Finite Group Databasebeta

← #10732183

Subgroups — #10732183

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×C4×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#4993C211yesmaximal, characteristic
3C2×#4985C211yesmaximal, characteristic, complemented
4C2×#4940C211yesmaximal, characteristic
5#71570C211yesmaximal, characteristic
6#71566C211yesmaximal, characteristic, complemented
7#71537C212yesmaximal
8#71536C212yesmaximal, complemented
9#71534C211yesmaximal, characteristic, complemented
10#69429C212yesmaximal
11#69413C212yesmaximal, complemented
12#15175—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C3×C42C2212yes—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C3×C4×C4⋊C4—24no—
25C2×C3×#95C2212yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2211yescharacteristic
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#4993—24no—
38#4985—24no—
39#4940—24no—
40C2×#400—31no—
41C2×#399—31no—
42C2×#388—31no—
43C2×C4×C4⋊C4D311yescharacteristic, complemented
44#1442—32no—
45#1441—31no—
46#1440—32no—
47#1439—31no—
48#1437—32no—
49#1436—32no—
50#1434—31no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4D414yes—
67C2×C3×C42—24no—
68C2×C3×C42—24no—
69C2×C3×C42—42no—
70C2×C3×C42—42no—
71C2×C3×C42D414yes—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—42no—
85C2×(C2×D3)⋊C4—42no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—42no—
92C3×#95—24no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—22no—
97C3⋊#95—24no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—42no—
104C22×C4⋊C4—31no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4C2×D311yescharacteristic
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C22⋊C4—31no—
109C22×C22⋊C4—31no—
110C22×C22⋊C4—31no—
111C22×C22⋊C4—32no—
112C22×C22⋊C4—32no—
113C22×C42C2×D311yescharacteristic
114C22×C42C2×D312yes—
115#400—64no—
116#399—64no—
117#388—64no—
118C4×C4⋊C4—24no—
119C2×#95—31no—
120C2×#95—31no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95C2×D312yes—
131C24×C3C2411yescharacteristic
132C23×D3—21no—
133C23×D3—21no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—21no—
137C23×D3—21no—
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—21no—
141C23×D3—21no—
142C23×D3#4112yes—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
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×C4C2×D414yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4#4114yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4114yes—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3#4112yes—
202C22×Dic3#4114yes—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—28no—
205C3×C42—28no—
206C3×C42—48no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—48no—
209(C2×D3)⋊C4—48no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—44no—
212Dic3⋊C4—48no—
213Dic3⋊C4—44no—
214C25—31no—
215C23×C4—31no—
216C23×C4—32no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—32no—
220C23×C4C22×D312yes—
221C23×C4C22×D312yes—
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—64no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—122no—
235C2×C4⋊C4D1214yes—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—64no—
244C2×C22⋊C4—64no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—122no—
249C2×C22⋊C4—122no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42#6714yes—
255#95—24no—
256#95—64no—
257#95—64no—
258#95—62no—
259#95—64no—
260#95—62no—
261#95—64no—
262#95—64no—
263#95—64no—
264#95—62no—
265#95—62no—
266#95—122no—
267C23×C3C22×D411yescharacteristic
268C23×C3C22×D411yescharacteristic
269C23×C3C2×#4111yescharacteristic
270C23×C3C2×#4112yes—
271C23×C3C2×#4112yes—
272C23×C3C2×#4111yescharacteristic
273C23×C3C2×#4111yescharacteristic
274C23×C3C2×#4112yes—
275C23×C3H(5,𝔽2)12yes—
276C23×C32−1+411yescharacteristic
277C23×C32−1+411yescharacteristic
278C22×D3—22no—
279C22×D3—24no—
280C22×D3—22no—
281C22×D3—24no—
282C22×D3—22no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—41no—
296C22×D3—42no—
297C22×D3—41no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
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—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
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—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4#12114yes—
329C2×C3×C4#12314yes—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C24—32no—
362C24—61no—
363C24—62no—
364C24—62no—
365C24—61no—
366C24—62no—
367C24—61no—
368C24—61no—
369C24—62no—
370C24—61no—
371C24—61no—
372C24C23×D311yescharacteristic
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
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—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—32no—
395C22×C4—34no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4C2×D1214yes—
426C22×C4C2×D1212yes—
427C22×C4#23514yes—
428C22×C4#23514yes—
429C22×C4#23512yes—
430C22×C4C2×#6712yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—128no—
436C4⋊C4—124no—
437C22⋊C4—68no—
438C22⋊C4—128no—
439C22⋊C4—128no—
440C22⋊C4—124no—
441C22⋊C4—124no—
442C42—28no—
443C42—48no—
444C22×C3C2×#12111yescharacteristic
445C22×C3C2×#12311yescharacteristic
446C22×C3#52511yescharacteristic
447C22×C3#52511yescharacteristic
448C22×C3#52912yes—
449C22×C3#52912yes—
450C22×C3#52912yes—
451C22×C3#53212yes—
452C22×C3#53211yescharacteristic
453C22×C3#53511yescharacteristic
454C22×C3#53611yescharacteristic
455C22×C3#53912yes—
456C22×C3#53912yes—
457C22×C3#54612yes—
458C22×C3#54811yescharacteristic
459C22×C3#54811yescharacteristic
460C22×C3#55112yes—
461C22×C3#55211yescharacteristic
462C22×C3#55212yes—
463C22×C3#55211yescharacteristic
464C22×C3#55312yes—
465C22×C3#55612yes—
466C22×C3#55912yes—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—82no—
473C2×D3—81no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—82no—
478C3×C4—24no—
479C3×C4—24no—
480C3×C4—44no—
481C3×C4—42no—
482C3×C4—44no—
483Dic3—42no—
484Dic3—44no—
485Dic3—82no—
486Dic3—81no—
487Dic3—81no—
488C23—64no—
489C23—64no—
490C23—62no—
491C23—62no—
492C23—62no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—122no—
502C23—121no—
503C23—122no—
504C23—122no—
505C23—121no—
506C23—122no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—122no—
511C23C22×D1211yescharacteristic
512C23C2×#23512yes—
513C23C2×#23512yes—
514C23C2×#23511yescharacteristic
515C23#100412yes—
516C23#100711yescharacteristic
517C23#100711yescharacteristic
518C23D3×#4111yescharacteristic
519C23D3×#4112yes—
520C23D3×#4111yescharacteristic
521C23C22×#6711yescharacteristic
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
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—42no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—121no—
558C2×C4—122no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—122no—
562C2×C4—121no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—121no—
580C2×C4#92914yes—
581C2×C4#93014yes—
582C2×C3#239511yescharacteristic
583C2×C3#253611yescharacteristic
584C2×C3#254411yescharacteristic
585C2×C3#255812yes—
586C2×C3#256012yes—
587C2×C3#257512yes—
588C2×C3#259612yes—
589C2×C3#259811yescharacteristic
590C2×C3#260411yescharacteristic
591C2×C3#262311yescharacteristic
592C2×C3#285411yescharacteristic
593D3—82no—
594C22—122no—
595C22—124no—
596C22—124no—
597C22—122no—
598C22—122no—
599C22—242no—
600C22—241no—
601C22—241no—
602C22—242no—
603C22—241no—
604C22—241no—
605C22#547712yes—
606C22#547912yes—
607C22#551512yes—
608C22#551512yes—
609C22#552812yes—
610C22#552812yes—
611C22#554012yes—
612C22#555412yes—
613C22#555612yes—
614C22#555912yes—
615C22#556812yes—
616C22#557611yescharacteristic
617C22#557611yescharacteristic
618C22#558111yescharacteristic
619C22#558112yes—
620C22#558111yescharacteristic
621C22C2×#92911yescharacteristic
622C22C2×#93011yescharacteristic
623C22#579111yescharacteristic
624C22#580511yescharacteristic
625C22#581611yescharacteristic
626C22#582811yescharacteristic
627C22#583211yescharacteristic
628C4—24no—
629C4—24no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—122no—
634C4—124no—
635C4—242no—
636C4—241no—
637C4—241no—
638C3#1517511yescharacteristic, complemented, Sylow
639C2—242no—
640C2#7374112yes—
641C2#7380412yes—
642C2#7380512yes—
643C2#7380912yes—
644C2#8208011yescharacteristic
645C2#8220011yescharacteristic
646C2#8227411yescharacteristic
647C2#8238111yescharacteristic
648C2#8241811yescharacteristic
649C2#8248511yescharacteristic
650C2#8249911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C4⋊C4C2
prime radicalC24×D3C23
hypercenterC2×C4×C4⋊C4D3
hypocenterC3#15175
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C4⋊C4C2semidirect
3. C2×#4985C2semidirect
6. #71566C2semidirect
8. #71536C2semidirect
9. #71534C2semidirect
11. #69413C2semidirect
43. C2×C4×C4⋊C4D3semidirect
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×C4×C4⋊C42
C2×#49932
C2×#49852
C2×#49402
#715702
#715662
#715372
#715362
#715342
#694292
#694132
#151753