Finite Group Databasebeta

← #10728189

Subgroups — #10728189

Order 768.

1622 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4989C211yesmaximal, characteristic
2C2×#4987C211yesmaximal, characteristic
3C2×#4981C211yesmaximal, characteristic, complemented
4C2×#4666C211yesmaximal, characteristic
5C3×#1441C211yesmaximal, characteristic, complemented
6#71568C211yesmaximal, characteristic, complemented
7#69428C212yesmaximal
8#69428C212yesmaximal
9#69423C211yesmaximal, characteristic, complemented
10#69406C212yesmaximal, complemented
11Dic3⋊#95C212yesmaximal, complemented
12#14840—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2212yes—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4989—24no—
37#4987—24no—
38#4981—24no—
39#4666—24no—
40C2×#398—31no—
41C2×#385—31no—
42C2×#380—31no—
43C2×C4×C22⋊C4—31no—
44#1445—31no—
45#1444—31no—
46#1441D311yescharacteristic, complemented
47#1439—32no—
48#1439—32no—
49#1436—32no—
50C4⋊#95—32no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—24no—
65C22×C4×D3—42no—
66C22×C4×D3D414yes—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—42no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—42no—
80C2×(C2×D3)⋊C4—42no—
81C2×(C2×D3)⋊C4—42no—
82C2×(C2×D3)⋊C4Q818yes—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—42no—
92C2×Dic3⋊C4—42no—
93C2×Dic3⋊C4—42no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—42no—
97C3×#95—22no—
98C3×#95—42no—
99C3×#95—42no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—42no—
107C3⋊#95D414yes—
108C24×C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C22⋊C4—32no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4—31no—
119C22×C42—32no—
120#398—64no—
121#385—64no—
122#380—64no—
123C4×C22⋊C4—64no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95C2×D311yescharacteristic
133C2×#95C2×D312yes—
134C2×#95C2×D312yes—
135C24×C3C2411yescharacteristic
136C23×D3—21no—
137C23×D3—21no—
138C23×D3—22no—
139C23×D3—21no—
140C23×D3—21no—
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—22no—
145C23×D3C2×D412yes—
146C23×D3C2×Q814yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×Q814yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4112yes—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3C2×D414yes—
202C22×Dic3C2×D412yes—
203C22×Dic3C2×Q814yes—
204C22×Dic3#4112yes—
205C22×Dic3#4114yes—
206C22×Dic3#4114yes—
207C2×C4×D3—28no—
208C2×C4×D3—28no—
209C2×C4×D3—44no—
210C2×C4×D3—48no—
211C2×C4×D3—48no—
212C3×C4⋊C4—44no—
213C3×C4⋊C4—44no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—216no—
216(C2×D3)⋊C4—44no—
217(C2×D3)⋊C4—48no—
218C4⋊Dic3—44no—
219Dic3⋊C4—48no—
220Dic3⋊C4—48no—
221Dic3⋊C4—44no—
222C4×Dic3—48no—
223C25—31no—
224C23×C4—32no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—34no—
228C23×C4—32no—
229C23×C4—31no—
230C23×C4—64no—
231C23×C4—62no—
232C23×C4—62no—
233C23×C4—122no—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D312yes—
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D311yescharacteristic
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—122no—
253C2×C4⋊C4—122no—
254C2×C4⋊C4—122no—
255C2×C22⋊C4—38no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—122no—
264C2×C22⋊C4—122no—
265C2×C22⋊C4—122no—
266C2×C42—64no—
267C2×C42—64no—
268C2×C42—122no—
269#95—22no—
270#95—34no—
271#95—42no—
272#95—42no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—62no—
277#95—64no—
278#95—62no—
279#95—122no—
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×Q812yes—
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4112yes—
286C23×C3C2×#4112yes—
287C23×C3C2×#4111yescharacteristic
288C23×C3H(5,𝔽2)12yes—
289C23×C3H(5,𝔽2)11yescharacteristic
290C23×C3H(5,𝔽2)11yescharacteristic
291C22×D3—22no—
292C22×D3—24no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—24no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—24no—
302C22×D3—24no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—42no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3C4⋊Q814yes—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4#12114yes—
345C2×Dic3—22no—
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3C22⋊Q818yes—
373C2×Dic3#12314yes—
374C4×D3—416no—
375C24—32no—
376C24—34no—
377C24—61no—
378C24—61no—
379C24—61no—
380C24—62no—
381C24—61no—
382C24—61no—
383C24—62no—
384C24—62no—
385C24—61no—
386C24C23×D311yescharacteristic
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—34no—
414C22×C4—32no—
415C22×C4—34no—
416C22×C4—34no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—68no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—68no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—124no—
445C22×C4—128no—
446C22×C4—128no—
447C22×C4#23512yes—
448C22×C4D3×D412yes—
449C22×C4D3×D412yes—
450C22×C4D3×Q814yes—
451C22×C4#23912yes—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—128no—
455C4⋊C4—124no—
456C4⋊C4—128no—
457C4⋊C4—124no—
458C22⋊C4—616no—
459C22⋊C4—68no—
460C22⋊C4—124no—
461C22⋊C4—128no—
462C42—128no—
463C22×C3C2×#12111yescharacteristic
464C22×C3C2×C22⋊Q812yes—
465C22×C3C2×#12311yescharacteristic
466C22×C3#52511yescharacteristic
467C22×C3#52912yes—
468C22×C3#52912yes—
469C22×C3C2×C4⋊Q811yescharacteristic
470C22×C3#53211yescharacteristic
471C22×C3#53312yes—
472C22×C3#53411yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3#53812yes—
475C22×C3#54012yes—
476C22×C3#54612yes—
477C22×C3#54611yescharacteristic
478C22×C3#55011yescharacteristic
479C22×C3#55112yes—
480C22×C3#55311yescharacteristic
481C22×C3#55412yes—
482C22×C3#55412yes—
483C22×C3#55412yes—
484C22×C3#55412yes—
485C22×C3#55511yescharacteristic
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—44no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C3×C4—24no—
498C3×C4—42no—
499C3×C4—44no—
500C3×C4—82no—
501C3×C4—81no—
502Dic3—24no—
503Dic3—28no—
504Dic3—42no—
505Dic3—81no—
506Dic3—82no—
507C23—34no—
508C23—64no—
509C23—64no—
510C23—62no—
511C23—62no—
512C23—64no—
513C23—64no—
514C23—62no—
515C23—64no—
516C23—62no—
517C23—62no—
518C23—64no—
519C23—62no—
520C23—122no—
521C23—122no—
522C23—122no—
523C23—121no—
524C23—121no—
525C23—122no—
526C23—121no—
527C23—122no—
528C23—122no—
529C23—121no—
530C23C2×#23511yescharacteristic
531C23C2×D3×D411yescharacteristic
532C23C2×D3×D411yescharacteristic
533C23#100411yescharacteristic
534C23#100412yes—
535C23C2×D3×Q812yes—
536C23C2×#23911yescharacteristic
537C23D3×#4111yescharacteristic
538C23D3×#4112yes—
539C23D3×#4112yes—
540C23#100911yescharacteristic
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—34no—
552C2×C4—38no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—62no—
585C2×C4—1216no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—121no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4#88414yes—
602C2×C3#239511yescharacteristic
603C2×C3#241211yescharacteristic
604C2×C3#241411yescharacteristic
605C2×C3#245012yes—
606C2×C3#245412yes—
607C2×C3#250211yescharacteristic
608C2×C3#252212yes—
609C2×C3#257412yes—
610C2×C3#258311yescharacteristic
611C2×C3#266111yescharacteristic
612C2×C3#285711yescharacteristic
613D3—44no—
614C22—64no—
615C22—64no—
616C22—64no—
617C22—124no—
618C22—122no—
619C22—122no—
620C22—124no—
621C22—124no—
622C22—122no—
623C22—122no—
624C22—124no—
625C22C2×#88411yescharacteristic
626C22#551411yescharacteristic
627C22#553211yescharacteristic
628C22#555612yes—
629C22#557812yes—
630C22#558911yescharacteristic
631C22#559911yescharacteristic
632C22#560711yescharacteristic
633C22#561012yes—
634C22#561512yes—
635C22D3×C22⋊Q812yes—
636C22#562711yescharacteristic
637C22#563011yescharacteristic
638C22#563612yes—
639C22#563612yes—
640C22D3×#12311yescharacteristic
641C22#565712yes—
642C22#566811yescharacteristic
643C22#569012yes—
644C22#569112yes—
645C22#570612yes—
646C22D3×C4⋊Q811yescharacteristic
647C22#573012yes—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—68no—
652C4—64no—
653C4—81no—
654C4—82no—
655C4—122no—
656C4—242no—
657C4—241no—
658C3#1484011yescharacteristic, complemented, Sylow
659C2—124no—
660C2#7401911yescharacteristic
661C2#7469011yescharacteristic
662C2#7508212yes—
663C2#7605212yes—
664C2#7666411yescharacteristic
665C2#7699511yescharacteristic
666C2#7725511yescharacteristic
667C2#7729112yes—
668C2#7776511yescharacteristic
669C2#7795312yes—
670C2#7812911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1441C2
prime radicalC24×D3C23
hypercenter#1441D3
hypocenterC3#14840
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. C2×#4981C2semidirect
5. C3×#1441C2semidirect
6. #71568C2semidirect
9. #69423C2semidirect
10. #69406C2semidirect
11. Dic3⋊#95C2semidirect
46. #1441D3semidirect
658. C3#14840semidirect

Maximal subgroups

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

subgroupindex
C2×#49892
C2×#49872
C2×#49812
C2×#46662
C3×#14412
#715682
#694282
#694232
#694062
Dic3⋊#952
#148403