Finite Group Databasebeta

← #10797147

Subgroups — #10797147

Order 768.

1782 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic
2C2×#5745C211yesmaximal, characteristic, complemented
3C2×C3×#418C211yesmaximal, characteristic, complemented
4C22×#833C211yesmaximal, characteristic, complemented
5#82213C211yesmaximal, characteristic, complemented
6#71998C211yesmaximal, characteristic, complemented
7#71587C211yesmaximal, characteristic, complemented
8#70218C218yesmaximal, complemented
9#21005—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C22×#214—21no—
15C2×#932—24no—
16C2×#932—22no—
17C2×#932C418yescomplemented
18C2×#932C2214yes—
19C22×#213—21no—
20C22×#208C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C22×#208C2211yescharacteristic
23#5745—24no—
24#5745—24no—
25C2×#907—22no—
26C2×D3×M16—22no—
27C2×#901C2214yescomplemented
28C3×#418—22no—
29C3×#418—22no—
30C2×C3×#102C2214yescomplemented
31C2×#833—24no—
32C2×#833C2214yescomplemented
33#5004—21no—
34#5002—22no—
35#5001—21no—
36#4966C2214yescomplemented
37#4552C2214yescomplemented
38#4530C2214yescomplemented
39C22×AGL(1,ℤ/8)—31no—
40C2×#568—31no—
41#2896—31no—
42C2×#418D311yescharacteristic, complemented
43C22×#100—31no—
44#1907—38no—
45#1742—31no—
46#1738—31no—
47C25×C3—21no—
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4—21no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4C2×C414yes—
57C22×C3×D4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C2×#235—21no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—21no—
64C2×#235C2×C414yes—
65C2×#235D414yes—
66C2×#235D412yes—
67C2×#235C2312yes—
68C22×D12D411yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6D411yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16D412yes—
74C2×C3×M16D412yes—
75C22×C3×C8C2312yes—
76C3×#117—21no—
77C3×#117—21no—
78C3×#117—21no—
79C3×#117—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×#214—28no—
84C2×#214—24no—
85#932—216no—
86#932—28no—
87#932C2×C4116yes—
88C2×#213—28no—
89C2×#213—24no—
90C2×#208—21no—
91C2×#208—22no—
92C2×#208—22no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208C2×C414yes—
98C2×#208D414yes—
99C2×#208C2312yes—
100C2×#208C2312yes—
101C2×#208C2312yes—
102C22×C3⋊C8—21no—
103C22×C3⋊C8—21no—
104C22×C3⋊C8—21no—
105#907—216no—
106D3×M16—216no—
107#901—28no—
108#901—28no—
109C2×ΓL(1,𝔽25)—24no—
110C2×D3×C8—24no—
111C3×#102—28no—
112#833—28no—
113#810—24no—
114#809—24no—
115C22×#41—31no—
116C23×D4C2×D311yescharacteristic
117C2×AGL(1,ℤ/8)—38no—
118C2×AGL(1,ℤ/8)—34no—
119C2×AGL(1,ℤ/8)—64no—
120C2×AGL(1,ℤ/8)—62no—
121C22×SD16—61no—
122C22×D8—61no—
123#568—64no—
124#568—64no—
125C2×#131—34no—
126C2×#131—62no—
127C22×M16—31no—
128C22×M16—31no—
129C22×M16—31no—
130C22×M16—62no—
131C22×M16C2×D311yescharacteristic
132C2×#117C2×D311yescharacteristic
133#419—61no—
134#418—22no—
135#418—22no—
136#418—61no—
137#417—34no—
138#416—62no—
139C2×#102C2×D314yescomplemented
140C2×#100—34no—
141C2×#100—64no—
142#343—34no—
143#341—34no—
144C24×C3—24no—
145C24×C3—22no—
146C24×C3—41no—
147C24×C3—41no—
148C24×C3—42no—
149C24×C3—84no—
150C24×C3#4111yescharacteristic
151C23×D3—21no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—41no—
167C2×C3×D4—41no—
168C2×C3×D4—42no—
169C2×C3×D4C22×C414yes—
170C22×C3×C4—22no—
171C22×C3×C4C22×C412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4#4112yes—
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×#67—24no—
182C2×#67—22no—
183C2×#67—24no—
184C2×#67—22no—
185C22×Dic3—21no—
186#235—24no—
187#235—24no—
188#235—24no—
189#235—24no—
190#235—24no—
191#235—22no—
192#235—22no—
193#235C22⋊C418yes—
194#235C22×C414yes—
195#235C2×D414yes—
196C2×D12—22no—
197C2×D12—21no—
198C2×D12C22⋊C414yes—
199C2×D12C2×D412yes—
200C2×C4×D3—24no—
201C2×C4×D3—24no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6C22⋊C414yes—
207C2×Dic6C2×D412yes—
208C3×M16—24no—
209C3×M16—24no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8C2×D414yes—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C22⋊C4—41no—
216C3×C22⋊C4—41no—
217C3×C42—41no—
218C3×C42—41no—
219#214—216no—
220#213—216no—
221#208—24no—
222#208—24no—
223#208—22no—
224#208—22no—
225#208—22no—
226#208—22no—
227#208—42no—
228#208C22×C414yes—
229#208C2×D414yes—
230C2×C3⋊C8—24no—
231C2×C3⋊C8—24no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—42no—
236ΓL(1,𝔽25)—216no—
237D3×C8—216no—
238C25—21no—
239C2×#41—34no—
240C2×#41—34no—
241C2×#41—32no—
242C2×#41—32no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—61no—
246C2×#41—61no—
247C22×Q8—31no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—31no—
257C22×D4—61no—
258C22×D4C4×D314yes—
259C22×D4C22×D312yes—
260C23×C4—61no—
261C23×C4C22×D311yescharacteristic
262AGL(1,ℤ/8)—316no—
263AGL(1,ℤ/8)—616no—
264AGL(1,ℤ/8)—68no—
265C2×SD16—68no—
266C2×SD16—64no—
267C2×D8—68no—
268C2×D8—64no—
269#131—68no—
270#131—68no—
271#131—616no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—34no—
275C2×M16—34no—
276C2×M16—32no—
277C2×M16—32no—
278C2×M16—32no—
279C2×M16—616no—
280C2×M16—62no—
281C2×M16—62no—
282C2×M16—61no—
283C2×M16—61no—
284C2×M16—61no—
285C2×M16—64no—
286C2×M16—62no—
287C2×M16—62no—
288C2×M16#6712yes—
289C2×M16#6712yes—
290C22×C8—61no—
291C22×C8—61no—
292C22×C8—64no—
293C22×C8—61no—
294C22×C8C22×D312yes—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299C2×C4⋊C4C22×D312yes—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302#103—64no—
303#102—28no—
304#102—64no—
305#100—68no—
306C23×C3—24no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—44no—
311C23×C3—44no—
312C23×C3—84no—
313C23×C3—84no—
314C23×C3—84no—
315C23×C3—88no—
316C23×C3—88no—
317C23×C3C4×D412yes—
318C23×C3#12111yescharacteristic
319C23×C3#12111yescharacteristic
320C23×C3#12312yes—
321C23×C3C2×#4111yescharacteristic
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—82no—
325C3×D4—24no—
326C3×D4—24no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—44no—
330C3×D4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—82no—
333C2×C3×C4C2×C22⋊C412yes—
334C2×C3×C4C2×C22⋊C412yes—
335C2×C3×C4C4×D412yes—
336C2×C3×C4C4×D412yes—
337C2×C3×C4C4×D412yes—
338C2×C3×C4#12012yes—
339C2×C3×C4#12012yes—
340C2×C3×C4#12112yes—
341C2×C3×C4#12112yes—
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12312yes—
345C2×C3×C4C23×C411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351#67—24no—
352#67—24no—
353#67—28no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356D12—22no—
357D12—24no—
358D12C2×C22⋊C414yes—
359C4×D3—24no—
360C4×D3—24no—
361C4×D3—28no—
362Dic6—22no—
363Dic6—24no—
364Dic6C2×C22⋊C414yes—
365C3×C8—24no—
366C3⋊C8—24no—
367C3⋊C8—24no—
368C3⋊C8—42no—
369C24—24no—
370C24—22no—
371C24—42no—
372C24—41no—
373C24—41no—
374C24—61no—
375C24—84no—
376C24#23711yescharacteristic
377#41—38no—
378#41—34no—
379#41—34no—
380#41—64no—
381#41—64no—
382#41—64no—
383#41—64no—
384#41—64no—
385#41—62no—
386#41—62no—
387C2×Q8—34no—
388C2×Q8—32no—
389C2×Q8—62no—
390C2×Q8—61no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—34no—
402C2×D4—32no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—61no—
416C2×D4C2×C4×D314yes—
417C22×C4—22no—
418C22×C4—61no—
419C22×C4—64no—
420C22×C4—64no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4C2×C4×D312yes—
424C22×C4#23512yes—
425C22×C4D3×D412yes—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D412yes—
429C22×C4#23711yescharacteristic
430C22×C4C2×#6711yescharacteristic
431C22×C4C2×#6711yescharacteristic
432C22×C4C23×D311yescharacteristic
433SD16—616no—
434D8—616no—
435M16—24no—
436M16—24no—
437M16—34no—
438M16—34no—
439M16—616no—
440M16—64no—
441M16—64no—
442M16—62no—
443M16—62no—
444M16—62no—
445M16—62no—
446M16—122no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—616no—
450C2×C8—64no—
451C2×C8—64no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—122no—
456C2×C8C2×#6714yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C42—41no—
462C42—41no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—44no—
469C22×C3—84no—
470C22×C3—88no—
471C22×C3—88no—
472C22×C3—84no—
473C22×C3—84no—
474C22×C3#38612yes—
475C22×C3#38612yes—
476C22×C3C2×C4×D411yescharacteristic
477C22×C3C2×#12111yescharacteristic
478C22×C3C2×#12311yescharacteristic
479C2×D3—24no—
480C2×D3—44no—
481C2×D3—84no—
482C3×C4—82no—
483C3×C4#38612yes—
484C3×C4#38612yes—
485C3×C4C22×C22⋊C411yescharacteristic
486C3×C4C2×C4×D411yescharacteristic
487C3×C4C2×#12011yescharacteristic
488C3×C4C2×#12111yescharacteristic
489Dic3—24no—
490C23—24no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—62no—
497C23—64no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—88no—
502C23—88no—
503C23—242no—
504C23#67⋊C412yes—
505C23#88312yes—
506C23#93811yescharacteristic
507C23#93811yescharacteristic
508C23C2×#23711yescharacteristic
509Q8—34no—
510Q8—64no—
511Q8—62no—
512D4—24no—
513D4—24no—
514D4—34no—
515D4—44no—
516D4—44no—
517D4—42no—
518D4—42no—
519D4—68no—
520D4—64no—
521D4—64no—
522D4—64no—
523D4—62no—
524C2×C4—42no—
525C2×C4—64no—
526C2×C4—62no—
527C2×C4—68no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—82no—
531C2×C4D3×C22⋊C412yes—
532C2×C4D3×C22⋊C412yes—
533C2×C4#67⋊C412yes—
534C2×C4#88312yes—
535C2×C4#88412yes—
536C2×C4#88412yes—
537C2×C4C4×#6712yes—
538C2×C4C4×#6712yes—
539C2×C4#93712yes—
540C2×C4#93712yes—
541C2×C4#93811yescharacteristic
542C2×C4#93811yescharacteristic
543C2×C4C22×C4×D311yescharacteristic
544C2×C4C2×#23511yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4C22×#6711yescharacteristic
549C8—24no—
550C8—64no—
551C8—64no—
552C8—122no—
553C2×C3—22no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—22no—
557C2×C3—84no—
558C2×C3C2×#38611yescharacteristic
559C2×C3#306812yes—
560D3—44no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—64no—
568C22—84no—
569C22—88no—
570C22—84no—
571C22—88no—
572C22—84no—
573C22—124no—
574C22—244no—
575C22#495612yes—
576C22#495612yes—
577C22C2×#67⋊C411yescharacteristic
578C22C2×#88311yescharacteristic
579C22C2×#93811yescharacteristic
580C4—64no—
581C4—82no—
582C4#495612yes—
583C4#495612yes—
584C4C2×D3×C22⋊C411yescharacteristic
585C4C2×#88411yescharacteristic
586C4C2×C4×#6711yescharacteristic
587C4C2×#93711yescharacteristic
588C3#2100511yescharacteristic, complemented, Sylow
589C2—22no—
590C2—22no—
591C2—21no—
592C2—21no—
593C2—84no—
594C2—124no—
595C2C2×#495611yescharacteristic
596C2#8396312yes—
subgroupG/H
centerC22C2×#938
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC22×C3C2×#121
prime radicalC22×#932C2
cosocleC22×C3×C4C24
hypercenterC2×#418D3
hypocenterC3#21005
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#5745C2semidirect
3. C2×C3×#418C2semidirect
4. C22×#833C2semidirect
5. #82213C2semidirect
6. #71998C2semidirect
7. #71587C2semidirect
8. #70218C2semidirect
17. C2×#932C4semidirect
27. C2×#901C22semidirect
30. C2×C3×#102C22semidirect
32. C2×#833C22semidirect
36. #4966C22semidirect
37. #4552C22semidirect
38. #4530C22semidirect
42. C2×#418D3semidirect
139. C2×#102C2×D3semidirect
588. C3#21005semidirect

Maximal subgroups

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

subgroupindex
C22×#9322
C2×#57452
C2×C3×#4182
C22×#8332
#822132
#719982
#715872
#702182
#210053