Finite Group Databasebeta

← #10944100

Subgroups — #10944100

Order 768.

1546 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×#5545C211yesmaximal, characteristic
2C2×#5019C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic, complemented
4C2×#4776C212yesmaximal
5#74193C211yesmaximal, characteristic, complemented
6C3×#1804C211yesmaximal, characteristic, complemented
7#70476C211yesmaximal, characteristic, complemented
8#70171C212yesmaximal, complemented
9#69556C211yesmaximal, characteristic, complemented
10#68896C212yesmaximal, complemented
11#68879C212yesmaximal, complemented
12#32914—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C22×#214C2212yes—
17C2×C4×#67—21no—
18C22×(C2×D3)⋊C4—21no—
19C22×#204C2212yes—
20#5545—24no—
21#5545—24no—
22C2×#897C2212yes—
23C2×#893C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×#67⋊C4—21no—
26C2×D3×C22⋊C4—21no—
27C2×C4×D12C2211yescharacteristic
28C2×#872C2211yescharacteristic
29C2×#869C2211yescharacteristic, complemented
30C2×C3×C4⋊C8C2211yescharacteristic, complemented
31C2×C3×#102C2212yes—
32C3×#382C2211yescharacteristic, complemented
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yescomplemented
35#5019—24no—
36#4973C2212yescomplemented
37C2×#810C2212yescomplemented
38C2×#809C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
40#4841—24no—
41#4776—28no—
42#4674—21no—
43#4674—21no—
44#4661—21no—
45#4661—21no—
46C3×#340C2212yescomplemented
47#4550C2212yescomplemented
48#4529C2212yescomplemented
49C2×#518—31no—
50C2×#461—31no—
51C2×#461—31no—
52C2×#450—32no—
53#2621—31no—
54#1877—32no—
55#1877—32no—
56#1873—32no—
57#1804—31no—
58#1804D311yescharacteristic, complemented
59#1798—31no—
60C24×D3—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C2×D3×D4—22no—
66C2×D3×D4—22no—
67C2×D3×D4—22no—
68C2×D3×D4—22no—
69C2×D3×D4—22no—
70C2×D3×D4D414yes—
71C22×D12C2311yescharacteristic
72C22×C4×D3—21no—
73C22×C4×D3—21no—
74C22×C4×D3C2311yescharacteristic
75C22×Dic6C2312yes—
76C22×C3×C8C2312yes—
77C3×C4×D4—22no—
78C3×C4×D4D414yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84D4×Dic3—22no—
85D4×Dic3—22no—
86C2×#214—24no—
87C2×#214—24no—
88C2×#214—24no—
89C4×#67—28no—
90C2×(C2×D3)⋊C4—28no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4D411yescharacteristic
102C2×Dic3⋊C4D411yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C2×#204—24no—
107C2×#204—24no—
108C2×#204—24no—
109#897—24no—
110#897—24no—
111#893—22no—
112#893—22no—
113#892—22no—
114#892—22no—
115#67⋊C4—28no—
116D3×C22⋊C4—28no—
117C4×D12—22no—
118C4×D12D414yes—
119#872—22no—
120#872—22no—
121#869—22no—
122#869—22no—
123C3×C4⋊C8—22no—
124C3×#102—24no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—22no—
128C3⋊#95—22no—
129(C2×D3)⋊C8—24no—
130#816—24no—
131#810—24no—
132#809—24no—
133C4⋊(C3⋊C8)—22no—
134C23×D4—31no—
135C22×SD16—32no—
136C22×SD16—32no—
137C2×C4⋊Q8—31no—
138C2×C22⋊Q8—32no—
139#518—64no—
140#518—64no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—31no—
149C22×C22⋊C4—61no—
150C22×C22⋊C4—61no—
151#461—64no—
152#461—64no—
153#450—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic, complemented
156C2×#103—32no—
157C2×#103—32no—
158C2×#102—32no—
159C2×#102C2×D312yes—
160C2×C22⋊C8—32no—
161#398—61no—
162#398—61no—
163#397—61no—
164#397—61no—
165#391—32no—
166#382C2×D311yescharacteristic, complemented
167#340—32no—
168#340C2×D312yescomplemented
169#336—32no—
170C24×C3—21no—
171C23×D3—21no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—82no—
176C23×D3—82no—
177C23×D3C2×D411yescharacteristic
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D412yes—
183C2×C3×D4#4112yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—82no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×#67—24no—
192C2×#67—24no—
193C2×#67—24no—
194C22×Dic3—21no—
195C22×Dic3—24no—
196C22×Dic3—22no—
197C22×Dic3—82no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201D3×D4—28no—
202D3×D4—28no—
203D3×D4—28no—
204D3×D4—28no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12C2×D412yes—
209C2×D12C2×D412yes—
210C2×D12#4112yes—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—24no—
216C2×C4×D3—24no—
217C2×C4×D3—82no—
218C2×C4×D3—82no—
219C2×C4×D3C2×D412yes—
220C2×C4×D3#4112yes—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4C2×D412yes—
234C3×C22⋊C4—24no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237#217—24no—
238#214—48no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—44no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3—42no—
248C4⋊Dic3—44no—
249C4⋊Dic3C2×D412yes—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252C4×Dic3—24no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C2×C3⋊C8—22no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258#204—48no—
259C25—61no—
260C22×Q8—32no—
261C22×D4—34no—
262C22×D4—31no—
263C22×D4—62no—
264C22×D4—61no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—31no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4—61no—
275C2×SD16—64no—
276C2×SD16—64no—
277C2×SD16—64no—
278C2×SD16—64no—
279C2×SD16—64no—
280C2×SD16—64no—
281C22×C8—32no—
282C22×C8C22×D312yes—
283C4⋊Q8—62no—
284C4⋊Q8—62no—
285C22⋊Q8—64no—
286C22⋊Q8—64no—
287C4×D4—22no—
288C4×D4—34no—
289C4×D4—68no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—68no—
293C4×D4—62no—
294C4×D4—62no—
295C4×D4—62no—
296C4×D4D1214yes—
297#117—62no—
298#117—62no—
299#117—62no—
300#117—62no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4C22×D312yes—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4—68no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—68no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—62no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42C22×D311yescharacteristic
322C4⋊C8—22no—
323C4⋊C8—62no—
324#103—64no—
325#103—64no—
326#102—24no—
327#102—64no—
328C22⋊C8—64no—
329#95—21no—
330#95—21no—
331#95—62no—
332#95—62no—
333C23×C3—22no—
334C23×C3—22no—
335C23×C3—22no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3C22×D411yescharacteristic
339C22×D3—21no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—21no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—216no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—88no—
350C22×D3—88no—
351C22×D3—84no—
352C22×D3—84no—
353C22×D3#12112yes—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4#12114yes—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—84no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374#67—216no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—84no—
387C2×Dic3#12112yes—
388D12—22no—
389D12—22no—
390D12—22no—
391D12#12114yes—
392C4×D3—22no—
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—84no—
397C4×D3—84no—
398Dic6—42no—
399Dic6—42no—
400Dic6—42no—
401Dic6—42no—
402C3×C8—42no—
403C3⋊C8—42no—
404C24—21no—
405C24—31no—
406C24—64no—
407C24—64no—
408C24—64no—
409C24—61no—
410C24—242no—
411C24—242no—
412C2×Q8—62no—
413C2×Q8—62no—
414C2×Q8—62no—
415C2×Q8—62no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—32no—
422C2×D4—32no—
423C2×D4—32no—
424C2×D4—64no—
425C2×D4—68no—
426C2×D4—68no—
427C2×D4—68no—
428C2×D4—64no—
429C2×D4—64no—
430C2×D4—68no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4C2×D1212yes—
435C2×D4D3×D412yes—
436C2×D4#23912yes—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—64no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—62no—
454C22×C4—82no—
455C22×C4—242no—
456C22×C4—242no—
457C22×C4—242no—
458C22×C4D3×D411yescharacteristic
459C22×C4D3×D411yescharacteristic
460C22×C4C23×D311yescharacteristic
461SD16—128no—
462SD16—128no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—62no—
467C2×C8—62no—
468C2×C8—62no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—32no—
473C4⋊C4—62no—
474C4⋊C4—62no—
475C4⋊C4—61no—
476C4⋊C4—124no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4C2×D1212yes—
480C22⋊C4—24no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—64no—
487C22⋊C4—124no—
488C42—64no—
489C42—61no—
490C42—61no—
491C42C2×D1212yes—
492C42D3×D412yes—
493C22×C3—24no—
494C22×C3—44no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—44no—
498C22×C3C2×#12011yescharacteristic
499C22×C3C2×#12111yescharacteristic
500C22×C3#52511yescharacteristic
501C22×C3#57712yes—
502C22×C3#57712yes—
503C2×D3—22no—
504C2×D3—24no—
505C2×D3—24no—
506C2×D3—42no—
507C2×D3—44no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—44no—
513C2×D3—816no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—82no—
518C3×C4C2×#12111yescharacteristic
519C3×C4C2×#12111yescharacteristic
520C3×C4#52511yescharacteristic
521C3×C4#52511yescharacteristic
522Dic3—22no—
523Dic3—41no—
524Dic3—42no—
525Dic3—41no—
526Dic3—82no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—32no—
531C23—42no—
532C23—42no—
533C23—616no—
534C23—61no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—62no—
539C23—61no—
540C23—62no—
541C23—122no—
542C23—122no—
543C23—248no—
544C23—244no—
545C23—244no—
546C23—248no—
547C23C2×D3×D411yescharacteristic
548Q8—122no—
549Q8—122no—
550Q8—122no—
551Q8—122no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4—34no—
556D4—616no—
557D4—62no—
558D4—62no—
559D4—62no—
560D4#89514yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—84no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—244no—
586C2×C4—244no—
587C2×C4—244no—
588C2×C4#88212yes—
589C2×C4#88212yes—
590C2×C4C22×D1211yescharacteristic
591C2×C4C2×D3×D411yescharacteristic
592C2×C4C2×D3×D411yescharacteristic
593C2×C4C2×#23911yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C8—42no—
597C8—122no—
598C2×C3—44no—
599C2×C3#284711yescharacteristic
600C2×C3#300911yescharacteristic
601C2×C3#300911yescharacteristic
602C2×C3#304212yes—
603C2×C3#312312yes—
604D3—42no—
605D3—44no—
606C22—24no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—62no—
612C22—64no—
613C22—64no—
614C22—122no—
615C22—121no—
616C22—124no—
617C22—122no—
618C22—121no—
619C22—122no—
620C22—124no—
621C22—2416no—
622C22C2×#88211yescharacteristic
623C22D3×#12111yescharacteristic
624C22#562711yescharacteristic
625C22#577012yes—
626C22#577512yes—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—62no—
631C4—82no—
632C4—121no—
633C4—121no—
634C4—122no—
635C4—242no—
636C4C2×#89511yescharacteristic
637C4#552411yescharacteristic
638C4D3×#12111yescharacteristic
639C4#562711yescharacteristic
640C3#3291411yescharacteristic, complemented, Sylow
641C2—44no—
642C2—122no—
643C2—124no—
644C2#7404811yescharacteristic
645C2#7883511yescharacteristic
646C2#7892911yescharacteristic
647C2#8053712yes—
648C2#8114712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenter#1804D3
hypocenterC3#32914
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#5019C2semidirect
3. C2×#4841C2semidirect
5. #74193C2semidirect
6. C3×#1804C2semidirect
7. #70476C2semidirect
8. #70171C2semidirect
9. #69556C2semidirect
10. #68896C2semidirect
11. #68879C2semidirect
29. C2×#869C22semidirect
30. C2×C3×C4⋊C8C22semidirect
32. C3×#382C22semidirect
34. C2×#816C22semidirect
36. #4973C22semidirect
37. C2×#810C22semidirect
39. C2×C4⋊(C3⋊C8)C22semidirect
46. C3×#340C22semidirect
47. #4550C22semidirect
48. #4529C22semidirect
58. #1804D3semidirect
155. C2×C4⋊C8C2×D3semidirect
166. #382C2×D3semidirect
168. #340C2×D3semidirect
640. C3#32914semidirect

Maximal subgroups

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

subgroupindex
C2×#55452
C2×#50192
C2×#48412
C2×#47762
#741932
C3×#18042
#704762
#701712
#695562
#688962
#688792
#329143