Finite Group Databasebeta

← #10827608

Subgroups — #10827608

Order 768.

1424 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
1C2×C3×#552C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic, complemented
3#82403C211yesmaximal, characteristic
4C2×#5001C211yesmaximal, characteristic, complemented
5#71743C212yesmaximal, complemented
6#71699C212yesmaximal, complemented
7#71697C211yesmaximal, characteristic, complemented
8#71633C212yesmaximal, complemented
9#71611C211yesmaximal, characteristic, complemented
10#71575C211yesmaximal, characteristic, complemented
11#71560C212yesmaximal, complemented
12#23982—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C3×#552—24no—
15C3×#552—24no—
16C2×C3×#126—21no—
17C2×C3×#125—21no—
18C2×C3×#125C2211yescharacteristic, complemented
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic, complemented
22C2×C3×C4×Q8C2211yescharacteristic, complemented
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic, complemented
27C22×#215C2212yes—
28C2×#937—21no—
29C2×#930C2212yes—
30C22×#208C2211yescharacteristic
31C2×#835C2212yescomplemented
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yescomplemented
35#5046—24no—
36#5001—24no—
37#4991—21no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41#4960—21no—
42#4955—21no—
43#4939C2212yes—
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C2×#552D311yescharacteristic, complemented
49C2×#450—31no—
50C2×#418—31no—
51#2408—31no—
52#2025—32no—
53#2000—32no—
54#1894—31no—
55#1880—32no—
56#1874—32no—
57#1870—31no—
58#1825—31no—
59C24×D3—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4D411yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×#67—22no—
66C22×D12—22no—
67C22×D12—24no—
68C22×D12—22no—
69C22×D12—21no—
70C22×D12—42no—
71C22×D12C2312yes—
72C3×#126—28no—
73C3×#125—28no—
74C3×#125—22no—
75C3×#125—22no—
76C3×#123—28no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8D414yescomplemented
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×#117—28no—
86C3×#117—22no—
87C3×#117D414yescomplemented
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×#217—21no—
109C2×#215—24no—
110C2×#215—24no—
111C2×#215—42no—
112#937—44no—
113#930—24no—
114#930—42no—
115C2×(C2×D3)⋊C4—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4—22no—
118C2×C4⋊Dic3C2312yes—
119C2×#208—22no—
120C2×#208—42no—
121C22×C3⋊C8C2312yes—
122C22×C3⋊C8C2312yes—
123#835—42no—
124C3⋊#95—21no—
125#830—22no—
126#830—42no—
127#809—24no—
128#809—24no—
129#809—24no—
130C23×D4—31no—
131C22×SD16—32no—
132C22×M16—31no—
133#552—24no—
134#552—24no—
135C2×#126—21no—
136C2×#125—21no—
137C2×#125C2×D311yescharacteristic, complemented
138C2×#123—21no—
139C2×C22⋊Q8C2×D311yescharacteristic
140C2×C22⋊Q8C2×D311yescharacteristic, complemented
141C2×#121—32no—
142C2×#120—61no—
143C2×C4×Q8C2×D311yescharacteristic, complemented
144C2×C4×D4C2×D311yescharacteristic
145C2×#117—21no—
146C2×#117C2×D311yescharacteristic
147C22×C4⋊C4C2×D311yescharacteristic, complemented
148#450—64no—
149#418—64no—
150C2×#103—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#394—61no—
157#392—61no—
158#390—32no—
159#380—31no—
160#379—61no—
161#340—32no—
162#340—32no—
163#340—32no—
164#336—32no—
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D411yescharacteristic
167C23×D3—22no—
168C23×D3—44no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—82no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8C2×D412yes—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—41no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2411yescharacteristic
200C2×#67—44no—
201C2×#67—44no—
202C2×#67—82no—
203C22×Dic3—22no—
204C2×D12—28no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—48no—
211C2×D12—44no—
212C2×D12—44no—
213C2×D12—42no—
214C2×D12#4114yes—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D412yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C42—24no—
234C3×C42—24no—
235C3×C42—21no—
236C3×C42—21no—
237#217—42no—
238#215—48no—
239(C2×D3)⋊C4—44no—
240(C2×D3)⋊C4—44no—
241(C2×D3)⋊C4—82no—
242C4⋊Dic3—22no—
243C4⋊Dic3—22no—
244#208—44no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C25—61no—
252C22×Q8C22×D311yescharacteristic
253C22×D4—32no—
254C22×D4—64no—
255C22×D4—62no—
256C22×D4—62no—
257C22×D4—62no—
258C22×D4—61no—
259C22×D4—122no—
260C22×D4#6711yescharacteristic
261C23×C4—21no—
262C23×C4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×M16—62no—
268C2×M16—122no—
269C22×C8—32no—
270C22×C8—32no—
271#126—28no—
272#125—28no—
273#125—22no—
274#125—22no—
275#123—28no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8#6714yescomplemented
280#121—64no—
281#121—122no—
282#120—124no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×D4—22no—
286C4×D4—22no—
287#117—28no—
288#117—22no—
289#117#6714yescomplemented
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4#6711yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—62no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4#6711yescharacteristic
311C2×C22⋊C4#6711yescharacteristic
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C22×D311yescharacteristic
315#103—122no—
316#102—64no—
317#102—64no—
318#102—64no—
319C22⋊C8—62no—
320C22⋊C8—122no—
321#95—61no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3#12012yes—
331C23×C3AGL(1,ℤ/8)12yes—
332C23×C3C22×D411yescharacteristic
333C22×D3—24no—
334C22×D3—48no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—88no—
342C22×D3—88no—
343C22×D3—84no—
344C22×D3—84no—
345C3×Q8—24no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×D4—44no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—48no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4#12012yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4112yes—
378C2×C3×C4H(5,𝔽2)12yes—
379#67—88no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383D12—24no—
384D12—24no—
385D12—48no—
386D12—42no—
387D12—42no—
388C3⋊C8—42no—
389C3⋊C8—42no—
390C24—62no—
391C24—124no—
392C24—122no—
393C24—122no—
394C24—122no—
395C24—242no—
396C24C2×#6711yescharacteristic
397C24C2×#6711yescharacteristic
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8C2×#6712yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—34no—
405C2×D4—41no—
406C2×D4—68no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—128no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—122no—
418C2×D4—242no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—62no—
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6712yes—
438C22×C4C2×#6712yes—
439C22×C4C23×D311yescharacteristic
440SD16—128no—
441M16—124no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4C2×#6712yes—
461C4⋊C4C2×#6712yes—
462C4⋊C4C2×#6712yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—124no—
469C22⋊C4—124no—
470C22⋊C4—122no—
471C22⋊C4—242no—
472C42—24no—
473C42—24no—
474C42—21no—
475C42—21no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3C2×#12011yescharacteristic
487C22×C3#54612yes—
488C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57712yes—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—816no—
495C2×D3—82no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—22no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—42no—
507C3×C4#52511yescharacteristic
508C3×C4#53511yescharacteristic
509C3×C4#54612yes—
510Dic3—82no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—21no—
519C23—64no—
520C23—128no—
521C23—122no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—244no—
528C23—244no—
529C23—248no—
530C23—248no—
531C23#94912yes—
532C23#95312yes—
533C23C22×#6711yescharacteristic
534Q8—24no—
535Q8—41no—
536Q8—41no—
537D4—44no—
538D4—64no—
539D4—64no—
540D4—128no—
541D4—122no—
542D4—122no—
543D4—248no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4#95312yes—
573C2×C4D3×#4112yes—
574C2×C4#100912yes—
575C2×C4C22×#6711yescharacteristic
576C2×C4C22×#6711yescharacteristic
577C8—122no—
578C8—122no—
579C2×C3—42no—
580C2×C3—42no—
581C2×C3#240411yescharacteristic
582C2×C3#300611yescharacteristic
583C2×C3#301011yescharacteristic
584C2×C3#331012yes—
585C2×C3#332312yes—
586D3—84no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—22no—
592C22—41no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—124no—
598C22—124no—
599C22—124no—
600C22—2416no—
601C22—242no—
602C22—242no—
603C22—242no—
604C22—242no—
605C22C2×#94911yescharacteristic
606C22#582211yescharacteristic
607C22#582912yes—
608C22#583712yes—
609C22C2×#95311yescharacteristic
610C4—22no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—242no—
619C4#582811yescharacteristic
620C4#582912yes—
621C4#583011yescharacteristic
622C3#2398211yescharacteristic, complemented, Sylow
623C2—42no—
624C2—42no—
625C2—244no—
626C2#8359912yes—
627C2#8360112yes—
628C2#8408211yescharacteristic
629C2#8409611yescharacteristic
630C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#552C2
socleC23×C3C22×D4
prime radical#82403C2
hypercenterC2×#552D3
hypocenterC3#23982
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×C3×#552C2semidirect
2. C2×#5046C2semidirect
4. C2×#5001C2semidirect
5. #71743C2semidirect
6. #71699C2semidirect
7. #71697C2semidirect
8. #71633C2semidirect
9. #71611C2semidirect
10. #71575C2semidirect
11. #71560C2semidirect
18. C2×C3×#125C22semidirect
21. C2×C3×C22⋊Q8C22semidirect
22. C2×C3×C4×Q8C22semidirect
26. C22×C3×C4⋊C4C22semidirect
31. C2×#835C22semidirect
34. C2×#830C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
48. C2×#552D3semidirect
80. C3×C22⋊Q8D4semidirect
87. C3×#117D4semidirect
137. C2×#125C2×D3semidirect
140. C2×C22⋊Q8C2×D3semidirect
143. C2×C4×Q8C2×D3semidirect
147. C22×C4⋊C4C2×D3semidirect
279. C22⋊Q8#67semidirect
289. #117#67semidirect
622. C3#23982semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×#50462
#824032
C2×#50012
#717432
#716992
#716972
#716332
#716112
#715752
#715602
#239823