Finite Group Databasebeta

← #10740100

Subgroups — #10740100

Order 768.

2018 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C22×#869C211yesmaximal, characteristic
3C2×#4979C211yesmaximal, characteristic
4C2×#4656C212yesmaximal
5C2×#4656C211yesmaximal, characteristic
6C3×#1436C211yesmaximal, characteristic
7#71536C211yesmaximal, characteristic
8#69424C211yesmaximal, characteristic
9#69420C212yesmaximal
10#69419C212yesmaximal
11#69407C212yesmaximal
12#15762—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C2×#888—24no—
27C2×#888—24no—
28C2×#888—24no—
29C2×#869—24no—
30C2×#869—28no—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41#4979—24no—
42#4656—28no—
43#4656—24no—
44C22×C4⋊Q8—31no—
45C22×C4⋊Q8—31no—
46C2×#395—31no—
47C2×#395—32no—
48C2×#395—31no—
49#1446—31no—
50#1444—32no—
51#1444—32no—
52#1436—31no—
53#1436—32no—
54#1436D311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C23×Dic3C2311yescharacteristic
66C22×Dic6—22no—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—24no—
71C22×Dic6—24no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—24no—
77C22×Dic6—42no—
78C22×Dic6—42no—
79C22×Dic6—42no—
80C22×Dic6D414yes—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D414yes—
84C2×C3×C42—22no—
85C2×C3×C42—24no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3—42no—
97C2×C4⋊Dic3D418yes—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—24no—
105C2×Dic3⋊C4—42no—
106C2×Dic3⋊C4D414yes—
107C2×C4×Dic3—22no—
108C2×C4×Dic3—22no—
109C2×C4×Dic3—22no—
110#888—232no—
111#869—232no—
112C3×#95—22no—
113C3×#95—24no—
114C3×#95Q818yescomplemented
115C3⋊#95—24no—
116C3⋊#95—22no—
117C3⋊#95—24no—
118C3⋊#95—24no—
119C3⋊#95—24no—
120C3⋊#95—42no—
121C3⋊#95Q818yescomplemented
122C23×Q8—31no—
123C23×Q8—31no—
124C23×Q8—31no—
125C2×C4⋊Q8—68no—
126C2×C4⋊Q8—64no—
127C2×C4⋊Q8—64no—
128C2×C4⋊Q8—64no—
129C2×C4⋊Q8—64no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C42—31no—
139C22×C42C2×D311yescharacteristic
140#395—64no—
141#395—68no—
142#395—64no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C2×#95—31no—
149C2×#95—32no—
150C2×#95C2×D312yes—
151C2×#95C2×D311yescharacteristic
152C2×#95C2×D312yes—
153C24×C3C2411yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—21no—
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×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D414yes—
179C22×C3×C4C2×Q814yes—
180C22×C3×C4C2×Q814yes—
181C22×C3×C4C2×Q814yes—
182C22×C3×C4C2×Q814yes—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D412yes—
215C22×Dic3C2×Q814yes—
216C22×Dic3C2×Q814yes—
217C2×Dic6—28no—
218C2×Dic6—28no—
219C2×Dic6—28no—
220C2×Dic6—28no—
221C2×Dic6—44no—
222C2×Dic6—48no—
223C2×Dic6—48no—
224C2×Dic6—48no—
225C2×Dic6—48no—
226C2×Dic6—48no—
227C2×Dic6—48no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C2×Dic6—44no—
231C2×Dic6—44no—
232C3×C4⋊C4—28no—
233C3×C42—28no—
234C4⋊Dic3—216no—
235C4⋊Dic3—28no—
236C4⋊Dic3—216no—
237C4⋊Dic3—48no—
238Dic3⋊C4—28no—
239Dic3⋊C4—28no—
240Dic3⋊C4—48no—
241C4×Dic3—28no—
242C22×Q8—34no—
243C22×Q8—64no—
244C22×Q8—64no—
245C22×Q8—62no—
246C22×Q8—62no—
247C22×Q8—64no—
248C22×Q8—62no—
249C22×Q8—62no—
250C22×Q8—62no—
251C22×Q8—62no—
252C22×Q8—62no—
253C22×Q8—62no—
254C22×Q8—122no—
255C22×Q8—122no—
256C22×Q8—122no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—32no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—32no—
263C23×C4C22×D312yes—
264C23×C4C22×D311yescharacteristic
265C23×C4C22×D311yescharacteristic
266C23×C4C22×D312yes—
267C23×C4C22×D311yescharacteristic
268C4⋊Q8—632no—
269C4⋊Q8—632no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—34no—
273C2×C4⋊C4—38no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—64no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4—64no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—64no—
290C2×C4⋊C4—64no—
291C2×C4⋊C4—122no—
292C2×C4⋊C4—122no—
293C2×C4⋊C4D1214yes—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—62no—
297C2×C42—62no—
298C2×C42—62no—
299#95—22no—
300#95—24no—
301#95—38no—
302#95—64no—
303#95—64no—
304#95—62no—
305#95—64no—
306#95—64no—
307#95—122no—
308#95Dic618yescomplemented
309C23×C3C22×D411yescharacteristic
310C23×C3C22×D411yescharacteristic
311C23×C3C22×D412yes—
312C23×C3C22×D411yescharacteristic
313C23×C3C22×Q812yes—
314C23×C3C22×Q812yes—
315C23×C3H(5,𝔽2)12yes—
316C23×C32−1+412yes—
317C23×C32−1+412yes—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4#12014yes—
345C2×C3×C4C4⋊Q814yescomplemented
346C2×C3×C4C4⋊Q818yescomplemented
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—42no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3#12014yes—
380C2×Dic3C4⋊Q814yescomplemented
381Dic6—416no—
382Dic6—416no—
383Dic6—416no—
384C24C23×D311yescharacteristic
385C2×Q8—68no—
386C2×Q8—68no—
387C2×Q8—68no—
388C2×Q8—68no—
389C2×Q8—124no—
390C2×Q8—128no—
391C2×Q8—128no—
392C2×Q8—128no—
393C2×Q8—128no—
394C2×Q8—128no—
395C2×Q8—124no—
396C2×Q8—124no—
397C2×Q8—128no—
398C2×Q8—124no—
399C2×Q8—124no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—32no—
420C22×C4—34no—
421C22×C4—32no—
422C22×C4—34no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—34no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4C2×Dic614yes—
454C22×C4C2×Dic614yes—
455C22×C4C2×Dic614yes—
456C22×C4C2×D1212yes—
457C22×C4C2×D1212yes—
458C22×C4C2×D1212yes—
459C22×C4D3×D412yes—
460C22×C4D3×D412yes—
461C22×C4D3×D414yes—
462C22×C4D3×Q814yes—
463C4⋊C4—28no—
464C4⋊C4—616no—
465C4⋊C4—68no—
466C4⋊C4—68no—
467C4⋊C4—616no—
468C4⋊C4—68no—
469C4⋊C4—128no—
470C4⋊C4—128no—
471C42—28no—
472C42—68no—
473C22×C3C2×#12011yescharacteristic
474C22×C3C2×#12011yescharacteristic
475C22×C3C2×C4⋊Q811yescharacteristic
476C22×C3C2×C4⋊Q811yescharacteristic
477C22×C3C2×C4⋊Q812yes—
478C22×C3#53411yescharacteristic
479C22×C3#53612yes—
480C22×C3#53611yescharacteristic
481C22×C3#53611yescharacteristic
482C22×C3#54412yes—
483C22×C3#54412yes—
484C22×C3D4212yes—
485C22×C3D4212yes—
486C22×C3D4×Q812yes—
487C22×C3D4×Q812yes—
488C22×C3D4×Q812yes—
489C22×C3D4×Q812yes—
490C22×C3D4×Q812yes—
491C22×C3D4×Q812yes—
492C22×C3Q8212yes—
493C22×C3#56312yes—
494C3×C4—24no—
495C3×C4—24no—
496C3×C4—28no—
497C3×C4—42no—
498C3×C4—44no—
499Dic3—24no—
500Dic3—24no—
501Dic3—42no—
502Dic3—42no—
503Dic3—44no—
504Dic3—44no—
505C23C22×Dic612yes—
506C23C22×D1211yescharacteristic
507C23C2×D3×D411yescharacteristic
508C23C2×D3×D412yes—
509C23C2×D3×D411yescharacteristic
510C23C2×D3×Q812yes—
511C23#100912yes—
512C23#101012yes—
513C23#101012yes—
514Q8—1216no—
515Q8—1216no—
516Q8—1216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—34no—
537C2×C4—34no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—121no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—121no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4#86914yescomplemented
578C2×C4#88214yes—
579C2×C4#88818yescomplemented
580C2×C3#240311yescharacteristic
581C2×C3#243411yescharacteristic
582C2×C3#243412yes—
583C2×C3#243411yescharacteristic
584C2×C3#255112yes—
585C2×C3#255112yes—
586C2×C3#267712yes—
587C2×C3#267712yes—
588C2×C3#273412yes—
589C22C2×#86911yescharacteristic
590C22C2×#88211yescharacteristic
591C22C2×#88812yes—
592C22#551912yes—
593C22#552611yescharacteristic
594C22D4×Dic612yes—
595C22D4×Dic612yes—
596C22D4×Dic612yes—
597C22D4×D1212yes—
598C22Q8×Dic612yes—
599C22Q8×D1212yes—
600C22D3×#12011yescharacteristic
601C22#560912yes—
602C22#563212yes—
603C22#563212yes—
604C22#564812yes—
605C22#565311yescharacteristic
606C22#567511yescharacteristic
607C22#568912yes—
608C22#571212yes—
609C22D3×C4⋊Q811yescharacteristic
610C4—24no—
611C4—24no—
612C4—28no—
613C4—42no—
614C4—44no—
615C4—64no—
616C4—64no—
617C4—124no—
618C4—122no—
619C4—122no—
620C4—124no—
621C3#1576211yescharacteristic, complemented, Sylow
622C2#7349011yescharacteristic
623C2#7371512yes—
624C2#7441212yes—
625C2#7488411yescharacteristic
626C2#7517912yes—
627C2#7539012yes—
628C2#7566712yes—
629C2#7606412yes—
630C2#7649911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×#1436C2
hypercenter#1436D3
hypocenterC3#15762
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
114. C3×#95Q8semidirect
121. C3⋊#95Q8semidirect
308. #95Dic6semidirect
345. C2×C3×C4C4⋊Q8semidirect
346. C2×C3×C4C4⋊Q8semidirect
380. C2×Dic3C4⋊Q8semidirect
577. C2×C4#869semidirect
579. C2×C4#888semidirect
621. C3#15762semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C22×#8692
C2×#49792
C2×#46562
C3×#14362
#715362
#694242
#694202
#694192
#694072
#157623