Finite Group Databasebeta

← #10717153

Subgroups — #10717153

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#877C211yesmaximal, characteristic
2C2×C3×#380C211yesmaximal, characteristic
3C22×C3⋊#95C211yesmaximal, characteristic
4C2×#4943C211yesmaximal, characteristic
5C2×#4942C211yesmaximal, characteristic
6#72127C211yesmaximal, characteristic
7#71568C212yesmaximal, complemented
8#71564C212yesmaximal, complemented
9#71544C211yesmaximal, characteristic
10#69414C212yesmaximal, complemented
11#69406C212yesmaximal, complemented
12#13878—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2212yes—
24C2×#877—24no—
25C2×#877—24no—
26C2×#877—24no—
27C3×#380—24no—
28C2×C3×#95C2212yes—
29C2×C3⋊#95C418yescomplemented
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2211yescharacteristic
40C2×C3⋊#95C2211yescharacteristic
41#4943—24no—
42#4942—24no—
43C22×#117—31no—
44C2×#381—31no—
45C2×#380—31no—
46C2×#380D311yescharacteristic
47C22×#95—31no—
48#1444—32no—
49#1442—32no—
50#1436—32no—
51#1436—32no—
52C22⋊#95—31no—
53C22⋊#95—31no—
54C25×C3C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4—22no—
57C23×C3×C4—24no—
58C23×C3×C4—42no—
59C23×C3×C4C2×C414yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—22no—
65C23×Dic3—22no—
66C23×Dic3—22no—
67C23×Dic3—22no—
68C23×Dic3—22no—
69C23×Dic3—22no—
70C23×Dic3—24no—
71C23×Dic3—42no—
72C23×Dic3C2×C414yes—
73C23×Dic3C2×C414yes—
74C23×Dic3D414yes—
75C23×Dic3Q814yes—
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2312yes—
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2311yescharacteristic
81C23×Dic3C2311yescharacteristic
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—42no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D414yes—
91C2×#217—22no—
92C2×#217—22no—
93C2×#217—22no—
94C2×#217—22no—
95C2×#217—22no—
96C2×#217D414yes—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—42no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—24no—
105C2×C4×Dic3—24no—
106#877—232no—
107C3×#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—42no—
114C3⋊#95—42no—
115C3⋊#95C2×C414yes—
116C3⋊#95C2×C414yes—
117C3⋊#95C2×C414yes—
118C3⋊#95C2×C414yes—
119C24×C4—31no—
120C24×C4—31no—
121C24×C4C2×D311yescharacteristic
122C2×#117—64no—
123C2×#117—64no—
124C2×#117—64no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4C2×D312yes—
128C22×C22⋊C4—31no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4C2×D311yescharacteristic
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C42—32no—
133#381—64no—
134#380—24no—
135#380—64no—
136C2×#95—38no—
137C2×#95—31no—
138C2×#95—31no—
139C2×#95—31no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C2×#95C2×D312yes—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3C22×C412yes—
154C24×C3C2×D412yes—
155C24×C3C2×D412yes—
156C24×C3C2×D412yes—
157C24×C3C2×Q812yes—
158C24×C3#4112yes—
159C24×C3#4112yes—
160C24×C3C2411yescharacteristic
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—28no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—28no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—44no—
184C22×C3×C4—48no—
185C22×C3×C4—48no—
186C22×C3×C4C22×C412yes—
187C22×C3×C4C22×C412yes—
188C22×C3×C4C2×D414yes—
189C22×C3×C4#4114yes—
190C22×Dic3—28no—
191C22×Dic3—28no—
192C22×Dic3—28no—
193C22×Dic3—28no—
194C22×Dic3—28no—
195C22×Dic3—28no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—22no—
207C22×Dic3—28no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—21no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—28no—
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—22no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—22no—
226C22×Dic3—44no—
227C22×Dic3—48no—
228C22×Dic3—48no—
229C22×Dic3C4⋊C4116yescomplemented
230C22×Dic3C22×C412yes—
231C22×Dic3C22×C412yes—
232C22×Dic3C22×C412yes—
233C22×Dic3C22×C412yes—
234C22×Dic3C2×D414yes—
235C22×Dic3C2×D412yes—
236C22×Dic3C2×D412yes—
237C22×Dic3C2×Q812yes—
238C22×Dic3C2×Q812yes—
239C22×Dic3#4114yes—
240C3×C4⋊C4—48no—
241C3×C22⋊C4—28no—
242C3×C22⋊C4—44no—
243#217—28no—
244#217—44no—
245Dic3⋊C4—216no—
246Dic3⋊C4—48no—
247C4×Dic3—216no—
248C25C22×D311yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4—34no—
253C23×C4—34no—
254C23×C4—34no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—31no—
258C23×C4—32no—
259C23×C4—31no—
260C23×C4—34no—
261C23×C4—32no—
262C23×C4—42no—
263C23×C4—64no—
264C23×C4—62no—
265C23×C4—62no—
266C23×C4—62no—
267C23×C4—62no—
268C23×C4—62no—
269C23×C4—62no—
270C23×C4—122no—
271C23×C4C4×D314yes—
272C23×C4C22×D312yes—
273C23×C4C22×D311yescharacteristic
274C23×C4C22×D312yes—
275C23×C4C22×D311yescharacteristic
276#117—632no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—64no—
285C2×C4⋊C4—122no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—34no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4#6714yes—
298C2×C42—64no—
299C2×C42—64no—
300C2×C42—64no—
301#95—24no—
302#95—34no—
303#95—34no—
304#95—34no—
305#95—34no—
306#95—64no—
307#95—64no—
308#95—64no—
309#95—64no—
310#95—64no—
311#95—122no—
312#95—122no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3C2×C4⋊C414yes—
330C23×C3C4×D414yes—
331C23×C3C4×D414yes—
332C23×C3#12014yes—
333C23×C3C22⋊Q814yes—
334C23×C3C22⋊Q814yes—
335C23×C3#12314yes—
336C23×C3C23×C411yescharacteristic
337C23×C3C22×D411yescharacteristic
338C23×C3C22×D411yescharacteristic
339C23×C3C22×D411yescharacteristic
340C23×C3C22×Q811yescharacteristic
341C23×C3C2×#4111yescharacteristic
342C23×C3C2×#4112yes—
343C23×C3C2×#4111yescharacteristic
344C23×C3C2×#4112yes—
345C23×C3H(5,𝔽2)12yes—
346C23×C32−1+412yes—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—416no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4#11714yescomplemented
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—232no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—416no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3C2×C4⋊C414yes—
410C2×Dic3C2×C4⋊C414yes—
411C2×Dic3#11714yescomplemented
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24C2×C4×D312yes—
417C24#23512yes—
418C24D3×D412yes—
419C24D3×D412yes—
420C24D3×Q812yes—
421C24#23912yes—
422C24C2×#6712yes—
423C24C23×D311yescharacteristic
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—28no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—316no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—34no—
455C22×C4—32no—
456C22×C4—34no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—68no—
461C22×C4—68no—
462C22×C4—68no—
463C22×C4—68no—
464C22×C4—68no—
465C22×C4—68no—
466C22×C4—68no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—62no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—68no—
479C22×C4—61no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—61no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—62no—
493C22×C4—62no—
494C22×C4—62no—
495C22×C4—62no—
496C22×C4—124no—
497C22×C4—128no—
498C22×C4—128no—
499C22×C4C2×C4×D312yes—
500C22×C4C2×C4×D312yes—
501C22×C4#23714yes—
502C22×C4C2×#6714yes—
503C4⋊C4—48no—
504C4⋊C4—616no—
505C4⋊C4—128no—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C22⋊C4—68no—
509C22⋊C4—124no—
510C42—616no—
511C22×C3—24no—
512C22×C3—24no—
513C22×C3—24no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3—28no—
517C22×C3—28no—
518C22×C3—28no—
519C22×C3—28no—
520C22×C3—24no—
521C22×C3#38518yes—
522C22×C3C22×C4⋊C411yescharacteristic
523C22×C3C2×#11711yescharacteristic
524C22×C3C2×#11711yescharacteristic
525C22×C3C2×C4×D411yescharacteristic
526C22×C3C2×C4×D411yescharacteristic
527C22×C3#51811yescharacteristic
528C22×C3#51911yescharacteristic
529C22×C3C2×#12011yescharacteristic
530C22×C3C2×C22⋊Q811yescharacteristic
531C22×C3C2×C22⋊Q811yescharacteristic
532C22×C3C2×#12311yescharacteristic
533C22×C3#53212yes—
534C22×C3#53312yes—
535C22×C3#53712yes—
536C22×C3#54412yes—
537C22×C3#54612yes—
538C22×C3#54612yes—
539C22×C3#54712yes—
540C22×C3#54712yes—
541C22×C3#55112yes—
542C22×C3#55112yes—
543C22×C3#55212yes—
544C22×C3#55212yes—
545C3×C4—24no—
546C3×C4—42no—
547C3×C4—44no—
548C3×C4—82no—
549Dic3—24no—
550Dic3—24no—
551Dic3—24no—
552Dic3—42no—
553Dic3—44no—
554Dic3—82no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23D3×C4⋊C414yes—
572C23#89314yes—
573C23#89414yes—
574C23#89614yes—
575C23C4×#6714yes—
576C23#93714yes—
577C23#94614yes—
578C23C22×C4×D311yescharacteristic
579C23C2×#23511yescharacteristic
580C23C2×D3×D411yescharacteristic
581C23C2×D3×D411yescharacteristic
582C23C2×#23712yes—
583C23#100412yes—
584C23C2×D3×Q811yescharacteristic
585C23C2×#23911yescharacteristic
586C23D3×#4112yes—
587C23#101012yes—
588C23C22×#6711yescharacteristic
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—34no—
600C2×C4—34no—
601C2×C4—34no—
602C2×C4—416no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—64no—
617C2×C4—62no—
618C2×C4—632no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—64no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—1216no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4#87714yescomplemented
654C2×C3—28no—
655C2×C3#228511yescharacteristic
656C2×C3C2×#38511yescharacteristic
657C2×C3#231611yescharacteristic
658C2×C3#234211yescharacteristic
659C2×C3#234211yescharacteristic
660C2×C3#234411yescharacteristic
661C2×C3#234411yescharacteristic
662C2×C3#243012yes—
663C2×C3#243812yes—
664C2×C3#243812yes—
665C2×C3#249812yes—
666C22—24no—
667C22—24no—
668C22—28no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—28no—
673C22—24no—
674C22—28no—
675C22—28no—
676C22#498918yes—
677C22C2×#87711yescharacteristic
678C22#548611yescharacteristic
679C22#549512yes—
680C22C2×D3×C4⋊C411yescharacteristic
681C22C2×#89311yescharacteristic
682C22C2×#89411yescharacteristic
683C22C2×#89611yescharacteristic
684C22#551611yescharacteristic
685C22D3×#11711yescharacteristic
686C22#553112yes—
687C22#559612yes—
688C22#559712yes—
689C22#561012yes—
690C22#561112yes—
691C22#562212yes—
692C22#562312yes—
693C22#563712yes—
694C22#564212yes—
695C22C2×C4×#6711yescharacteristic
696C22C2×#93711yescharacteristic
697C22#580612yes—
698C22C2×#94611yescharacteristic
699C22#582412yes—
700C4—24no—
701C4—42no—
702C4—44no—
703C4—64no—
704C4—64no—
705C4—64no—
706C4—82no—
707C4—122no—
708C4—124no—
709C4—242no—
710C3#1387811yescharacteristic, complemented, Sylow
711C2—28no—
712C2#7386411yescharacteristic
713C2#7386911yescharacteristic
714C2#7387811yescharacteristic
715C2#7389712yes—
716C2#7392612yes—
717C2#7400911yescharacteristic
718C2#8209811yescharacteristic
719C2C2×#498911yescharacteristic
720C2#8223811yescharacteristic
721C2#8265112yes—
722C2#8276212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#380C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#380D3
hypocenterC3#13878
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
7. #71568C2semidirect
8. #71564C2semidirect
10. #69414C2semidirect
11. #69406C2semidirect
29. C2×C3⋊#95C4semidirect
229. C22×Dic3C4⋊C4semidirect
371. C2×C3×C4#117semidirect
411. C2×Dic3#117semidirect
653. C2×C4#877semidirect
710. C3#13878semidirect

Maximal subgroups

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

subgroupindex
C22×#8772
C2×C3×#3802
C22×C3⋊#952
C2×#49432
C2×#49422
#721272
#715682
#715642
#715442
#694142
#694062
#138783