Finite Group Databasebeta

← #10732206

Subgroups — #10732206

Order 768.

2446 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C22×#944C211yesmaximal, characteristic, complemented
3C22×#897C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic, complemented
5C2×#4929C211yesmaximal, characteristic, complemented
6C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#71563C211yesmaximal, characteristic, complemented
8#69428C214yesmaximal
9#69420C214yesmaximal, complemented
10#15178—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#946—24no—
17C2×#946—24no—
18C2×#946—24no—
19C2×#944—24no—
20C2×#944—24no—
21C2×#944—24no—
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×C4⋊Dic3C2212yes—
26C22×C4⋊Dic3C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2214yes—
29C22×C4×Dic3C2211yescharacteristic
30C2×#897—24no—
31C2×#897—24no—
32C2×#897—24no—
33C3×#391—24no—
34C2×C3×#95C2214yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2214yes—
37C2×C3⋊#95C2214yes—
38C4×(C2×D3)⋊C4—24no—
39#4929—24no—
40C22×C4⋊Q8—31no—
41C22×C22⋊Q8—31no—
42C22×C22⋊Q8—31no—
43C2×#391—31no—
44C2×#391D311yescharacteristic, complemented
45C2×C4×C22⋊C4—31no—
46#1444—34no—
47#1439—34no—
48C4⋊#95—31no—
49C24×D3C2311yescharacteristic
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8D414yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3—22no—
71C22×C4×D3D414yes—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—22no—
78C22×Dic6D414yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C42—22no—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85#946—232no—
86#944—232no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—24no—
94C2×(C2×D3)⋊C4—24no—
95C2×(C2×D3)⋊C4Q818yescomplemented
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—24no—
100C2×C4⋊Dic3—24no—
101C2×C4⋊Dic3D414yes—
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—28no—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4—24no—
108C2×C4×Dic3—22no—
109C2×C4×Dic3—22no—
110C2×C4×Dic3D414yes—
111#897—232no—
112C3×#95—28no—
113C3⋊#95—28no—
114C3⋊#95—24no—
115C3⋊#95—28no—
116C23×Q8—31no—
117C23×Q8C2×D311yescharacteristic
118C24×C4—31no—
119C2×C4⋊Q8—64no—
120C2×C4⋊Q8—64no—
121C2×C4⋊Q8—64no—
122C2×C22⋊Q8—64no—
123C2×C22⋊Q8—64no—
124C2×C22⋊Q8—64no—
125C2×C22⋊Q8—64no—
126C2×C22⋊Q8—64no—
127C2×C22⋊Q8—64no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—34no—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C42—31no—
137C22×C42C2×D311yescharacteristic
138#391—24no—
139#391—64no—
140C4×C22⋊C4—64no—
141C2×#95—34no—
142C2×#95—34no—
143C2×#95—32no—
144C2×#95C2×D314yes—
145C24×C3C2411yescharacteristic
146C23×D3—21no—
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—21no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3C2×D412yes—
156C23×D3C2×Q814yes—
157C2×C3×Q8—28no—
158C2×C3×Q8—28no—
159C2×C3×Q8—28no—
160C2×C3×Q8—44no—
161C2×C3×Q8—44no—
162C2×C3×Q8—44no—
163C2×C3×Q8—44no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×Q814yes—
189C22×C3×C4#4112yes—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—21no—
209C22×Dic3—24no—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D414yes—
214C22×Dic3C2×Q814yes—
215C22×Dic3#4114yes—
216C2×C4×D3—28no—
217C2×C4×D3—28no—
218C2×C4×D3—28no—
219C2×C4×D3—44no—
220C2×C4×D3—44no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×Dic6—28no—
224C2×Dic6—28no—
225C2×Dic6—28no—
226C2×Dic6—44no—
227C2×Dic6—44no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C3×C4⋊C4—28no—
231C3×C42—44no—
232(C2×D3)⋊C4—216no—
233(C2×D3)⋊C4—216no—
234(C2×D3)⋊C4—216no—
235C4⋊Dic3—216no—
236C4⋊Dic3—28no—
237Dic3⋊C4—232no—
238Dic3⋊C4—216no—
239C4×Dic3—28no—
240C25—31no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—34no—
248C22×Q8—62no—
249C22×Q8—62no—
250C22×Q8—62no—
251C22×Q8—62no—
252C22×Q8—62no—
253C22×Q8—62no—
254C22×Q8D1214yes—
255C23×C4—32no—
256C23×C4—32no—
257C23×C4—32no—
258C23×C4—31no—
259C23×C4—34no—
260C23×C4—62no—
261C23×C4—62no—
262C23×C4—62no—
263C23×C4—62no—
264C23×C4—62no—
265C23×C4—62no—
266C23×C4C22×D312yes—
267C23×C4C22×D312yes—
268C23×C4C22×D312yes—
269C23×C4C22×D311yescharacteristic
270C4⋊Q8—632no—
271C22⋊Q8—632no—
272C22⋊Q8—632no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—64no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—68no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—64no—
287C2×C4⋊C4#6714yes—
288C2×C22⋊C4—38no—
289C2×C22⋊C4—64no—
290C2×C22⋊C4—64no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—34no—
301C2×C42—62no—
302C2×C42—62no—
303#95—28no—
304#95—68no—
305#95—64no—
306#95—68no—
307C23×C3C22×D411yescharacteristic
308C23×C3C22×D411yescharacteristic
309C23×C3C22×D411yescharacteristic
310C23×C3C22×D411yescharacteristic
311C23×C3C22×D411yescharacteristic
312C23×C3C22×D411yescharacteristic
313C23×C3C22×Q812yes—
314C23×C3C2×#4111yescharacteristic
315C23×C3C2×#4112yes—
316C23×C32−1+412yes—
317C23×C32−1+412yes—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—24no—
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3C4⋊Q814yescomplemented
341C3×Q8—416no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—28no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4#12014yes—
362C2×C3×C4#12014yes—
363C2×C3×C4#12014yes—
364C2×C3×C4#12114yes—
365C2×C3×C4#12114yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12714yes—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—22no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—22no—
381C2×Dic3—28no—
382C2×Dic3—42no—
383C2×Dic3—41no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—42no—
389C2×Dic3—41no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—44no—
393C2×Dic3C22⋊Q818yescomplemented
394C4×D3—416no—
395Dic6—416no—
396C24—32no—
397C24—34no—
398C24—61no—
399C24—61no—
400C24—61no—
401C24—61no—
402C24—61no—
403C24—62no—
404C24—62no—
405C24—62no—
406C24—61no—
407C24C23×D311yescharacteristic
408C2×Q8—28no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—68no—
416C2×Q8—68no—
417C2×Q8—68no—
418C2×Q8—124no—
419C2×Q8—124no—
420C2×Q8—124no—
421C2×Q8—124no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—34no—
439C22×C4—34no—
440C22×C4—34no—
441C22×C4—34no—
442C22×C4—34no—
443C22×C4—32no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—62no—
451C22×C4—61no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—62no—
463C22×C4—61no—
464C22×C4—68no—
465C22×C4—68no—
466C22×C4—68no—
467C22×C4—124no—
468C22×C4—124no—
469C22×C4—124no—
470C22×C4—124no—
471C22×C4C2×D1214yes—
472C22×C4C2×D1212yes—
473C22×C4#23512yes—
474C22×C4D3×D412yes—
475C22×C4D3×D412yes—
476C22×C4D3×D412yes—
477C22×C4D3×D412yes—
478C22×C4D3×Q814yes—
479C22×C4C2×#6714yes—
480C22×C4C2×#6712yes—
481C4⋊C4—28no—
482C4⋊C4—616no—
483C4⋊C4—616no—
484C4⋊C4—632no—
485C4⋊C4—68no—
486C22⋊C4—616no—
487C22⋊C4—616no—
488C22⋊C4—616no—
489C42—44no—
490C42—68no—
491C22×C3C2×#12011yescharacteristic
492C22×C3C2×#12011yescharacteristic
493C22×C3C2×#12011yescharacteristic
494C22×C3C2×#12111yescharacteristic
495C22×C3C2×#12111yescharacteristic
496C22×C3C2×#12111yescharacteristic
497C22×C3C2×C22⋊Q812yes—
498C22×C3C2×#12711yescharacteristic
499C22×C3C2×C4⋊Q811yescharacteristic
500C22×C3#53211yescharacteristic
501C22×C3#53312yes—
502C22×C3#53611yescharacteristic
503C22×C3#53611yescharacteristic
504C22×C3#53612yes—
505C22×C3#54112yes—
506C22×C3#54712yes—
507C22×C3#54712yes—
508C22×C3#54812yes—
509C22×C3#54812yes—
510C22×C3D4×Q812yes—
511C22×C3D4×Q812yes—
512C22×C3D4×Q812yes—
513C22×C3D4×Q812yes—
514C2×D3—24no—
515C2×D3—24no—
516C2×D3—24no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—44no—
520C2×D3—44no—
521C2×D3—44no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—44no—
525C3×C4—44no—
526C3×C4—44no—
527C3×C4—44no—
528C3×C4#39218yes—
529Dic3—28no—
530Dic3—44no—
531Dic3—42no—
532Dic3—44no—
533C23—34no—
534C23—64no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—64no—
539C23—62no—
540C23—62no—
541C23—62no—
542C23—64no—
543C23—64no—
544C23—64no—
545C23—64no—
546C23—122no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—122no—
556C23C22×D1211yescharacteristic
557C23C2×#23511yescharacteristic
558C23C2×D3×D411yescharacteristic
559C23C2×D3×D411yescharacteristic
560C23C2×D3×D411yescharacteristic
561C23C2×D3×D411yescharacteristic
562C23C2×D3×Q812yes—
563C23#100712yes—
564C23D3×#4112yes—
565C23#101012yes—
566C23C22×#6711yescharacteristic
567Q8—416no—
568Q8—1216no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—38no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—68no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—121no—
606C2×C4—122no—
607C2×C4—121no—
608C2×C4—122no—
609C2×C4—121no—
610C2×C4—122no—
611C2×C4—121no—
612C2×C4—122no—
613C2×C4—124no—
614C2×C4—1216no—
615C2×C4#88214yes—
616C2×C4#88214yes—
617C2×C4#88414yes—
618C2×C4#88414yes—
619C2×C4#93014yes—
620C2×C4#93714yes—
621C2×C4#94014yes—
622C2×C3C2×#39211yescharacteristic
623C2×C3#240912yes—
624C2×C3#242912yes—
625C2×C3#242912yes—
626C2×C3#243211yescharacteristic
627C2×C3#243211yescharacteristic
628C2×C3#243411yescharacteristic
629C2×C3#243411yescharacteristic
630C2×C3#254212yes—
631C2×C3#262111yescharacteristic
632C2×C3#285711yescharacteristic
633D3—44no—
634C22—64no—
635C22—64no—
636C22—64no—
637C22—122no—
638C22—124no—
639C22—124no—
640C22—124no—
641C22—124no—
642C22—122no—
643C22—122no—
644C22—122no—
645C22C2×#88211yescharacteristic
646C22C2×#88211yescharacteristic
647C22C2×#88411yescharacteristic
648C22C2×#88411yescharacteristic
649C22Q8×D1212yes—
650C22#557612yes—
651C22#557812yes—
652C22#558312yes—
653C22D3×C22⋊Q812yes—
654C22#562812yes—
655C22#563112yes—
656C22#563212yes—
657C22#566811yescharacteristic
658C22#567511yescharacteristic
659C22#567712yes—
660C22D3×C4⋊Q811yescharacteristic
661C22#572411yescharacteristic
662C22#572712yes—
663C22C2×#93011yescharacteristic
664C22C2×#93711yescharacteristic
665C22C2×#94011yescharacteristic
666C22Q8×#6712yes—
667C22#582412yes—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—68no—
672C4—122no—
673C4—124no—
674C4—124no—
675C4#496018yes—
676C3#1517811yescharacteristic, complemented, Sylow
677C2—124no—
678C2#7649811yescharacteristic
679C2#7649911yescharacteristic
680C2#7661912yes—
681C2#7662512yes—
682C2#7664111yescharacteristic
683C2#7666411yescharacteristic
684C2C2×#496011yescharacteristic
685C2#8249612yes—
686C2#8278012yes—
687C2#8295911yescharacteristic
688C2#8323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#391C2
prime radicalC24×D3C23
hypercenterC2×#391D3
hypocenterC3#15178
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×#944C2semidirect
4. C2×C3×#391C2semidirect
5. C2×#4929C2semidirect
7. #71563C2semidirect
9. #69420C2semidirect
44. C2×#391D3semidirect
95. C2×(C2×D3)⋊C4Q8semidirect
340. C22×D3C4⋊Q8semidirect
393. C2×Dic3C22⋊Q8semidirect
676. C3#15178semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C22×#9442
C22×#8972
C2×C3×#3912
C2×#49292
C2×C4×(C2×D3)⋊C42
#715632
#694282
#694202
#151783