Finite Group Databasebeta

← #10718680

Subgroups — #10718680

Order 768.

2574 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#947C211yesmaximal, characteristic
2C22×C4×D12C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C2×#5229C211yesmaximal, characteristic, complemented
5C2×#4988C211yesmaximal, characteristic
6#71563C211yesmaximal, characteristic, complemented
7C4×C3⋊#95C211yesmaximal, characteristic, complemented
8#69429C214yesmaximal, complemented
9(C2×D3)⋊#95C214yesmaximal, complemented
10#14027—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C23×C4×D3C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#947—24no—
17C2×#947—24no—
18C2×#947C418yescomplemented
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2214yes—
21C22×(C2×D3)⋊C4C2214yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×C4×D12—24no—
26C2×C4×D12—24no—
27C2×C4×D12—24no—
28C3×#391—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31#5229—24no—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2214yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2214yes—
36#4988—24no—
37C22×#124—31no—
38C22×C4×D4—31no—
39C2×#391—31no—
40C2×#391D311yescharacteristic, complemented
41C2×#390—31no—
42#1440—34no—
43C4⋊#95—31no—
44C22⋊#95—34no—
45C4×#95—31no—
46C24×D3C2312yes—
47C22×C3×Q8—22no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8C2×C414yescomplemented
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12C2×C414yes—
67C22×D12D414yes—
68C22×D12D414yes—
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—24no—
72C22×C4×D3—28no—
73C22×C4×D3—44no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42D414yes—
80#947—232no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—28no—
83C2×(C2×D3)⋊C4—28no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—28no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—44no—
89C2×(C2×D3)⋊C4C2×C4116yescomplemented
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3D414yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3C2×C414yescomplemented
99C4×D12—232no—
100C3×#95—24no—
101C3×#95—24no—
102C3⋊#95—24no—
103C3⋊#95—28no—
104C3⋊#95—24no—
105C3⋊#95—44no—
106C23×Q8C2×D311yescharacteristic
107C23×D4—31no—
108C24×C4—32no—
109C2×#124—38no—
110C2×#124—64no—
111C2×#124—64no—
112C2×C4×D4—64no—
113C2×C4×D4—64no—
114C2×C4×D4—64no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—34no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—34no—
120C22×C42—31no—
121C22×C42—31no—
122C22×C42C2×D311yescharacteristic
123#391—24no—
124#391—64no—
125#390—64no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—34no—
129C2×#95—34no—
130C2×#95C2×D312yes—
131C2×#95C2×D312yes—
132C24×C3C2411yescharacteristic
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—24no—
136C23×D3—24no—
137C23×D3—22no—
138C23×D3—22no—
139C23×D3—24no—
140C23×D3—24no—
141C23×D3C22×C414yes—
142C23×D3C2×D414yes—
143C23×D3C2×D414yes—
144C2×C3×Q8—28no—
145C2×C3×Q8—28no—
146C2×C3×Q8—28no—
147C2×C3×Q8—44no—
148C2×C3×Q8—44no—
149C2×C3×Q8—44no—
150C2×C3×Q8—44no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
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—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4C22×C414yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4#4112yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4114yes—
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—24no—
191C22×Dic3—22no—
192C22×Dic3—24no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3C22×C414yes—
197C22×Dic3C2×D414yes—
198C22×Dic3#4114yes—
199C2×D12—28no—
200C2×D12—28no—
201C2×D12—28no—
202C2×D12—28no—
203C2×D12—28no—
204C2×D12—28no—
205C2×D12C22⋊C4116yes—
206C2×C4×D3—216no—
207C2×C4×D3—416no—
208C2×C4×D3—416no—
209C2×C4×D3—48no—
210C2×C4×D3—48no—
211C3×C4⋊C4—44no—
212C3×C42—28no—
213(C2×D3)⋊C4—216no—
214(C2×D3)⋊C4—232no—
215(C2×D3)⋊C4—416no—
216C4⋊Dic3—28no—
217C4×Dic3—28no—
218C4×Dic3—44no—
219C25—32no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8C4×D314yescomplemented
227C22×D4—34no—
228C22×D4—34no—
229C22×D4—34no—
230C22×D4—62no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4—62no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—32no—
237C23×C4—64no—
238C23×C4—64no—
239C23×C4—64no—
240C23×C4—68no—
241C23×C4—124no—
242C23×C4C22×D312yes—
243C23×C4C22×D312yes—
244C23×C4C22×D311yescharacteristic
245C23×C4C22×D311yescharacteristic
246C23×C4C22×D311yescharacteristic
247#124—632no—
248C4×D4—632no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—34no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C22⋊C4—316no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—68no—
259C2×C22⋊C4—68no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—68no—
262C2×C22⋊C4—68no—
263C2×C22⋊C4—124no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—34no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—62no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42#6714yes—
273#95—24no—
274#95—24no—
275#95—64no—
276#95—68no—
277#95—64no—
278#95—124no—
279C23×C3C23×C411yescharacteristic
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C22×D411yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4112yes—
287C23×C3C2×#4112yes—
288C23×C3H(5,𝔽2)12yes—
289C23×C32−1+412yes—
290C22×D3—24no—
291C22×D3—24no—
292C22×D3—28no—
293C22×D3—28no—
294C22×D3—24no—
295C22×D3—24no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—28no—
299C22×D3—28no—
300C22×D3—28no—
301C22×D3—28no—
302C22×D3—42no—
303C22×D3—44no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—44no—
308C22×D3—44no—
309C22×D3—44no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3C2×C22⋊C418yes—
313C3×Q8—416no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4C2×C22⋊C414yes—
336C2×C3×C4C4×D414yes—
337C2×C3×C4C4×D414yes—
338C2×C3×C4#12014yes—
339C2×C3×C4#12114yes—
340C2×C3×C4#12114yes—
341C2×C3×C4#12314yes—
342C2×C3×C4#12414yescomplemented
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—28no—
350C2×Dic3—24no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—44no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362D12—232no—
363C4×D3—432no—
364C24—34no—
365C24—34no—
366C24—34no—
367C24—62no—
368C24—62no—
369C24—62no—
370C24—64no—
371C24—64no—
372C24—64no—
373C24—64no—
374C24—62no—
375C24C23×D311yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×D4—316no—
384C2×D4—68no—
385C2×D4—68no—
386C2×D4—68no—
387C2×D4—68no—
388C2×D4—68no—
389C2×D4—68no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—34no—
409C22×C4—34no—
410C22×C4—34no—
411C22×C4—616no—
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—64no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—1216no—
429C22×C4—1216no—
430C22×C4—128no—
431C22×C4—128no—
432C22×C4C2×C4×D314yes—
433C22×C4C2×C4×D312yes—
434C22×C4#23512yes—
435C22×C4D3×D412yes—
436C22×C4D3×D412yes—
437C22×C4D3×D412yes—
438C22×C4#23712yes—
439C22×C4#23914yes—
440C22×C4C2×#6712yes—
441C22×C4C2×#6712yes—
442C22×C4C2×#6712yes—
443C4⋊C4—44no—
444C4⋊C4—68no—
445C22⋊C4—632no—
446C22⋊C4—616no—
447C22⋊C4—1216no—
448C42—28no—
449C42—68no—
450C42—124no—
451C22×C3C22×C22⋊C411yescharacteristic
452C22×C3C2×C4×D411yescharacteristic
453C22×C3C2×C4×D411yescharacteristic
454C22×C3C4×#4112yes—
455C22×C3#52012yes—
456C22×C3C2×#12011yescharacteristic
457C22×C3C2×#12111yescharacteristic
458C22×C3C2×#12111yescharacteristic
459C22×C3C2×#12311yescharacteristic
460C22×C3C2×#12411yescharacteristic
461C22×C3#52912yes—
462C22×C3#53211yescharacteristic
463C22×C3#53511yescharacteristic
464C22×C3#53611yescharacteristic
465C22×C3#54212yes—
466C22×C3#54612yes—
467C22×C3#54612yes—
468C22×C3#54612yes—
469C22×C3#54712yes—
470C22×C3#54712yes—
471C22×C3#54812yes—
472C22×C3#55612yes—
473C22×C3#55912yes—
474C2×D3—216no—
475C2×D3—28no—
476C2×D3—44no—
477C2×D3—48no—
478C2×D3—48no—
479C2×D3—44no—
480C2×D3—44no—
481C2×D3—48no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—44no—
485C3×C4—24no—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4—82no—
489C3×C4#38618yes—
490Dic3—44no—
491Dic3—44no—
492Dic3—44no—
493C23—38no—
494C23—68no—
495C23—68no—
496C23—64no—
497C23—64no—
498C23—64no—
499C23—64no—
500C23—64no—
501C23—64no—
502C23—68no—
503C23—68no—
504C23—68no—
505C23—68no—
506C23—124no—
507C23—124no—
508C23—124no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—124no—
514C23—124no—
515C23—124no—
516C23C22×C4×D311yescharacteristic
517C23C2×#23511yescharacteristic
518C23C2×D3×D411yescharacteristic
519C23C2×D3×D411yescharacteristic
520C23C2×D3×D411yescharacteristic
521C23C2×#23711yescharacteristic
522C23C2×#23912yes—
523C23#100712yes—
524C23D3×#4112yes—
525C23#100912yes—
526C23C22×#6711yescharacteristic
527Q8—416no—
528D4—632no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—68no—
558C2×C4—1232no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—124no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4D3×C22⋊C414yes—
571C2×C4#67⋊C414yes—
572C2×C4#88314yes—
573C2×C4#88414yes—
574C2×C4C4×#6714yes—
575C2×C4#93714yes—
576C2×C4#93814yes—
577C2×C4#94714yescomplemented
578C2×C3C2×#38611yescharacteristic
579C2×C3C4×#12411yescharacteristic
580C2×C3#232611yescharacteristic
581C2×C3#233912yes—
582C2×C3#236712yes—
583C2×C3#242712yes—
584C2×C3#243211yescharacteristic
585C2×C3#243511yescharacteristic
586C2×C3#244712yes—
587C2×C3#254411yescharacteristic
588C2×C3#254711yescharacteristic
589D3—48no—
590C22—616no—
591C22—68no—
592C22—124no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—124no—
597C22—124no—
598C22—128no—
599C22—128no—
600C22—128no—
601C22C2×D3×C22⋊C411yescharacteristic
602C22C2×#67⋊C411yescharacteristic
603C22C2×#88311yescharacteristic
604C22C2×#88411yescharacteristic
605C22C4×#23912yes—
606C22#557412yes—
607C22#558012yes—
608C22#558212yes—
609C22#563012yes—
610C22#563012yes—
611C22#563112yes—
612C22#563312yes—
613C22#567612yes—
614C22#568112yes—
615C22#572612yes—
616C22#573212yes—
617C22C2×C4×#6711yescharacteristic
618C22C2×#93711yescharacteristic
619C22C2×#93811yescharacteristic
620C22C2×#94711yescharacteristic
621C22#581611yescharacteristic
622C22#583011yescharacteristic
623C22#583211yescharacteristic
624C4—24no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—124no—
629C4—124no—
630C4—124no—
631C4#495618yes—
632C3#1402711yescharacteristic, complemented, Sylow
633C2—128no—
634C2#7649312yes—
635C2#7650512yes—
636C2#7661312yes—
637C2#7662412yes—
638C2C2×#495611yescharacteristic
639C2C4×#94711yescharacteristic
640C2#8245511yescharacteristic
641C2#8275911yescharacteristic
642C2#8276111yescharacteristic
643C2#8296911yescharacteristic
644C2#8324011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#391C2
socle, cosocleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#391D3
hypocenterC3#14027
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×C3×#391C2semidirect
4. C2×#5229C2semidirect
6. #71563C2semidirect
7. C4×C3⋊#95C2semidirect
8. #69429C2semidirect
9. (C2×D3)⋊#95C2semidirect
18. C2×#947C4semidirect
40. C2×#391D3semidirect
53. C22×C3×Q8C2×C4semidirect
89. C2×(C2×D3)⋊C4C2×C4semidirect
98. C2×C4×Dic3C2×C4semidirect
226. C22×Q8C4×D3semidirect
342. C2×C3×C4#124semidirect
577. C2×C4#947semidirect
632. C3#14027semidirect

Maximal subgroups

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

subgroupindex
C22×#9472
C22×C4×D122
C2×C3×#3912
C2×#52292
C2×#49882
#715632
C4×C3⋊#952
#694292
(C2×D3)⋊#952
#140273