Finite Group Databasebeta

← #10826282

Subgroups — #10826282

Order 768.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#216C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic
3C3×#2409C211yesmaximal, characteristic
4C2×#5228C211yesmaximal, characteristic, complemented
5C2×#5050C212yesmaximal, complemented
6#71700C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic
8#69691C212yesmaximal, complemented
9#69690C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69514C211yesmaximal, characteristic, complemented
12#23856—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C22×#216—24no—
19C22×#216—28no—
20C22×#216C2212yes—
21C22×#216C2212yes—
22C22×#216C2212yes—
23C2×D4×Dic3—21no—
24C2×#924C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27D4×Dic6—24no—
28D4×Dic6—24no—
29C2×#888—21no—
30C2×#878—21no—
31C2×#878—21no—
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#395—21no—
35C3×#393—21no—
36C3×#391C2212yes—
37C3×#386—21no—
38C3×#385—21no—
39C3×#380C2211yescharacteristic
40#5228—24no—
41C2×#835C2212yes—
42C2×#830C2212yes—
43C2×#830C2211yescharacteristic
44#5050—28no—
45C2×#810C2212yes—
46C2×#810C2212yes—
47C2×#810C2212yes—
48#4529C2212yes—
49#4524C2212yes—
50C23×Q16—31no—
51C2×D4×Q8—31no—
52C2×#451—31no—
53C2×#451—32no—
54#2409D311yescharacteristic
55#2016—32no—
56#2001—32no—
57#1892—31no—
58#1878—32no—
59#1869—31no—
60#1787—31no—
61C22×C3×Q8—21no—
62C22×C3×Q8—24no—
63C22×C3×Q8—22no—
64C22×C3×Q8—22no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2312yes—
67C22×C3×D4D411yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—21no—
72C23×Dic3—21no—
73C22×Dic6—24no—
74C22×Dic6—21no—
75C22×Dic6—21no—
76C22×Dic6—22no—
77C22×Dic6—22no—
78C22×Dic6—22no—
79C22×Dic6—24no—
80C22×Dic6—42no—
81C22×Dic6—42no—
82C22×Dic6D414yes—
83C22×Dic6C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—42no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#216—232no—
101C2×#216—24no—
102C2×#216—24no—
103C2×#216—24no—
104C2×#216—24no—
105C2×#216—24no—
106C2×#216—24no—
107C2×#216—24no—
108C2×#216—42no—
109C2×#216D418yescomplemented
110D4×Dic3—28no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114#924—22no—
115#924—22no—
116#924—22no—
117#924D414yes—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8—22no—
123C22×C3⋊C8—24no—
124C22×C3⋊C8C2312yes—
125C22×C3⋊C8C2312yes—
126#888—28no—
127#878—28no—
128#878—28no—
129#869—22no—
130#869—22no—
131C4×Dic6—22no—
132C4×Dic6—22no—
133C3×#95—21no—
134C3×#95—22no—
135C3×#95—22no—
136#835—24no—
137#830—22no—
138#830—24no—
139#810—24no—
140#810—24no—
141#810—42no—
142C23×Q8—31no—
143C23×Q8—31no—
144C23×Q8C2×D311yescharacteristic
145C22×Q16—32no—
146C22×Q16—32no—
147C22×Q16—32no—
148C22×Q16—68no—
149C22×Q16—64no—
150C23×C8—31no—
151D4×Q8—64no—
152D4×Q8—64no—
153C2×C4⋊Q8—31no—
154C2×C4⋊Q8—61no—
155C2×C22⋊Q8—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—61no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8C2×D312yes—
160C2×C4×Q8—31no—
161C2×C4×D4—61no—
162C2×C4×D4C2×D311yescharacteristic
163#451—68no—
164#451—64no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103—32no—
168C2×#103—32no—
169C2×C22⋊C8—32no—
170C2×C22⋊C8—31no—
171#395—21no—
172#393—21no—
173#391C2×D312yes—
174#386—21no—
175#385—21no—
176#380C2×D311yescharacteristic
177#340—32no—
178#336—32no—
179C24×C3C2×D411yescharacteristic
180C24×C3C2×D411yescharacteristic
181C2×C3×Q8—22no—
182C2×C3×Q8—28no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—44no—
188C2×C3×Q8—48no—
189C2×C3×Q8—44no—
190C2×C3×Q8—42no—
191C2×C3×Q8C2×D414yes—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4Q1614yescomplemented
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—44no—
204C22×C3×C4—42no—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—44no—
219C22×Dic3—44no—
220C2×Dic6—21no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—28no—
227C2×Dic6—24no—
228C2×Dic6—24no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—44no—
232C2×Dic6—48no—
233C2×Dic6—48no—
234C2×Dic6—44no—
235C2×Dic6—44no—
236C2×Dic6—44no—
237C2×Dic6—44no—
238C2×Dic6C2×D412yes—
239C2×Dic6C2×D412yes—
240C2×Dic6C2×D414yes—
241C2×Dic6C2×D412yes—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—44no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—44no—
249C3×C22⋊C4—42no—
250C3×C42—21no—
251C3×C42—21no—
252#217—24no—
253#217—24no—
254#216—216no—
255#216—216no—
256#216—48no—
257C4⋊Dic3—24no—
258C4⋊Dic3—24no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264C4×Dic3—24no—
265C2×C3⋊C8—28no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8C2×D414yes—
272C22×Q8—21no—
273C22×Q8—24no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8—34no—
277C22×Q8—31no—
278C22×Q8—31no—
279C22×Q8—31no—
280C22×Q8—42no—
281C22×Q8—62no—
282C22×Q8—62no—
283C22×Q8—64no—
284C22×Q8—64no—
285C22×Q8—62no—
286C22×Q8—61no—
287C22×Q8—61no—
288C22×Q8—122no—
289C22×Q8—122no—
290C22×Q8C22×D312yes—
291C22×D4#6711yescharacteristic
292C23×C4—21no—
293C23×C4—61no—
294C23×C4—61no—
295C23×C4C22×D311yescharacteristic
296C23×C4C22×D311yescharacteristic
297C2×Q16—38no—
298C2×Q16—64no—
299C2×Q16—64no—
300C2×Q16—64no—
301C2×Q16—64no—
302C2×Q16—632no—
303C2×Q16—64no—
304C2×Q16—64no—
305C2×Q16—64no—
306C2×Q16—122no—
307C22×C8—32no—
308C22×C8—32no—
309C22×C8—64no—
310C22×C8—62no—
311C4⋊Q8—62no—
312C4⋊Q8—62no—
313C4⋊Q8—68no—
314C22⋊Q8—24no—
315C22⋊Q8—34no—
316C22⋊Q8—42no—
317C22⋊Q8—68no—
318C22⋊Q8—68no—
319C22⋊Q8—62no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—68no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4C22×D312yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4#6711yescharacteristic
341C2×C22⋊C4#6711yescharacteristic
342C2×C42—61no—
343C2×C42C22×D311yescharacteristic
344#103—64no—
345#103—64no—
346#103—64no—
347#103—122no—
348C22⋊C8—64no—
349C22⋊C8—62no—
350#95—21no—
351#95—22no—
352#95—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—22no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3#12012yes—
360C23×C3C2×Q1612yes—
361C23×C3C2×Q1612yescomplemented
362C23×C3#13712yes—
363C23×C3C22×D411yescharacteristic
364C3×Q8—24no—
365C3×Q8—24no—
366C3×Q8—48no—
367C3×Q8—42no—
368C3×Q8—42no—
369C3×D4—28no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—48no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4#12012yes—
391C2×C3×C4C2×Q1612yescomplemented
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×D412yes—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4H(5,𝔽2)12yes—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—48no—
402C2×Dic3—48no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406Dic6—22no—
407Dic6—24no—
408Dic6—22no—
409Dic6—22no—
410Dic6—24no—
411Dic6—48no—
412Dic6—48no—
413Dic6—41no—
414Dic6—41no—
415Dic6—42no—
416Dic6#12014yes—
417C3⋊C8—24no—
418C3⋊C8—42no—
419C24C2×#6711yescharacteristic
420C24C2×#6711yescharacteristic
421C2×Q8—22no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—34no—
428C2×Q8—32no—
429C2×Q8—32no—
430C2×Q8—32no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—64no—
436C2×Q8—68no—
437C2×Q8—64no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—62no—
442C2×Q8—62no—
443C2×Q8—62no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—124no—
447C2×Q8—124no—
448C2×Q8—124no—
449C2×Q8—128no—
450C2×Q8—124no—
451C2×Q8—128no—
452C2×Q8—124no—
453C2×Q8D3×D414yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4#21614yescomplemented
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—61no—
468C22×C4—64no—
469C22×C4—64no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—124no—
474C22×C4—124no—
475C22×C4D3×D412yes—
476C22×C4D3×D412yes—
477C22×C4C2×#6711yescharacteristic
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6711yescharacteristic
481C22×C4C23×D311yescharacteristic
482Q16—616no—
483Q16—616no—
484Q16—128no—
485C2×C8—34no—
486C2×C8—68no—
487C2×C8—62no—
488C2×C8—62no—
489C2×C8—62no—
490C2×C8—62no—
491C2×C8—62no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—32no—
496C4⋊C4—44no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—61no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C42—21no—
510C42—21no—
511C42—64no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3—22no—
517C22×C3—24no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—42no—
521C22×C3#45114yescomplemented
522C22×C3C2×#12011yescharacteristic
523C22×C3D4212yes—
524C22×C3C22×Q1611yescharacteristic
525C22×C3C2×#13711yescharacteristic
526C22×C3#57812yes—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4—44no—
531C3×C4—41no—
532C3×C4—82no—
533C3×C4C2×#12011yescharacteristic
534C3×C4#53411yescharacteristic
535C3×C4D4212yes—
536Dic3—24no—
537Dic3—42no—
538Dic3—42no—
539Dic3—42no—
540C23—22no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—22no—
546C23#93712yes—
547C23C2×#21612yes—
548C23C2×#21612yescomplemented
549C23#95112yes—
550C23C22×#6711yescharacteristic
551Q8—24no—
552Q8—24no—
553Q8—34no—
554Q8—48no—
555Q8—42no—
556Q8—42no—
557Q8—62no—
558Q8—64no—
559Q8—62no—
560Q8—62no—
561Q8—64no—
562Q8—128no—
563Q8—128no—
564Q8—121no—
565Q8—121no—
566Q8—122no—
567D4—28no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—62no—
589C2×C4—64no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—128no—
594C2×C4—128no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4#93712yes—
599C2×C4C2×#21612yescomplemented
600C2×C4C2×D3×D411yescharacteristic
601C2×C4C2×D3×D412yes—
602C2×C4C2×D3×D411yescharacteristic
603C2×C4#100412yes—
604C8—64no—
605C8—122no—
606C2×C3—24no—
607C2×C3—42no—
608C2×C3#240311yescharacteristic
609C2×C3C2×#45111yescharacteristic
610C2×C3#301111yescharacteristic
611C2×C3#327012yes—
612C2×C3D4×Q1612yes—
613C22—24no—
614C22—24no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22#505014yescomplemented
623C22C2×#93711yescharacteristic
624C22D4×#6712yes—
625C22C22×#21611yescharacteristic
626C22C2×#95111yescharacteristic
627C22#583812yes—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—44no—
632C4—41no—
633C4—64no—
634C4—82no—
635C4—122no—
636C4—122no—
637C4—122no—
638C4D3×#12011yescharacteristic
639C4#558911yescharacteristic
640C4#559012yes—
641C3#2385611yescharacteristic, complemented, Sylow
642C2—24no—
643C2—42no—
644C2#8260411yescharacteristic
645C2C2×#505011yescharacteristic
646C2#8341111yescharacteristic
647C2#8354612yes—
648C2D4×#21612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2409C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2409D3
hypocenterC3#23856
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#216C2semidirect
4. C2×#5228C2semidirect
5. C2×#5050C2semidirect
6. #71700C2semidirect
8. #69691C2semidirect
9. #69690C2semidirect
10. #69525C2semidirect
11. #69514C2semidirect
109. C2×#216D4semidirect
194. C2×C3×D4Q16semidirect
361. C23×C3C2×Q16semidirect
391. C2×C3×C4C2×Q16semidirect
456. C2×D4#216semidirect
521. C22×C3#451semidirect
548. C23C2×#216semidirect
599. C2×C4C2×#216semidirect
622. C22#5050semidirect
641. C3#23856semidirect

Maximal subgroups

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

subgroupindex
C23×#2162
C2×D4×Dic62
C3×#24092
C2×#52282
C2×#50502
#717002
#716102
#696912
#696902
#695252
#695142
#238563