Finite Group Databasebeta

← #10943990

Subgroups — #10943990

Order 768.

1576 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
1C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4806C212yesmaximal, complemented
5#74375C211yesmaximal, characteristic
6C3×#1803C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic, complemented
8#70240C212yesmaximal, complemented
9#69588C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68880C212yesmaximal, complemented
12#32903—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#215C2212yes—
17C22×#204C2212yes—
18C4×D3×Q8—24no—
19C4×D3×Q8—24no—
20C2×#895—21no—
21C2×#895—21no—
22C2×#895C2212yes—
23C2×D3×C4⋊C4—21no—
24C2×D3×C4⋊C4C2211yescharacteristic
25C2×#887—21no—
26C2×#887C2211yescharacteristic
27C2×#874C2211yescharacteristic
28C2×C42×D3—21no—
29C2×C42×D3C2211yescharacteristic
30C2×C4×Dic6—21no—
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C3×#384C2211yescharacteristic
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37#5027—24no—
38#4988—21no—
39#4988—21no—
40#4988C2212yes—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4848—24no—
45#4806—28no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C22×C4×Q8—31no—
50C2×#460—31no—
51C2×#460—31no—
52C2×#448—32no—
53#2856—31no—
54#1880—32no—
55#1880—32no—
56#1874—32no—
57#1803—31no—
58#1803D311yescharacteristic, complemented
59#1797—31no—
60C22×C3×Q8C2311yescharacteristic
61C2×D3×Q8—22no—
62C2×D3×Q8—22no—
63C2×D3×Q8—22no—
64C2×D3×Q8—22no—
65C2×D3×Q8—22no—
66C2×D3×Q8D414yescomplemented
67C22×D12—22no—
68C22×D12—21no—
69C22×D12—21no—
70C22×D12C2312yes—
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×C4×D3D411yescharacteristic
74C22×C4×D3D411yescharacteristic
75C22×C4×D3C2311yescharacteristic
76C22×Dic6—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2312yes—
79C3×C4×Q8—22no—
80C3×C4×Q8D414yescomplemented
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88Q8×Dic3—22no—
89Q8×Dic3—22no—
90C2×#215—24no—
91C2×#215—24no—
92C2×#215—24no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3D411yescharacteristic
105C2×C4×Dic3D411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×#204—24no—
109C2×#204—24no—
110C2×#204—24no—
111#895—24no—
112#895—24no—
113#895—44no—
114#895—44no—
115D3×C4⋊C4—28no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—22no—
118#887—28no—
119#887—22no—
120#887—22no—
121#874—22no—
122#874—22no—
123C42×D3—28no—
124C42×D3—22no—
125C42×D3—22no—
126C4×Dic6—28no—
127C4×Dic6—22no—
128C4×Dic6D414yescomplemented
129C3×C4⋊C8—22no—
130C3×#103—24no—
131#821—24no—
132(C2×D3)⋊C8—24no—
133#810—24no—
134#809—24no—
135C4⋊(C3⋊C8)—22no—
136C23×Q8—31no—
137C22×SD16—32no—
138C22×SD16—32no—
139C2×#127—31no—
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×Q8—61no—
147C2×C4×Q8—61no—
148C2×C4×Q8—64no—
149C2×C4×Q8—64no—
150C2×C4×Q8C2×D311yescharacteristic
151C22×C4⋊C4—31no—
152C22×C4⋊C4—61no—
153C22×C42—31no—
154C22×C42—61no—
155#460—64no—
156#460—64no—
157#448—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—32no—
161C2×#103C2×D312yes—
162C2×#102—32no—
163C2×#102—32no—
164C2×C22⋊C8—32no—
165#390—32no—
166#390—61no—
167#390—61no—
168#384C2×D311yescharacteristic
169#340—32no—
170#340C2×D312yes—
171#336—32no—
172C23×D3—22no—
173C23×D3C2×D411yescharacteristic
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8#4112yes—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191D3×Q8—28no—
192D3×Q8—28no—
193D3×Q8—28no—
194D3×Q8—28no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—22no—
201C2×D12—42no—
202C2×D12—42no—
203C2×D12—44no—
204C2×D12—44no—
205C2×D12—44no—
206C2×D12—42no—
207C2×D12—42no—
208C2×D12—42no—
209C2×D12—42no—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—24no—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—41no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3SD1614yescomplemented
226C2×C4×D3C2×D412yes—
227C2×C4×D3#4112yes—
228C2×Dic6—24no—
229C2×Dic6—24no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—24no—
234C2×Dic6C2×D412yes—
235C2×Dic6C2×D412yes—
236C2×Dic6#4112yes—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C2×C3×C8—22no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4C2×D412yes—
247C3×C42—24no—
248C3×C42C2×D412yes—
249C3×C42C2×D412yes—
250#215—48no—
251(C2×D3)⋊C4—44no—
252(C2×D3)⋊C4—44no—
253(C2×D3)⋊C4—44no—
254C4⋊Dic3—24no—
255C4⋊Dic3—21no—
256C4⋊Dic3C2×D412yes—
257Dic3⋊C4—24no—
258Dic3⋊C4—24no—
259Dic3⋊C4—24no—
260C4×Dic3—24no—
261C4×Dic3—24no—
262C4×Dic3—22no—
263C4×Dic3—24no—
264C4×Dic3—22no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270#204—48no—
271C22×Q8—34no—
272C22×Q8—31no—
273C22×Q8—61no—
274C22×Q8—62no—
275C22×Q8—62no—
276C22×Q8—62no—
277C22×Q8—62no—
278C22×Q8—62no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—32no—
281C22×D4—62no—
282C22×D4—61no—
283C22×D4—61no—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—31no—
287C23×C4—61no—
288C23×C4—61no—
289C2×SD16—64no—
290C2×SD16—64no—
291C2×SD16—64no—
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C22×C8—32no—
296C22×C8C22×D312yes—
297#127—62no—
298#127—62no—
299#121—64no—
300#121—64no—
301#121—124no—
302#121—124no—
303C4×Q8—22no—
304C4×Q8—34no—
305C4×Q8—68no—
306C4×Q8—62no—
307C4×Q8—68no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8D1214yescomplemented
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—68no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—62no—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—62no—
327C2×C22⋊C4—62no—
328C2×C22⋊C4—62no—
329C2×C42—21no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—68no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—62no—
338C2×C42—62no—
339C2×C42C22×D311yescharacteristic
340C4⋊C8—22no—
341C4⋊C8—62no—
342#103—24no—
343#103—64no—
344#102—64no—
345#102—64no—
346C22⋊C8—64no—
347C23×C3C22×D411yescharacteristic
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3—42no—
355C22×D3—42no—
356C22×D3—42no—
357C22×D3—42no—
358C22×D3—42no—
359C22×D3#12112yes—
360C22×D3C2×SD1612yes—
361C22×D3AGL(1,ℤ/8)12yes—
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8#12114yescomplemented
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4#12012yes—
378C2×C3×C4#12012yes—
379C2×C3×C4C2×SD1612yes—
380C2×C3×C4#13712yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3#12112yes—
401C2×Dic3C2×SD1612yes—
402D12—42no—
403D12—42no—
404D12—42no—
405D12—42no—
406D12—84no—
407D12—88no—
408D12—84no—
409C4×D3—216no—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—28no—
415C4×D3—44no—
416C4×D3—48no—
417Dic6—216no—
418Dic6—22no—
419Dic6—22no—
420Dic6—22no—
421Dic6#12114yescomplemented
422C3×C8—42no—
423C3⋊C8—42no—
424C24—31no—
425C24—62no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—32no—
430C2×Q8—32no—
431C2×Q8—32no—
432C2×Q8—64no—
433C2×Q8—64no—
434C2×Q8—64no—
435C2×Q8—68no—
436C2×Q8—68no—
437C2×Q8—68no—
438C2×Q8—68no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8C2×D1212yes—
443C2×Q8D3×D412yes—
444C2×Q8#23912yes—
445C2×D4—62no—
446C2×D4—62no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—122no—
452C2×D4—122no—
453C2×D4—124no—
454C2×D4—124no—
455C2×D4—124no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—122no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—34no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—31no—
469C22×C4—64no—
470C22×C4—64no—
471C22×C4—64no—
472C22×C4—62no—
473C22×C4—64no—
474C22×C4—61no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—62no—
483C22×C4—121no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4D3×D411yescharacteristic
487C22×C4D3×D411yescharacteristic
488C22×C4C23×D311yescharacteristic
489SD16—128no—
490SD16—128no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—62no—
495C2×C8—62no—
496C2×C8—62no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—32no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—61no—
509C4⋊C4C2×D1212yes—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—124no—
513C42—24no—
514C42—64no—
515C42—64no—
516C42—62no—
517C42—62no—
518C42—64no—
519C42—61no—
520C42—61no—
521C42C2×D1212yes—
522C42D3×D412yes—
523C22×C3C2×#12011yescharacteristic
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3C22×SD1611yescharacteristic
527C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
528C22×C3C2×#13711yescharacteristic
529C22×C3#57511yescharacteristic
530C2×D3—22no—
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—22no—
534C2×D3—22no—
535C2×D3—22no—
536C2×D3—82no—
537C2×D3—82no—
538C2×D3—82no—
539C2×D3—82no—
540C2×D3—82no—
541C2×D3—82no—
542C2×D3—82no—
543C2×D3#46014yescomplemented
544C3×C4—24no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—82no—
548C3×C4#44814yescomplemented
549C3×C4C2×#12111yescharacteristic
550C3×C4C2×#12111yescharacteristic
551C3×C4#52511yescharacteristic
552C3×C4#52511yescharacteristic
553Dic3—22no—
554Dic3—24no—
555Dic3—22no—
556Dic3—41no—
557Dic3—42no—
558C23—32no—
559C23—32no—
560C23—32no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—122no—
566C23—122no—
567C23—122no—
568C23—122no—
569C23—122no—
570C23—122no—
571C23—122no—
572C23C2×D3×D411yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—22no—
576Q8—34no—
577Q8—616no—
578Q8—62no—
579Q8—62no—
580Q8—62no—
581Q8#89514yescomplemented
582D4—122no—
583D4—122no—
584D4—122no—
585D4—122no—
586D4—244no—
587D4—248no—
588D4—244no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—616no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—62no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—68no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—124no—
620C2×C4—128no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4#88212yes—
624C2×C4#88212yes—
625C2×C4D3×SD1612yes—
626C2×C4#91812yes—
627C2×C4C22×D1211yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×D3×D411yescharacteristic
630C2×C4C2×#23911yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—42no—
634C8—122no—
635C2×C3#284711yescharacteristic
636C2×C3C2×#44811yescharacteristic
637C2×C3#300611yescharacteristic
638C2×C3C2×#46011yescharacteristic
639C2×C3#303711yescharacteristic
640C2×C3#310311yescharacteristic
641C2×C3#311211yescharacteristic
642D3—24no—
643D3—162no—
644C22—34no—
645C22—62no—
646C22—62no—
647C22—62no—
648C22—62no—
649C22—62no—
650C22—62no—
651C22—242no—
652C22—242no—
653C22—242no—
654C22—242no—
655C22—242no—
656C22—242no—
657C22—242no—
658C22C2×#88211yescharacteristic
659C22D3×#12111yescharacteristic
660C22#562711yescharacteristic
661C22C2×D3×SD1611yescharacteristic
662C22C2×#91811yescharacteristic
663C22D3×AGL(1,ℤ/8)11yescharacteristic
664C22#577711yescharacteristic
665C4—22no—
666C4—24no—
667C4—42no—
668C4—62no—
669C4—64no—
670C4—62no—
671C4—82no—
672C4—121no—
673C4—122no—
674C4#480614yescomplemented
675C4C2×#89511yescharacteristic
676C4#552411yescharacteristic
677C4D3×#12111yescharacteristic
678C4#562711yescharacteristic
679C3#3290311yescharacteristic, complemented, Sylow
680C2—64no—
681C2—482no—
682C2#7404811yescharacteristic
683C2C2×#480611yescharacteristic
684C2#7888411yescharacteristic
685C2D3×#46011yescharacteristic
686C2#8047611yescharacteristic
687C2#8099611yescharacteristic
688C2#8109411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radical#74375C2
hypercenter#1803D3
hypocenterC3#32903
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×C4×D3×Q8C2semidirect
2. C2×#5027C2semidirect
3. C2×#4848C2semidirect
4. C2×#4806C2semidirect
6. C3×#1803C2semidirect
7. #70475C2semidirect
8. #70240C2semidirect
9. #69588C2semidirect
10. #68909C2semidirect
11. #68880C2semidirect
58. #1803D3semidirect
66. C2×D3×Q8D4semidirect
80. C3×C4×Q8D4semidirect
128. C4×Dic6D4semidirect
225. C2×C4×D3SD16semidirect
312. C4×Q8D12semidirect
365. C3×Q8#121semidirect
421. Dic6#121semidirect
543. C2×D3#460semidirect
548. C3×C4#448semidirect
581. Q8#895semidirect
674. C4#4806semidirect
679. C3#32903semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×Q82
C2×#50272
C2×#48482
C2×#48062
#743752
C3×#18032
#704752
#702402
#695882
#689092
#688802
#329033