Finite Group Databasebeta

← #10729047

Subgroups — #10729047

Order 768.

2150 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×#897C211yesmaximal, characteristic
2C2×#4989C212yesmaximal
3C2×D3×#95C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic, complemented
5#71568C211yesmaximal, characteristic, complemented
6#71565C212yesmaximal, complemented
7(C2×D3)⋊#95C211yesmaximal, characteristic
8(C2×D3)⋊#95C212yesmaximal
9#69424C211yesmaximal, characteristic, complemented
10#69424C211yesmaximal, characteristic, complemented
11#69410C212yesmaximal, complemented
12#14914—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2212yes—
21C22×C4⋊Dic3C2212yes—
22C22×Dic3⋊C4C2212yes—
23C2×#897—24no—
24C2×#897—28no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4989—28no—
39D3×#95—24no—
40C22×C22⋊Q8—31no—
41C2×#385—32no—
42C22×#95—31no—
43#1446—31no—
44#1446—31no—
45#1444—31no—
46#1444D311yescharacteristic, complemented
47#1441—32no—
48#1441—32no—
49C22⋊#95—32no—
50C22⋊#95—31no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—22no—
64C22×C4×D3—24no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3—24no—
71C22×C4×D3—42no—
72C22×C4×D3—42no—
73C22×C4×D3—42no—
74C22×C4×D3—42no—
75C22×C4×D3D414yes—
76C22×C4×D3Q814yescomplemented
77C22×Dic6—22no—
78C22×Dic6—24no—
79C22×Dic6—22no—
80C22×Dic6—22no—
81C22×Dic6—42no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—42no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4D418yes—
92C2×(C2×D3)⋊C4Q818yescomplemented
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—42no—
99#897—232no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—42no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—42no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C3⋊#95—42no—
113C23×Q8—31no—
114C24×C4—31no—
115C24×C4—31no—
116C24×C4—31no—
117C2×C22⋊Q8—68no—
118C2×C22⋊Q8—64no—
119C22×C4⋊C4—32no—
120C22×C4⋊C4—32no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C4⋊C4C2×D312yes—
123C22×C22⋊C4—32no—
124C22×C22⋊C4—32no—
125#385—68no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—64no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D311yescharacteristic
139C2×#95C2×D312yes—
140C24×C3C2411yescharacteristic
141C23×D3—21no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3C2×D414yes—
148C23×D3C2×D412yes—
149C23×D3C2×Q814yes—
150C23×D3C2×Q812yes—
151C23×D3#4112yes—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
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—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D414yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×Q814yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4112yes—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D414yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3C2×Q812yes—
209C22×Dic3#4112yes—
210C2×C4×D3—28no—
211C2×C4×D3—28no—
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—44no—
215C2×C4×D3—48no—
216C2×C4×D3—48no—
217C2×C4×D3—48no—
218C2×C4×D3—48no—
219C2×C4×D3—48no—
220C2×C4×D3—44no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3—48no—
224C2×C4×D3—44no—
225C2×Dic6—28no—
226C2×Dic6—48no—
227C2×Dic6—48no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—48no—
232(C2×D3)⋊C4—216no—
233(C2×D3)⋊C4—216no—
234C4⋊Dic3—216no—
235Dic3⋊C4—48no—
236C25—31no—
237C22×Q8—64no—
238C22×Q8—62no—
239C22×Q8—62no—
240C22×Q8—62no—
241C22×Q8—122no—
242C23×C4—32no—
243C23×C4—32no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—34no—
248C23×C4—34no—
249C23×C4—62no—
250C23×C4—64no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4—64no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4—122no—
259C23×C4—122no—
260C23×C4—122no—
261C23×C4—122no—
262C23×C4C22×D312yes—
263C23×C4C22×D311yescharacteristic
264C23×C4C22×D312yes—
265C23×C4C22×D311yescharacteristic
266C23×C4C22×D311yescharacteristic
267C22⋊Q8—632no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—64no—
278C2×C4⋊C4—122no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—38no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285#95—22no—
286#95—22no—
287#95—42no—
288#95—62no—
289#95—62no—
290#95—64no—
291#95—64no—
292#95—62no—
293#95—62no—
294#95—122no—
295#95—122no—
296#95—122no—
297#95—122no—
298C23×C3C22×D411yescharacteristic
299C23×C3C22×D412yes—
300C23×C3C22×Q811yescharacteristic
301C23×C3C22×Q812yes—
302C23×C3C2×#4111yescharacteristic
303C23×C3C2×#4111yescharacteristic
304C23×C3C2×#4111yescharacteristic
305C23×C3C2×#4111yescharacteristic
306C23×C3C2×#4111yescharacteristic
307C23×C3H(5,𝔽2)12yes—
308C23×C3H(5,𝔽2)12yes—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—22no—
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—22no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—24no—
320C22×D3—24no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3#12014yes—
328C22×D3C22⋊Q818yes—
329C22×D3C22⋊Q814yes—
330C22×D3C4⋊Q814yes—
331C22×D3C4⋊Q818yes—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
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#12314yes—
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—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C4×D3—416no—
391C4×D3—416no—
392C4×D3—416no—
393Dic6—416no—
394C24—34no—
395C24—32no—
396C24—34no—
397C24—32no—
398C24—32no—
399C24—61no—
400C24—61no—
401C24—62no—
402C24—61no—
403C24—61no—
404C24—62no—
405C24C23×D311yescharacteristic
406C2×Q8—68no—
407C2×Q8—128no—
408C2×Q8—128no—
409C2×Q8—124no—
410C2×Q8—124no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
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—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—34no—
434C22×C4—32no—
435C22×C4—34no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—68no—
439C22×C4—68no—
440C22×C4—68no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—68no—
466C22×C4—124no—
467C22×C4—128no—
468C22×C4—128no—
469C22×C4—128no—
470C22×C4—128no—
471C22×C4—128no—
472C22×C4—124no—
473C22×C4—124no—
474C22×C4—128no—
475C22×C4—124no—
476C22×C4—124no—
477C22×C4#23512yes—
478C22×C4D3×D414yes—
479C22×C4D3×D412yes—
480C22×C4#23712yes—
481C22×C4D3×Q814yes—
482C22×C4D3×Q812yes—
483C22×C4#23912yes—
484C4⋊C4—28no—
485C4⋊C4—48no—
486C4⋊C4—616no—
487C4⋊C4—128no—
488C22⋊C4—616no—
489C22⋊C4—616no—
490C22×C3C2×#12011yescharacteristic
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3C2×C22⋊Q812yes—
493C22×C3C2×#12311yescharacteristic
494C22×C3#52511yescharacteristic
495C22×C3#52911yescharacteristic
496C22×C3#52911yescharacteristic
497C22×C3C2×C4⋊Q811yescharacteristic
498C22×C3C2×C4⋊Q812yes—
499C22×C3#53311yescharacteristic
500C22×C3#53311yescharacteristic
501C22×C3#53411yescharacteristic
502C22×C3#53511yescharacteristic
503C22×C3#53812yes—
504C22×C3#54312yes—
505C22×C3#54612yes—
506C22×C3#54612yes—
507C22×C3#54612yes—
508C22×C3#54612yes—
509C22×C3#55412yes—
510C22×C3#55412yes—
511C22×C3#55412yes—
512C22×C3#55412yes—
513C2×D3—24no—
514C2×D3—24no—
515C2×D3—24no—
516C2×D3—24no—
517C2×D3—24no—
518C2×D3—28no—
519C2×D3—28no—
520C2×D3—24no—
521C2×D3—28no—
522C2×D3—28no—
523C2×D3#39518yes—
524C3×C4—24no—
525C3×C4—42no—
526C3×C4—42no—
527C3×C4—44no—
528C3×C4—44no—
529Dic3—42no—
530Dic3—42no—
531Dic3—44no—
532Dic3—42no—
533Dic3—44no—
534C23—34no—
535C23—38no—
536C23—34no—
537C23—38no—
538C23—34no—
539C23—64no—
540C23—64no—
541C23—62no—
542C23—64no—
543C23—64no—
544C23—62no—
545C23—62no—
546C23—62no—
547C23—64no—
548C23—62no—
549C23—64no—
550C23—64no—
551C23—62no—
552C23—62no—
553C23—64no—
554C23—62no—
555C23—64no—
556C23—64no—
557C23C2×#23511yescharacteristic
558C23C2×D3×D411yescharacteristic
559C23C2×D3×D412yes—
560C23C2×#23711yescharacteristic
561C23#100412yes—
562C23C2×D3×Q811yescharacteristic
563C23C2×D3×Q812yes—
564C23C2×#23911yescharacteristic
565C23D3×#4111yescharacteristic
566C23D3×#4111yescharacteristic
567C23#100912yes—
568Q8—1216no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—1216no—
611C2×C4—1216no—
612C2×C4—1216no—
613C2×C4—121no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—121no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4#89414yes—
630C2×C3C2×#39511yescharacteristic
631C2×C3#240411yescharacteristic
632C2×C3#240511yescharacteristic
633C2×C3#245012yes—
634C2×C3#245012yes—
635C2×C3#250211yescharacteristic
636C2×C3#254912yes—
637C2×C3#254912yes—
638C2×C3#258311yescharacteristic
639C2×C3#262611yescharacteristic
640C2×C3#285511yescharacteristic
641D3—28no—
642C22—38no—
643C22—64no—
644C22—64no—
645C22—64no—
646C22—68no—
647C22—68no—
648C22—64no—
649C22—68no—
650C22—64no—
651C22—68no—
652C22—64no—
653C22C2×#89411yescharacteristic
654C22#551112yes—
655C22#555012yes—
656C22#555711yescharacteristic
657C22#557811yescharacteristic
658C22D3×#12011yescharacteristic
659C22#559112yes—
660C22#560611yescharacteristic
661C22#561012yes—
662C22D3×C22⋊Q811yescharacteristic
663C22D3×C22⋊Q812yes—
664C22#563012yes—
665C22#563412yes—
666C22#563512yes—
667C22#565311yescharacteristic
668C22#565712yes—
669C22#569012yes—
670C22#569111yescharacteristic
671C22#571511yescharacteristic
672C22D3×C4⋊Q811yescharacteristic
673C22D3×C4⋊Q812yes—
674C22#572811yescharacteristic
675C22#573012yes—
676C4—24no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—122no—
682C4—122no—
683C4—124no—
684C4—124no—
685C4—122no—
686C3#1491411yescharacteristic, complemented, Sylow
687C2—68no—
688C2#7432911yescharacteristic
689C2#7459911yescharacteristic
690C2#7461012yes—
691C2#7624412yes—
692C2#7628711yescharacteristic
693C2#7675411yescharacteristic
694C2D3×#39511yescharacteristic
695C2#7725312yes—
696C2#7795312yes—
697C2#7796111yescharacteristic
698C2#7811011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1444C2
prime radicalC24×D3C23
hypercenter#1444D3
hypocenterC3#14914
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
4. C3×#1444C2semidirect
5. #71568C2semidirect
6. #71565C2semidirect
9. #69424C2semidirect
10. #69424C2semidirect
11. #69410C2semidirect
46. #1444D3semidirect
76. C22×C4×D3Q8semidirect
92. C2×(C2×D3)⋊C4Q8semidirect
686. C3#14914semidirect

Maximal subgroups

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

subgroupindex
C22×#8972
C2×#49892
C2×D3×#952
C3×#14442
#715682
#715652
(C2×D3)⋊#952
#694242
#694102
#149143