Finite Group Databasebeta

← #10826286

Subgroups — #10826286

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
1C2×Q8×#67C211yesmaximal, characteristic
2C23×#216C211yesmaximal, characteristic, complemented
3C2×#4809C212yesmaximal, complemented
4C2×#4739C211yesmaximal, characteristic, complemented
5#74017C211yesmaximal, characteristic
6C3×#1787C211yesmaximal, characteristic, complemented
7#70283C212yesmaximal, complemented
8#70273C212yesmaximal, complemented
9#70239C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic
11#69514C211yesmaximal, characteristic, complemented
12#23856—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×D3×Q8C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16Q8×#67—24no—
17Q8×#67—24no—
18C2×#946—21no—
19C2×#946C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#944—21no—
22C2×#944C2211yescharacteristic
23C22×#216—24no—
24C22×#216—28no—
25C22×#216C2212yes—
26C22×#216C2212yes—
27C2×C4×#67—21no—
28C2×C4×#67C2211yescharacteristic
29C2×#924—21no—
30C2×#924C2211yescharacteristic
31C23×C3⋊C8C2211yescharacteristic
32C22×Dic12C2212yes—
33C2×#897C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C3×#380C2211yescharacteristic
38C2×(C2×D3)⋊C8C2212yes—
39C2×#816C2212yes—
40#4973C2212yes—
41C2×#810C2212yes—
42#4958—21no—
43#4950—21no—
44#4809—28no—
45#4739—24no—
46#4673—21no—
47#4656—21no—
48#4550C2212yes—
49#4529C2212yes—
50C23×Q16—31no—
51C2×D4×Q8—31no—
52C2×#451—32no—
53C2×#451—31no—
54#2409—31no—
55#2016—32no—
56#2001—32no—
57#1892—31no—
58#1878—32no—
59#1869—31no—
60#1787D311yescharacteristic, complemented
61C22×C3×Q8—21no—
62C22×C3×Q8—22no—
63C22×C3×Q8—24no—
64C22×C3×Q8—42no—
65C22×C3×Q8D414yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4C2311yescharacteristic
70C22×#67D411yescharacteristic
71C23×Dic3—21no—
72C2×D3×Q8—22no—
73C2×D3×Q8—22no—
74C2×D3×Q8—24no—
75C2×D3×Q8—42no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—22no—
79C22×Dic6—24no—
80C22×Dic6—21no—
81C22×Dic6—22no—
82C22×Dic6—21no—
83C22×Dic6—42no—
84C22×Dic6C2312yes—
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—21no—
89C2×#217—21no—
90C2×#217D411yescharacteristic
91#946—28no—
92#946—22no—
93#946—22no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96#944—28no—
97#944—22no—
98#944—22no—
99C2×#216—232no—
100C2×#216—24no—
101C2×#216—24no—
102C2×#216—24no—
103C2×#216—24no—
104C2×#216—24no—
105C2×#216D418yescomplemented
106C4×#67—28no—
107C4×#67—22no—
108C4×#67—22no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—22no—
111C2×(C2×D3)⋊C4D411yescharacteristic
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—22no—
114C2×C4⋊Dic3C2311yescharacteristic
115#924—28no—
116#924—22no—
117#924D414yes—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4D411yescharacteristic
123C2×C4×Dic3—21no—
124C2×C4×Dic3C2311yescharacteristic
125C22×C3⋊C8—22no—
126C22×C3⋊C8—24no—
127C22×C3⋊C8C2312yes—
128C2×Dic12—24no—
129C2×Dic12—24no—
130C2×Dic12—42no—
131#897—24no—
132#897—42no—
133C3×#103—24no—
134C3×#103—42no—
135C3×C22⋊C8—22no—
136C3×#95—21no—
137C3⋊#95—22no—
138C3⋊#95—22no—
139(C2×D3)⋊C8—24no—
140#816—24no—
141#810—24no—
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—32no—
158C2×C22⋊Q8—61no—
159C2×C22⋊Q8—61no—
160C2×C4×Q8—31no—
161C2×C4×D4—31no—
162C2×C4×D4—61no—
163#451—64no—
164#451—68no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103C2×D312yes—
168C2×#103C2×D312yes—
169C2×C22⋊C8—32no—
170C2×C22⋊C8C2×D311yescharacteristic
171#395—61no—
172#393—61no—
173#391—32no—
174#386—61no—
175#385—61no—
176#380C2×D311yescharacteristic
177#340—32no—
178#336—32no—
179C24×C3C2×D411yescharacteristic
180C23×D3C2×D411yescharacteristic
181C2×C3×Q8—24no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—22no—
185C2×C3×Q8—28no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—44no—
189C2×C3×Q8—44no—
190C2×C3×Q8—48no—
191C2×C3×Q8C2×D412yes—
192C2×C3×Q8C2×D412yes—
193C2×C3×Q8C2×D414yes—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—24no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—44no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2411yescharacteristic
205C2×#67—22no—
206C2×#67—22no—
207C2×#67Q1614yescomplemented
208C22×Dic3—22no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—44no—
212C22×Dic3—42no—
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216D3×Q8—48no—
217D3×Q8—44no—
218D3×Q8—44no—
219C2×C4×D3—24no—
220C2×C4×D3—22no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—22no—
224C2×C4×D3—44no—
225C2×Dic6—28no—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—24no—
233C2×Dic6—21no—
234C2×Dic6—22no—
235C2×Dic6—48no—
236C2×Dic6—44no—
237C2×Dic6—44no—
238C2×Dic6—44no—
239C2×Dic6—42no—
240C2×Dic6C2×D414yes—
241C2×Dic6C2×D412yes—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C2×C3×C8—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C22⋊C4—42no—
248#217—22no—
249#217—24no—
250#216—216no—
251#216—216no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—44no—
255C4⋊Dic3—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—44no—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—24no—
260Dic3⋊C4—24no—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—22no—
264C4×Dic3—24no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C2×C3⋊C8—28no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8C2×D414yes—
271Dic12—48no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—24no—
275C22×Q8—31no—
276C22×Q8—32no—
277C22×Q8—42no—
278C22×Q8—61no—
279C22×Q8—64no—
280C22×Q8—62no—
281C22×Q8—62no—
282C22×Q8—61no—
283C22×Q8—64no—
284C22×Q8—62no—
285C22×Q8—62no—
286C22×Q8—122no—
287C22×Q8—122no—
288C22×Q8D1214yes—
289C22×Q8C22×D311yescharacteristic
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C23×C4—21no—
293C23×C4—31no—
294C23×C4—61no—
295C23×C4—61no—
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—64no—
309C22×C8—62no—
310C22×C8C22×D312yes—
311C4⋊Q8—62no—
312C4⋊Q8—68no—
313C4⋊Q8—62no—
314C22⋊Q8—34no—
315C22⋊Q8—62no—
316C22⋊Q8—68no—
317C22⋊Q8—68no—
318C22⋊Q8—62no—
319C22⋊Q8—62no—
320C22⋊Q8—64no—
321C22⋊Q8—122no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×D4—68no—
325C4×D4—62no—
326C4×D4—62no—
327C2×C4⋊C4—31no—
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—62no—
335C2×C4⋊C4C22×D312yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—62no—
342C2×C42—31no—
343C2×C42—61no—
344#103—24no—
345#103—42no—
346#103—64no—
347#103—64no—
348C22⋊C8—22no—
349C22⋊C8—64no—
350#95—21no—
351#95—62no—
352#95—62no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3#12012yes—
356C23×C3C2×Q1612yes—
357C23×C3#13712yes—
358C23×C3C22×D411yescharacteristic
359C22×D3—21no—
360C22×D3—22no—
361C22×D3—21no—
362C22×D3—22no—
363C22×D3C2×Q1612yescomplemented
364C3×Q8—22no—
365C3×Q8—24no—
366C3×Q8—22no—
367C3×Q8—22no—
368C3×Q8—48no—
369C3×Q8—42no—
370C3×Q8#12014yes—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—24no—
377C2×C3×C4—22no—
378C2×C3×C4—48no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4#12012yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D412yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4H(5,𝔽2)12yes—
388#67—28no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—24no—
397C2×Dic3—22no—
398C2×Dic3—48no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3C2×Q1612yescomplemented
404C4×D3—48no—
405C4×D3—41no—
406C4×D3—41no—
407C4×D3—42no—
408Dic6—24no—
409Dic6—24no—
410Dic6—24no—
411Dic6—48no—
412Dic6—48no—
413Dic6—41no—
414Dic6—42no—
415Dic6—41no—
416Dic6—42no—
417C3×C8—42no—
418C3⋊C8—24no—
419C24—31no—
420C24D3×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—32no—
429C2×Q8—34no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—64no—
434C2×Q8—68no—
435C2×Q8—61no—
436C2×Q8—62no—
437C2×Q8—62no—
438C2×Q8—62no—
439C2×Q8—62no—
440C2×Q8—62no—
441C2×Q8—62no—
442C2×Q8—62no—
443C2×Q8—124no—
444C2×Q8—124no—
445C2×Q8—128no—
446C2×Q8—124no—
447C2×Q8—128no—
448C2×Q8—124no—
449C2×Q8—124no—
450C2×Q8—122no—
451C2×Q8C2×D1212yes—
452C2×Q8C2×D1212yes—
453C2×Q8D3×D414yes—
454C2×D4—34no—
455C2×D4—62no—
456C2×D4—62no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—44no—
467C22×C4—64no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—62no—
473C22×C4—61no—
474C22×C4—62no—
475C22×C4—124no—
476C22×C4—124no—
477C22×C4—122no—
478C22×C4C2×D1212yes—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D412yes—
481C22×C4C23×D311yescharacteristic
482Q16—616no—
483Q16—616no—
484Q16—128no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—34no—
489C2×C8—68no—
490C2×C8—62no—
491C2×C8—62no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—32no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—62no—
500C4⋊C4—64no—
501C4⋊C4—61no—
502C4⋊C4—124no—
503C22⋊C4—42no—
504C22⋊C4—64no—
505C22⋊C4—62no—
506C22⋊C4—64no—
507C22⋊C4—62no—
508C22⋊C4—124no—
509C42—64no—
510C42—61no—
511C42—61no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3#45114yescomplemented
517C22×C3C2×#12011yescharacteristic
518C22×C3D4212yes—
519C22×C3C22×Q1611yescharacteristic
520C22×C3C2×#13711yescharacteristic
521C22×C3#57812yes—
522C2×D3—22no—
523C2×D3—24no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—42no—
527C3×C4—22no—
528C3×C4—24no—
529C3×C4—42no—
530C3×C4—82no—
531C3×C4C2×#12011yescharacteristic
532C3×C4#53411yescharacteristic
533C3×C4D4212yes—
534Dic3—22no—
535Dic3—42no—
536Dic3—44no—
537Dic3—41no—
538Dic3—41no—
539Dic3—42no—
540C23—22no—
541C23—22no—
542C23—32no—
543C23—61no—
544C23—61no—
545C23—62no—
546C23—62no—
547C23#88212yes—
548C23#91512yes—
549C23D3×Q1612yes—
550C23C2×D3×D411yescharacteristic
551Q8—22no—
552Q8—24no—
553Q8—22no—
554Q8—22no—
555Q8—48no—
556Q8—42no—
557Q8—64no—
558Q8—64no—
559Q8—64no—
560Q8—128no—
561Q8—128no—
562Q8—121no—
563Q8—121no—
564Q8—122no—
565Q8—122no—
566Q8#88214yes—
567D4—68no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—128no—
591C2×C4—128no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4#88212yes—
600C2×C4C22×D1211yescharacteristic
601C2×C4C2×D3×D412yes—
602C2×C4C2×D3×D411yescharacteristic
603C2×C4#100412yes—
604C8—42no—
605C8—64no—
606C2×C3—24no—
607C2×C3#240311yescharacteristic
608C2×C3C2×#45111yescharacteristic
609C2×C3#301111yescharacteristic
610C2×C3#327012yes—
611C2×C3D4×Q1612yes—
612D3—42no—
613C22—24no—
614C22—22no—
615C22—22no—
616C22—24no—
617C22—64no—
618C22—62no—
619C22—121no—
620C22—121no—
621C22—122no—
622C22#480914yescomplemented
623C22C2×#88211yescharacteristic
624C22#559012yes—
625C22C2×#91511yescharacteristic
626C22C2×D3×Q1611yescharacteristic
627C22#577912yes—
628C4—22no—
629C4—24no—
630C4—42no—
631C4—62no—
632C4—82no—
633C4—121no—
634C4—122no—
635C4—121no—
636C4—122no—
637C4—124no—
638C4C2×#88211yescharacteristic
639C4#549311yescharacteristic
640C4#559012yes—
641C3#2385611yescharacteristic, complemented, Sylow
642C2—24no—
643C2—122no—
644C2#7385211yescharacteristic
645C2C2×#480911yescharacteristic
646C2#7896111yescharacteristic
647C2#8006412yes—
648C2#8021112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#1787D3
hypocenterC3#23856
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C23×#216C2semidirect
3. C2×#4809C2semidirect
4. C2×#4739C2semidirect
6. C3×#1787C2semidirect
7. #70283C2semidirect
8. #70273C2semidirect
9. #70239C2semidirect
11. #69514C2semidirect
60. #1787D3semidirect
105. C2×#216D4semidirect
207. C2×#67Q16semidirect
363. C22×D3C2×Q16semidirect
403. C2×Dic3C2×Q16semidirect
516. C22×C3#451semidirect
622. C22#4809semidirect
641. C3#23856semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#672
C23×#2162
C2×#48092
C2×#47392
#740172
C3×#17872
#702832
#702732
#702392
#699522
#695142
#238563