Finite Group Databasebeta

← #10822367

Subgroups — #10822367

Order 768.

1916 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5822C212yesmaximal, complemented
2C2×#5748C211yesmaximal, characteristic, complemented
3C3×#2879C211yesmaximal, characteristic, complemented
4#83762C212yesmaximal, complemented
5C2×C3⋊#338C211yesmaximal, characteristic, complemented
6#70013C218yesmaximal, complemented
7#23434—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C2212yes—
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×#951—22no—
13#5822—24no—
14#5822—24no—
15#5822—24no—
16#5822C2218yescomplemented
17C2×#950—24no—
18C2×#949—22no—
19C2×#942—22no—
20C2×#932—22no—
21C22×#208C2212yes—
22#5748—24no—
23#5748—22no—
24#5748C2214yescomplemented
25C2×#909—22no—
26C2×#908—22no—
27C2×#904—22no—
28C3×#408—22no—
29C3×#408C2214yescomplemented
30C3×#407—22no—
31#5129—24no—
32#5126—24no—
33#5110C2218yescomplemented
34C3⋊#338C2214yescomplemented
35C2×#575—32no—
36C2×#575—31no—
37#2879—32no—
38#2879D311yescharacteristic, complemented
39#2017—38no—
40C2×#338—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—22no—
46C2×C3×#41—22no—
47C2×C3×#41D412yes—
48C2×C3×#41C2312yes—
49C22×C3×Q8D411yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4D411yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235D414yes—
66C2×#235C2314yes—
67C22×D12D412yes—
68C22×C4×D3—22no—
69C22×Dic6D412yes—
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C2×C3×M16—21no—
75C2×C3×M16—21no—
76C2×C3×M16D412yes—
77C2×C3×M16C2312yes—
78C22×C3×C8D412yes—
79C22×C3×C8C2312yes—
80#951—28no—
81#951—28no—
82#950—216no—
83#950—216no—
84#949—28no—
85#949—28no—
86C2×#216—24no—
87#942—28no—
88#942—28no—
89C2×#215—24no—
90C2×#214—24no—
91#932—28no—
92#932—28no—
93C2×#213—24no—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208D414yes—
99C2×#208C2314yes—
100C22×C3⋊C8D414yes—
101#909—28no—
102#909—28no—
103#908—28no—
104#908—28no—
105C2×Dic12—22no—
106#904—28no—
107#904—28no—
108C2×D24—22no—
109C2×#204—24no—
110C3×C22⋊C8—28no—
111(C2×D3)⋊C8—216no—
112C22×#41—32no—
113C22×#41C2×D311yescharacteristic
114#575—38no—
115#575—34no—
116#575—64no—
117#575—64no—
118#575—64no—
119#575—64no—
120#575—62no—
121C2×#137—62no—
122C2×#137—62no—
123C2×#137—62no—
124C2×AGL(1,ℤ/8)—62no—
125C2×AGL(1,ℤ/8)—62no—
126C2×AGL(1,ℤ/8)—62no—
127C2×#135—64no—
128C2×#135—62no—
129C22×M16—32no—
130C22×M16C2×D311yescharacteristic
131C22×M16C2×D311yescharacteristic
132#408—22no—
133#408—38no—
134#408—64no—
135#408C2×D314yescomplemented
136#407—22no—
137#407—64no—
138#338—34no—
139C24×C3—21no—
140C24×C3#4111yescharacteristic
141C23×D3—22no—
142C3×#41—24no—
143C3×#41—24no—
144C3×#41—24no—
145C3×#41—24no—
146C3×#41—41no—
147C3×#41—41no—
148C3×#41—42no—
149C3×#41—42no—
150C3×#41—41no—
151C3×#41—41no—
152C2×C3×Q8—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8C2×D412yes—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—42no—
182C2×#67—42no—
183C2×#67—42no—
184C2×#67—42no—
185C22×Dic3—22no—
186#235—24no—
187#235—24no—
188#235—24no—
189#235—24no—
190#235—24no—
191#235—24no—
192#235—24no—
193#235—24no—
194#235—42no—
195#235—42no—
196#235—42no—
197#235—42no—
198#235—42no—
199#235—42no—
200#235—42no—
201#235—42no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12C2×D414yes—
208C2×C4×D3—24no—
209C2×C4×D3—24no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6C2×D414yes—
220C3×M16—24no—
221C3×M16—24no—
222C3×M16—41no—
223C3×M16—42no—
224C3×M16—41no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—24no—
228C2×C3×C8—22no—
229C2×C3×C8C2×D414yes—
230#216—216no—
231#215—216no—
232#214—216no—
233#213—216no—
234#208—24no—
235#208—24no—
236#208—24no—
237#208—24no—
238C2×C3⋊C8—24no—
239C2×C3⋊C8—24no—
240C2×C3⋊C8C2×D418yes—
241Dic12—28no—
242D24—28no—
243#204—216no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—34no—
251C2×#41—34no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—62no—
256C2×#41—62no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41D1212yes—
263C2×#41C22×D312yes—
264C22×Q8—32no—
265C22×Q8D1211yescharacteristic
266C22×D4—21no—
267C22×D4—32no—
268C22×D4—62no—
269C22×D4D1211yescharacteristic
270C23×C4—21no—
271C23×C4—62no—
272C23×C4C22×D311yescharacteristic
273#137—68no—
274#137—68no—
275#137—68no—
276#137—68no—
277#137—68no—
278#137—68no—
279AGL(1,ℤ/8)—68no—
280AGL(1,ℤ/8)—68no—
281AGL(1,ℤ/8)—68no—
282AGL(1,ℤ/8)—68no—
283AGL(1,ℤ/8)—68no—
284AGL(1,ℤ/8)—68no—
285#135—68no—
286#135—616no—
287#135—616no—
288#135—68no—
289C2×Q16—62no—
290C2×Q16—64no—
291C2×SD16—64no—
292C2×SD16—64no—
293C2×SD16—64no—
294C2×D8—62no—
295C2×D8—64no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—21no—
301C2×M16—21no—
302C2×M16—34no—
303C2×M16—34no—
304C2×M16—62no—
305C2×M16—62no—
306C2×M16—62no—
307C2×M16—62no—
308C2×M16#6712yes—
309C2×M16C22×D312yes—
310C22×C8—34no—
311C22×C8#6712yes—
312C22×C8C22×D312yes—
313C22⋊C8—28no—
314C22⋊C8—616no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—81no—
318C23×C3—82no—
319C23×C3#12112yes—
320C23×C3#12112yes—
321C23×C3#12111yescharacteristic
322C23×C3#12111yescharacteristic
323C23×C3C2×#4111yescharacteristic
324C22×D3—24no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—84no—
328C3×Q8—24no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×D4—24no—
332C3×D4—44no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—41no—
336C3×D4—41no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—44no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4#12012yes—
345C2×C3×C4#12012yes—
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12012yes—
349C2×C3×C4#12112yes—
350C2×C3×C4#12112yes—
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12111yescharacteristic
353C2×C3×C4#12711yescharacteristic
354C2×C3×C4#12712yes—
355C2×C3×C4#12711yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C22×D412yes—
358C2×C3×C4C22×D412yes—
359C2×C3×C4C22×D411yescharacteristic
360#67—44no—
361#67—44no—
362#67—44no—
363#67—44no—
364C2×Dic3—24no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367D12—24no—
368D12—24no—
369D12—42no—
370D12—42no—
371C4×D3—44no—
372C4×D3—44no—
373C4×D3—44no—
374C4×D3—44no—
375Dic6—24no—
376Dic6—24no—
377Dic6—42no—
378Dic6—42no—
379C3×C8—24no—
380C3×C8—42no—
381C3⋊C8—28no—
382C24—21no—
383C24—62no—
384C24#23511yescharacteristic
385#41—24no—
386#41—24no—
387#41—24no—
388#41—24no—
389#41—42no—
390#41—42no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—64no—
396#41—64no—
397#41—64no—
398#41—64no—
399#41—64no—
400#41—64no—
401#41—64no—
402#41—64no—
403#41—122no—
404#41—122no—
405#41—122no—
406#41—122no—
407#41—122no—
408#41—122no—
409#41—122no—
410#41—122no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—34no—
415C2×Q8—62no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8C2×D1212yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—34no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—64no—
430C2×D4—64no—
431C2×D4—62no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—122no—
437C2×D4—122no—
438C2×D4—122no—
439C2×D4—122no—
440C2×D4C2×D1212yes—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—62no—
447C22×C4—64no—
448C22×C4—64no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4C2×D1211yescharacteristic
454C22×C4C2×D1211yescharacteristic
455C22×C4#23511yescharacteristic
456C22×C4D3×D412yes—
457C22×C4D3×D412yes—
458C22×C4D3×D412yes—
459C22×C4D3×D412yes—
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C23×D311yescharacteristic
463Q16—68no—
464Q16—616no—
465SD16—616no—
466SD16—616no—
467SD16—616no—
468D8—68no—
469D8—616no—
470M16—24no—
471M16—24no—
472M16—42no—
473M16—41no—
474M16—41no—
475M16—64no—
476M16—64no—
477M16—64no—
478M16—64no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—38no—
484C2×C8—64no—
485C2×C8—64no—
486C2×C8C2×#6714yes—
487C22×C3—42no—
488C22×C3—41no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—44no—
493C22×C3—42no—
494C22×C3—84no—
495C22×C3#39211yescharacteristic
496C22×C3#39212yes—
497C22×C3#39211yescharacteristic
498C22×C3C2×#12112yes—
499C22×C3C2×#12111yescharacteristic
500C2×D3—48no—
501C2×D3—44no—
502C2×D3—84no—
503C2×D3—84no—
504C3×C4—42no—
505C3×C4#39211yescharacteristic
506C3×C4#39212yes—
507C3×C4#39211yescharacteristic
508C3×C4C2×#12012yes—
509C3×C4C2×#12011yescharacteristic
510C3×C4C2×#12711yescharacteristic
511Dic3—44no—
512C23—22no—
513C23—42no—
514C23—64no—
515C23—81no—
516C23—82no—
517C23—122no—
518C23—122no—
519C23—244no—
520C23#88412yes—
521C23#88412yes—
522C23#93011yescharacteristic
523C23#93011yescharacteristic
524C23C2×#23511yescharacteristic
525Q8—24no—
526Q8—41no—
527Q8—41no—
528Q8—64no—
529Q8—64no—
530Q8—122no—
531Q8—122no—
532D4—24no—
533D4—42no—
534D4—42no—
535D4—44no—
536D4—41no—
537D4—41no—
538D4—64no—
539D4—64no—
540D4—124no—
541D4—124no—
542D4—124no—
543D4—124no—
544D4—122no—
545D4—122no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—64no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—124no—
555C2×C4—124no—
556C2×C4—124no—
557C2×C4—124no—
558C2×C4#88212yes—
559C2×C4#88212yes—
560C2×C4#88212yes—
561C2×C4#88212yes—
562C2×C4#88412yes—
563C2×C4#88412yes—
564C2×C4#93011yescharacteristic
565C2×C4#93011yescharacteristic
566C2×C4#93711yescharacteristic
567C2×C4#93711yescharacteristic
568C2×C4#93712yes—
569C2×C4#94012yes—
570C2×C4#94011yescharacteristic
571C2×C4#94011yescharacteristic
572C2×C4C22×D1211yescharacteristic
573C2×C4C2×D3×D412yes—
574C2×C4C2×D3×D412yes—
575C2×C4C22×#6711yescharacteristic
576C8—24no—
577C8—42no—
578C8—68no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—22no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—82no—
585C2×C3C2×#39211yescharacteristic
586D3—84no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—44no—
593C22—42no—
594C22—84no—
595C22—128no—
596C22—124no—
597C22—244no—
598C22—244no—
599C22#496012yes—
600C22#496011yescharacteristic
601C22#496011yescharacteristic
602C22C2×#88412yes—
603C22C2×#93011yescharacteristic
604C4—42no—
605C4—124no—
606C4#496012yes—
607C4#496011yescharacteristic
608C4#496011yescharacteristic
609C4C2×#88212yes—
610C4C2×#93711yescharacteristic
611C4C2×#94011yescharacteristic
612C3#2343411yescharacteristic, complemented, Sylow
613C2—21no—
614C2—22no—
615C2—22no—
616C2—21no—
617C2—21no—
618C2—82no—
619C2—244no—
620C2C2×#496011yescharacteristic
subgroupG/H
centerC4C2×#940
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2879C2
socleC2×C3C2×#392
hypercenter#2879D3
hypocenterC3#23434
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
1. C2×#5822C2semidirect
2. C2×#5748C2semidirect
3. C3×#2879C2semidirect
4. #83762C2semidirect
5. C2×C3⋊#338C2semidirect
6. #70013C2semidirect
16. #5822C22semidirect
24. #5748C22semidirect
29. C3×#408C22semidirect
33. #5110C22semidirect
34. C3⋊#338C22semidirect
38. #2879D3semidirect
135. #408C2×D3semidirect
612. C3#23434semidirect

Maximal subgroups

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

subgroupindex
C2×#58222
C2×#57482
C3×#28792
#837622
C2×C3⋊#3382
#700132
#234343