Finite Group Databasebeta

← #10876322

Subgroups — #10876322

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4987C211yesmaximal, characteristic
2#73862C211yesmaximal, characteristic
3C3×#1823C211yesmaximal, characteristic, complemented
4#71553C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#69927C211yesmaximal, characteristic, complemented
7#69505C211yesmaximal, characteristic, complemented
8#68891C212yesmaximal, complemented
9#68877C212yesmaximal, complemented
10#68873C212yesmaximal, complemented
11#68831C212yesmaximal, complemented
12#27602—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4—21no—
16C22×C4⋊Dic3C2211yescharacteristic
17C2×#892C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×C3×C22⋊C8C2211yescharacteristic
20C3×#380C2211yescharacteristic
21#5212C2211yescharacteristic
22C2×C3⋊#95—21no—
23C2×#830C2211yescharacteristic
24C2×(C2×D3)⋊C8C2212yes—
25C2×(C2×D3)⋊C8C2212yes—
26C2×#818C2212yescomplemented
27#4987—21no—
28#4987—21no—
29#4987C2212yes—
30#4987C2212yes—
31#4987C2211yescharacteristic
32#4987C2211yescharacteristic
33C2×#808C2212yescomplemented
34#4955—21no—
35Dic3×C22⋊C4—21no—
36#4666—21no—
37#4664—21no—
38C3×#340C2212yescomplemented
39C3×#340C2212yescomplemented
40#4551C2212yescomplemented
41#4529C2212yescomplemented
42#4529C2212yescomplemented
43#4529C2212yescomplemented
44#2280—31no—
45C2×#380—31no—
46#2077—32no—
47#2076—32no—
48#1854—32no—
49#1853—31no—
50#1852—31no—
51#1825—32no—
52#1823—31no—
53#1823D311yescharacteristic, complemented
54#1779—31no—
55C24×D3D411yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—24no—
63C22×C4×D3—42no—
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×C3×C8C2312yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×#217—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—44no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—42no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C22×C3⋊C8C2312yes—
92#892—24no—
93#892—42no—
94C3×C22⋊C8—22no—
95C3×#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#830—22no—
102(C2×D3)⋊C8—24no—
103(C2×D3)⋊C8—42no—
104#818—42no—
105#808—42no—
106C24×C4—31no—
107C2×#117—31no—
108C2×#117—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C22⋊C4—61no—
112C2×#107—32no—
113C2×#106—32no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—32no—
116C2×C22⋊C8—32no—
117C2×C22⋊C8C2×D311yescharacteristic
118#381—61no—
119#380—31no—
120#380—31no—
121#380—32no—
122#380—31no—
123#380—32no—
124#380—61no—
125#380—61no—
126#380C2×D311yescharacteristic
127#379—61no—
128C4×C22⋊C4—61no—
129C4×C22⋊C4—61no—
130C2×#95—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340C2×D312yescomplemented
136#340C2×D312yescomplemented
137C24×C3C2×D411yescharacteristic
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—41no—
144C23×D3—41no—
145C23×D3C2×D411yescharacteristic
146C23×D3C2×D411yescharacteristic
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—41no—
165C22×Dic3—42no—
166C22×Dic3—42no—
167C22×Dic3—42no—
168C22×Dic3C2×D411yescharacteristic
169C22×Dic3C2×D411yescharacteristic
170C2×C4×D3—21no—
171C2×C4×D3—21no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—21no—
177C2×C4×D3—21no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—48no—
182C2×C4×D3—48no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×C4×D3#4114yes—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—42no—
194C3×C22⋊C4—42no—
195#217—42no—
196(C2×D3)⋊C4—42no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—42no—
201(C2×D3)⋊C4—42no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—42no—
207C4×Dic3—42no—
208C4×Dic3—42no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C25—31no—
213C23×C4—21no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—31no—
218C23×C4—31no—
219C23×C4—64no—
220C23×C4—62no—
221C23×C4—62no—
222C23×C4—61no—
223C23×C4—122no—
224C23×C4C22×D311yescharacteristic
225C22×C8—32no—
226C22×C8C22×D312yes—
227#117—64no—
228#117—122no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—122no—
237C2×C4⋊C4C22×D312yes—
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—124no—
249C2×C42—61no—
250C2×C42—61no—
251#107—122no—
252#106—122no—
253C22⋊C8—22no—
254C22⋊C8—64no—
255C22⋊C8—62no—
256C22⋊C8—122no—
257#95—21no—
258#95—62no—
259#95—62no—
260#95—61no—
261#95—61no—
262#95—61no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3#12312yes—
268C23×C3#13512yes—
269C23×C3AGL(1,ℤ/8)12yes—
270C23×C3C22×D411yescharacteristic
271C22×D3—28no—
272C22×D3—21no—
273C22×D3—21no—
274C22×D3—21no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—21no—
284C22×D3—44no—
285C22×D3—44no—
286C22×D3—44no—
287C22×D3—44no—
288C22×D3—44no—
289C22×D3—44no—
290C22×D3#12312yes—
291C22×D3#13712yes—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—84no—
306C2×C3×C4#12312yes—
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4112yes—
310C2×C3×C4H(5,𝔽2)12yes—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—44no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—84no—
327C2×Dic3#12312yes—
328C4×D3—24no—
329C4×D3—24no—
330C4×D3—44no—
331C4×D3—44no—
332C4×D3—42no—
333C4×D3—42no—
334C3×C8—42no—
335C3⋊C8—42no—
336C24—31no—
337C24—31no—
338C24—62no—
339C24—62no—
340C24—62no—
341C24—62no—
342C24—62no—
343C24—121no—
344C24—121no—
345C24D3×D411yescharacteristic
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—31no—
352C22×C4—34no—
353C22×C4—31no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—62no—
373C22×C4—121no—
374C22×C4—121no—
375C22×C4—128no—
376C22×C4—128no—
377C22×C4—121no—
378C22×C4—122no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4#23512yes—
382C22×C4D3×D411yescharacteristic
383C22×C4#23712yes—
384C22×C4C23×D311yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—122no—
401C22⋊C4—42no—
402C22⋊C4—122no—
403C22⋊C4—122no—
404C22⋊C4—124no—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C22⋊C4—122no—
409C42—122no—
410C42—122no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3#48214yes—
418C22×C3C2×#12311yescharacteristic
419C22×C3C2×#12311yescharacteristic
420C22×C3#53411yescharacteristic
421C22×C3C2×#13511yescharacteristic
422C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
423C22×C3C2×#13711yescharacteristic
424C22×C3#57511yescharacteristic
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—416no—
432C2×D3—41no—
433C2×D3—41no—
434C2×D3—41no—
435C2×D3—41no—
436C2×D3—41no—
437C2×D3—41no—
438C2×D3—41no—
439C2×D3—41no—
440C3×C4—22no—
441C3×C4—82no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4#55112yes—
445C3×C4#55112yes—
446Dic3—22no—
447Dic3—41no—
448Dic3—81no—
449Dic3—81no—
450Dic3—82no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—32no—
456C23—32no—
457C23—68no—
458C23—61no—
459C23—61no—
460C23—61no—
461C23—61no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—124no—
471C23—124no—
472C23—124no—
473C23—124no—
474C23—124no—
475C23—124no—
476C23#88312yes—
477C23#91212yes—
478C23#91412yes—
479C23C2×D3×D411yescharacteristic
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—32no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—84no—
502C2×C4—124no—
503C2×C4—124no—
504C2×C4—124no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—122no—
516C2×C4—244no—
517C2×C4#88312yes—
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×#23711yescharacteristic
520C2×C4D3×#4112yes—
521C2×C4#100912yes—
522C8—42no—
523C8—122no—
524C2×C3—24no—
525C2×C3#248311yescharacteristic
526C2×C3C2×#48211yescharacteristic
527C2×C3#309111yescharacteristic
528C2×C3#309211yescharacteristic
529C2×C3#309511yescharacteristic
530C2×C3#319011yescharacteristic
531C2×C3#319411yescharacteristic
532D3—42no—
533D3—42no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—62no—
541C22—62no—
542C22—62no—
543C22—62no—
544C22—62no—
545C22—62no—
546C22—1216no—
547C22—121no—
548C22—121no—
549C22—121no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22—121no—
555C22#478414yes—
556C22C2×#88311yescharacteristic
557C22D3×#12311yescharacteristic
558C22#565311yescharacteristic
559C22C2×#91211yescharacteristic
560C22C2×#91411yescharacteristic
561C22D3×#13711yescharacteristic
562C22#577711yescharacteristic
563C4—22no—
564C4—62no—
565C4—81no—
566C4—81no—
567C4—82no—
568C4—121no—
569C4—241no—
570C4—242no—
571C4—241no—
572C4#552712yes—
573C4#565812yes—
574C3#2760211yescharacteristic, complemented, Sylow
575C2—24no—
576C2—122no—
577C2—122no—
578C2#7413011yescharacteristic
579C2C2×#478411yescharacteristic
580C2#7896511yescharacteristic
581C2#8128111yescharacteristic
582C2#8130411yescharacteristic
583C2#8142711yescharacteristic
584C2#8144311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenter#1823D3
hypocenterC3#27602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. C3×#1823C2semidirect
4. #71553C2semidirect
6. #69927C2semidirect
7. #69505C2semidirect
8. #68891C2semidirect
9. #68877C2semidirect
10. #68873C2semidirect
11. #68831C2semidirect
26. C2×#818C22semidirect
33. C2×#808C22semidirect
38. C3×#340C22semidirect
39. C3×#340C22semidirect
40. #4551C22semidirect
41. #4529C22semidirect
42. #4529C22semidirect
43. #4529C22semidirect
53. #1823D3semidirect
135. #340C2×D3semidirect
136. #340C2×D3semidirect
574. C3#27602semidirect

Maximal subgroups

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

subgroupindex
C2×#49872
#738622
C3×#18232
#715532
#699552
#699272
#695052
#688912
#688772
#688732
#688312
#276023