Finite Group Databasebeta

← #10732240

Subgroups — #10732240

Order 768.

1990 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#5212C211yesmaximal, characteristic
4C2×#4929C211yesmaximal, characteristic, complemented
5C2×#4929C211yesmaximal, characteristic, complemented
6C2×#4935C211yesmaximal, characteristic, complemented
7#71546C214yesmaximal
8#71535C211yesmaximal, characteristic, complemented
9#69411C214yesmaximal, complemented
10#15182—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×#217C2212yes—
17C22×#217C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2214yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C2×#924—24no—
23C2×#924—24no—
24C2×#924—24no—
25C22×Dic3⋊C4C2212yes—
26C3×C4×C22⋊C4—24no—
27C2×C3×#95C2212yes—
28#5212—24no—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2214yes—
31C2×C3⋊#95C2214yes—
32C2×C3⋊#95C2214yes—
33#4935—24no—
34#4929—24no—
35#4929—24no—
36C22×C22⋊Q8—31no—
37C2×#391—31no—
38C2×#391—31no—
39C2×#384—31no—
40C2×#380—31no—
41C2×C4×C22⋊C4D311yescharacteristic
42#1441—34no—
43#1439—34no—
44C4⋊#95—31no—
45C25×C3C2311yescharacteristic
46C23×C3×C4—22no—
47C23×C3×C4—22no—
48C23×C3×C4—22no—
49C23×C3×C4—22no—
50C23×C3×C4—22no—
51C23×C3×C4—22no—
52C23×C3×C4D414yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6D414yes—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4Q818yescomplemented
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42D414yes—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217—44no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—28no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—28no—
92C2×C4⋊Dic3—44no—
93C2×C4⋊Dic3D414yes—
94#924—232no—
95C2×Dic3⋊C4—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C3×#95—24no—
99C3⋊#95—28no—
100C3⋊#95—24no—
101C3⋊#95—28no—
102C3⋊#95—44no—
103C23×Q8—31no—
104C24×C4C2×D311yescharacteristic
105C2×C22⋊Q8—64no—
106C2×C22⋊Q8—64no—
107C2×C22⋊Q8—64no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—34no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—32no—
115C22×C22⋊C4C2×D312yes—
116C22×C42C2×D311yescharacteristic
117C22×C42C2×D311yescharacteristic
118#391—64no—
119#391—64no—
120#384—64no—
121#380—64no—
122C4×C22⋊C4—24no—
123C2×#95—34no—
124C2×#95—34no—
125C2×#95—32no—
126C2×#95—34no—
127C2×#95C2×D312yes—
128C24×C3—21no—
129C24×C3—21no—
130C24×C3—21no—
131C24×C3—22no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3—21no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3C2×D412yes—
138C24×C3C2×Q814yes—
139C24×C3C2411yescharacteristic
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—28no—
157C22×C3×C4—28no—
158C22×C3×C4—28no—
159C22×C3×C4—44no—
160C22×C3×C4—44no—
161C22×C3×C4—44no—
162C22×C3×C4—44no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×Q814yes—
170C22×C3×C4C2×Q814yes—
171C22×C3×C4#4114yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—24no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—24no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3C2×D414yes—
196C22×Dic3C2×D414yes—
197C2×Dic6—28no—
198C2×Dic6—28no—
199C2×Dic6—28no—
200C2×Dic6—44no—
201C2×Dic6—44no—
202C2×Dic6—44no—
203C2×Dic6—44no—
204C3×C22⋊C4—216no—
205C3×C42—28no—
206C3×C42—44no—
207#217—216no—
208#217—48no—
209C4⋊Dic3—28no—
210C4⋊Dic3—416no—
211C4⋊Dic3—44no—
212C4⋊Dic3—44no—
213Dic3⋊C4—216no—
214C25C22×D311yescharacteristic
215C22×Q8—34no—
216C22×Q8—62no—
217C22×Q8—62no—
218C22×Q8—62no—
219C22×Q8—62no—
220C22×Q8—62no—
221C22×Q8—62no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—32no—
229C23×C4—32no—
230C23×C4—32no—
231C23×C4—32no—
232C23×C4D1214yes—
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D312yes—
236C23×C4C22×D311yescharacteristic
237C22⋊Q8—632no—
238C2×C4⋊C4—34no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—68no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—68no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—124no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—124no—
260C2×C22⋊C4Dic618yescomplemented
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42#6714yes—
267#95—24no—
268#95—68no—
269#95—64no—
270#95—68no—
271#95—124no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—24no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C4⋊Q814yes—
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D411yescharacteristic
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D411yescharacteristic
299C23×C3C22×Q812yes—
300C23×C3C2×#4111yescharacteristic
301C23×C3C2×#4112yes—
302C23×C3C2×#4111yescharacteristic
303C23×C3C2×#4111yescharacteristic
304C23×C3H(5,𝔽2)12yes—
305C23×C32−1+412yes—
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—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—416no—
327C2×C3×C4#12014yes—
328C2×C3×C4#12114yes—
329C2×C3×C4#12114yes—
330C2×C3×C4#12114yes—
331C2×C3×C4C22⋊Q818yescomplemented
332C2×C3×C4#12314yes—
333C2×C3×C4#12314yes—
334C2×C3×C4#12414yes—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—28no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—44no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—44no—
354C2×Dic3—44no—
355C2×Dic3—44no—
356C2×Dic3—44no—
357Dic6—416no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24C2×Dic614yes—
368C24C2×D1212yes—
369C24C23×D311yescharacteristic
370C2×Q8—68no—
371C2×Q8—68no—
372C2×Q8—68no—
373C2×Q8—124no—
374C2×Q8—124no—
375C2×Q8—124no—
376C2×Q8—124no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—64no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—64no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—62no—
422C22×C4C2×Dic614yes—
423C22×C4C2×Dic614yes—
424C22×C4C2×D1212yes—
425C22×C4C2×D1212yes—
426C22×C4#23514yes—
427C22×C4#23512yes—
428C22×C4D3×D412yes—
429C22×C4D3×D412yes—
430C22×C4#23712yes—
431C22×C4#23712yes—
432C22×C4C2×#6714yes—
433C22×C4C2×#6712yes—
434C4⋊C4—616no—
435C4⋊C4—68no—
436C4⋊C4—1216no—
437C4⋊C4—124no—
438C4⋊C4—124no—
439C22⋊C4—216no—
440C22⋊C4—616no—
441C22⋊C4—128no—
442C42—28no—
443C42—44no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3C2×#12011yescharacteristic
456C22×C3C2×#12111yescharacteristic
457C22×C3C2×#12111yescharacteristic
458C22×C3C2×#12111yescharacteristic
459C22×C3C2×C22⋊Q812yes—
460C22×C3C2×#12311yescharacteristic
461C22×C3C2×#12311yescharacteristic
462C22×C3C2×#12411yescharacteristic
463C22×C3C2×C4⋊Q811yescharacteristic
464C22×C3#53211yescharacteristic
465C22×C3#53312yes—
466C22×C3#53511yescharacteristic
467C22×C3#53611yescharacteristic
468C22×C3#53712yes—
469C22×C3#54212yes—
470C22×C3#54612yes—
471C22×C3D4×Q812yes—
472C22×C3D4×Q812yes—
473C22×C3#55012yes—
474C22×C3#55212yes—
475C22×C3#55412yes—
476C22×C3#55412yes—
477C22×C3#55612yes—
478C3×C4—28no—
479C3×C4—42no—
480C3×C4—44no—
481C3×C4#39418yes—
482Dic3—44no—
483Dic3—44no—
484Dic3—82no—
485Dic3—82no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#86914yes—
509C23C22×Dic612yes—
510C23C22×D1211yescharacteristic
511C23C2×#23511yescharacteristic
512C23C2×#23512yes—
513C23C2×D3×D411yescharacteristic
514C23C2×D3×D411yescharacteristic
515C23C2×#23711yescharacteristic
516C23C2×#23711yescharacteristic
517C23#100912yes—
518C23#101012yes—
519C23C22×#6711yescharacteristic
520Q8—1216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—416no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—68no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—124no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—124no—
560C2×C4—122no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4#88214yes—
565C2×C4#88314yes—
566C2×C4#88514yes—
567C2×C4#88614yes—
568C2×C4#92418yescomplemented
569C2×C4#93014yes—
570C2×C4#93814yes—
571C2×C4#93914yes—
572C2×C3—44no—
573C2×C3C2×#39411yescharacteristic
574C2×C3#241712yes—
575C2×C3#242312yes—
576C2×C3#243111yescharacteristic
577C2×C3#243411yescharacteristic
578C2×C3#249212yes—
579C2×C3#249911yescharacteristic
580C2×C3#250211yescharacteristic
581C2×C3#253812yes—
582C2×C3#262011yescharacteristic
583C2×C3#285711yescharacteristic
584C22—24no—
585C22—24no—
586C22—24no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—42no—
594C22—44no—
595C22C2×#86911yescharacteristic
596C22#547012yes—
597C22#547711yescharacteristic
598C22C2×#88211yescharacteristic
599C22C2×#88311yescharacteristic
600C22C2×#88511yescharacteristic
601C22C2×#88611yescharacteristic
602C22#552511yescharacteristic
603C22#552611yescharacteristic
604C22#553412yes—
605C22D4×Dic612yes—
606C22D4×Dic612yes—
607C22#554112yes—
608C22#554212yes—
609C22#554312yes—
610C22#554312yes—
611C22#555012yes—
612C22#555312yes—
613C22C2×#92412yes—
614C22C2×#93011yescharacteristic
615C22C2×#93811yescharacteristic
616C22C2×#93911yescharacteristic
617C22#583312yes—
618C4—28no—
619C4—42no—
620C4—44no—
621C4—124no—
622C4—124no—
623C4—242no—
624C4—242no—
625C4#496118yes—
626C3#1518211yescharacteristic, complemented, Sylow
627C2—44no—
628C2#7348412yes—
629C2#7349011yescharacteristic
630C2#7350812yes—
631C2#7351411yescharacteristic
632C2#7351711yescharacteristic
633C2#7358611yescharacteristic
634C2#8207611yescharacteristic
635C2C2×#496111yescharacteristic
636C2#8228711yescharacteristic
637C2#8235312yes—
638C2#8237612yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C22⋊C4C2
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#15182
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#4929C2semidirect
5. C2×#4929C2semidirect
6. C2×#4935C2semidirect
8. #71535C2semidirect
9. #69411C2semidirect
70. C2×C3×C22⋊C4Q8semidirect
260. C2×C22⋊C4Dic6semidirect
331. C2×C3×C4C22⋊Q8semidirect
568. C2×C4#924semidirect
626. C3#15182semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C2×C3×C4×C22⋊C42
C2×#52122
C2×#49292
C2×#49352
#715462
#715352
#694112
#151823