Finite Group Databasebeta

← #10732185

Subgroups — #10732185

Order 768.

1418 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
1C2×C3×#388C211yesmaximal, characteristic
2C2×#4986C211yesmaximal, characteristic, complemented
3C2×#4949C211yesmaximal, characteristic
4C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#71565C211yesmaximal, characteristic, complemented
6#71549C212yesmaximal
7#71546C211yesmaximal, characteristic
8#71545C212yesmaximal
9#71534C211yesmaximal, characteristic, complemented
10#69413C212yesmaximal, complemented
11#69406C212yesmaximal, complemented
12#15175—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2212yes—
24C3×#388—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37#4986—24no—
38#4949—24no—
39C4×C4⋊Dic3—24no—
40C2×#400—31no—
41C2×#399—31no—
42C2×#388D311yescharacteristic
43C2×C4×C4⋊C4—31no—
44#1442—32no—
45#1441—31no—
46#1440—32no—
47#1439—31no—
48#1437—32no—
49#1436—32no—
50#1434—31no—
51C25×C3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C42—24no—
74C2×C3×C42D414yes—
75C2×#217—24no—
76C2×#217—24no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—42no—
80C2×#217—42no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3D414yes—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—42no—
89C2×C4×Dic3—24no—
90C2×C4×Dic3—42no—
91C2×C4×Dic3—42no—
92C3×#95—22no—
93C3×#95—22no—
94C3⋊#95—24no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—42no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—31no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C22⋊C4—32no—
109C22×C22⋊C4—32no—
110C22×C22⋊C4C2×D311yescharacteristic
111C22×C22⋊C4C2×D311yescharacteristic
112C22×C22⋊C4C2×D311yescharacteristic
113C22×C42—32no—
114C22×C42C2×D311yescharacteristic
115#400—64no—
116#399—64no—
117#388—24no—
118C4×C4⋊C4—64no—
119C2×#95—31no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95C2×D311yescharacteristic
130C2×#95C2×D311yescharacteristic
131C24×C3—21no—
132C24×C3—21no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—21no—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3#4112yes—
142C24×C3C2411yescharacteristic
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4114yes—
175C22×Dic3—22no—
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—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
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×Dic3C2×D414yes—
201C22×Dic3#4114yes—
202C22×Dic3#4114yes—
203C3×C4⋊C4—44no—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—44no—
207C3×C42—28no—
208#217—48no—
209#217—48no—
210C4⋊Dic3—28no—
211C4⋊Dic3—28no—
212Dic3⋊C4—48no—
213C4×Dic3—48no—
214C25C22×D311yescharacteristic
215C23×C4—32no—
216C23×C4—32no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4C22×D311yescharacteristic
220C23×C4C22×D311yescharacteristic
221C23×C4C22×D311yescharacteristic
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—122no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—64no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—122no—
249C2×C22⋊C4—122no—
250C2×C42—24no—
251C2×C42—64no—
252C2×C42—122no—
253C2×C42—122no—
254C2×C42#6714yes—
255#95—22no—
256#95—22no—
257#95—64no—
258#95—64no—
259#95—64no—
260#95—64no—
261#95—64no—
262#95—62no—
263#95—62no—
264#95—64no—
265#95—64no—
266#95—122no—
267C23×C3—22no—
268C23×C3—24no—
269C23×C3—22no—
270C23×C3—24no—
271C23×C3—22no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—41no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3C22×D411yescharacteristic
291C23×C3C22×D411yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3C2×#4112yes—
297C23×C3C2×#4112yes—
298C23×C3H(5,𝔽2)12yes—
299C23×C32−1+411yescharacteristic
300C23×C32−1+411yescharacteristic
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4#12114yes—
332C2×C3×C4#12314yes—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—42no—
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×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24#23512yes—
372C24C23×D311yescharacteristic
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4#23512yes—
427C22×C4D3×D412yes—
428C22×C4#23714yes—
429C22×C4#23712yes—
430C22×C4C2×#6712yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C4⋊C4—44no—
434C4⋊C4—68no—
435C4⋊C4—68no—
436C4⋊C4—128no—
437C22⋊C4—28no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—128no—
441C22⋊C4—128no—
442C42—28no—
443C42—128no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—82no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—82no—
455C22×C3C2×#12111yescharacteristic
456C22×C3C2×#12311yescharacteristic
457C22×C3#52511yescharacteristic
458C22×C3#52511yescharacteristic
459C22×C3#52912yes—
460C22×C3#52912yes—
461C22×C3#52912yes—
462C22×C3#53211yescharacteristic
463C22×C3#53212yes—
464C22×C3#53511yescharacteristic
465C22×C3#53611yescharacteristic
466C22×C3#53912yes—
467C22×C3#53912yes—
468C22×C3#54612yes—
469C22×C3#54811yescharacteristic
470C22×C3#54811yescharacteristic
471C22×C3#55112yes—
472C22×C3#55212yes—
473C22×C3#55211yescharacteristic
474C22×C3#55211yescharacteristic
475C22×C3#55312yes—
476C22×C3#55612yes—
477C22×C3#55912yes—
478C3×C4—24no—
479C3×C4—24no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—81no—
483C3×C4—81no—
484Dic3—44no—
485Dic3—44no—
486Dic3—44no—
487Dic3—82no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23C2×#23511yescharacteristic
512C23C2×#23511yescharacteristic
513C23C2×D3×D411yescharacteristic
514C23C2×#23712yes—
515C23C2×#23711yescharacteristic
516C23D3×#4112yes—
517C23D3×#4112yes—
518C23#100912yes—
519C23#101011yescharacteristic
520C23#101011yescharacteristic
521C23C22×#6711yescharacteristic
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
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#93514yes—
581C2×C4#93814yes—
582C2×C3—82no—
583C2×C3#239511yescharacteristic
584C2×C3#253611yescharacteristic
585C2×C3#254411yescharacteristic
586C2×C3#255812yes—
587C2×C3#256012yes—
588C2×C3#257512yes—
589C2×C3#259612yes—
590C2×C3#259811yescharacteristic
591C2×C3#260411yescharacteristic
592C2×C3#262311yescharacteristic
593C2×C3#285411yescharacteristic
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22#552712yes—
606C22#553012yes—
607C22#553812yes—
608C22#554111yescharacteristic
609C22#554111yescharacteristic
610C22#555211yescharacteristic
611C22#555211yescharacteristic
612C22#555712yes—
613C22#559612yes—
614C22#560012yes—
615C22#566512yes—
616C22#566512yes—
617C22#567612yes—
618C22#568112yes—
619C22#570212yes—
620C22#570812yes—
621C22#579111yescharacteristic
622C22C2×#93511yescharacteristic
623C22C2×#93811yescharacteristic
624C22#582511yescharacteristic
625C22#582811yescharacteristic
626C22#583011yescharacteristic
627C22#583211yescharacteristic
628C4—24no—
629C4—24no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—124no—
635C4—124no—
636C4—124no—
637C4—242no—
638C3#1517511yescharacteristic, complemented, Sylow
639C2—82no—
640C2#7560112yes—
641C2#7560212yes—
642C2#7560912yes—
643C2#7563712yes—
644C2#8228411yescharacteristic
645C2#8234611yescharacteristic
646C2#8237511yescharacteristic
647C2#8269911yescharacteristic
648C2#8286611yescharacteristic
649C2#8296911yescharacteristic
650C2#8308811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#388C2
prime radicalC25×C3C23
hypercenterC2×#388D3
hypocenterC3#15175
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#4986C2semidirect
4. C2×C4×C4⋊Dic3C2semidirect
5. #71565C2semidirect
9. #71534C2semidirect
10. #69413C2semidirect
11. #69406C2semidirect
638. C3#15175semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3882
C2×#49862
C2×#49492
C2×C4×C4⋊Dic32
#715652
#715492
#715462
#715452
#715342
#694132
#694062
#151753