Finite Group Databasebeta

← #10729293

Subgroups — #10729293

Order 768.

1902 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
1C2×C3×#381C211yesmaximal, characteristic
2C2×#5211C211yesmaximal, characteristic
3C22×C3⋊#95C211yesmaximal, characteristic
4C2×#4950C211yesmaximal, characteristic
5C2×#4946C211yesmaximal, characteristic
6C2×#4942C211yesmaximal, characteristic
7#71544C211yesmaximal, characteristic
8#69417C212yesmaximal, complemented
9#69411C212yesmaximal, complemented
10#69410C212yesmaximal, complemented
11#69409C212yesmaximal, complemented
12#14935—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C3×#381—24no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27#5211—24no—
28C2×C3⋊#95—24no—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39#4950—24no—
40#4946—24no—
41#4942—24no—
42C2×#385—31no—
43C2×#385—31no—
44C2×#381—31no—
45C2×#381D311yescharacteristic
46C2×#380—31no—
47C22×#95—31no—
48#1443—32no—
49#1441—32no—
50#1441—32no—
51#1441—32no—
52C22⋊#95—31no—
53C25×C3C2311yescharacteristic
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4—22no—
60C23×C3×C4Q814yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×Dic3—22no—
66C23×Dic3—22no—
67C23×Dic3—22no—
68C23×Dic3—24no—
69C23×Dic3—24no—
70C23×Dic3—22no—
71C23×Dic3—22no—
72C23×Dic3—42no—
73C23×Dic3—42no—
74C23×Dic3D414yes—
75C23×Dic3C2312yes—
76C23×Dic3C2311yescharacteristic
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2312yes—
80C23×Dic3C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D414yes—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217—22no—
91C2×#217—22no—
92C2×#217D414yes—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—42no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—42no—
102C2×Dic3⋊C4—42no—
103C3×#95—24no—
104C3×#95—24no—
105C3⋊#95—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—42no—
115C24×C4—31no—
116C24×C4—31no—
117C24×C4C2×D311yescharacteristic
118C22×C4⋊C4—31no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—32no—
121C22×C4⋊C4—31no—
122C22×C22⋊C4—31no—
123C22×C22⋊C4—31no—
124C22×C22⋊C4C2×D311yescharacteristic
125C22×C22⋊C4C2×D311yescharacteristic
126#385—64no—
127#385—64no—
128#381—24no—
129#381—64no—
130#380—64no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—31no—
140C2×#95—32no—
141C2×#95—64no—
142C2×#95C2×D312yes—
143C2×#95C2×D312yes—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3C2×D412yes—
149C24×C3C2×D412yes—
150C24×C3C2×D412yes—
151C24×C3C2×Q812yes—
152C24×C3#4112yes—
153C24×C3#4112yes—
154C24×C3#4112yes—
155C24×C3C2411yescharacteristic
156C22×C3×C4—28no—
157C22×C3×C4—28no—
158C22×C3×C4—28no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
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—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×Q814yes—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—28no—
191C22×Dic3—28no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—28no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—44no—
219C22×Dic3—48no—
220C22×Dic3—48no—
221C22×Dic3—48no—
222C22×Dic3—48no—
223C22×Dic3—44no—
224C22×Dic3—44no—
225C22×Dic3C2×D412yes—
226C22×Dic3C2×D412yes—
227C22×Dic3C2×D412yes—
228C22×Dic3C2×D412yes—
229C22×Dic3#4114yes—
230C22×Dic3#4114yes—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—44no—
233#217—28no—
234#217—44no—
235C4⋊Dic3—48no—
236Dic3⋊C4—48no—
237Dic3⋊C4—44no—
238Dic3⋊C4—44no—
239C25C22×D311yescharacteristic
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—32no—
251C23×C4—34no—
252C23×C4—32no—
253C23×C4—64no—
254C23×C4—64no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4—62no—
260C23×C4—122no—
261C23×C4—122no—
262C23×C4Dic614yes—
263C23×C4C22×D311yescharacteristic
264C23×C4C22×D311yescharacteristic
265C23×C4C22×D312yes—
266C23×C4C22×D312yes—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—122no—
275C2×C4⋊C4—122no—
276C2×C4⋊C4—122no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—34no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4#6714yes—
289#95—24no—
290#95—24no—
291#95—64no—
292#95—64no—
293#95—64no—
294#95—64no—
295#95—64no—
296#95—64no—
297#95—62no—
298#95—62no—
299#95—64no—
300#95—122no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3#12114yes—
318C23×C3#12114yes—
319C23×C3#12114yes—
320C23×C3C22⋊Q814yes—
321C23×C3C22⋊Q814yes—
322C23×C3C22⋊Q814yes—
323C23×C3#12614yes—
324C23×C3C22×D411yescharacteristic
325C23×C3C22×D411yescharacteristic
326C23×C3C22×D411yescharacteristic
327C23×C3C22×Q811yescharacteristic
328C23×C3C2×#4111yescharacteristic
329C23×C3C2×#4111yescharacteristic
330C23×C3C2×#4111yescharacteristic
331C23×C3C2×#4112yes—
332C23×C3C2×#4112yes—
333C23×C3H(5,𝔽2)12yes—
334C23×C32−1+412yes—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
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—416no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—416no—
375C2×Dic3—416no—
376C2×Dic3—42no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—42no—
381C2×Dic3—41no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3#12314yes—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24C2×Dic612yes—
401C24#23512yes—
402C24D3×D412yes—
403C24D3×D412yes—
404C24#23712yes—
405C24#23712yes—
406C24C2×#6712yes—
407C24C23×D311yescharacteristic
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
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—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—32no—
434C22×C4—34no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—34no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—68no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—68no—
448C22×C4—68no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—62no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—61no—
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—128no—
477C22×C4—128no—
478C22×C4—128no—
479C22×C4—124no—
480C22×C4—124no—
481C22×C4C2×Dic614yes—
482C22×C4C2×#6714yes—
483C4⋊C4—128no—
484C4⋊C4—124no—
485C4⋊C4—128no—
486C4⋊C4—124no—
487C22⋊C4—28no—
488C22⋊C4—44no—
489C22⋊C4—68no—
490C22⋊C4—124no—
491C22×C3—24no—
492C22×C3—24no—
493C22×C3—24no—
494C22×C3—24no—
495C22×C3—28no—
496C22×C3—28no—
497C22×C3—28no—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—24no—
501C22×C3#39618yes—
502C22×C3C2×#12111yescharacteristic
503C22×C3C2×#12111yescharacteristic
504C22×C3C2×#12111yescharacteristic
505C22×C3C2×C22⋊Q811yescharacteristic
506C22×C3C2×C22⋊Q811yescharacteristic
507C22×C3C2×C22⋊Q811yescharacteristic
508C22×C3C2×#12311yescharacteristic
509C22×C3#52511yescharacteristic
510C22×C3C2×#12611yescharacteristic
511C22×C3#52912yes—
512C22×C3#53312yes—
513C22×C3#53411yescharacteristic
514C22×C3#53611yescharacteristic
515C22×C3#53912yes—
516C22×C3#54412yes—
517C22×C3#54612yes—
518C22×C3#54612yes—
519C22×C3#54712yes—
520C22×C3#54712yes—
521C22×C3#55112yes—
522C22×C3#55112yes—
523C22×C3#55212yes—
524C22×C3#55212yes—
525C3×C4—44no—
526C3×C4—44no—
527C3×C4—81no—
528C3×C4—81no—
529Dic3—24no—
530Dic3—42no—
531Dic3—42no—
532Dic3—42no—
533Dic3—44no—
534Dic3—82no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23#87814yes—
552C23#87814yes—
553C23#87914yes—
554C23#88414yes—
555C23#92414yes—
556C23#93814yes—
557C23#93914yes—
558C23C22×Dic611yescharacteristic
559C23C2×#23511yescharacteristic
560C23C2×D3×D411yescharacteristic
561C23C2×D3×D411yescharacteristic
562C23C2×#23711yescharacteristic
563C23C2×#23711yescharacteristic
564C23D3×#4112yes—
565C23D3×#4112yes—
566C23#100912yes—
567C23#101012yes—
568C23C22×#6711yescharacteristic
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—34no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—1216no—
610C2×C4—1216no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—121no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C3—28no—
631C2×C3C2×#39611yescharacteristic
632C2×C3#241211yescharacteristic
633C2×C3#241311yescharacteristic
634C2×C3#242512yes—
635C2×C3#243812yes—
636C2×C3#243812yes—
637C2×C3#248611yescharacteristic
638C2×C3#249312yes—
639C2×C3#253411yescharacteristic
640C2×C3#262711yescharacteristic
641C2×C3#285511yescharacteristic
642C22—28no—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22—24no—
652C22#494818yes—
653C22C2×#87811yescharacteristic
654C22C2×#87811yescharacteristic
655C22C2×#87911yescharacteristic
656C22C2×#88411yescharacteristic
657C22#551712yes—
658C22#551912yes—
659C22#552712yes—
660C22#553112yes—
661C22#564811yescharacteristic
662C22#564912yes—
663C22#565012yes—
664C22D3×#12311yescharacteristic
665C22#565311yescharacteristic
666C22#565412yes—
667C22#565512yes—
668C22#565611yescharacteristic
669C22#565812yes—
670C22#566512yes—
671C22C2×#92411yescharacteristic
672C22C2×#93811yescharacteristic
673C22C2×#93911yescharacteristic
674C22#582412yes—
675C22#582912yes—
676C4—44no—
677C4—44no—
678C4—64no—
679C4—81no—
680C4—81no—
681C4—122no—
682C4—124no—
683C4—122no—
684C4—122no—
685C4—242no—
686C3#1493511yescharacteristic, complemented, Sylow
687C2—28no—
688C2#7406712yes—
689C2#7407511yescharacteristic
690C2#7407712yes—
691C2#7409011yescharacteristic
692C2#7411711yescharacteristic
693C2#7413111yescharacteristic
694C2C2×#494811yescharacteristic
695C2#8224712yes—
696C2#8284211yescharacteristic
697C2#8284712yes—
698C2#8289511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#381C2
prime radicalC25×C3C23
hypercenterC2×#381D3
hypocenterC3#14935
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
8. #69417C2semidirect
9. #69411C2semidirect
10. #69410C2semidirect
11. #69409C2semidirect
686. C3#14935semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3812
C2×#52112
C22×C3⋊#952
C2×#49502
C2×#49462
C2×#49422
#715442
#694172
#694112
#694102
#694092
#149353