Finite Group Databasebeta

← #10944081

Subgroups — #10944081

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×#5027C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×#4806C212yesmaximal, complemented
5#74244C211yesmaximal, characteristic
6C3×#1805C211yesmaximal, characteristic, complemented
7#70476C211yesmaximal, characteristic, complemented
8#70240C212yesmaximal, complemented
9#69559C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68881C212yesmaximal, complemented
12#32912—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C22×#215C2212yes—
17C22×#204C2212yes—
18#5572—24no—
19#5572—24no—
20C2×#895C2212yes—
21C2×#892—21no—
22C2×#892C2211yescharacteristic
23C2×#887—21no—
24C2×#887C2211yescharacteristic
25C2×#874C2211yescharacteristic
26C2×#872—21no—
27C2×#872C2211yescharacteristic
28C2×C4×Dic6—21no—
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C3×#382C2211yescharacteristic
33C2×#821C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5027—24no—
36#4988C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4848—24no—
41#4806—28no—
42#4672—21no—
43#4672—21no—
44#4671—21no—
45#4671—21no—
46C3×#340C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C2×#519—31no—
50C2×#460—31no—
51C2×#460—31no—
52C2×#448—32no—
53#2619—31no—
54#1880—32no—
55#1880—32no—
56#1875—32no—
57#1805—31no—
58#1805D311yescharacteristic, complemented
59#1798—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×D3C2311yescharacteristic
73C22×Dic6—21no—
74C22×Dic6C2311yescharacteristic
75C22×C3×C8C2312yes—
76C3×C4×Q8—22no—
77C3×C4×Q8D414yescomplemented
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C42—21no—
82C2×C3×C42C2311yescharacteristic
83Q8×Dic3—22no—
84Q8×Dic3—22no—
85C2×#215—24no—
86C2×#215—24no—
87C2×#215—24no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4D411yescharacteristic
99C2×Dic3⋊C4D411yescharacteristic
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C2×#204—24no—
107C2×#204—24no—
108C2×#204—24no—
109#895—24no—
110#895—24no—
111#892—28no—
112#892—22no—
113#892—22no—
114#887—28no—
115#887—22no—
116#887—22no—
117#874—22no—
118#874—22no—
119#872—28no—
120#872—22no—
121#872—22no—
122C4×Dic6—28no—
123C4×Dic6—22no—
124C4×Dic6D414yescomplemented
125C3×C4⋊C8—22no—
126C3×#103—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—
134C23×Q8—31no—
135C22×SD16—32no—
136C22×SD16—32no—
137C2×#127—31no—
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—64no—
152#460—64no—
153#448—68no—
154C2×C4⋊C8—31no—
155C2×C4⋊C8C2×D311yescharacteristic
156C2×#103—32no—
157C2×#103C2×D312yes—
158C2×#102—32no—
159C2×#102—32no—
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—
170C23×D3—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—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3C2×D411yescharacteristic
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189D3×Q8—28no—
190D3×Q8—28no—
191D3×Q8—28no—
192D3×Q8—28no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—44no—
200C2×D12—42no—
201C2×D12—42no—
202C2×D12—82no—
203C2×D12—82no—
204C2×D12—82no—
205C2×D12—82no—
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—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C2×D412yes—
232C3×C42—24no—
233C3×C42C2×D412yes—
234C3×C42C2×D412yes—
235#215—48no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—44no—
240(C2×D3)⋊C4—82no—
241(C2×D3)⋊C4—82no—
242C4⋊Dic3—24no—
243C4⋊Dic3—21no—
244C4⋊Dic3C2×D412yes—
245Dic3⋊C4—24no—
246Dic3⋊C4—22no—
247Dic3⋊C4—24no—
248Dic3⋊C4—22no—
249C4×Dic3—24no—
250C4×Dic3—24no—
251C4×Dic3—24no—
252C4×Dic3—24no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C2×C3⋊C8—22no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258#204—48no—
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—32no—
269C22×D4—62no—
270C22×D4—61no—
271C22×D4—61no—
272C23×C4—31no—
273C23×C4—61no—
274C2×SD16—64no—
275C2×SD16—64no—
276C2×SD16—64no—
277C2×SD16—64no—
278C2×SD16—64no—
279C2×SD16—64no—
280C22×C8—32no—
281C22×C8C22×D312yes—
282#127—62no—
283#127—62no—
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×Q8D1214yescomplemented
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—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4C22×D312yes—
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—24no—
327#103—64no—
328#102—64no—
329#102—64no—
330C22⋊C8—64no—
331#95—21no—
332#95—21no—
333C23×C3C22×D411yescharacteristic
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3#12112yes—
346C3×Q8—22no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8#12114yes—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4#12012yes—
361C2×C3×C4#12012yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—42no—
379C2×Dic3#12112yes—
380D12—42no—
381D12—42no—
382D12—42no—
383D12—42no—
384D12—84no—
385D12—84no—
386D12—88no—
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—31no—
400C24—62no—
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×Q8C2×D1212yes—
418C2×Q8D3×D412yes—
419C2×Q8#23912yes—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—124no—
427C2×D4—122no—
428C2×D4—122no—
429C2×D4—242no—
430C2×D4—242no—
431C2×D4—242no—
432C2×D4—242no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—31no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D411yescharacteristic
452C22×C4C23×D311yescharacteristic
453SD16—128no—
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—22no—
464C4⋊C4—22no—
465C4⋊C4—32no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—64no—
471C4⋊C4—61no—
472C4⋊C4C2×D1212yes—
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—
486C42C2×D1212yes—
487C42D3×D412yes—
488C22×C3C2×#12011yescharacteristic
489C22×C3C2×#12111yescharacteristic
490C22×C3#52511yescharacteristic
491C22×C3#57712yes—
492C22×C3#57712yes—
493C2×D3—22no—
494C2×D3—24no—
495C2×D3—42no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—82no—
499C2×D3—82no—
500C2×D3—82no—
501C2×D3—82no—
502C2×D3—82no—
503C2×D3—82no—
504C2×D3—82no—
505C3×C4—24no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—82no—
510C3×C4C2×#12111yescharacteristic
511C3×C4C2×#12111yescharacteristic
512C3×C4#52511yescharacteristic
513C3×C4#52511yescharacteristic
514Dic3—22no—
515Dic3—24no—
516Dic3—41no—
517Dic3—41no—
518Dic3—42no—
519C23—32no—
520C23—61no—
521C23—62no—
522C23—62no—
523C23—61no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—34no—
536Q8—616no—
537Q8—62no—
538Q8—62no—
539Q8—62no—
540Q8#89514yes—
541D4—122no—
542D4—122no—
543D4—122no—
544D4—122no—
545D4—244no—
546D4—244no—
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—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—616no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—122no—
575C2×C4#88212yes—
576C2×C4#88212yes—
577C2×C4C22×D1211yescharacteristic
578C2×C4C2×D3×D411yescharacteristic
579C2×C4C2×D3×D411yescharacteristic
580C2×C4C2×#23911yescharacteristic
581C2×C4D3×#4111yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C8—42no—
584C8—122no—
585C2×C3#284711yescharacteristic
586C2×C3#301011yescharacteristic
587C2×C3#301011yescharacteristic
588C2×C3#304312yes—
589C2×C3#312612yes—
590D3—42no—
591D3—162no—
592C22—62no—
593C22—64no—
594C22—122no—
595C22—121no—
596C22—121no—
597C22—242no—
598C22—242no—
599C22—242no—
600C22—242no—
601C22—242no—
602C22—242no—
603C22—242no—
604C22C2×#88211yescharacteristic
605C22D3×#12111yescharacteristic
606C22#562711yescharacteristic
607C22#577012yes—
608C22#577812yes—
609C4—24no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—62no—
614C4—64no—
615C4—82no—
616C4—121no—
617C4—122no—
618C4—121no—
619C4C2×#89511yescharacteristic
620C4#552411yescharacteristic
621C4D3×#12111yescharacteristic
622C4#562711yescharacteristic
623C3#3291211yescharacteristic, complemented, Sylow
624C2—122no—
625C2—482no—
626C2#7404811yescharacteristic
627C2#7883611yescharacteristic
628C2#7896011yescharacteristic
629C2#8060012yes—
630C2#8121012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radical#74244C2
hypercenter#1805D3
hypocenterC3#32912
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#5572C2semidirect
2. C2×#5027C2semidirect
3. C2×#4848C2semidirect
4. C2×#4806C2semidirect
6. C3×#1805C2semidirect
7. #70476C2semidirect
8. #70240C2semidirect
9. #69559C2semidirect
10. #68909C2semidirect
11. #68881C2semidirect
58. #1805D3semidirect
66. C2×D3×Q8D4semidirect
77. C3×C4×Q8D4semidirect
124. C4×Dic6D4semidirect
295. C4×Q8D12semidirect
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×#50272
C2×#48482
C2×#48062
#742442
C3×#18052
#704762
#702402
#695592
#689092
#688812
#329123