Finite Group Databasebeta

← #10751743

Subgroups — #10751743

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic, complemented
3C2×#4989C211yesmaximal, characteristic
4C2×#4981C211yesmaximal, characteristic, complemented
5#72134C211yesmaximal, characteristic, complemented
6#71570C212yesmaximal
7#71568C212yesmaximal, complemented
8#69428C212yesmaximal
9(C2×D3)⋊#95C211yesmaximal, characteristic
10#69423C211yesmaximal, characteristic, complemented
11#69409C212yesmaximal, complemented
12#16606—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C2×#946—24no—
18C2×#946—24no—
19C2×#946—24no—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2212yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2211yescharacteristic
28C3×#395—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4989—24no—
41#4981—24no—
42C22×C22⋊Q8—31no—
43C2×#398—31no—
44C2×#395D311yescharacteristic, complemented
45C2×#385—31no—
46#1446—31no—
47#1445—31no—
48#1444—32no—
49#1441—32no—
50#1439—32no—
51#1439—32no—
52C22⋊#95—31no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C22×C3×Q8—42no—
58C22×C3×Q8D414yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2312yes—
68C23×Dic3C2311yescharacteristic
69C22×C4×D3—22no—
70C22×C4×D3—22no—
71C22×C4×D3—42no—
72C22×C4×D3—42no—
73C22×C4×D3D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4—42no—
80#946—232no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—42no—
89C2×(C2×D3)⋊C4—42no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4Q818yescomplemented
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3D414yes—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—42no—
102C2×Dic3⋊C4—42no—
103C3×#95—24no—
104C3×#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C3⋊#95—42no—
113C3⋊#95—42no—
114C23×Q8C2×D311yescharacteristic
115C24×C4—31no—
116C2×C22⋊Q8—64no—
117C2×C22⋊Q8—64no—
118C2×C22⋊Q8—64no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4—32no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4C2×D311yescharacteristic
124C22×C4⋊C4C2×D312yes—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—31no—
128C22×C22⋊C4—31no—
129#398—64no—
130#395—24no—
131#385—64no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—31no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95C2×D311yescharacteristic
142C2×#95C2×D312yes—
143C24×C3C2411yescharacteristic
144C23×D3—21no—
145C23×D3—21no—
146C23×D3—21no—
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3—21no—
153C23×D3C2×D412yes—
154C23×D3C2×Q814yes—
155C2×C3×Q8—28no—
156C2×C3×Q8—48no—
157C2×C3×Q8—44no—
158C2×C3×Q8—44no—
159C2×C3×Q8—48no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
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—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D414yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×Q814yes—
190C22×C3×C4#4112yes—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—21no—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×Q814yes—
219C22×Dic3#4112yes—
220C2×C4×D3—28no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3—48no—
224C2×C4×D3—48no—
225C3×C4⋊C4—48no—
226C3×C4⋊C4—44no—
227(C2×D3)⋊C4—28no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—48no—
230(C2×D3)⋊C4—44no—
231C4⋊Dic3—28no—
232Dic3⋊C4—216no—
233Dic3⋊C4—48no—
234Dic3⋊C4—44no—
235C25—31no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8#6714yes—
241C23×C4—32no—
242C23×C4—32no—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—34no—
246C23×C4—31no—
247C23×C4—62no—
248C23×C4—62no—
249C23×C4—122no—
250C23×C4—122no—
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D312yes—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256C22⋊Q8—632no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—122no—
273C2×C4⋊C4—122no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—122no—
283C2×C22⋊C4—122no—
284C2×C22⋊C4—122no—
285#95—24no—
286#95—22no—
287#95—64no—
288#95—62no—
289#95—64no—
290#95—62no—
291#95—62no—
292#95—122no—
293#95—122no—
294#95—122no—
295#95—122no—
296C23×C3C22×D411yescharacteristic
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D411yescharacteristic
299C23×C3C22×Q812yes—
300C23×C3C2×#4111yescharacteristic
301C23×C3C2×#4111yescharacteristic
302C23×C3H(5,𝔽2)12yes—
303C23×C3H(5,𝔽2)12yes—
304C23×C3H(5,𝔽2)11yescharacteristic
305C23×C32−1+412yes—
306C23×C32−1+411yescharacteristic
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3C4⋊Q814yes—
330C3×Q8—416no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4#12014yes—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C4×D3—416no—
392C24—32no—
393C24—34no—
394C24—61no—
395C24—62no—
396C24—62no—
397C24—61no—
398C24—62no—
399C24—61no—
400C24—61no—
401C24—61no—
402C24—61no—
403C24C23×D311yescharacteristic
404C2×Q8—28no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—48no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—34no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—32no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—61no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—61no—
458C22×C4—68no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—124no—
465C22×C4—124no—
466C22×C4—128no—
467C22×C4—128no—
468C22×C4D3×D412yes—
469C22×C4D3×D412yes—
470C22×C4D3×Q814yes—
471C22×C4#23912yes—
472C22×C4C2×#6714yes—
473C22×C4C2×#6712yes—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—616no—
477C4⋊C4—68no—
478C4⋊C4—128no—
479C4⋊C4—124no—
480C22⋊C4—616no—
481C22⋊C4—68no—
482C22⋊C4—128no—
483C22⋊C4—124no—
484C22×C3C2×#12011yescharacteristic
485C22×C3#52511yescharacteristic
486C22×C3C2×C4⋊Q811yescharacteristic
487C22×C3#53412yes—
488C22×C3#53411yescharacteristic
489C22×C3#53411yescharacteristic
490C22×C3#53512yes—
491C22×C3#53611yescharacteristic
492C22×C3#53711yescharacteristic
493C22×C3#53912yes—
494C22×C3#54212yes—
495C22×C3#54312yes—
496C22×C3#54412yes—
497C22×C3#54611yescharacteristic
498C22×C3#54711yescharacteristic
499C22×C3#54811yescharacteristic
500C22×C3D4×Q812yes—
501C22×C3D4×Q812yes—
502C22×C3#55011yescharacteristic
503C22×C3#55412yes—
504C22×C3#55412yes—
505C22×C3#56212yes—
506C22×C3#56312yes—
507C2×D3—24no—
508C2×D3—24no—
509C2×D3—24no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—44no—
514C2×D3—44no—
515C2×D3—44no—
516C2×D3—44no—
517C2×D3—42no—
518C3×C4—24no—
519C3×C4—44no—
520C3×C4—42no—
521C3×C4—44no—
522C3×C4—81no—
523Dic3—42no—
524Dic3—42no—
525Dic3—44no—
526Dic3—44no—
527Dic3—81no—
528C23—34no—
529C23—64no—
530C23—64no—
531C23—62no—
532C23—64no—
533C23—64no—
534C23—64no—
535C23—62no—
536C23—64no—
537C23—62no—
538C23—62no—
539C23—62no—
540C23—62no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—121no—
547C23—121no—
548C23—122no—
549C23—121no—
550C23—121no—
551C23C2×D3×D411yescharacteristic
552C23C2×D3×D411yescharacteristic
553C23#100412yes—
554C23#100412yes—
555C23C2×D3×Q812yes—
556C23C2×#23911yescharacteristic
557C23#100712yes—
558C23#100711yescharacteristic
559C23D3×#4111yescharacteristic
560C23#100911yescharacteristic
561C23C22×#6711yescharacteristic
562Q8—416no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—1216no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—121no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4#93714yes—
624C2×C3#240711yescharacteristic
625C2×C3#243411yescharacteristic
626C2×C3#258311yescharacteristic
627C2×C3#261311yescharacteristic
628C2×C3#263811yescharacteristic
629C2×C3#265911yescharacteristic
630C2×C3#269212yes—
631C2×C3#272012yes—
632C2×C3#272912yes—
633C2×C3#273212yes—
634C2×C3#281411yescharacteristic
635D3—44no—
636C22—64no—
637C22—64no—
638C22—64no—
639C22—124no—
640C22—124no—
641C22—122no—
642C22—122no—
643C22—124no—
644C22—124no—
645C22—122no—
646C22—122no—
647C22#558911yescharacteristic
648C22#558911yescharacteristic
649C22#559212yes—
650C22#562711yescharacteristic
651C22#562911yescharacteristic
652C22#563011yescharacteristic
653C22#563111yescharacteristic
654C22#563412yes—
655C22#563612yes—
656C22#563812yes—
657C22#564712yes—
658C22D3×C4⋊Q811yescharacteristic
659C22#572411yescharacteristic
660C22#572712yes—
661C22#572912yes—
662C22#573012yes—
663C22#573312yes—
664C22#573512yes—
665C22C2×#93711yescharacteristic
666C22#580412yes—
667C22Q8×#6712yes—
668C22#581711yescharacteristic
669C22#583411yescharacteristic
670C4—24no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—81no—
675C4—124no—
676C4—122no—
677C4—124no—
678C4—122no—
679C4—241no—
680C3#1660611yescharacteristic, complemented, Sylow
681C2—124no—
682C2#7720611yescharacteristic
683C2#7721512yes—
684C2#7722112yes—
685C2#7725511yescharacteristic
686C2#7725811yescharacteristic
687C2#7727411yescharacteristic
688C2#8260711yescharacteristic
689C2#8280712yes—
690C2#8281211yescharacteristic
691C2#8323211yescharacteristic
692C2#8326712yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#395C2
prime radicalC24×D3C23
hypercenterC2×#395D3
hypocenterC3#16606
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×C3×#395C2semidirect
4. C2×#4981C2semidirect
5. #72134C2semidirect
7. #71568C2semidirect
10. #69423C2semidirect
11. #69409C2semidirect
44. C2×#395D3semidirect
91. C2×(C2×D3)⋊C4Q8semidirect
680. C3#16606semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C2×C3×#3952
C2×#49892
C2×#49812
#721342
#715702
#715682
#694282
(C2×D3)⋊#952
#694232
#694092
#166063