Finite Group Databasebeta

← #10717751

Subgroups — #10717751

Order 768.

1846 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#926C211yesmaximal, characteristic
2C2×#4945C212yesmaximal
3C3×C22⋊#95C211yesmaximal, characteristic
4#72127C211yesmaximal, characteristic
5#71549C212yesmaximal
6#69411C212yesmaximal, complemented
7#69407C212yesmaximal, complemented
8Dic3⋊#95C211yesmaximal, characteristic, complemented
9#69403C211yesmaximal, characteristic, complemented
10#69401C211yesmaximal, characteristic, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#13943—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×#217C2212yes—
16C22×#217C2211yescharacteristic
17C22×#217C2211yescharacteristic
18C2×#926—28no—
19C2×#926C418yescomplemented
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4945—28no—
39C22×#117—31no—
40C2×#388—32no—
41#1441—32no—
42#1440—32no—
43#1436—32no—
44C4⋊#95—31no—
45#1434—31no—
46#1434—31no—
47C22⋊#95—31no—
48C22⋊#95D311yescharacteristic
49C4×#95—31no—
50C25×C3C2311yescharacteristic
51C23×C3×C4—24no—
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—24no—
55C23×C3×C4C2×C414yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—42no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217—42no—
73C2×#217C2×C414yes—
74C2×#217C2×C414yes—
75C2×#217D418yes—
76#926—232no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3C2×C414yes—
80C2×C4⋊Dic3C2×C414yes—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—42no—
84C2×C4×Dic3—24no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—42no—
89C2×C4×Dic3C2×C414yes—
90C2×C4×Dic3C2×C414yes—
91C3×#95—22no—
92C3×#95—24no—
93C3×#95—22no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—42no—
102C3⋊#95—42no—
103C24×C4C2×D311yescharacteristic
104C2×#117—38no—
105C2×#117—68no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—32no—
109C22×C22⋊C4—31no—
110C22×C22⋊C4—31no—
111C22×C22⋊C4—32no—
112C22×C22⋊C4C2×D312yes—
113C22×C42—31no—
114C22×C42—31no—
115C22×C42—32no—
116#388—68no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—31no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D311yescharacteristic
128C2×#95C2×D312yes—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—21no—
132C24×C3—21no—
133C24×C3—22no—
134C24×C3—21no—
135C24×C3—21no—
136C24×C3—21no—
137C24×C3—21no—
138C24×C3C22×C412yes—
139C24×C3C2×D414yes—
140C24×C3C2411yescharacteristic
141C22×C3×C4—216no—
142C22×C3×C4—216no—
143C22×C3×C4—28no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—28no—
150C22×C3×C4—28no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
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×C4C22×C412yes—
165C22×C3×C4C22×C412yes—
166C22×C3×C4#4114yes—
167C22×C3×C4#4112yes—
168C22×C3×C4#4112yes—
169C22×C3×C4#4114yes—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3C22×C412yes—
198C22×Dic3C22×C412yes—
199C22×Dic3C22×C412yes—
200C22×Dic3C22×C412yes—
201C22×Dic3C2×D414yes—
202C22×Dic3C2×D414yes—
203C22×Dic3#4114yes—
204C3×C22⋊C4—48no—
205#217—28no—
206#217—48no—
207#217C22⋊C4116yes—
208C4⋊Dic3—28no—
209C4⋊Dic3—28no—
210Dic3⋊C4—48no—
211C4×Dic3—28no—
212C4×Dic3—28no—
213C4×Dic3—48no—
214C25C22×D311yescharacteristic
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—32no—
224C23×C4—32no—
225C23×C4C4×D314yes—
226C23×C4C22×D312yes—
227C23×C4C22×D312yes—
228C23×C4C22×D311yescharacteristic
229C23×C4C22×D311yescharacteristic
230#117—632no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—122no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—34no—
241C2×C22⋊C4—38no—
242C2×C22⋊C4—34no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4—122no—
248C2×C42—34no—
249C2×C42—34no—
250C2×C42—64no—
251C2×C42—64no—
252C2×C42—64no—
253C2×C42—64no—
254C2×C42—122no—
255#95—22no—
256#95—24no—
257#95—22no—
258#95—64no—
259#95—64no—
260#95—64no—
261#95—64no—
262#95—62no—
263#95—64no—
264#95—62no—
265#95—122no—
266#95—122no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—24no—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C2×C22⋊C414yes—
290C23×C3C23×C411yescharacteristic
291C23×C3C22×D412yes—
292C23×C3C2×#4112yes—
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3C2×#4112yes—
297C23×C3H(5,𝔽2)11yescharacteristic
298C23×C3H(5,𝔽2)11yescharacteristic
299C23×C32−1+411yescharacteristic
300C23×C32−1+411yescharacteristic
301C2×C3×C4—232no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4#11714yes—
325C2×C3×C4#11714yes—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3C2×C22⋊C414yes—
359C2×Dic3C2×C22⋊C414yes—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24C2×C4×D312yes—
370C24D3×D414yes—
371C24C23×D311yescharacteristic
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—32no—
396C22×C4—32no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—32no—
400C22×C4—32no—
401C22×C4—34no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4C2×C4×D312yes—
430C22×C4C2×C4×D312yes—
431C22×C4#23514yes—
432C22×C4#23714yes—
433C22×C4#23712yes—
434C22×C4#23712yes—
435C4⋊C4—68no—
436C4⋊C4—68no—
437C4⋊C4—128no—
438C22⋊C4—316no—
439C22⋊C4—48no—
440C22⋊C4—68no—
441C22⋊C4—128no—
442C42—68no—
443C42—68no—
444C42—128no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—24no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3C22×C22⋊C411yescharacteristic
457C22×C3C2×#11711yescharacteristic
458C22×C3C2×#11711yescharacteristic
459C22×C3C4×#4111yescharacteristic
460C22×C3C4×#4111yescharacteristic
461C22×C3#52011yescharacteristic
462C22×C3#52011yescharacteristic
463C22×C3#52512yes—
464C22×C3#52512yes—
465C22×C3#52912yes—
466C22×C3#52912yes—
467C22×C3#54011yescharacteristic
468C22×C3#54111yescharacteristic
469C22×C3#54211yescharacteristic
470C22×C3#54211yescharacteristic
471C22×C3#54612yes—
472C22×C3#54612yes—
473C22×C3#54712yes—
474C22×C3#54712yes—
475C22×C3#55212yes—
476C22×C3#55312yes—
477C22×C3#55512yes—
478C22×C3#55612yes—
479C3×C4—24no—
480C3×C4—24no—
481C3×C4—44no—
482C3×C4—82no—
483Dic3—24no—
484Dic3—24no—
485Dic3—42no—
486Dic3—42no—
487Dic3—44no—
488Dic3—82no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23D3×C22⋊C414yes—
512C23C22×C4×D311yescharacteristic
513C23C2×#23512yes—
514C23C2×D3×D412yes—
515C23C2×#23712yes—
516C23C2×#23711yescharacteristic
517C23C2×#23711yescharacteristic
518C23#100411yescharacteristic
519C23#100711yescharacteristic
520C23D3×#4112yes—
521C23#100911yescharacteristic
522C23#101011yescharacteristic
523C2×C4—232no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—34no—
536C2×C4—34no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
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—64no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—64no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4#87714yes—
581C2×C4#87714yes—
582C2×C3—44no—
583C2×C3#228011yescharacteristic
584C2×C3#229811yescharacteristic
585C2×C3#231411yescharacteristic
586C2×C3#233911yescharacteristic
587C2×C3#233911yescharacteristic
588C2×C3#234611yescharacteristic
589C2×C3#234711yescharacteristic
590C2×C3#247012yes—
591C2×C3#247412yes—
592C2×C3#247612yes—
593C2×C3#247812yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22C4×#23511yescharacteristic
606C22C2×#87711yescharacteristic
607C22C2×#87711yescharacteristic
608C22C2×D3×C22⋊C411yescharacteristic
609C22#549012yes—
610C22#550411yescharacteristic
611C22#550712yes—
612C22#552211yescharacteristic
613C22#552311yescharacteristic
614C22#553812yes—
615C22#554212yes—
616C22#555712yes—
617C22#556512yes—
618C22#558612yes—
619C22#558612yes—
620C22#560311yescharacteristic
621C22#561612yes—
622C22#561911yescharacteristic
623C22#562511yescharacteristic
624C22#563912yes—
625C22#564311yescharacteristic
626C22#565412yes—
627C22#565512yes—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—64no—
632C4—64no—
633C4—82no—
634C4—122no—
635C4—122no—
636C4—124no—
637C4—242no—
638C3#1394311yescharacteristic, complemented, Sylow
639C2—44no—
640C2#7336811yescharacteristic
641C2#7344011yescharacteristic
642C2#7346011yescharacteristic
643C2#7381711yescharacteristic
644C2#7386811yescharacteristic
645C2#7388111yescharacteristic
646C2#7412111yescharacteristic
647C2#7443912yes—
648C2#7444912yes—
649C2#7472912yes—
650C2#7473012yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C22⋊#95C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#13943
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #69411C2semidirect
7. #69407C2semidirect
8. Dic3⋊#95C2semidirect
9. #69403C2semidirect
10. #69401C2semidirect
11. Dic3×#95C2semidirect
19. C2×#926C4semidirect
638. C3#13943semidirect

Maximal subgroups

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

subgroupindex
C22×#9262
C2×#49452
C3×C22⋊#952
#721272
#715492
#694112
#694072
Dic3⋊#952
#694032
#694012
Dic3×#952
#139433