—
167C24×C3—416no—
168C24×C3C2×D418yes—
169C24×C3#4114yes—
170C24×C3C2414yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—24no—
176C23×D3—24no—
177C23×D3—24no—
178C23×D3—24no—
179C23×D3—24no—
180C23×D3C2414yes—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4C2412yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2414yes—
191C2×#67—216no—
192C2×#67—216no—
193C2×#67—216no—
194C2×#67—216no—
195C2×#67—216no—
196C2×#67—48no—
197C2×#67—48no—
198C2×#67—48no—
199C2×#67—48no—
200C2×#67—432no—
201C2×#67—432no—
202C2×#67—432no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3C2×D418yes—
211C22×Dic3C2×D418yes—
212C22×Dic3C2414yes—
213C22×Dic3C2414yes—
214C22×Dic3C2414yes—
215C2×D12—48no—
216C2×D12—44no—
217C2×D12—416no—
218C2×Dic6—216no—
219C2×Dic6—44no—
220C2×Dic6—416no—
221C3×C22⋊C4—216no—
222C3×C22⋊C4—48no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—44no—
225#217—216no—
226#217—216no—
227#217C2×D4132yescomplemented
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—216no—
230(C2×D3)⋊C4—48no—
231Dic3⋊C4—48no—
232C4×Dic3—216no—
233C25—24no—
234C25—28no—
235C25—32no—
236C25—44no—
237C25—42no—
238C25—42no—
239C25C22×D311yescharacteristic
240C25C22×D312yes—
241C25C22×D314yes—
242C22×Q8—34no—
243C22×Q8—68no—
244C22×Q8—122no—
245C22×D4—38no—
246C22×D4—38no—
247C22×D4—38no—
248C22×D4—34no—
249C22×D4—68no—
250C22×D4—68no—
251C22×D4—68no—
252C22×D4—68no—
253C22×D4—68no—
254C22×D4—124no—
255C22×D4—122no—
256C22×D4—124no—
257C22×D4—124no—
258C22×D4—124no—
259C22×D4—124no—
260C23×C4—32no—
261C23×C4—32no—
262C23×C4—32no—
263C23×C4C22×D312yes—
264C23×C4C22×D311yescharacteristic
265C23×C4C22×D311yescharacteristic
266#124—664no—
267#120—664no—
268C2×C4⋊C4—38no—
269C2×C4⋊C4—124no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—316no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—68no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—68no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—124no—
286C2×C22⋊C4C22×D318yes—
287C2×C22⋊C4C22×D318yes—
288C2×C22⋊C4C22×D314yes—
289C2×C22⋊C4C22×D314yes—
290C2×C42—38no—
291C2×C42—68no—
292#95—34no—
293#95—38no—
294#95—34no—
295C23×C3—216no—
296C23×C3—28no—
297C23×C3—216no—
298C23×C3—216no—
299C23×C3—216no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—416no—
311C23×C3—432no—
312C23×C3—432no—
313C23×C3—416no—
314C23×C3—4128no—
315C23×C3—416no—
316C23×C3—48no—
317C23×C3—48no—
318C23×C3—48no—
319C23×C3—44no—
320C23×C3—48no—
321C23×C3—48no—
322C23×C3—44no—
323C23×C3C22×D418yes—
324C23×C3C22×D414yes—
325C23×C3C2×#4114yes—
326C23×C3C2×#4112yes—
327C23×C3H(5,𝔽2)12yes—
328C23×C3H(5,𝔽2)14yes—
329C23×C3H(5,𝔽2)12yes—
330C23×C3C2511yescharacteristic
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—24no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—48no—
339C22×D3—48no—
340C22×D3—48no—
341C22×D3—48no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—48no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—48no—
353C2×C3×C4—44no—
354C2×C3×C4—44no—
355#67—432no—
356#67—432no—
357#67—432no—
358C2×Dic3—216no—
359C2×Dic3—216no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—48no—
368C2×Dic3C22×D418yes—
369C2×Dic3C22×D418yes—
370D12—416no—
371Dic6—416no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—28no—
383C24—216no—
384C24—34no—
385C24—44no—
386C24—48no—
387C24—44no—
388C24—432no—
389C24—416no—
390C24—416no—
391C24—432no—
392C24—416no—
393C24—64no—
394C24—64no—
395C24—64no—
396C24—64no—
397C24—64no—
398C24—64no—
399C24—64no—
400C24#23514yes—
401C24D3×D418yes—
402C24C23×D312yes—
403C24C23×D314yes—
404C24C23×D311yescharacteristic
405C2×Q8—616no—
406C2×Q8—124no—
407C2×Q8—1216no—
408C2×D4—616no—
409C2×D4—616no—
410C2×D4—616no—
411C2×D4—616no—
412C2×D4—616no—
413C2×D4—128no—
414C2×D4—124no—
415C2×D4—1216no—
416C2×D4—128no—
417C2×D4—128no—
418C2×D4—128no—
419C2×D4—128no—
420C2×D4—1232no—
421C2×D4—1232no—
422C2×D4—1232no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—38no—
431C22×C4—38no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—34no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4C23×D312yes—
443C22×C4C23×D312yes—
444C22×C4C23×D314yes—
445C4⋊C4—128no—
446C22⋊C4—216no—
447C22⋊C4—332no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—616no—
452C22⋊C4—616no—
453C22⋊C4—616no—
454C22⋊C4—616no—
455C22⋊C4—128no—
456C42—616no—
457C22×C3—28no—
458C22×C3—24no—
459C22×C3—28no—
460C22×C3—28no—
461C22×C3—28no—
462C22×C3—464no—
463C22×C3—432no—
464C22×C3—416no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3#53518yes—
473C22×C3#53814yes—
474C22×C3D4214yes—
475C22×C3#54618yes—
476C22×C3H(3,𝔽4)14yes—
477C22×C3C23×D412yes—
478C22×C3C22×#4111yescharacteristic
479C22×C3C2×H(5,𝔽2)11yescharacteristic
480C22×C3C2×H(5,𝔽2)12yes—
481C22×C3C2×H(5,𝔽2)11yescharacteristic
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—44no—
485C2×D3—44no—
486C2×D3—44no—
487C2×D3—44no—
488C2×D3—44no—
489C2×D3—816no—
490C3×C4—44no—
491C3×C4—42no—
492C3×C4—42no—
493Dic3—28no—
494Dic3—28no—
495Dic3—44no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—216no—
507C23—216no—
508C23—28no—
509C23—216no—
510C23—216no—
511C23—416no—
512C23—432no—
513C23—432no—
514C23—416no—
515C23—4128no—
516C23—416no—
517C23—48no—
518C23—48no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—44no—
523C23—48no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—64no—
530C23—64no—
531C23—128no—
532C23—128no—
533C23—128no—
534C23—128no—
535C23—128no—
536C23—128no—
537C23—128no—
538C23C2×#23514yes—
539C23C2×#23512yes—
540C23C2×D3×D418yes—
541C23C2×D3×D414yes—
542C23#100414yes—
543C23#100412yes—
544C23#100412yes—
545C23C24×D311yescharacteristic
546Q8—1216no—
547D4—1216no—
548D4—1232no—
549D4—1232no—
550D4—1232no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—38no—
559C2×C4—38no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—616no—
568C2×C4—616no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—128no—
573C2×C3—48no—
574C2×C3—44no—
575C2×C3#267018yescomplemented
576C2×C3C2×#53512yes—
577C2×C3C2×#53811yescharacteristic
578C2×C3C2×D4211yescharacteristic
579C2×C3C2×#54612yes—
580C2×C3C2×H(3,𝔽4)11yescharacteristic
581D3—84no—
582C22—28no—
583C22—28no—
584C22—24no—
585C22—28no—
586C22—28no—
587C22—464no—
588C22—416no—
589C22—432no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—124no—
598C22—124no—
599C22—124no—
600C22—124no—
601C22—124no—
602C22—124no—
603C22—124no—
604C22—2416no—
605C22#549018yes—
606C22#549414yes—
607C22#559014yes—
608C22#559218yes—
609C22#559414yes—
610C22C22×#23511yescharacteristic
611C22C22×D3×D412yes—
612C22C2×#100411yescharacteristic
613C22C2×#100412yes—
614C22C2×#100411yescharacteristic
615C4—44no—
616C4—42no—
617C4—42no—
618C4—68no—
619C4—68no—
620C4—124no—
621C3C2×#267011yescharacteristic, complemented, Sylow
622C2—48no—
623C2—44no—
624C2—244no—
625C2#7383718yesdirect product factor
626C2C2×#549411yescharacteristic
627C2C2×#549012yes—
628C2C2×#559212yes—
629C2C2×#559411yescharacteristic
630C2C2×#559011yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#379C2
socleC24×C3C24
hypercenterC2×#379D3
hypocenterC3C2×#2670
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#953C2semidirect
2. C22×#885C2semidirect
3. C2×C3×#379C2semidirect
4. C2×#4959C2semidirect
5. C2×#4960C2semidirect
6. C2×#4960C2semidirect
7. C2×#4947C2semidirect
8. C2×#4957C2semidirect
9. C2×#4945C2semidirect
10. #73837C2direct (N × H)
16. C23×D12C22semidirect
17. C23×Dic6C22semidirect
23. C2×#953C22semidirect
27. C22×(C2×D3)⋊C4C22semidirect
28. C22×(C2×D3)⋊C4C22semidirect
29. C22×(C2×D3)⋊C4C22semidirect
30. C22×Dic3⋊C4C22semidirect
31. C22×C4×Dic3C22semidirect
33. C2×#885C22semidirect
34. C3×#379C22semidirect
35. C2×C3⋊#95C22semidirect
36. C2×C3⋊#95C22semidirect
37. C2×C3⋊#95C22semidirect
38. #4960C22semidirect
39. #4960C22semidirect
40. #4959C22semidirect
41. #4957C22semidirect
42. #4947C22semidirect
43. #4945C22semidirect
53. C2×#379D3semidirect
83. C22×D12C23semidirect
86. C22×Dic6C23semidirect
98. C2×#217D4semidirect
105. C2×(C2×D3)⋊C4C23semidirect
106. C2×(C2×D3)⋊C4C23semidirect
107. C2×(C2×D3)⋊C4C23semidirect
109. C2×Dic3⋊C4C23semidirect
111. C2×C4×Dic3C23semidirect
113. C3⋊#95C23semidirect
114. C3⋊#95C23semidirect
115. C3⋊#95C23semidirect
144. #379C2×D3semidirect
227. #217C2×D4semidirect
575. C2×C3#2670semidirect
621. C3C2×#2670semidirect
625. C2#73837direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9532
C22×#8852
C2×C3×#3792
C2×#49592
C2×#49602
C2×#49472
C2×#49572
C2×#49452
#738372
C2×#26703