Finite Group Databasebeta

← #10732205

Subgroups — #10732205

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×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×#944C211yesmaximal, characteristic, complemented
3C22×#924C211yesmaximal, characteristic
4C2×#5229C211yesmaximal, characteristic, complemented
5C2×C4×#217C211yesmaximal, characteristic
6C2×#4973C211yesmaximal, characteristic, complemented
7#71568C214yesmaximal, complemented
8#71563C211yesmaximal, characteristic, complemented
9#71547C214yesmaximal
10#15178—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C2×C3×C22⋊Q8—24no—
15C2×C3×C22⋊Q8—24no—
16C2×C3×C22⋊Q8—24no—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×#217C2212yes—
21C22×#217C2212yes—
22C2×#944—24no—
23C2×#944—24no—
24C2×#944—24no—
25C22×C4⋊Dic3C2211yescharacteristic
26C2×#924—24no—
27C2×#924—24no—
28C2×#924—24no—
29C22×Dic3⋊C4C2214yes—
30C22×Dic3⋊C4C2212yes—
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2211yescharacteristic
33#5229—24no—
34C4×#217—24no—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2214yes—
37C2×C3⋊#95C2214yes—
38C2×C3⋊#95C2214yes—
39#4973—24no—
40C22×C4⋊Q8—31no—
41C22×C22⋊Q8—31no—
42C22×C22⋊Q8D311yescharacteristic
43C2×#391—31no—
44C2×#391—31no—
45C2×C4×C22⋊C4—31no—
46#1444—34no—
47#1439—34no—
48C4⋊#95—31no—
49C25×C3C2311yescharacteristic
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×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4—22no—
60C23×C3×C4—22no—
61C23×C3×C4—22no—
62C23×C3×C4—22no—
63C23×C3×C4D414yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—22no—
78C22×Dic6D414yes—
79C3×C22⋊Q8—232no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4D414yes—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×#217—24no—
90C2×#217—24no—
91C2×#217—24no—
92C2×#217—24no—
93C2×#217—24no—
94C2×#217Q818yescomplemented
95#944—232no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3D414yes—
99#924—232no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—28no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—28no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3—22no—
109C2×C4×Dic3—22no—
110C2×C4×Dic3—22no—
111C2×C4×Dic3D414yes—
112C3⋊#95—28no—
113C3⋊#95—24no—
114C3⋊#95—28no—
115C3⋊#95—28no—
116C23×Q8—31no—
117C23×Q8C2×D311yescharacteristic
118C24×C4C2×D311yescharacteristic
119C2×C4⋊Q8—64no—
120C2×C4⋊Q8—64no—
121C2×C4⋊Q8—64no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—64no—
126C2×C22⋊Q8—64no—
127C2×C22⋊Q8—64no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—34no—
131C22×C4⋊C4C2×D312yes—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4C2×D312yes—
136C22×C42—31no—
137C22×C42—31no—
138#391—64no—
139#391—64no—
140C4×C22⋊C4—64no—
141C2×#95—34no—
142C2×#95—34no—
143C2×#95—34no—
144C2×#95—32no—
145C24×C3—21no—
146C24×C3—21no—
147C24×C3—21no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—22no—
151C24×C3—21no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3C2×D412yes—
155C24×C3C2×Q814yes—
156C24×C3C2411yescharacteristic
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—21no—
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×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—28no—
181C22×C3×C4—28no—
182C22×C3×C4—28no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4—44no—
186C22×C3×C4—44no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D414yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4#4112yes—
197C22×Dic3—22no—
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×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—24no—
215C22×Dic3—22no—
216C22×Dic3—22no—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×D414yes—
219C22×Dic3C2×D414yes—
220C22×Dic3C2×Q814yes—
221C22×Dic3C2×Q814yes—
222C22×Dic3#4114yes—
223C2×Dic6—28no—
224C2×Dic6—28no—
225C2×Dic6—28no—
226C2×Dic6—44no—
227C2×Dic6—44no—
228C2×Dic6—44no—
229C2×Dic6—44no—
230C3×C4⋊C4—216no—
231C3×C4⋊C4—28no—
232C3×C22⋊C4—216no—
233#217—216no—
234#217—216no—
235C4⋊Dic3—28no—
236Dic3⋊C4—216no—
237Dic3⋊C4—232no—
238C4×Dic3—28no—
239C4×Dic3—44no—
240C25C22×D311yescharacteristic
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×Q8#6714yes—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—32no—
262C23×C4—32no—
263C23×C4—32no—
264C23×C4—32no—
265C23×C4#6714yes—
266C23×C4C22×D312yes—
267C23×C4C22×D312yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C4⋊Q8—632no—
271C22⋊Q8—232no—
272C22⋊Q8—632no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4—68no—
282C2×C4⋊C4—68no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—64no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—68no—
287C2×C4⋊C4#6714yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—64no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—64no—
297C2×C42—34no—
298C2×C42—62no—
299C2×C42—62no—
300C2×C42—62no—
301C2×C42—62no—
302C2×C42—62no—
303#95—68no—
304#95—64no—
305#95—68no—
306#95—68no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—24no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3C4⋊Q814yescomplemented
330C23×C3C22×D411yescharacteristic
331C23×C3C22×D411yescharacteristic
332C23×C3C22×D411yescharacteristic
333C23×C3C22×D411yescharacteristic
334C23×C3C22×D411yescharacteristic
335C23×C3C22×D411yescharacteristic
336C23×C3C22×Q812yes—
337C23×C3C2×#4111yescharacteristic
338C23×C3C2×#4112yes—
339C23×C32−1+412yes—
340C23×C32−1+412yes—
341C3×Q8—416no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—28no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—416no—
362C2×C3×C4#12014yes—
363C2×C3×C4#12014yes—
364C2×C3×C4#12014yes—
365C2×C3×C4#12114yes—
366C2×C3×C4#12114yes—
367C2×C3×C4#12114yes—
368C2×C3×C4#12714yes—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—28no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—44no—
394C2×Dic3C22⋊Q818yescomplemented
395Dic6—416no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24D3×Q814yes—
406C24C2×#6712yes—
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—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—34no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—34no—
445C22×C4—34no—
446C22×C4—34no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4D3×D412yes—
472C22×C4D3×D412yes—
473C22×C4D3×D412yes—
474C22×C4#23712yes—
475C22×C4C2×#6714yes—
476C22×C4C2×#6714yes—
477C22×C4C2×#6712yes—
478C22×C4C2×#6712yes—
479C22×C4C2×#6712yes—
480C22×C4C2×#6712yes—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—616no—
484C4⋊C4—632no—
485C4⋊C4—68no—
486C22⋊C4—216no—
487C22⋊C4—616no—
488C22⋊C4—616no—
489C42—68no—
490C42—124no—
491C22×C3—24no—
492C22×C3—24no—
493C22×C3—24no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—44no—
497C22×C3—44no—
498C22×C3—44no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—44no—
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#12011yescharacteristic
504C22×C3C2×#12011yescharacteristic
505C22×C3C2×#12111yescharacteristic
506C22×C3C2×#12111yescharacteristic
507C22×C3C2×#12111yescharacteristic
508C22×C3C2×C22⋊Q812yes—
509C22×C3C2×#12711yescharacteristic
510C22×C3C2×C4⋊Q811yescharacteristic
511C22×C3#53211yescharacteristic
512C22×C3#53312yes—
513C22×C3#53611yescharacteristic
514C22×C3#53611yescharacteristic
515C22×C3#53612yes—
516C22×C3#54112yes—
517C22×C3#54712yes—
518C22×C3#54712yes—
519C22×C3#54812yes—
520C22×C3#54812yes—
521C22×C3D4×Q812yes—
522C22×C3D4×Q812yes—
523C22×C3D4×Q812yes—
524C22×C3D4×Q812yes—
525C3×C4—42no—
526C3×C4—44no—
527C3×C4—44no—
528C3×C4#39218yes—
529Dic3—28no—
530Dic3—44no—
531Dic3—44no—
532Dic3—44no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23#94414yescomplemented
556C23C2×D3×D411yescharacteristic
557C23C2×D3×D411yescharacteristic
558C23C2×D3×D411yescharacteristic
559C23C2×#23711yescharacteristic
560C23C2×D3×Q812yes—
561C23#100712yes—
562C23D3×#4112yes—
563C23#101012yes—
564C23C22×#6711yescharacteristic
565C23C22×#6711yescharacteristic
566C23C22×#6711yescharacteristic
567Q8—416no—
568Q8—1216no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—38no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—416no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—64no—
603C2×C4—68no—
604C2×C4—124no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4#93714yes—
616C2×C4#93714yes—
617C2×C4#93814yes—
618C2×C4#93914yes—
619C2×C4#93914yes—
620C2×C4#94014yes—
621C2×C4#95314yes—
622C2×C3—44no—
623C2×C3C2×#39211yescharacteristic
624C2×C3#240912yes—
625C2×C3#242912yes—
626C2×C3#242912yes—
627C2×C3#243211yescharacteristic
628C2×C3#243211yescharacteristic
629C2×C3#243411yescharacteristic
630C2×C3#243411yescharacteristic
631C2×C3#254212yes—
632C2×C3#262111yescharacteristic
633C2×C3#285711yescharacteristic
634C22—24no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22#562212yes—
646C22#562512yes—
647C22D3×C22⋊Q812yes—
648C22#562812yes—
649C22#563112yes—
650C22#563212yes—
651C22#563212yes—
652C22#563312yes—
653C22C2×#93711yescharacteristic
654C22C2×#93711yescharacteristic
655C22C2×#93811yescharacteristic
656C22C2×#93911yescharacteristic
657C22C2×#93911yescharacteristic
658C22C2×#94011yescharacteristic
659C22C2×#94411yescharacteristic
660C22Q8×#6712yes—
661C22Q8×#6712yes—
662C22#581611yescharacteristic
663C22#582412yes—
664C22#582511yescharacteristic
665C22#583112yes—
666C22#583211yescharacteristic
667C22C2×#95311yescharacteristic
668C4—42no—
669C4—44no—
670C4—44no—
671C4—68no—
672C4—124no—
673C4—124no—
674C4—124no—
675C4#522318yes—
676C3#1517811yescharacteristic, complemented, Sylow
677C2—44no—
678C2#8275112yes—
679C2#8275712yes—
680C2#8275812yes—
681C2#8275911yescharacteristic
682C2#8276011yescharacteristic
683C2#8278012yes—
684C2#8278611yescharacteristic
685C2#8281111yescharacteristic
686C2C2×#522311yescharacteristic
687C2#8406511yescharacteristic
688C2#8409311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C22⋊Q8C2
prime radicalC25×C3C23
hypercenterC22×C22⋊Q8D3
hypocenterC3#15178
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#944C2semidirect
4. C2×#5229C2semidirect
6. C2×#4973C2semidirect
7. #71568C2semidirect
8. #71563C2semidirect
94. C2×#217Q8semidirect
329. C23×C3C4⋊Q8semidirect
394. C2×Dic3C22⋊Q8semidirect
555. C23#944semidirect
676. C3#15178semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×#9442
C22×#9242
C2×#52292
C2×C4×#2172
C2×#49732
#715682
#715632
#715472
#151783