Finite Group Databasebeta

← #10729283

Subgroups — #10729283

Order 768.

1614 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#890C211yesmaximal, characteristic, complemented
2C2×#4945C211yesmaximal, characteristic
3C2×#4951C212yesmaximal
4C3×#1439C211yesmaximal, characteristic
5#71566C211yesmaximal, characteristic, complemented
6#71547C212yesmaximal
7#71546C211yesmaximal, characteristic
8#69413C212yesmaximal, complemented
9#69410C211yesmaximal, characteristic, complemented
10#69407C212yesmaximal, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#14933—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×#217C2211yescharacteristic
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#890—28no—
26C2×#890—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4951—28no—
39#4945—24no—
40C22×#125—31no—
41C2×#388—32no—
42C2×#388—31no—
43#1442—32no—
44#1441—31no—
45#1441—31no—
46#1439—31no—
47#1439—32no—
48#1439D311yescharacteristic
49#1436—32no—
50C4×#95—31no—
51C25×C3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—24no—
75C2×#217—24no—
76C2×#217—24no—
77C2×#217—42no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3—24no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3D418yes—
83C2×Dic3⋊C4—24no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—24no—
91C2×C4×Dic3—24no—
92C2×C4×Dic3—42no—
93C2×C4×Dic3D414yes—
94#890—232no—
95C3×#95—22no—
96C3×#95—24no—
97C3⋊#95—24no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C2×#125—64no—
107C2×#125—68no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C22⋊C4—31no—
114C22×C22⋊C4—31no—
115C22×C22⋊C4—32no—
116C22×C22⋊C4C2×D312yes—
117C22×C22⋊C4C2×D311yescharacteristic
118C22×C42—31no—
119C22×C42—32no—
120#388—64no—
121#388—68no—
122C2×#95—31no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D312yes—
133C24×C3—21no—
134C24×C3—22no—
135C24×C3—21no—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3#4112yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
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—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4114yes—
168C22×C3×C4#4112yes—
169C22×Dic3—22no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3C2×D412yes—
200C22×Dic3C2×D414yes—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D414yes—
203C22×Dic3#4112yes—
204C22×Dic3#4114yes—
205C3×C4⋊C4—28no—
206C3×C22⋊C4—216no—
207C3×C22⋊C4—44no—
208#217—28no—
209#217—48no—
210#217—44no—
211C4⋊Dic3—28no—
212C4⋊Dic3—216no—
213Dic3⋊C4—216no—
214Dic3⋊C4—44no—
215C4×Dic3—28no—
216C4×Dic3—48no—
217C25C22×D311yescharacteristic
218C23×C4—31no—
219C23×C4—32no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—31no—
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D312yes—
226C23×C4C22×D312yes—
227C23×C4C22×D311yescharacteristic
228#125—632no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—38no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—64no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4—122no—
256C2×C42—34no—
257C2×C42—64no—
258C2×C42—64no—
259C2×C42—64no—
260C2×C42—122no—
261#95—22no—
262#95—24no—
263#95—64no—
264#95—64no—
265#95—64no—
266#95—62no—
267#95—62no—
268#95—62no—
269#95—64no—
270#95—64no—
271#95—122no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—41no—
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D412yes—
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4111yescharacteristic
299C23×C3C2×#4112yes—
300C23×C3C2×#4111yescharacteristic
301C23×C3C2×#4111yescharacteristic
302C23×C3C2×#4112yes—
303C23×C3H(5,𝔽2)11yescharacteristic
304C23×C32−1+411yescharacteristic
305C23×C32−1+412yes—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
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—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4#12414yes—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—22no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3#12014yes—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24#23712yes—
377C24C23×D311yescharacteristic
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
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—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—32no—
398C22×C4—34no—
399C22×C4—32no—
400C22×C4—34no—
401C22×C4—32no—
402C22×C4—34no—
403C22×C4—61no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4#23512yes—
434C22×C4D3×D414yes—
435C22×C4D3×D412yes—
436C22×C4#23714yes—
437C22×C4#23712yes—
438C4⋊C4—28no—
439C4⋊C4—616no—
440C4⋊C4—616no—
441C4⋊C4—68no—
442C4⋊C4—124no—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—68no—
446C22⋊C4—124no—
447C22⋊C4—128no—
448C42—68no—
449C42—128no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#52512yes—
463C22×C3#52511yescharacteristic
464C22×C3C2×#12411yescharacteristic
465C22×C3#52912yes—
466C22×C3#52912yes—
467C22×C3#53211yescharacteristic
468C22×C3#53211yescharacteristic
469C22×C3#53212yes—
470C22×C3#53611yescharacteristic
471C22×C3#53711yescharacteristic
472C22×C3#53912yes—
473C22×C3#54112yes—
474C22×C3#54612yes—
475C22×C3#54712yes—
476C22×C3#54812yes—
477C22×C3#54812yes—
478C22×C3#55211yescharacteristic
479C22×C3#55212yes—
480C22×C3#55212yes—
481C22×C3#55211yescharacteristic
482C22×C3#55311yescharacteristic
483C22×C3#55311yescharacteristic
484C3×C4—24no—
485C3×C4—44no—
486C3×C4—44no—
487C3×C4—81no—
488Dic3—24no—
489Dic3—42no—
490Dic3—42no—
491Dic3—44no—
492Dic3—44no—
493Dic3—81no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23C2×#23511yescharacteristic
518C23C2×D3×D411yescharacteristic
519C23C2×D3×D412yes—
520C23C2×#23711yescharacteristic
521C23C2×#23711yescharacteristic
522C23C2×#23712yes—
523C23#100711yescharacteristic
524C23D3×#4111yescharacteristic
525C23D3×#4112yes—
526C23#100911yescharacteristic
527C23#101012yes—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—34no—
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—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—121no—
579C2×C4—122no—
580C2×C4—121no—
581C2×C4—122no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—122no—
587C2×C4#88514yes—
588C2×C3—82no—
589C2×C3#238811yescharacteristic
590C2×C3#243011yescharacteristic
591C2×C3#243611yescharacteristic
592C2×C3#246012yes—
593C2×C3#247512yes—
594C2×C3#251912yes—
595C2×C3#255812yes—
596C2×C3#256211yescharacteristic
597C2×C3#261111yescharacteristic
598C2×C3#266411yescharacteristic
599C2×C3#286111yescharacteristic
600C22—44no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—82no—
611C22C2×#88511yescharacteristic
612C22#551311yescharacteristic
613C22#553011yescharacteristic
614C22#555212yes—
615C22#555712yes—
616C22#558612yes—
617C22D3×#12011yescharacteristic
618C22#559612yes—
619C22#559812yes—
620C22#560012yes—
621C22#560611yescharacteristic
622C22#561811yescharacteristic
623C22#562512yes—
624C22#562911yescharacteristic
625C22#563112yes—
626C22#563911yescharacteristic
627C22#564811yescharacteristic
628C22#565012yes—
629C22#566512yes—
630C22#566811yescharacteristic
631C22#567612yes—
632C22#570812yes—
633C22#571611yescharacteristic
634C4—24no—
635C4—44no—
636C4—44no—
637C4—64no—
638C4—81no—
639C4—124no—
640C4—124no—
641C4—122no—
642C4—122no—
643C4—241no—
644C3#1493311yescharacteristic, complemented, Sylow
645C2—82no—
646C2#7394011yescharacteristic
647C2#7444612yes—
648C2#7470111yescharacteristic
649C2#7560112yes—
650C2#7642611yescharacteristic
651C2#7666511yescharacteristic
652C2#7686411yescharacteristic
653C2#7691412yes—
654C2#7730912yes—
655C2#7753711yescharacteristic
656C2#7805911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1439C2
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#14933
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#890C2semidirect
5. #71566C2semidirect
8. #69413C2semidirect
9. #69410C2semidirect
10. #69407C2semidirect
11. Dic3×#95C2semidirect
644. C3#14933semidirect

Maximal subgroups

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

subgroupindex
C22×#8902
C2×#49452
C2×#49512
C3×#14392
#715662
#715472
#715462
#694132
#694102
#694072
Dic3×#952
#149333