Finite Group Databasebeta

← #10718666

Subgroups — #10718666

Order 768.

2358 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×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C22×#947C211yesmaximal, characteristic
3C2×#5229C211yesmaximal, characteristic, complemented
4C2×#4988C211yesmaximal, characteristic
5C2×#4939C211yesmaximal, characteristic
6#71570C214yesmaximal, complemented
7#71563C211yesmaximal, characteristic, complemented
8#71537C214yesmaximal, complemented
9C4×C3⋊#95C211yesmaximal, characteristic, complemented
10#14026—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C2×C3×C4×Q8—24no—
14C2×C3×C4×Q8—24no—
15C2×C3×C4×Q8—24no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2212yes—
19C22×C3×C42C2211yescharacteristic
20C2×#947—24no—
21C2×#947—24no—
22C2×#947C418yescomplemented
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2214yes—
25C22×(C2×D3)⋊C4C2214yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4×Dic3C2211yescharacteristic
29C22×C4×Dic3C2211yescharacteristic
30#5229—24no—
31C2×C3⋊#95C2214yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2214yes—
35#4988—24no—
36#4939—24no—
37C22×#124—31no—
38C22×C4×Q8D311yescharacteristic, complemented
39C2×#391—31no—
40C2×#390—31no—
41C2×#390—31no—
42#1439—34no—
43#1437—34no—
44C4⋊#95—31no—
45C4×#95—31no—
46C24×D3C2312yes—
47C22×C3×Q8—22no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8C2×C414yes—
52C22×C3×Q8D414yes—
53C22×C3×Q8D414yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
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×D12—22no—
67C22×D12—22no—
68C22×D12C2×C414yes—
69C3×C4×Q8—232no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C42—24no—
77C2×C3×C42—24no—
78C2×C3×C42—24no—
79C2×C3×C42—22no—
80C2×C3×C42—22no—
81C2×C3×C42D414yes—
82#947—232no—
83C2×(C2×D3)⋊C4—28no—
84C2×(C2×D3)⋊C4—28no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—28no—
88C2×(C2×D3)⋊C4—28no—
89C2×(C2×D3)⋊C4—28no—
90C2×(C2×D3)⋊C4—44no—
91C2×(C2×D3)⋊C4—44no—
92C2×(C2×D3)⋊C4C2×C4116yes—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3C2×C414yes—
102C3⋊#95—28no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—44no—
106C23×Q8C2×D311yescharacteristic
107C23×D4—31no—
108C2×#124—38no—
109C2×#124—64no—
110C2×#124—64no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4C2×D312yes—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—34no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—34no—
120C22×C22⋊C4—32no—
121C22×C42—31no—
122C22×C42—31no—
123C22×C42C2×D312yes—
124C22×C42C2×D311yescharacteristic
125#391—64no—
126#390—64no—
127#390—64no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—34no—
131C2×#95—34no—
132C24×C3C2411yescharacteristic
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3—24no—
138C23×D3—24no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—24no—
142C23×D3—24no—
143C23×D3C22×C414yes—
144C2×C3×Q8—28no—
145C2×C3×Q8—28no—
146C2×C3×Q8—28no—
147C2×C3×Q8—28no—
148C2×C3×Q8—28no—
149C2×C3×Q8—28no—
150C2×C3×Q8C22⋊C4116yescomplemented
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4C22×C414yes—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D414yes—
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—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3C22×C414yes—
198C22×Dic3#4114yes—
199C2×D12—28no—
200C2×D12—28no—
201C2×D12—28no—
202C2×D12—44no—
203C2×D12—44no—
204C2×D12—44no—
205C2×D12—44no—
206C3×C4⋊C4—216no—
207C3×C4⋊C4—28no—
208C3×C42—216no—
209C3×C42—28no—
210(C2×D3)⋊C4—232no—
211(C2×D3)⋊C4—416no—
212(C2×D3)⋊C4—48no—
213(C2×D3)⋊C4—48no—
214C4⋊Dic3—44no—
215C4×Dic3—28no—
216C4×Dic3—44no—
217C25—32no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8C4×D314yes—
223C22×Q8D1214yes—
224C22×Q8#6714yes—
225C22×D4—34no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4—62no—
229C22×D4—62no—
230C22×D4—62no—
231C22×D4—62no—
232C23×C4—32no—
233C23×C4—32no—
234C23×C4—32no—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240#124—632no—
241C4×Q8—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4#6714yes—
251C2×C22⋊C4—316no—
252C2×C22⋊C4—68no—
253C2×C22⋊C4—68no—
254C2×C22⋊C4—68no—
255C2×C22⋊C4—68no—
256C2×C22⋊C4—68no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—124no—
260C2×C22⋊C4—124no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
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—64no—
274#95—68no—
275#95—64no—
276#95—124no—
277C23×C3C23×C411yescharacteristic
278C23×C3C22×D411yescharacteristic
279C23×C3C22×D411yescharacteristic
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4112yes—
285C23×C3C2×#4112yes—
286C23×C3H(5,𝔽2)12yes—
287C23×C32−1+412yes—
288C22×D3—24no—
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—28no—
292C22×D3—28no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—44no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3—44no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3—44no—
308C22×D3—44no—
309C22×D3—44no—
310C22×D3—44no—
311C3×Q8—232no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—28no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—44no—
332C2×C3×C4C2×C22⋊C418yes—
333C2×C3×C4C2×C22⋊C414yes—
334C2×C3×C4C4×D414yes—
335C2×C3×C4C4×D414yes—
336C2×C3×C4#12014yes—
337C2×C3×C4#12114yes—
338C2×C3×C4#12114yes—
339C2×C3×C4#12314yes—
340C2×C3×C4#12414yescomplemented
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—44no—
359C2×Dic3—44no—
360D12—416no—
361C24—34no—
362C24—62no—
363C24—62no—
364C24—62no—
365C24—64no—
366C24—64no—
367C24—64no—
368C24—64no—
369C24—62no—
370C24—62no—
371C24—62no—
372C24C23×D311yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8(C2×D3)⋊C4116yescomplemented
380C2×D4—68no—
381C2×D4—68no—
382C2×D4—68no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4—124no—
386C2×D4—124no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—34no—
403C22×C4—34no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—62no—
421C22×C4C2×C4×D314yes—
422C22×C4C2×C4×D312yes—
423C22×C4C2×D1214yes—
424C22×C4C2×D1212yes—
425C22×C4#23512yes—
426C22×C4#23512yes—
427C22×C4#23914yes—
428C22×C4C2×#6714yes—
429C22×C4C2×#6712yes—
430C22×C4C2×#6714yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C22×C4C2×#6712yes—
434C22×C4C2×#6712yes—
435C4⋊C4—216no—
436C4⋊C4—28no—
437C4⋊C4—124no—
438C22⋊C4—632no—
439C22⋊C4—1216no—
440C22⋊C4—128no—
441C22⋊C4—128no—
442C42—216no—
443C42—28no—
444C42—68no—
445C42—124no—
446C22×C3C22×C22⋊C411yescharacteristic
447C22×C3C2×C4×D411yescharacteristic
448C22×C3C2×C4×D411yescharacteristic
449C22×C3C4×#4112yes—
450C22×C3#52012yes—
451C22×C3C2×#12011yescharacteristic
452C22×C3C2×#12111yescharacteristic
453C22×C3C2×#12111yescharacteristic
454C22×C3C2×#12311yescharacteristic
455C22×C3C2×#12411yescharacteristic
456C22×C3#52912yes—
457C22×C3#53211yescharacteristic
458C22×C3#53511yescharacteristic
459C22×C3#53611yescharacteristic
460C22×C3#54212yes—
461C22×C3#54612yes—
462C22×C3#54812yes—
463C22×C3#54812yes—
464C22×C3#54812yes—
465C22×C3#55012yes—
466C22×C3#55012yes—
467C22×C3#55612yes—
468C22×C3#55912yes—
469C2×D3—44no—
470C2×D3—48no—
471C2×D3—48no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—84no—
475C2×D3—82no—
476C2×D3—82no—
477C2×D3—84no—
478C2×D3—82no—
479C2×D3—82no—
480C3×C4—24no—
481C3×C4—28no—
482C3×C4—42no—
483C3×C4—44no—
484C3×C4#38618yes—
485Dic3—44no—
486Dic3—44no—
487Dic3—82no—
488C23—64no—
489C23—64no—
490C23—64no—
491C23—68no—
492C23—68no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—124no—
499C23—124no—
500C23—124no—
501C23—124no—
502C23—122no—
503C23—124no—
504C23—122no—
505C23—124no—
506C23—124no—
507C23—124no—
508C23—124no—
509C23—122no—
510C23—124no—
511C23C22×C4×D311yescharacteristic
512C23C22×D1211yescharacteristic
513C23C2×#23511yescharacteristic
514C23C2×#23511yescharacteristic
515C23C2×#23912yes—
516C23#100712yes—
517C23D3×#4112yes—
518C23#100912yes—
519C23C22×#6711yescharacteristic
520C23C22×#6711yescharacteristic
521C23C22×#6711yescharacteristic
522Q8—232no—
523D4—1216no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—64no—
545C2×C4—64no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4C2×(C2×D3)⋊C418yes—
564C2×C4C2×(C2×D3)⋊C414yes—
565C2×C4C4×#6714yes—
566C2×C4C4×#6714yes—
567C2×C4#92914yes—
568C2×C4#93014yes—
569C2×C4#93014yes—
570C2×C4#94714yescomplemented
571C2×C4#95314yes—
572C2×C3C2×#38611yescharacteristic
573C2×C3C4×#12411yescharacteristic
574C2×C3#232611yescharacteristic
575C2×C3#234112yes—
576C2×C3#236712yes—
577C2×C3#242612yes—
578C2×C3#243111yescharacteristic
579C2×C3#243611yescharacteristic
580C2×C3#244812yes—
581C2×C3#254511yescharacteristic
582C2×C3#254811yescharacteristic
583D3—84no—
584C22—124no—
585C22—124no—
586C22—128no—
587C22—128no—
588C22—124no—
589C22—244no—
590C22—242no—
591C22—242no—
592C22—244no—
593C22—242no—
594C22—242no—
595C22#556812yes—
596C22C4×#23912yes—
597C22#557412yes—
598C22#557612yes—
599C22#557712yes—
600C22#558012yes—
601C22#558212yes—
602C22#558512yes—
603C22C22×(C2×D3)⋊C411yescharacteristic
604C22C2×C4×#6711yescharacteristic
605C22C2×C4×#6711yescharacteristic
606C22C2×#92911yescharacteristic
607C22C2×#93011yescharacteristic
608C22C2×#93011yescharacteristic
609C22C2×#94711yescharacteristic
610C22#581611yescharacteristic
611C22#581712yes—
612C22#581712yes—
613C22#582912yes—
614C22#583011yescharacteristic
615C22#583211yescharacteristic
616C22#583412yes—
617C22C2×#95311yescharacteristic
618C4—28no—
619C4—24no—
620C4—44no—
621C4—42no—
622C4—124no—
623C4—124no—
624C4—242no—
625C4#521318yes—
626C3#1402611yescharacteristic, complemented, Sylow
627C2—244no—
628C2C4×#94711yescharacteristic
629C2#8245511yescharacteristic
630C2#8245712yes—
631C2#8246612yes—
632C2#8246912yes—
633C2#8247411yescharacteristic
634C2#8247611yescharacteristic
635C2#8249512yes—
636C2C2×#521311yescharacteristic
637C2#8406611yescharacteristic
638C2#8409211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×C4×Q8C2
socle, cosocleC24×C3C24
prime radicalC23×D12C22
hypercenterC22×C4×Q8D3
hypocenterC3#14026
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×Q8C2semidirect
3. C2×#5229C2semidirect
6. #71570C2semidirect
7. #71563C2semidirect
8. #71537C2semidirect
9. C4×C3⋊#95C2semidirect
22. C2×#947C4semidirect
38. C22×C4×Q8D3semidirect
150. C2×C3×Q8C22⋊C4semidirect
340. C2×C3×C4#124semidirect
379. C2×Q8(C2×D3)⋊C4semidirect
570. C2×C4#947semidirect
626. C3#14026semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C22×#9472
C2×#52292
C2×#49882
C2×#49392
#715702
#715632
#715372
C4×C3⋊#952
#140263