Finite Group Databasebeta

← #10944075

Subgroups — #10944075

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5572C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×#5170C212yesmaximal, complemented
4#83189C211yesmaximal, characteristic
5C2×#5075C211yesmaximal, characteristic, complemented
6#72048C212yesmaximal, complemented
7#71627C211yesmaximal, characteristic, complemented
8#70486C211yesmaximal, characteristic, complemented
9#70476C211yesmaximal, characteristic, complemented
10#70213C212yesmaximal, complemented
11#70201C212yesmaximal, complemented
12#32912—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C2×#938C2212yes—
19C22×#214C2212yes—
20#5572—24no—
21#5572—24no—
22C2×#892—21no—
23C2×#892C2211yescharacteristic
24C2×#887—21no—
25C2×#887C2211yescharacteristic
26C2×#872—21no—
27C2×#872C2211yescharacteristic
28C2×C4×Dic6—21no—
29C2×C4×Dic6C2211yescharacteristic
30C3×#460—24no—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×#835C2212yes—
34#5223—21no—
35#5223—21no—
36#5222—21no—
37#5222—21no—
38#5221C2212yes—
39C2×#832C2212yes—
40#5170—28no—
41C2×(C2×D3)⋊C8C2212yes—
42C2×#816C2212yes—
43#5075—24no—
44#4936C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#519—31no—
50C2×#460—31no—
51C2×#460D311yescharacteristic, complemented
52C2×#448—32no—
53#2619—31no—
54#1880—32no—
55#1880—32no—
56#1875—32no—
57#1805—31no—
58#1805—31no—
59#1798—31no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2312yes—
64C22×#67—22no—
65C2×D3×Q8—22no—
66C2×D3×Q8—22no—
67C2×D3×Q8—22no—
68C2×D3×Q8—22no—
69C2×D3×Q8—22no—
70C2×D3×Q8D414yescomplemented
71C22×C4×D3—21no—
72C22×C4×D3C2311yescharacteristic
73C22×Dic6—21no—
74C22×Dic6C2311yescharacteristic
75C2×C3×SD16—24no—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C22×C3×C8C2312yes—
79C3×#127—22no—
80C3×#127—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8D414yescomplemented
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—22no—
90Q8×Dic3—22no—
91Q8×Dic3—22no—
92#938—24no—
93#938—24no—
94C2×#214—24no—
95C2×#214—24no—
96C2×#214—24no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2312yes—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4D411yescharacteristic
106C2×Dic3⋊C4D411yescharacteristic
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113#892—28no—
114#892—22no—
115#892—22no—
116#887—28no—
117#887—22no—
118#887—22no—
119#872—28no—
120#872—22no—
121#872—22no—
122C4×Dic6—28no—
123C4×Dic6—22no—
124C4×Dic6D414yescomplemented
125C3×C4⋊C8—22no—
126C3×#102—24no—
127#835—24no—
128#832—24no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131(C2×D3)⋊C8—24no—
132#816—24no—
133C4⋊(C3⋊C8)—22no—
134C23×Q8—31no—
135C22×SD16—32no—
136C22×SD16C2×D312yes—
137C2×#127C2×D311yescharacteristic
138C2×#121—32no—
139#519—64no—
140#519—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—61no—
145C2×C4×Q8—61no—
146C2×C4×Q8C2×D311yescharacteristic
147C2×#117—31no—
148C2×#117—31no—
149C2×#117—61no—
150C2×#117—61no—
151#460—24no—
152#460—64no—
153#448—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103—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#382—31no—
167#340—32no—
168#340—32no—
169#336—32no—
170C24×C3—22no—
171C23×D3C2×D411yescharacteristic
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×D412yes—
177C2×C3×Q8#4112yes—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—82no—
187C2×C3×D4—82no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—44no—
194C2×#67—82no—
195C2×#67—82no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202D3×Q8—28no—
203D3×Q8—28no—
204D3×Q8—28no—
205D3×Q8—28no—
206C2×C4×D3—24no—
207C2×C4×D3—24no—
208C2×C4×D3—24no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3C2×D412yes—
214C2×C4×D3#4112yes—
215C2×Dic6—24no—
216C2×Dic6—24no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—24no—
221C2×Dic6C2×D412yes—
222C2×Dic6C2×D412yes—
223C2×Dic6#4112yes—
224C3×SD16—48no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4C2×D412yes—
231C3×C42—24no—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234#217—44no—
235#217—82no—
236#217—82no—
237#214—48no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241C4⋊Dic3—24no—
242C4⋊Dic3—21no—
243C4⋊Dic3—22no—
244C4⋊Dic3—22no—
245C4⋊Dic3C2×D412yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—22no—
248Dic3⋊C4—24no—
249Dic3⋊C4—22no—
250C4×Dic3—24no—
251C4×Dic3—24no—
252C4×Dic3—24no—
253C4×Dic3—24no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C22×Q8—34no—
260C22×Q8—31no—
261C22×Q8—61no—
262C22×Q8—62no—
263C22×Q8—62no—
264C22×Q8—62no—
265C22×Q8—62no—
266C22×Q8—62no—
267C22×Q8C22×D311yescharacteristic
268C22×D4—21no—
269C22×D4—21no—
270C22×D4—62no—
271C22×D4C22×D312yes—
272C23×C4—31no—
273C23×C4—61no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—64no—
278C2×SD16—64no—
279C2×SD16—64no—
280C22×C8—32no—
281C22×C8C22×D312yes—
282#127—22no—
283#127—22no—
284#121—64no—
285#121—64no—
286C4×Q8—22no—
287C4×Q8—34no—
288C4×Q8—68no—
289C4×Q8—62no—
290C4×Q8—68no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8#6714yescomplemented
296#117—68no—
297#117—62no—
298#117—68no—
299#117—62no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C22⋊C4—31no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—62no—
316C2×C22⋊C4—62no—
317C2×C42—21no—
318C2×C42—31no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42C22×D311yescharacteristic
324C4⋊C8—22no—
325C4⋊C8—62no—
326#103—64no—
327#103—64no—
328#102—24no—
329#102—64no—
330C22⋊C8—64no—
331#95—61no—
332#95—61no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3C22×D411yescharacteristic
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3#12112yes—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8#12114yes—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—84no—
355C3×D4—84no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—42no—
363C2×C3×C4#12012yes—
364C2×C3×C4#12012yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371#67—88no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—22no—
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#12112yes—
387C4×D3—216no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—22no—
391C4×D3—22no—
392Dic6—216no—
393Dic6—22no—
394Dic6—22no—
395Dic6—22no—
396Dic6#12114yes—
397C3×C8—42no—
398C3⋊C8—42no—
399C24—22no—
400C24—31no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—32no—
405C2×Q8—32no—
406C2×Q8—32no—
407C2×Q8—64no—
408C2×Q8—64no—
409C2×Q8—64no—
410C2×Q8—68no—
411C2×Q8—68no—
412C2×Q8—68no—
413C2×Q8—68no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8D3×D412yes—
418C2×Q8#23712yes—
419C2×Q8C2×#6712yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—124no—
431C2×D4—242no—
432C2×D4—242no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—31no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—62no—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D411yescharacteristic
452C22×C4C23×D311yescharacteristic
453SD16—48no—
454SD16—128no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—32no—
464C4⋊C4—64no—
465C4⋊C4—64no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—64no—
469C4⋊C4—61no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4C2×#6712yes—
473C22⋊C4—64no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C22⋊C4—124no—
477C22⋊C4—242no—
478C22⋊C4—242no—
479C42—24no—
480C42—64no—
481C42—64no—
482C42—64no—
483C42—64no—
484C42—61no—
485C42—61no—
486C42D3×D412yes—
487C42C2×#6712yes—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3C2×#12011yescharacteristic
496C22×C3C2×#12111yescharacteristic
497C22×C3#52511yescharacteristic
498C22×C3#57712yes—
499C22×C3#57712yes—
500C2×D3—22no—
501C2×D3—24no—
502C2×D3—42no—
503C2×D3—41no—
504C2×D3—41no—
505C3×C4—24no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4C2×#12111yescharacteristic
510C3×C4C2×#12111yescharacteristic
511C3×C4#52511yescharacteristic
512C3×C4#52511yescharacteristic
513Dic3—22no—
514Dic3—24no—
515Dic3—41no—
516Dic3—41no—
517Dic3—42no—
518Dic3—82no—
519C23—32no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—62no—
528C23—62no—
529C23—61no—
530C23—61no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—34no—
536Q8—616no—
537Q8—62no—
538Q8—62no—
539Q8—62no—
540Q8#93814yes—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—84no—
546D4—84no—
547D4—248no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—32no—
555C2×C4—42no—
556C2×C4—616no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4#93712yes—
576C2×C4#93712yes—
577C2×C4C2×D3×D411yescharacteristic
578C2×C4C2×D3×D411yescharacteristic
579C2×C4C2×#23711yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C2×C4D3×#4111yescharacteristic
582C2×C4C22×#6711yescharacteristic
583C8—42no—
584C8—122no—
585C2×C3—162no—
586C2×C3#284711yescharacteristic
587C2×C3#301011yescharacteristic
588C2×C3#301011yescharacteristic
589C2×C3#304312yes—
590C2×C3#312612yes—
591D3—42no—
592C22—64no—
593C22—62no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—121no—
602C22—121no—
603C22—122no—
604C22D3×#12111yescharacteristic
605C22#560611yescharacteristic
606C22#576212yes—
607C22#577512yes—
608C22C2×#93711yescharacteristic
609C4—24no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—62no—
614C4—64no—
615C4—121no—
616C4—122no—
617C4—121no—
618C4—242no—
619C4D3×#12111yescharacteristic
620C4#560611yescharacteristic
621C4C2×#93811yescharacteristic
622C4#582811yescharacteristic
623C3#3291211yescharacteristic, complemented, Sylow
624C2—122no—
625C2—162no—
626C2#8036512yes—
627C2#8038512yes—
628C2#8272311yescharacteristic
629C2#8386011yescharacteristic
630C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radical#83189C2
hypercenterC2×#460D3
hypocenterC3#32912
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×#5572C2semidirect
2. C2×C3×#460C2semidirect
3. C2×#5170C2semidirect
5. C2×#5075C2semidirect
6. #72048C2semidirect
7. #71627C2semidirect
8. #70486C2semidirect
9. #70476C2semidirect
10. #70213C2semidirect
11. #70201C2semidirect
51. C2×#460D3semidirect
70. C2×D3×Q8D4semidirect
82. C3×C4×Q8D4semidirect
124. C4×Dic6D4semidirect
295. C4×Q8#67semidirect
623. C3#32912semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C2×C3×#4602
C2×#51702
#831892
C2×#50752
#720482
#716272
#704862
#704762
#702132
#702012
#329123