Finite Group Databasebeta

← #10731739

Subgroups — #10731739

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C2×#4947C212yesmaximal
3C2×#4942C212yesmaximal
4C2×Dic3×C22⋊C4C211yesmaximal, characteristic
5C3×#1439C211yesmaximal, characteristic
6#71567C211yesmaximal, characteristic, complemented
7#71546C211yesmaximal, characteristic
8#69424C212yesmaximal, complemented
9#69412C211yesmaximal, characteristic, complemented
10Dic3⋊#95C212yesmaximal, complemented
11Dic3⋊#95C211yesmaximal, characteristic, complemented
12#15125—31nomaximal, Sylow
13C24×Dic3C2211yescharacteristic
14C23×Dic6C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×#888—28no—
26C2×#888—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4947—28no—
39#4942—28no—
40Dic3×C22⋊C4—24no—
41C22×C4⋊Q8—31no—
42C2×#393—32no—
43C2×#380—32no—
44C2×C4×C22⋊C4—31no—
45#1446—32no—
46#1442—31no—
47#1442—31no—
48#1439—31no—
49#1439D311yescharacteristic
50C4⋊#95—32no—
51C4⋊#95—31no—
52C25×C3C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—22no—
58C23×Dic3—24no—
59C23×Dic3—24no—
60C23×Dic3—22no—
61C23×Dic3Q814yescomplemented
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C22×Dic6—24no—
69C22×Dic6—24no—
70C22×Dic6—24no—
71C22×Dic6—24no—
72C22×Dic6—24no—
73C22×Dic6—42no—
74C22×Dic6—42no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—42no—
82C2×C3×C22⋊C4Q814yescomplemented
83C2×#217—22no—
84C2×#217—22no—
85C2×#217—24no—
86C2×#217—24no—
87C2×#217—42no—
88C2×#217D414yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4D418yes—
97C2×C4×Dic3—24no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—24no—
100C2×C4×Dic3D414yes—
101#888—232no—
102C3×#95—22no—
103C3×#95—42no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—42no—
113C23×Q8—32no—
114C24×C4—31no—
115C2×C4⋊Q8—64no—
116C2×C4⋊Q8—68no—
117C22×C4⋊C4—32no—
118C22×C4⋊C4—31no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4C2×D311yescharacteristic
121C22×C22⋊C4—31no—
122C22×C22⋊C4—32no—
123C22×C22⋊C4C2×D312yes—
124C22×C22⋊C4C2×D311yescharacteristic
125C22×C42—31no—
126C22×C42—31no—
127#393—68no—
128#380—68no—
129C4×C22⋊C4—64no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—31no—
138C2×#95—32no—
139C2×#95C2×D311yescharacteristic
140C2×#95C2×D312yes—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3C2×D412yes—
150C24×C3C2×Q812yes—
151C24×C3C2×Q812yes—
152C24×C3C2411yescharacteristic
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×Q812yes—
175C22×C3×C4C2×Q812yes—
176C22×C3×C4#4112yes—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—28no—
194C22×Dic3—216no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—44no—
203C22×Dic3—44no—
204C22×Dic3—48no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D414yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×Q812yes—
212C22×Dic3C2×Q812yes—
213C22×Dic3#4112yes—
214C22×Dic3#4114yes—
215C22×Dic3#4112yes—
216C2×Dic6—216no—
217C2×Dic6—48no—
218C2×Dic6—48no—
219C2×Dic6—48no—
220C2×Dic6—48no—
221C2×Dic6—48no—
222C2×Dic6—48no—
223C3×C4⋊C4—28no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—48no—
226#217—28no—
227#217—48no—
228C4⋊Dic3—28no—
229Dic3⋊C4—216no—
230Dic3⋊C4—48no—
231C4×Dic3—28no—
232C4×Dic3—44no—
233C25C22×D311yescharacteristic
234C22×Q8—64no—
235C22×Q8—64no—
236C22×Q8—64no—
237C22×Q8—64no—
238C22×Q8—64no—
239C22×Q8—122no—
240C22×Q8—122no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—34no—
244C23×C4—32no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—32no—
248C23×C4—62no—
249C23×C4—64no—
250C23×C4—64no—
251C23×C4—62no—
252C23×C4C22×D312yes—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256C4⋊Q8—632no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—64no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4Dic614yescomplemented
279C2×C42—34no—
280C2×C42—64no—
281C2×C42—62no—
282C2×C42—64no—
283#95—22no—
284#95—42no—
285#95—62no—
286#95—64no—
287#95—64no—
288#95—64no—
289#95—64no—
290#95—62no—
291#95—64no—
292#95—62no—
293#95—122no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C4⋊Q814yescomplemented
317C23×C3C22×D411yescharacteristic
318C23×C3C22×D412yes—
319C23×C3C22×D411yescharacteristic
320C23×C3C22×Q811yescharacteristic
321C23×C3C22×Q811yescharacteristic
322C23×C3C2×#4111yescharacteristic
323C23×C3C2×#4111yescharacteristic
324C23×C3C2×#4111yescharacteristic
325C23×C3C2×#4112yes—
326C23×C3H(5,𝔽2)12yes—
327C23×C32−1+412yes—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4C22⋊Q814yes—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—41no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—416no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3#12014yes—
382C2×Dic3#12114yes—
383C2×Dic3C22⋊Q814yes—
384C2×Dic3C22⋊Q818yes—
385C2×Dic3#12414yes—
386C2×Dic3#12418yes—
387Dic6—432no—
388C24—21no—
389C24—22no—
390C24—21no—
391C24—22no—
392C24—21no—
393C24—22no—
394C24—21no—
395C24—22no—
396C24C2×Dic612yes—
397C24D3×D412yes—
398C24D3×Q812yes—
399C24C23×D311yescharacteristic
400C2×Q8—616no—
401C2×Q8—128no—
402C2×Q8—128no—
403C2×Q8—128no—
404C2×Q8—128no—
405C2×Q8—128no—
406C2×Q8—128no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—32no—
432C22×C4—32no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—34no—
436C22×C4—32no—
437C22×C4—68no—
438C22×C4—616no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—124no—
463C22×C4—128no—
464C22×C4—124no—
465C22×C4C2×Dic612yes—
466C22×C4C2×Dic612yes—
467C22×C4D3×D414yes—
468C22×C4D3×D412yes—
469C22×C4#23712yes—
470C4⋊C4—28no—
471C4⋊C4—616no—
472C4⋊C4—68no—
473C4⋊C4—128no—
474C22⋊C4—28no—
475C22⋊C4—48no—
476C22⋊C4—68no—
477C22⋊C4—128no—
478C42—68no—
479C42—124no—
480C22×C3—24no—
481C22×C3—28no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—42no—
491C22×C3C2×#12011yescharacteristic
492C22×C3C2×#12111yescharacteristic
493C22×C3C2×C22⋊Q811yescharacteristic
494C22×C3C2×C22⋊Q811yescharacteristic
495C22×C3C2×C22⋊Q812yes—
496C22×C3C2×#12411yescharacteristic
497C22×C3C2×#12412yes—
498C22×C3C2×C4⋊Q811yescharacteristic
499C22×C3#53211yescharacteristic
500C22×C3#53311yescharacteristic
501C22×C3#53311yescharacteristic
502C22×C3#53511yescharacteristic
503C22×C3#53611yescharacteristic
504C22×C3#54212yes—
505C22×C3#54412yes—
506C22×C3D4212yes—
507C22×C3#54612yes—
508C22×C3#54712yes—
509C22×C3#54812yes—
510C22×C3D4×Q812yes—
511C22×C3#55412yes—
512C22×C3#55512yes—
513C22×C3#55612yes—
514C3×C4—24no—
515C3×C4—42no—
516C3×C4—44no—
517C3×C4—82no—
518Dic3—24no—
519Dic3—42no—
520Dic3—44no—
521Dic3—42no—
522Dic3—82no—
523Dic3#39318yescomplemented
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#88814yescomplemented
547C23C22×Dic611yescharacteristic
548C23C2×D3×D411yescharacteristic
549C23C2×D3×D411yescharacteristic
550C23C2×D3×D412yes—
551C23C2×#23711yescharacteristic
552C23#100412yes—
553C23C2×D3×Q811yescharacteristic
554C23D3×#4112yes—
555C23D3×#4111yescharacteristic
556C23D3×#4111yescharacteristic
557C23#101012yes—
558Q8—1232no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—34no—
569C2×C4—38no—
570C2×C4—34no—
571C2×C4—34no—
572C2×C4—38no—
573C2×C4—34no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—1216no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—121no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4#87814yes—
618C2×C3—44no—
619C2×C3C2×#39311yescharacteristic
620C2×C3#240811yescharacteristic
621C2×C3#240911yescharacteristic
622C2×C3#243912yes—
623C2×C3#244012yes—
624C2×C3#249112yes—
625C2×C3#249212yes—
626C2×C3#254711yescharacteristic
627C2×C3#256311yescharacteristic
628C2×C3#262811yescharacteristic
629C2×C3#285711yescharacteristic
630C22—24no—
631C22—28no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22C2×#87811yescharacteristic
642C22C2×#88811yescharacteristic
643C22#551711yescharacteristic
644C22D4×Dic612yes—
645C22#553912yes—
646C22D3×#12011yescharacteristic
647C22#559012yes—
648C22#559611yescharacteristic
649C22#559812yes—
650C22D3×#12111yescharacteristic
651C22#561012yes—
652C22#561612yes—
653C22#562211yescharacteristic
654C22#562312yes—
655C22D3×C22⋊Q811yescharacteristic
656C22D3×C22⋊Q812yes—
657C22#564811yescharacteristic
658C22#565512yes—
659C22#566212yes—
660C22D3×#12412yes—
661C22D3×#12411yescharacteristic
662C22#570212yes—
663C22#571511yescharacteristic
664C4—24no—
665C4—38no—
666C4—42no—
667C4—44no—
668C4—64no—
669C4—82no—
670C4—122no—
671C4—124no—
672C4—122no—
673C4—242no—
674C3#1512511yescharacteristic, complemented, Sylow
675C2—44no—
676C2#7400111yescharacteristic
677C2#7441612yes—
678C2#7468911yescharacteristic
679C2#7554712yes—
680C2#7636111yescharacteristic
681C2#7666911yescharacteristic
682C2D3×#39311yescharacteristic
683C2#7693512yes—
684C2#7728812yes—
685C2#7751411yescharacteristic
686C2#7804911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1439C2
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#15125
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
6. #71567C2semidirect
8. #69424C2semidirect
9. #69412C2semidirect
10. Dic3⋊#95C2semidirect
11. Dic3⋊#95C2semidirect
61. C23×Dic3Q8semidirect
82. C2×C3×C22⋊C4Q8semidirect
278. C2×C22⋊C4Dic6semidirect
316. C23×C3C4⋊Q8semidirect
523. Dic3#393semidirect
546. C23#888semidirect
674. C3#15125semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49472
C2×#49422
C2×Dic3×C22⋊C42
C3×#14392
#715672
#715462
#694242
#694122
Dic3⋊#952
#151253