Finite Group Databasebeta

← #10944091

Subgroups — #10944091

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5546C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4781C212yesmaximal, complemented
5#74244C211yesmaximal, characteristic, complemented
6C3×#1804C211yesmaximal, characteristic, complemented
7#70476C211yesmaximal, characteristic, complemented
8#70172C212yesmaximal, complemented
9#69559C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68880C212yesmaximal, complemented
12#32913—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×D4×Dic3C2211yescharacteristic
16C22×#213C2212yes—
17C2×C4×#67—21no—
18C22×Dic3⋊C4—21no—
19C22×#204C2212yes—
20#5546—24no—
21#5546—24no—
22C2×#895C2212yes—
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×#887C2211yescharacteristic
25C2×#67⋊C4—21no—
26C2×#877—21no—
27C2×#874C2211yescharacteristic
28C2×#872C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2212yes—
32C3×#382C2211yescharacteristic
33C2×#821C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5015—24no—
36#4988C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4848—24no—
41#4781—28no—
42#4672—21no—
43#4672—21no—
44#4671—21no—
45#4671—21no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#520—31no—
50C2×#460—31no—
51C2×#459—31no—
52C2×#449—32no—
53#2619—31no—
54#1880—32no—
55#1879—32no—
56#1874—32no—
57#1805—31no—
58#1804D311yescharacteristic, complemented
59#1798—31no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C22×#67—21no—
63C23×Dic3—21no—
64C23×Dic3—21no—
65C2×#237—22no—
66C2×#237—22no—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—22no—
70C2×#237D414yescomplemented
71C22×D12—22no—
72C22×D12—21no—
73C22×D12—21no—
74C22×D12C2312yes—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2312yes—
79C3×C4×D4—22no—
80C3×C4×D4D414yescomplemented
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86D4×Dic3—22no—
87D4×Dic3—22no—
88C2×#213—24no—
89C2×#213—24no—
90C2×#213—24no—
91C4×#67—28no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—28no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4D411yescharacteristic
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×#204—24no—
109C2×#204—24no—
110C2×#204—24no—
111#895—24no—
112#895—24no—
113D3×C4⋊C4—22no—
114D3×C4⋊C4—22no—
115#887—22no—
116#887—22no—
117#67⋊C4—28no—
118#877—28no—
119#874—22no—
120#874—22no—
121#872—22no—
122#872—22no—
123C4×Dic6—22no—
124C4×Dic6D414yescomplemented
125C3×C4⋊C8—22no—
126C3×#102—24no—
127C3×#95—21no—
128C3×#95—21no—
129#821—24no—
130(C2×D3)⋊C8—24no—
131#810—24no—
132#809—24no—
133C4⋊(C3⋊C8)—22no—
134C22×#41—31no—
135C22×SD16—32no—
136C22×D8—32no—
137C2×#127—31no—
138C2×#121—32no—
139#520—64no—
140#520—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—61no—
145C2×C4×D4—61no—
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—61no—
149C22×C4⋊C4—31no—
150C22×C4⋊C4—61no—
151#460—64no—
152#459—64no—
153#449—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#102—32no—
158C2×#102—32no—
159C2×#102C2×D312yes—
160C2×C22⋊C8—32no—
161#394—61no—
162#394—61no—
163#392—61no—
164#392—61no—
165#390—32no—
166#382C2×D311yescharacteristic
167#340—32no—
168#340C2×D312yes—
169#336—32no—
170C24×C3—21no—
171C23×D3—22no—
172C23×D3C2×D411yescharacteristic
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4C2×D412yes—
177C2×C3×D4C2×D412yes—
178C2×C3×D4#4112yes—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—82no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C2×#67—24no—
187C2×#67—24no—
188C2×#67—24no—
189C22×Dic3—24no—
190C22×Dic3—24no—
191C22×Dic3—21no—
192C22×Dic3—24no—
193C22×Dic3—21no—
194C22×Dic3—24no—
195C22×Dic3—82no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199#237—28no—
200#237—28no—
201#237—28no—
202#237—28no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—44no—
210C2×D12—42no—
211C2×D12—42no—
212C2×D12—82no—
213C2×D12—82no—
214C2×D12—82no—
215C2×D12—82no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—24no—
221C2×C4×D3—82no—
222C2×C4×D3C2×D412yes—
223C2×C4×D3#4112yes—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C2×D412yes—
228C2×Dic6C2×D412yes—
229C2×Dic6#4112yes—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—24no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240#217—24no—
241#213—48no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—44no—
245(C2×D3)⋊C4—82no—
246(C2×D3)⋊C4—82no—
247C4⋊Dic3—21no—
248C4⋊Dic3C2×D412yes—
249Dic3⋊C4—24no—
250Dic3⋊C4—24no—
251Dic3⋊C4—24no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254C4×Dic3—24no—
255C4×Dic3—24no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
261#204—48no—
262C2×#41—34no—
263C2×#41—62no—
264C2×#41—62no—
265C2×#41—62no—
266C2×#41—62no—
267C2×#41—62no—
268C22×Q8—31no—
269C22×D4—32no—
270C22×D4—62no—
271C22×D4—61no—
272C22×D4—61no—
273C22×D4—61no—
274C22×D4C22×D311yescharacteristic
275C23×C4—21no—
276C23×C4—31no—
277C23×C4—61no—
278C23×C4—61no—
279C23×C4—61no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×D8—64no—
284C2×D8—64no—
285C2×D8—64no—
286C22×C8—32no—
287C22×C8C22×D312yes—
288#127—62no—
289#127—62no—
290#121—64no—
291#121—64no—
292C4×Q8—34no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×D4—22no—
297C4×D4—68no—
298C4×D4—62no—
299C4×D4—68no—
300C4×D4—62no—
301C4×D4D1214yescomplemented
302#117—68no—
303#117—62no—
304#117—62no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—68no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4C22×D312yes—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—62no—
321C2×C22⋊C4—62no—
322C2×C22⋊C4—62no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C4⋊C8—22no—
328C4⋊C8—62no—
329#103—64no—
330#102—24no—
331#102—64no—
332#102—64no—
333C22⋊C8—64no—
334#95—21no—
335#95—21no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3C22×D411yescharacteristic
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
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—216no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—42no—
387C2×Dic3—84no—
388C2×Dic3#12112yes—
389D12—42no—
390D12—42no—
391D12—42no—
392D12—42no—
393D12—84no—
394D12—84no—
395D12—88no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—84no—
401Dic6—22no—
402Dic6—22no—
403Dic6—22no—
404Dic6#12114yes—
405C3×C8—42no—
406C3⋊C8—42no—
407C24—21no—
408C24—31no—
409C24—62no—
410#41—68no—
411#41—68no—
412#41—68no—
413#41—68no—
414C2×Q8—32no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—64no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—62no—
432C2×D4—124no—
433C2×D4—122no—
434C2×D4—122no—
435C2×D4—242no—
436C2×D4—242no—
437C2×D4—242no—
438C2×D4—242no—
439C2×D4C2×D1212yes—
440C2×D4D3×D412yes—
441C2×D4#23912yes—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—31no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—64no—
461C22×C4—82no—
462C22×C4—242no—
463C22×C4—242no—
464C22×C4D3×D411yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4C23×D311yescharacteristic
467SD16—128no—
468D8—128no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—62no—
473C2×C8—62no—
474C2×C8—62no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—32no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—64no—
484C4⋊C4—61no—
485C4⋊C4C2×D1212yes—
486C22⋊C4—24no—
487C22⋊C4—64no—
488C22⋊C4—62no—
489C22⋊C4—62no—
490C22⋊C4—124no—
491C22⋊C4—242no—
492C22⋊C4—242no—
493C42—64no—
494C42—64no—
495C42—61no—
496C42—61no—
497C42C2×D1212yes—
498C42D3×D412yes—
499C22×C3—24no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—44no—
504C22×C3C2×#12011yescharacteristic
505C22×C3C2×#12111yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3#57612yes—
508C22×C3#57712yes—
509C2×D3—22no—
510C2×D3—24no—
511C2×D3—42no—
512C2×D3—41no—
513C2×D3—41no—
514C2×D3—82no—
515C2×D3—82no—
516C2×D3—82no—
517C2×D3—82no—
518C2×D3—82no—
519C2×D3—82no—
520C2×D3—82no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—82no—
525C3×C4C2×#12111yescharacteristic
526C3×C4C2×#12111yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#52511yescharacteristic
529Dic3—24no—
530Dic3—22no—
531Dic3—42no—
532Dic3—41no—
533Dic3—41no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—32no—
538C23—42no—
539C23—42no—
540C23—61no—
541C23—62no—
542C23—62no—
543C23—61no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23—122no—
551C23C2×D3×D411yescharacteristic
552Q8—34no—
553Q8—62no—
554Q8—62no—
555Q8—62no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—616no—
560D4—122no—
561D4—122no—
562D4—122no—
563D4—122no—
564D4—244no—
565D4—244no—
566D4—248no—
567D4#89514yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—616no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—61no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—84no—
592C2×C4—122no—
593C2×C4—244no—
594C2×C4—244no—
595C2×C4#88212yes—
596C2×C4#88212yes—
597C2×C4C22×D1211yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23911yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C8—42no—
604C8—122no—
605C2×C3—44no—
606C2×C3#284711yescharacteristic
607C2×C3#300811yescharacteristic
608C2×C3#301011yescharacteristic
609C2×C3#304512yes—
610C2×C3#311912yes—
611D3—42no—
612D3—162no—
613C22—24no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—44no—
618C22—62no—
619C22—64no—
620C22—121no—
621C22—121no—
622C22—122no—
623C22—242no—
624C22—242no—
625C22—242no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22C2×#88211yescharacteristic
631C22D3×#12111yescharacteristic
632C22#562711yescharacteristic
633C22#577012yes—
634C22#577412yes—
635C4—22no—
636C4—22no—
637C4—42no—
638C4—62no—
639C4—64no—
640C4—82no—
641C4—121no—
642C4—122no—
643C4—121no—
644C4C2×#89511yescharacteristic
645C4#552411yescharacteristic
646C4D3×#12111yescharacteristic
647C4#562711yescharacteristic
648C3#3291311yescharacteristic, complemented, Sylow
649C2—44no—
650C2—122no—
651C2—482no—
652C2#7404811yescharacteristic
653C2#7883611yescharacteristic
654C2#7892811yescharacteristic
655C2#8053812yes—
656C2#8114412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
hypercenter#1804D3
hypocenterC3#32913
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
1. C2×#5546C2semidirect
2. C2×#5015C2semidirect
3. C2×#4848C2semidirect
4. C2×#4781C2semidirect
5. #74244C2semidirect
6. C3×#1804C2semidirect
7. #70476C2semidirect
8. #70172C2semidirect
9. #69559C2semidirect
10. #68909C2semidirect
11. #68880C2semidirect
58. #1804D3semidirect
70. C2×#237D4semidirect
80. C3×C4×D4D4semidirect
124. C4×Dic6D4semidirect
301. C4×D4D12semidirect
648. C3#32913semidirect

Maximal subgroups

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

subgroupindex
C2×#55462
C2×#50152
C2×#48482
C2×#47812
#742442
C3×#18042
#704762
#701722
#695592
#689092
#688802
#329133