Finite Group Databasebeta

← #10717331

Subgroups — #10717331

Order 768.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C2×#5211C211yesmaximal, characteristic
3C2×#4932C211yesmaximal, characteristic, complemented
4#72127C211yesmaximal, characteristic
5#71546C212yesmaximal
6#71545C212yesmaximal
7#71535C211yesmaximal, characteristic, complemented
8#71534C211yesmaximal, characteristic, complemented
9#71534C211yesmaximal, characteristic, complemented
10#69416C212yesmaximal, complemented
11#69407C212yesmaximal, complemented
12#13895—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×C4⋊Dic3C2212yes—
21C22×C4×Dic3C2212yes—
22C3×C4×C22⋊C4C418yescomplemented
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25#5211—24no—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4932—24no—
39C2×#381—31no—
40C2×C4×C22⋊C4D311yescharacteristic
41C2×#376—31no—
42#1443—32no—
43#1439—32no—
44#1437—32no—
45#1436—32no—
46C4⋊#95—31no—
47#1434—31no—
48#1434—31no—
49C22⋊#95—31no—
50C25×C3C2311yescharacteristic
51C23×C3×C4—24no—
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—42no—
55C23×C3×C4C2×C414yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4C2×C414yes—
70C2×C3×C22⋊C4C2×C414yes—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42—42no—
74C2×C3×C42C2×C414yes—
75C2×C3×C42C2×C414yes—
76C2×#217—24no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—42no—
81C2×#217—42no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3—42no—
85C2×C4×Dic3—24no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C3×#95C2×C414yes—
89C3×#95C2×C414yes—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—24no—
93C3⋊#95—24no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—42no—
98C3⋊#95—42no—
99C3⋊#95—42no—
100C3⋊#95—42no—
101C3⋊#95D414yes—
102C24×C4C2×D311yescharacteristic
103C22×C4⋊C4—32no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4—32no—
106C22×C22⋊C4C2×D311yescharacteristic
107C22×C22⋊C4C2×D311yescharacteristic
108C22×C42—32no—
109C22×C42C2×D311yescharacteristic
110C22×C42C2×D311yescharacteristic
111#381—64no—
112C4×C22⋊C4Dic318yescomplemented
113#376—64no—
114C2×#95—31no—
115C2×#95—31no—
116C2×#95—31no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95C2×D311yescharacteristic
127C2×#95C2×D311yescharacteristic
128C24×C3—21no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—21no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3—21no—
135C24×C3—22no—
136C24×C3—21no—
137C24×C3C22×C412yes—
138C24×C3#4114yes—
139C24×C3C2411yescharacteristic
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—28no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—28no—
163C22×C3×C4—44no—
164C22×C3×C4—48no—
165C22×C3×C4—48no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C22×C412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4114yes—
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
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×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D414yes—
204C22×Dic3#4114yes—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—28no—
207C3×C42—28no—
208C3×C42—44no—
209#217—48no—
210#217—48no—
211C4⋊Dic3—48no—
212C4×Dic3—216no—
213C25C22×D311yescharacteristic
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4—32no—
220C23×C4—32no—
221C23×C4—42no—
222C23×C4C2×Dic314yes—
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227C23×C4C22×D311yescharacteristic
228C23×C4C22×D311yescharacteristic
229C2×C4⋊C4—64no—
230C2×C4⋊C4—64no—
231C2×C4⋊C4—122no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—122no—
240C2×C22⋊C4—122no—
241C2×C22⋊C4C2×Dic314yes—
242C2×C22⋊C4C2×Dic314yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—64no—
247C2×C42—64no—
248C2×C42—64no—
249C2×C42C2×Dic314yes—
250C2×C42C2×Dic314yes—
251#95—34no—
252#95—64no—
253#95—64no—
254#95—64no—
255#95—64no—
256#95—62no—
257#95—62no—
258#95—62no—
259#95—122no—
260#95—122no—
261#95—122no—
262#95—122no—
263#95C2×Dic314yes—
264#95C2×Dic314yes—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—24no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3#11714yes—
288C23×C3C23×C411yescharacteristic
289C23×C3C22×D411yescharacteristic
290C23×C3C2×#4112yes—
291C23×C3C2×#4112yes—
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3H(5,𝔽2)12yes—
297C23×C3H(5,𝔽2)11yescharacteristic
298C23×C32−1+411yescharacteristic
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—416no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4#11714yes—
330C2×C3×C4#11714yes—
331C2×C3×C4C4×D414yes—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
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—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24#23514yes—
370C24C22×Dic312yes—
371C24C23×D311yescharacteristic
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—28no—
395C22×C4—34no—
396C22×C4—34no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—62no—
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×C4#23514yes—
428C22×C4#23512yes—
429C22×C4D3×D412yes—
430C22×C4#23712yes—
431C22×C4C22×Dic312yes—
432C22×C4C22×Dic312yes—
433C22×C4C22×Dic312yes—
434C22×C4C22×Dic312yes—
435C22×C4C22×Dic312yes—
436C22×C4C22×Dic312yes—
437C4⋊C4—128no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—128no—
441C22⋊C4—128no—
442C42—28no—
443C42—44no—
444C42—616no—
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×C3C2×#11711yescharacteristic
457C22×C3C2×#11711yescharacteristic
458C22×C3C2×#11711yescharacteristic
459C22×C3C2×C4×D411yescharacteristic
460C22×C3C4×#4111yescharacteristic
461C22×C3#51811yescharacteristic
462C22×C3#52011yescharacteristic
463C22×C3#52512yes—
464C22×C3#52912yes—
465C22×C3#52912yes—
466C22×C3#52912yes—
467C22×C3#54012yes—
468C22×C3#54212yes—
469C22×C3#54612yes—
470C22×C3#54711yescharacteristic
471C22×C3#55011yescharacteristic
472C22×C3#55112yes—
473C22×C3#55112yes—
474C22×C3#55112yes—
475C22×C3#55212yes—
476C22×C3#55312yes—
477C22×C3#55511yescharacteristic
478C22×C3#55611yescharacteristic
479C3×C4—24no—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485Dic3—44no—
486Dic3—44no—
487Dic3—82no—
488Dic3—82no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23#92614yes—
512C23C2×#23512yes—
513C23C2×#23512yes—
514C23C2×#23511yescharacteristic
515C23C2×#23511yescharacteristic
516C23C2×D3×D411yescharacteristic
517C23C2×#23711yescharacteristic
518C23#100412yes—
519C23D3×#4112yes—
520C23#100911yescharacteristic
521C23#101011yescharacteristic
522C23C23×Dic311yescharacteristic
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—122no—
560C2×C4—122no—
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—122no—
581C2×C4#92614yes—
582C2×C4#92614yes—
583C2×C4D4×Dic314yes—
584C2×C3—44no—
585C2×C3#228311yescharacteristic
586C2×C3#229011yescharacteristic
587C2×C3#231911yescharacteristic
588C2×C3#234211yescharacteristic
589C2×C3#234311yescharacteristic
590C2×C3#235311yescharacteristic
591C2×C3#235411yescharacteristic
592C2×C3#247012yes—
593C2×C3#247112yes—
594C2×C3#252612yes—
595C2×C3#252712yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22#547912yes—
608C22#547912yes—
609C22#549012yes—
610C22#549512yes—
611C22#552712yes—
612C22#553112yes—
613C22#553812yes—
614C22#554012yes—
615C22#554211yescharacteristic
616C22#554712yes—
617C22#555111yescharacteristic
618C22#555311yescharacteristic
619C22#555612yes—
620C22#556112yes—
621C22#556212yes—
622C22#556311yescharacteristic
623C22C2×#92611yescharacteristic
624C22C2×#92611yescharacteristic
625C22C2×#92611yescharacteristic
626C22C2×D4×Dic311yescharacteristic
627C22#579811yescharacteristic
628C22Dic3×#4111yescharacteristic
629C22#582711yescharacteristic
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—124no—
637C4—124no—
638C4—242no—
639C4—242no—
640C3#1389511yescharacteristic, complemented, Sylow
641C2—44no—
642C2#7350012yes—
643C2#7353012yes—
644C2#7353112yes—
645C2#7359312yes—
646C2#8206011yescharacteristic
647C2#8209711yescharacteristic
648C2#8223711yescharacteristic
649C2#8231811yescharacteristic
650C2#8232511yescharacteristic
651C2#8233611yescharacteristic
652C2#8233711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×C4×C22⋊C4C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#13895
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#4932C2semidirect
7. #71535C2semidirect
8. #71534C2semidirect
9. #71534C2semidirect
10. #69416C2semidirect
11. #69407C2semidirect
22. C3×C4×C22⋊C4C4semidirect
112. C4×C22⋊C4Dic3semidirect
640. C3#13895semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C2×#52112
C2×#49322
#721272
#715462
#715452
#715352
#715342
#694162
#694072
#138953