Finite Group Databasebeta

← #10876176

Subgroups — #10876176

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2#82060C211yesmaximal, characteristic, complemented
3#72128C211yesmaximal, characteristic
4#72128C211yesmaximal, characteristic
5#71556C211yesmaximal, characteristic, complemented
6#71555C211yesmaximal, characteristic, complemented
7#71531C211yesmaximal, characteristic, complemented
8#69654C212yesmaximal, complemented
9#69641C212yesmaximal, complemented
10#69637C212yesmaximal, complemented
11#69636C212yesmaximal, complemented
12#27588—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4—21no—
16C22×C3×C22⋊C4—21no—
17C22×C3×C42C2211yescharacteristic
18C3×#390—21no—
19C3×#390—21no—
20C3×#390C2212yes—
21C3×#390C2212yes—
22C2×#832C2212yes—
23C2×#832C2212yes—
24C2×#832C2212yes—
25C2×#832C2212yes—
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28#5211—21no—
29C4×#217—21no—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#808C2212yescomplemented
33C2×#807C2212yescomplemented
34#4937—21no—
35#4936C2211yescharacteristic, complemented
36#4936C2211yescharacteristic, complemented
37C4×C4⋊Dic3C2211yescharacteristic, complemented
38C4×C4⋊Dic3C2211yescharacteristic, complemented
39#4932—21no—
40#4529C2212yescomplemented
41#4529C2212yescomplemented
42#4524C2212yescomplemented
43#4524C2212yescomplemented
44C2×#390D311yescharacteristic
45#2283—31no—
46#2092—32no—
47#2073—32no—
48#1926—32no—
49#1925—32no—
50#1853—31no—
51#1852—31no—
52#1817—31no—
53#1786—31no—
54#1786—31no—
55C25×C3—21no—
56C22×C3×D4—24no—
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4—42no—
61C22×C3×D4C2312yes—
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—44no—
74C2×C3×C22⋊C4—44no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217—21no—
82C2×#217—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4×Dic3—21no—
88C2×C4×Dic3—21no—
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2312yes—
91#832—24no—
92#832—24no—
93#832—42no—
94#832—42no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103#830—22no—
104#830—22no—
105#808—42no—
106#807—42no—
107C23×D4C2×D311yescharacteristic
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42C2×D311yescharacteristic
112C2×#107—32no—
113C2×#106—32no—
114C2×#102—32no—
115C2×#102—32no—
116C2×#102—32no—
117C2×#102—32no—
118C2×C22⋊C8—31no—
119C2×C22⋊C8—31no—
120#390—21no—
121#390—21no—
122#390C2×D312yes—
123#390C2×D312yes—
124#383—61no—
125#382—31no—
126#382—31no—
127#381—61no—
128#380—31no—
129#380—31no—
130C4×C4⋊C4—31no—
131C4×C4⋊C4—31no—
132C4×C22⋊C4—61no—
133#376—61no—
134#340—32no—
135#340—32no—
136#336—32no—
137#336—32no—
138C24×C3—24no—
139C24×C3—22no—
140C24×C3—42no—
141C24×C3—42no—
142C24×C3—42no—
143C24×C3—42no—
144C24×C3—42no—
145C24×C3—42no—
146C24×C3C2×D411yescharacteristic
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—48no—
154C2×C3×D4—48no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4#4114yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—41no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4#4112yes—
176C22×C3×C4#4112yes—
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—42no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—44no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—44no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—21no—
195#217—42no—
196#217—42no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4×Dic3—42no—
202C4×Dic3—42no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C25—21no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—42no—
215C22×D4C22×D312yes—
216C23×C4—21no—
217C23×C4#6711yescharacteristic
218C23×C4#6711yescharacteristic
219C23×C4C22×D311yescharacteristic
220C22×C8—32no—
221C22×C8—32no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22×D312yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—61no—
240C2×C42—61no—
241C2×C42#6711yescharacteristic
242C2×C42#6711yescharacteristic
243C2×C42C22×D311yescharacteristic
244C2×C42C22×D311yescharacteristic
245#107—122no—
246#106—122no—
247#102—64no—
248#102—64no—
249#102—122no—
250#102—122no—
251C22⋊C8—62no—
252C22⋊C8—62no—
253#95—61no—
254#95—61no—
255#95—61no—
256#95—61no—
257#95—61no—
258#95—61no—
259#95—61no—
260#95—61no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—24no—
266C23×C3—48no—
267C23×C3—48no—
268C23×C3—48no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—84no—
276C23×C3—84no—
277C23×C3—84no—
278C23×C3—84no—
279C23×C3#12312yes—
280C23×C3#13512yes—
281C23×C3AGL(1,ℤ/8)12yes—
282C23×C3C22×D411yescharacteristic
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—48no—
286C3×D4—42no—
287C3×D4—42no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—84no—
309C2×C3×C4#12312yes—
310C2×C3×C4AGL(1,ℤ/8)12yes—
311C2×C3×C4C2×#4112yes—
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)12yes—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C3⋊C8—42no—
328C3⋊C8—42no—
329C24—24no—
330C24—22no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24C2×#6711yescharacteristic
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4#23514yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4#23712yes—
366C22×C4#23712yes—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C23×D311yescharacteristic
373C2×C8—62no—
374C2×C8—62no—
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—122no—
391C22⋊C4—122no—
392C42—22no—
393C42—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—122no—
399C42—122no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—816no—
410C22×C3—82no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3#48214yes—
415C22×C3C2×#12311yescharacteristic
416C22×C3#52511yescharacteristic
417C22×C3#53511yescharacteristic
418C22×C3C2×#13511yescharacteristic
419C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
420C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
421C22×C3#57511yescharacteristic
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—82no—
428C3×C4#55112yes—
429C3×C4#55112yes—
430Dic3—81no—
431Dic3—81no—
432Dic3—81no—
433Dic3—81no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—24no—
439C23—48no—
440C23—48no—
441C23—48no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23#93212yes—
453C23#93512yes—
454C23#95012yes—
455C23C22×#6711yescharacteristic
456D4—24no—
457D4—24no—
458D4—48no—
459D4—42no—
460D4—42no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—84no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4#93212yes—
495C2×C4#93512yes—
496C2×C4C2×#23512yes—
497C2×C4C2×#23711yescharacteristic
498C2×C4C2×#23711yescharacteristic
499C2×C4#100412yes—
500C8—122no—
501C8—122no—
502C2×C3—24no—
503C2×C3—84no—
504C2×C3#248211yescharacteristic
505C2×C3C2×#48211yescharacteristic
506C2×C3#309411yescharacteristic
507C2×C3#310511yescharacteristic
508C2×C3#311411yescharacteristic
509C2×C3#321111yescharacteristic
510C2×C3#321711yescharacteristic
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—816no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22#503514yes—
526C22#579111yescharacteristic
527C22C2×#93211yescharacteristic
528C22C2×#93211yescharacteristic
529C22C2×#93511yescharacteristic
530C22#580511yescharacteristic
531C22C2×#95011yescharacteristic
532C22#582211yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4—241no—
540C4—241no—
541C4—241no—
542C4—241no—
543C4#549512yes—
544C4#549512yes—
545C3#2758811yescharacteristic, complemented, Sylow
546C2—24no—
547C2—84no—
548C2#8211311yescharacteristic
549C2#8230911yescharacteristic
550C2#8231111yescharacteristic
551C2C2×#503511yescharacteristic
552C2#8261611yescharacteristic
553C2#8294611yescharacteristic
554C2#8316711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#390C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#27588
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. #82060C2semidirect
5. #71556C2semidirect
6. #71555C2semidirect
7. #71531C2semidirect
8. #69654C2semidirect
9. #69641C2semidirect
10. #69637C2semidirect
11. #69636C2semidirect
32. C2×#808C22semidirect
33. C2×#807C22semidirect
35. #4936C22semidirect
36. #4936C22semidirect
37. C4×C4⋊Dic3C22semidirect
38. C4×C4⋊Dic3C22semidirect
40. #4529C22semidirect
41. #4529C22semidirect
42. #4524C22semidirect
43. #4524C22semidirect
545. C3#27588semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
#820602
#721282
#715562
#715552
#715312
#696542
#696412
#696372
#696362
#275883