Finite Group Databasebeta

← #10717713

Subgroups — #10717713

Order 768.

1862 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#877C211yesmaximal, characteristic
2C2×C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×#4651C211yesmaximal, characteristic, complemented
4C3×#1437C211yesmaximal, characteristic
5#71547C212yesmaximal
6#71545C211yesmaximal, characteristic
7#71544C212yesmaximal
8#71536C211yesmaximal, characteristic, complemented
9#69409C212yesmaximal, complemented
10#69405C211yesmaximal, characteristic, complemented
11#69401C212yesmaximal, complemented
12#13938—31nomaximal, Sylow
13C24×Dic3C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×#877—24no—
26C2×#877—24no—
27C2×#877C418yescomplemented
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C4×Dic3⋊C4—24no—
40#4651—24no—
41C22×#117—31no—
42C2×#383—31no—
43C2×C4×C4⋊C4—31no—
44#1441—32no—
45#1439—32no—
46#1437—31no—
47#1437D311yescharacteristic
48#1436—31no—
49#1436—31no—
50#1434—32no—
51C22⋊#95—32no—
52C25×C3C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—24no—
58C23×Dic3—22no—
59C23×Dic3—22no—
60C23×Dic3—24no—
61C23×Dic3C2×C414yes—
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—42no—
73C2×C3×C22⋊C4C2×C414yes—
74C2×C3×C42—22no—
75C2×C3×C42—42no—
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—42no—
81C2×#217C2×C414yes—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4C2×C414yes—
90C2×Dic3⋊C4C2×C414yes—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—42no—
95C2×C4×Dic3C2×C414yes—
96C2×C4×Dic3C2×C414yes—
97C2×C4×Dic3D414yes—
98C2×C4×Dic3D414yes—
99#877—232no—
100C3×#95—22no—
101C3×#95—24no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—42no—
110C3⋊#95—42no—
111C24×C4—31no—
112C2×#117—38no—
113C2×#117—64no—
114C2×#117—64no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—32no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4C2×D312yes—
121C22×C22⋊C4C2×D311yescharacteristic
122C22×C42—31no—
123C22×C42—31no—
124C22×C42—31no—
125C22×C42C2×D311yescharacteristic
126#383—64no—
127C4×C4⋊C4—64no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D312yes—
138C2×#95C2×D311yescharacteristic
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3C22×C412yes—
147C24×C3#4114yes—
148C24×C3C2411yescharacteristic
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
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—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4114yes—
173C22×Dic3—216no—
174C22×Dic3—216no—
175C22×Dic3—28no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—28no—
184C22×Dic3—28no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3C22×C412yes—
203C22×Dic3C22×C412yes—
204C22×Dic3C22×C412yes—
205C22×Dic3C22×C412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D412yes—
210C22×Dic3#4114yes—
211C22×Dic3#4114yes—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—48no—
214C3×C42—44no—
215#217—28no—
216#217—48no—
217Dic3⋊C4—216no—
218Dic3⋊C4—28no—
219Dic3⋊C4—28no—
220C4×Dic3—28no—
221C4×Dic3—44no—
222C4×Dic3C22⋊C4116yescomplemented
223C25C22×D311yescharacteristic
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—34no—
228C23×C4—31no—
229C23×C4—32no—
230C23×C4—32no—
231C23×C4—62no—
232C23×C4—62no—
233C23×C4—64no—
234C23×C4—64no—
235C23×C4C22×D312yes—
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239#117—632no—
240C2×C4⋊C4—34no—
241C2×C4⋊C4—34no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—64no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—122no—
260C2×C22⋊C4C4×D314yes—
261C2×C42—22no—
262C2×C42—34no—
263C2×C42—34no—
264C2×C42—34no—
265C2×C42—34no—
266C2×C42—42no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—62no—
270C2×C42—122no—
271#95—22no—
272#95—24no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—62no—
280#95—122no—
281#95—122no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3#11714yes—
303C23×C3C23×C411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C22×D411yescharacteristic
306C23×C3C2×#4112yes—
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3C2×#4112yes—
310C23×C3H(5,𝔽2)12yes—
311C23×C32−1+412yes—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4C2×C22⋊C414yes—
336C2×Dic3—232no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3C2×C22⋊C414yes—
369C2×Dic3C2×C22⋊C414yes—
370C2×Dic3#11714yes—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24C2×C4×D312yes—
379C24#23514yes—
380C24C23×D311yescharacteristic
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—32no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—34no—
410C22×C4—68no—
411C22×C4—68no—
412C22×C4—616no—
413C22×C4—616no—
414C22×C4—68no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4C2×C4×D312yes—
440C22×C4C2×C4×D312yes—
441C22×C4D3×D412yes—
442C22×C4D3×D412yes—
443C22×C4#23714yes—
444C4⋊C4—68no—
445C4⋊C4—616no—
446C4⋊C4—68no—
447C22⋊C4—28no—
448C22⋊C4—48no—
449C22⋊C4—68no—
450C22⋊C4—128no—
451C42—316no—
452C42—44no—
453C42—68no—
454C42—124no—
455C22×C3—28no—
456C22×C3—24no—
457C22×C3—42no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—42no—
464C22×C3C22×C22⋊C411yescharacteristic
465C22×C3C2×#11711yescharacteristic
466C22×C3C2×#11711yescharacteristic
467C22×C3C4×#4112yes—
468C22×C3#52012yes—
469C22×C3#52512yes—
470C22×C3#52512yes—
471C22×C3#52512yes—
472C22×C3#53211yescharacteristic
473C22×C3#53211yescharacteristic
474C22×C3#53511yescharacteristic
475C22×C3#53611yescharacteristic
476C22×C3#53712yes—
477C22×C3#55112yes—
478C22×C3#55112yes—
479C22×C3#55212yes—
480C22×C3#55212yes—
481C22×C3#55312yes—
482C22×C3#55312yes—
483C22×C3#55612yes—
484C22×C3#55612yes—
485C3×C4—24no—
486C3×C4—44no—
487C3×C4—42no—
488C3×C4—82no—
489Dic3—24no—
490Dic3—24no—
491Dic3—24no—
492Dic3—42no—
493Dic3—44no—
494Dic3—82no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23#87214yes—
516C23C22×C4×D311yescharacteristic
517C23C2×#23512yes—
518C23C2×D3×D411yescharacteristic
519C23C2×D3×D411yescharacteristic
520C23C2×#23712yes—
521C23#100412yes—
522C23D3×#4112yes—
523C23D3×#4112yes—
524C23#101012yes—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—34no—
535C2×C4—34no—
536C2×C4—34no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—632no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—121no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—121no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4D3×C22⋊C414yes—
584C2×C3—44no—
585C2×C3#228011yescharacteristic
586C2×C3#229711yescharacteristic
587C2×C3#231111yescharacteristic
588C2×C3#234312yes—
589C2×C3#234712yes—
590C2×C3#259812yes—
591C2×C3#259912yes—
592C2×C3#260112yes—
593C2×C3#260112yes—
594C22—24no—
595C22—28no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22C2×#87211yescharacteristic
604C22C2×D3×C22⋊C411yescharacteristic
605C22#549512yes—
606C22D3×#11711yescharacteristic
607C22#553112yes—
608C22C4×#23712yes—
609C22#554112yes—
610C22#554612yes—
611C22#554712yes—
612C22#554712yes—
613C22#555612yes—
614C22#559211yescharacteristic
615C22#559712yes—
616C22#560112yes—
617C22#560612yes—
618C22#564811yescharacteristic
619C22#566112yes—
620C22#566811yescharacteristic
621C22#567112yes—
622C22#570212yes—
623C22#571611yescharacteristic
624C4—24no—
625C4—42no—
626C4—44no—
627C4—64no—
628C4—64no—
629C4—64no—
630C4—82no—
631C4—124no—
632C4—122no—
633C4—242no—
634C3#1393811yescharacteristic, complemented, Sylow
635C2—44no—
636C2#7345512yes—
637C2#7356111yescharacteristic
638C2#7466211yescharacteristic
639C2#7477612yes—
640C2#7547412yes—
641C2#7564212yes—
642C2#7564312yes—
643C2#7629011yescharacteristic
644C2#7636912yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1437C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenter#1437D3
hypocenterC3#13938
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic3⋊C4C2semidirect
3. C2×#4651C2semidirect
8. #71536C2semidirect
9. #69409C2semidirect
10. #69405C2semidirect
11. #69401C2semidirect
27. C2×#877C4semidirect
222. C4×Dic3C22⋊C4semidirect
634. C3#13938semidirect

Maximal subgroups

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

subgroupindex
C22×#8772
C2×C4×Dic3⋊C42
C2×#46512
C3×#14372
#715472
#715452
#715442
#715362
#694092
#694052
#694012
#139383