Finite Group Databasebeta

← #10732239

Subgroups — #10732239

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×#897C211yesmaximal, characteristic
2C2×#4987C211yesmaximal, characteristic
3C2×#4973C211yesmaximal, characteristic, complemented
4C2×#4975C211yesmaximal, characteristic, complemented
5C2×#4929C211yesmaximal, characteristic, complemented
6C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7C3×C4⋊#95C211yesmaximal, characteristic, complemented
8#69428C214yesmaximal
9#69410C214yesmaximal, complemented
10#15182—31nomaximal, Sylow
11C23×C4×D3C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2214yes—
22C22×C4×Dic3C2211yescharacteristic
23C2×#897—24no—
24C2×#897—24no—
25C2×#897—24no—
26C2×C3×#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31#4987—24no—
32#4975—24no—
33#4973—24no—
34C4×(C2×D3)⋊C4—24no—
35#4929—24no—
36C22×C22⋊Q8—31no—
37C2×#391—31no—
38C2×#391—31no—
39C2×#384—31no—
40C2×#380—31no—
41C2×C4×C22⋊C4—31no—
42#1441—34no—
43#1439—34no—
44C4⋊#95D311yescharacteristic, complemented
45C24×D3C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C23×Dic3C2311yescharacteristic
53C23×Dic3C2312yes—
54C22×C4×D3—22no—
55C22×C4×D3—22no—
56C22×C4×D3—22no—
57C22×C4×D3—22no—
58C22×C4×D3—22no—
59C22×C4×D3—22no—
60C22×C4×D3D414yes—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6D414yes—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—44no—
84C2×(C2×D3)⋊C4Q818yescomplemented
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×Dic3⋊C4—28no—
92C2×Dic3⋊C4—28no—
93C2×Dic3⋊C4—44no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3D414yes—
97#897—232no—
98C3×#95—44no—
99C3⋊#95—28no—
100C3⋊#95—24no—
101C3⋊#95—28no—
102C3⋊#95—24no—
103C23×Q8—31no—
104C24×C4—31no—
105C2×C22⋊Q8—64no—
106C2×C22⋊Q8—64no—
107C2×C22⋊Q8—64no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—34no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C22⋊C4—32no—
114C22×C22⋊C4—32no—
115C22×C22⋊C4—32no—
116C22×C42—31no—
117C22×C42C2×D311yescharacteristic
118#391—64no—
119#391—64no—
120#384—64no—
121#380—64no—
122C4×C22⋊C4—64no—
123C2×#95—34no—
124C2×#95—34no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95C2×D314yes—
128C24×C3C2411yescharacteristic
129C23×D3—21no—
130C23×D3—21no—
131C23×D3—21no—
132C23×D3—22no—
133C23×D3—21no—
134C23×D3—21no—
135C23×D3—21no—
136C23×D3—22no—
137C23×D3—22no—
138C23×D3C2×D412yes—
139C23×D3C2×Q814yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×Q814yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×C3×C4#4112yes—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3C2×D412yes—
186C22×Dic3C2×D414yes—
187C22×Dic3C2×D414yes—
188C22×Dic3C2×Q814yes—
189C22×Dic3#4114yes—
190C2×C4×D3—28no—
191C2×C4×D3—28no—
192C2×C4×D3—28no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×C4×D3—44no—
197C2×Dic6—28no—
198C2×Dic6—28no—
199C2×Dic6—28no—
200C2×Dic6—44no—
201C2×Dic6—44no—
202C2×Dic6—44no—
203C2×Dic6—44no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—44no—
206C3×C42—44no—
207(C2×D3)⋊C4—216no—
208(C2×D3)⋊C4—216no—
209(C2×D3)⋊C4—48no—
210C4⋊Dic3—216no—
211C4⋊Dic3—44no—
212Dic3⋊C4—416no—
213C4×Dic3—28no—
214C25—31no—
215C22×Q8—34no—
216C22×Q8—62no—
217C22×Q8—62no—
218C22×Q8—62no—
219C22×Q8—62no—
220C22×Q8—62no—
221C22×Q8—62no—
222C23×C4—32no—
223C23×C4—32no—
224C23×C4—31no—
225C23×C4—34no—
226C23×C4—32no—
227C23×C4—62no—
228C23×C4—62no—
229C23×C4—62no—
230C23×C4—62no—
231C23×C4—62no—
232C23×C4—62no—
233C23×C4C22×D312yes—
234C23×C4C22×D312yes—
235C23×C4C22×D312yes—
236C23×C4C22×D311yescharacteristic
237C22⋊Q8—632no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—68no—
247C2×C4⋊C4—68no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—124no—
252C2×C4⋊C4D1214yes—
253C2×C22⋊C4—38no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—124no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—34no—
265C2×C42—62no—
266C2×C42—62no—
267#95—44no—
268#95—64no—
269#95—64no—
270#95—68no—
271#95—68no—
272C23×C3C22×D411yescharacteristic
273C23×C3C22×D411yescharacteristic
274C23×C3C22×D411yescharacteristic
275C23×C3C22×D411yescharacteristic
276C23×C3C22×Q812yes—
277C23×C3C2×#4111yescharacteristic
278C23×C3C2×#4111yescharacteristic
279C23×C3C2×#4111yescharacteristic
280C23×C3C2×#4112yes—
281C23×C3H(5,𝔽2)12yes—
282C23×C32−1+412yes—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—24no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—24no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—24no—
292C22×D3—24no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3C4⋊Q814yes—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—28no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4#12014yes—
324C2×C3×C4#12114yes—
325C2×C3×C4#12114yes—
326C2×C3×C4#12114yes—
327C2×C3×C4#12314yes—
328C2×C3×C4#12314yes—
329C2×C3×C4#12414yes—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—22no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—44no—
354C2×Dic3—44no—
355C2×Dic3C22⋊Q818yescomplemented
356C4×D3—416no—
357Dic6—416no—
358C24—32no—
359C24—34no—
360C24—61no—
361C24—61no—
362C24—61no—
363C24—61no—
364C24—61no—
365C24—62no—
366C24—61no—
367C24—62no—
368C24—62no—
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—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—34no—
393C22×C4—34no—
394C22×C4—34no—
395C22×C4—34no—
396C22×C4—32no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—68no—
414C22×C4—68no—
415C22×C4—68no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—124no—
422C22×C4—124no—
423C22×C4—124no—
424C22×C4—124no—
425C22×C4C2×D1214yes—
426C22×C4C2×D1212yes—
427C22×C4#23512yes—
428C22×C4#23512yes—
429C22×C4D3×D412yes—
430C22×C4D3×D412yes—
431C22×C4D3×D412yes—
432C22×C4D3×Q814yes—
433C22×C4#23912yes—
434C4⋊C4—28no—
435C4⋊C4—44no—
436C4⋊C4—616no—
437C4⋊C4—1216no—
438C4⋊C4—124no—
439C22⋊C4—616no—
440C22⋊C4—616no—
441C22⋊C4—128no—
442C42—44no—
443C42—68no—
444C22×C3C2×#12011yescharacteristic
445C22×C3C2×#12111yescharacteristic
446C22×C3C2×#12111yescharacteristic
447C22×C3C2×#12111yescharacteristic
448C22×C3C2×C22⋊Q812yes—
449C22×C3C2×#12311yescharacteristic
450C22×C3C2×#12311yescharacteristic
451C22×C3C2×#12411yescharacteristic
452C22×C3C2×C4⋊Q811yescharacteristic
453C22×C3#53211yescharacteristic
454C22×C3#53312yes—
455C22×C3#53511yescharacteristic
456C22×C3#53611yescharacteristic
457C22×C3#53712yes—
458C22×C3#54212yes—
459C22×C3#54612yes—
460C22×C3D4×Q812yes—
461C22×C3D4×Q812yes—
462C22×C3#55012yes—
463C22×C3#55212yes—
464C22×C3#55412yes—
465C22×C3#55412yes—
466C22×C3#55612yes—
467C2×D3—24no—
468C2×D3—24no—
469C2×D3—24no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—42no—
475C2×D3—44no—
476C2×D3—42no—
477C2×D3—44no—
478C3×C4—44no—
479C3×C4—44no—
480C3×C4—82no—
481C3×C4#39418yes—
482Dic3—28no—
483Dic3—44no—
484Dic3—42no—
485Dic3—82no—
486C23—34no—
487C23—64no—
488C23—62no—
489C23—62no—
490C23—62no—
491C23—64no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—64no—
496C23—64no—
497C23—64no—
498C23—64no—
499C23—122no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23C22×D1211yescharacteristic
510C23C2×#23511yescharacteristic
511C23C2×#23511yescharacteristic
512C23C2×D3×D411yescharacteristic
513C23C2×D3×D411yescharacteristic
514C23C2×D3×D411yescharacteristic
515C23#100412yes—
516C23C2×D3×Q812yes—
517C23C2×#23911yescharacteristic
518C23D3×#4112yes—
519C23#101012yes—
520Q8—1216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—38no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—64no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—121no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4—122no—
557C2×C4—121no—
558C2×C4—122no—
559C2×C4—1216no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—124no—
564C2×C4—124no—
565C2×C4#87514yes—
566C2×C4#88214yes—
567C2×C4#88414yes—
568C2×C4#88414yes—
569C2×C4#89414yes—
570C2×C4#89414yes—
571C2×C4#89514yes—
572C2×C3C2×#39411yescharacteristic
573C2×C3#241712yes—
574C2×C3#242312yes—
575C2×C3#243111yescharacteristic
576C2×C3#243411yescharacteristic
577C2×C3#249212yes—
578C2×C3#249911yescharacteristic
579C2×C3#250211yescharacteristic
580C2×C3#253812yes—
581C2×C3#262011yescharacteristic
582C2×C3#285711yescharacteristic
583D3—44no—
584C22—64no—
585C22—64no—
586C22—64no—
587C22—122no—
588C22—124no—
589C22—122no—
590C22—122no—
591C22—124no—
592C22—122no—
593C22—124no—
594C22—124no—
595C22C2×#87511yescharacteristic
596C22C2×#88211yescharacteristic
597C22C2×#88411yescharacteristic
598C22C2×#88411yescharacteristic
599C22C2×#89411yescharacteristic
600C22C2×#89411yescharacteristic
601C22C2×#89511yescharacteristic
602C22#551112yes—
603C22#553312yes—
604C22#555412yes—
605C22#556112yes—
606C22Q8×D1212yes—
607C22#557812yes—
608C22#559912yes—
609C22#560812yes—
610C22D3×C22⋊Q812yes—
611C22#563212yes—
612C22#566811yescharacteristic
613C22#567511yescharacteristic
614C22#569412yes—
615C22#571511yescharacteristic
616C22D3×C4⋊Q811yescharacteristic
617C22#573012yes—
618C4—44no—
619C4—44no—
620C4—68no—
621C4—82no—
622C4—122no—
623C4—124no—
624C4—242no—
625C4#467218yes—
626C3#1518211yescharacteristic, complemented, Sylow
627C2—124no—
628C2C2×#467211yescharacteristic
629C2#7378912yes—
630C2#7393612yes—
631C2#7416511yescharacteristic
632C2#7432911yescharacteristic
633C2#7621212yes—
634C2#7621612yes—
635C2#7631311yescharacteristic
636C2#7645711yescharacteristic
637C2#7649911yescharacteristic
638C2#7666411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C4⋊#95C2
prime radicalC24×D3C23
hypercenterC4⋊#95D3
hypocenterC3#15182
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#4973C2semidirect
4. C2×#4975C2semidirect
5. C2×#4929C2semidirect
7. C3×C4⋊#95C2semidirect
9. #69410C2semidirect
44. C4⋊#95D3semidirect
84. C2×(C2×D3)⋊C4Q8semidirect
355. C2×Dic3C22⋊Q8semidirect
626. C3#15182semidirect

Maximal subgroups

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

subgroupindex
C22×#8972
C2×#49872
C2×#49732
C2×#49752
C2×#49292
C2×C4×(C2×D3)⋊C42
C3×C4⋊#952
#694282
#694102
#151823