Finite Group Databasebeta

← #10731735

Subgroups — #10731735

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic, complemented
2C2×C3×#393C211yesmaximal, characteristic
3C2×#4947C211yesmaximal, characteristic
4C2×#4942C212yesmaximal
5C2×Dic3×C22⋊C4C211yesmaximal, characteristic
6#72134C211yesmaximal, characteristic, complemented
7#71547C212yesmaximal
8#69424C211yesmaximal, characteristic, complemented
9#69413C212yesmaximal, complemented
10Dic3⋊#95C211yesmaximal, characteristic, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#15125—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C2×#944—28no—
21C2×#944—24no—
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C3×#393—24no—
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4947—24no—
40#4942—28no—
41Dic3×C22⋊C4—24no—
42C22×C4⋊Q8—31no—
43C2×#393—31no—
44C2×#393D311yescharacteristic
45C2×#380—32no—
46C2×C4×C22⋊C4—31no—
47#1446—31no—
48#1446—31no—
49#1442—32no—
50#1439—32no—
51C4⋊#95—32no—
52C4⋊#95—31no—
53C25×C3C2311yescharacteristic
54C22×C3×Q8—24no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—42no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—22no—
64C23×Dic3—24no—
65C23×Dic3—24no—
66C23×Dic3—22no—
67C23×Dic3Q814yescomplemented
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C22×Dic6—22no—
75C22×Dic6—24no—
76C22×Dic6—22no—
77C22×Dic6—22no—
78C22×Dic6—42no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—42no—
84C2×C3×C22⋊C4D414yes—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217—24no—
88C2×#217—24no—
89C2×#217—42no—
90C2×#217Q814yescomplemented
91#944—232no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4D418yes—
101C2×C4×Dic3—24no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3D414yes—
105C3×#95—24no—
106C3×#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—42no—
115C3⋊#95—42no—
116C23×Q8—31no—
117C23×Q8C2×D311yescharacteristic
118C24×C4—31no—
119C2×C4⋊Q8—64no—
120C2×C4⋊Q8—68no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—32no—
124C22×C22⋊C4—31no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4C2×D312yes—
127C22×C22⋊C4C2×D311yescharacteristic
128C22×C42—31no—
129C22×C42—31no—
130#393—24no—
131#393—64no—
132#380—68no—
133C4×C22⋊C4—64no—
134C2×#95—32no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—31no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95C2×D312yes—
144C2×#95C2×D311yescharacteristic
145C24×C3—21no—
146C24×C3—22no—
147C24×C3—21no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3C2×D412yes—
154C24×C3C2×Q812yes—
155C24×C3C2×Q812yes—
156C24×C3C2411yescharacteristic
157C2×C3×Q8—28no—
158C2×C3×Q8—48no—
159C2×C3×Q8—48no—
160C2×C3×Q8—44no—
161C2×C3×Q8—44no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4112yes—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—28no—
203C22×Dic3—216no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—44no—
212C22×Dic3—44no—
213C22×Dic3—48no—
214C22×Dic3C2×D414yes—
215C22×Dic3C2×D412yes—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×Q812yes—
219C22×Dic3C2×Q812yes—
220C22×Dic3C2×Q812yes—
221C22×Dic3C2×Q812yes—
222C22×Dic3#4112yes—
223C22×Dic3#4114yes—
224C22×Dic3#4112yes—
225C2×Dic6—28no—
226C2×Dic6—48no—
227C2×Dic6—48no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—48no—
232#217—28no—
233#217—48no—
234Dic3⋊C4—216no—
235Dic3⋊C4—216no—
236Dic3⋊C4—48no—
237C4×Dic3—28no—
238C4×Dic3—44no—
239C25C22×D311yescharacteristic
240C22×Q8—24no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—64no—
246C22×Q8—62no—
247C22×Q8—62no—
248C22×Q8—62no—
249C22×Q8—122no—
250C23×C4—31no—
251C23×C4—31no—
252C23×C4—34no—
253C23×C4—32no—
254C23×C4—31no—
255C23×C4—31no—
256C23×C4—32no—
257C23×C4—62no—
258C23×C4—64no—
259C23×C4—64no—
260C23×C4—62no—
261C23×C4C22×D312yes—
262C23×C4C22×D312yes—
263C23×C4C22×D311yescharacteristic
264C23×C4C22×D311yescharacteristic
265C4⋊Q8—632no—
266C2×C4⋊C4—38no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—122no—
286C2×C22⋊C4#6714yes—
287C2×C42—34no—
288C2×C42—64no—
289C2×C42—62no—
290C2×C42—64no—
291#95—24no—
292#95—22no—
293#95—64no—
294#95—62no—
295#95—64no—
296#95—62no—
297#95—64no—
298#95—64no—
299#95—62no—
300#95—122no—
301#95—122no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3C4⋊Q814yescomplemented
325C23×C3C22×D411yescharacteristic
326C23×C3C22×D412yes—
327C23×C3C22×D411yescharacteristic
328C23×C3C22×Q811yescharacteristic
329C23×C3C22×Q811yescharacteristic
330C23×C3C2×#4111yescharacteristic
331C23×C3C2×#4111yescharacteristic
332C23×C3C2×#4112yes—
333C23×C3C2×#4111yescharacteristic
334C23×C3H(5,𝔽2)12yes—
335C23×C32−1+412yes—
336C3×Q8—416no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4#12114yes—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—41no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—416no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3#12014yes—
391C2×Dic3C22⋊Q814yes—
392C2×Dic3C22⋊Q814yes—
393C2×Dic3C22⋊Q818yes—
394C2×Dic3#12414yes—
395C2×Dic3#12418yes—
396Dic6—416no—
397C24—22no—
398C24—21no—
399C24—21no—
400C24—22no—
401C24—21no—
402C24—22no—
403C24—21no—
404C24—22no—
405C24D3×Q812yes—
406C24D3×Q812yes—
407C24C2×#6712yes—
408C24C23×D311yescharacteristic
409C2×Q8—28no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—68no—
415C2×Q8—128no—
416C2×Q8—128no—
417C2×Q8—124no—
418C2×Q8—124no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—32no—
439C22×C4—34no—
440C22×C4—32no—
441C22×C4—34no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—68no—
450C22×C4—616no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—61no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—124no—
475C22×C4—128no—
476C22×C4—124no—
477C22×C4D3×D414yes—
478C22×C4D3×D412yes—
479C22×C4#23712yes—
480C22×C4C2×#6712yes—
481C22×C4C2×#6712yes—
482C4⋊C4—616no—
483C4⋊C4—616no—
484C4⋊C4—128no—
485C22⋊C4—28no—
486C22⋊C4—48no—
487C22⋊C4—68no—
488C22⋊C4—128no—
489C42—68no—
490C42—124no—
491C22×C3—24no—
492C22×C3—28no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—44no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#12111yescharacteristic
504C22×C3C2×C22⋊Q811yescharacteristic
505C22×C3C2×C22⋊Q811yescharacteristic
506C22×C3C2×C22⋊Q812yes—
507C22×C3C2×#12411yescharacteristic
508C22×C3C2×#12412yes—
509C22×C3C2×C4⋊Q811yescharacteristic
510C22×C3#53211yescharacteristic
511C22×C3#53311yescharacteristic
512C22×C3#53311yescharacteristic
513C22×C3#53511yescharacteristic
514C22×C3#53611yescharacteristic
515C22×C3#54212yes—
516C22×C3#54412yes—
517C22×C3D4212yes—
518C22×C3#54612yes—
519C22×C3#54712yes—
520C22×C3#54812yes—
521C22×C3D4×Q812yes—
522C22×C3#55412yes—
523C22×C3#55512yes—
524C22×C3#55612yes—
525C3×C4—24no—
526C3×C4—44no—
527C3×C4—42no—
528C3×C4—82no—
529Dic3—24no—
530Dic3—42no—
531Dic3—42no—
532Dic3—44no—
533Dic3—82no—
534Dic3#39318yescomplemented
535C23—24no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23#94414yescomplemented
558C23C2×D3×D412yes—
559C23C2×D3×D411yescharacteristic
560C23C2×#23711yescharacteristic
561C23#100412yes—
562C23C2×D3×Q811yescharacteristic
563C23C2×D3×Q811yescharacteristic
564C23D3×#4112yes—
565C23D3×#4111yescharacteristic
566C23D3×#4111yescharacteristic
567C23#101012yes—
568C23C22×#6711yescharacteristic
569Q8—416no—
570Q8—1216no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—34no—
581C2×C4—38no—
582C2×C4—34no—
583C2×C4—34no—
584C2×C4—38no—
585C2×C4—34no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—62no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—64no—
611C2×C4—1216no—
612C2×C4—122no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—121no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—121no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—121no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4#93914yes—
630C2×C3—44no—
631C2×C3C2×#39311yescharacteristic
632C2×C3#240811yescharacteristic
633C2×C3#240911yescharacteristic
634C2×C3#243912yes—
635C2×C3#244012yes—
636C2×C3#249112yes—
637C2×C3#249212yes—
638C2×C3#254711yescharacteristic
639C2×C3#256311yescharacteristic
640C2×C3#262811yescharacteristic
641C2×C3#285711yescharacteristic
642C22—28no—
643C22—24no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22D3×#12011yescharacteristic
654C22#559112yes—
655C22#559211yescharacteristic
656C22#562211yescharacteristic
657C22#562211yescharacteristic
658C22#562312yes—
659C22D3×C22⋊Q811yescharacteristic
660C22D3×C22⋊Q812yes—
661C22D3×C22⋊Q811yescharacteristic
662C22#563212yes—
663C22#563612yes—
664C22#566912yes—
665C22#567012yes—
666C22#567112yes—
667C22D3×#12412yes—
668C22D3×#12411yescharacteristic
669C22#567511yescharacteristic
670C22#567712yes—
671C22D4×#6712yes—
672C22C2×#93911yescharacteristic
673C22C2×#94411yescharacteristic
674C22#582412yes—
675C22#583211yescharacteristic
676C4—24no—
677C4—38no—
678C4—42no—
679C4—44no—
680C4—64no—
681C4—82no—
682C4—122no—
683C4—124no—
684C4—122no—
685C4—242no—
686C3#1512511yescharacteristic, complemented, Sylow
687C2—44no—
688C2#7678011yescharacteristic
689C2#7678212yes—
690C2#7679312yes—
691C2#7679911yescharacteristic
692C2#7681011yescharacteristic
693C2D3×#39311yescharacteristic
694C2#8258211yescharacteristic
695C2#8276312yes—
696C2#8281111yescharacteristic
697C2#8296712yes—
698C2#8297311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#393C2
prime radicalC25×C3C23
hypercenterC2×#393D3
hypocenterC3#15125
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×#944C2semidirect
6. #72134C2semidirect
8. #69424C2semidirect
9. #69413C2semidirect
10. Dic3⋊#95C2semidirect
11. Dic3⋊#95C2semidirect
67. C23×Dic3Q8semidirect
90. C2×#217Q8semidirect
324. C23×C3C4⋊Q8semidirect
534. Dic3#393semidirect
557. C23#944semidirect
686. C3#15125semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×C3×#3932
C2×#49472
C2×#49422
C2×Dic3×C22⋊C42
#721342
#715472
#694242
#694132
Dic3⋊#952
#151253