Finite Group Databasebeta

← #10828466

Subgroups — #10828466

Order 768.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5035C212yesmaximal, complemented
3#82069C211yesmaximal, characteristic, complemented
4#71783C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#71762C211yesmaximal, characteristic, complemented
7#71762C211yesmaximal, characteristic, complemented
8#71678C212yesmaximal, complemented
9#71607C212yesmaximal, complemented
10#71606C212yesmaximal, complemented
11#71531C211yesmaximal, characteristic, complemented
12#24062—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#532—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#832C2212yes—
26C2×#832C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5210—21no—
30#5210—21no—
31C2×#830C2212yes—
32#5035—28no—
33C2×#808C2212yes—
34C2×#807C2212yes—
35C4×Dic3⋊C4—21no—
36C4×Dic3⋊C4—21no—
37#4928C2211yescharacteristic
38#4928C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45#4524C2212yes—
46C2×#532D311yescharacteristic
47C2×#482—32no—
48#2296—31no—
49#2058—31no—
50#2057—31no—
51#2041—31no—
52#2041—31no—
53#1875—32no—
54#1864—32no—
55#1862—32no—
56#1817—31no—
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C3×C4⋊Q8—24no—
70C3×#127—24no—
71C3×#124—28no—
72C3×#121—28no—
73C3×#121—24no—
74C3×#121D418yes—
75C3×C4×D4—28no—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—24no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89#926—22no—
90#926—22no—
91#926—24no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104#832—24no—
105#832—24no—
106#832—24no—
107#832—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112#830—24no—
113#808—24no—
114#807—24no—
115C22×#41—21no—
116#532—28no—
117C2×C4⋊Q8C2×D311yescharacteristic
118C2×#127C2×D311yescharacteristic
119C2×#124—21no—
120C2×#121—21no—
121C2×#121C2×D312yes—
122C2×C4×D4—21no—
123C2×C4×D4C2×D312yes—
124C2×#117—31no—
125C2×#117—31no—
126C22×C42C2×D311yescharacteristic
127#482—68no—
128C2×#107—32no—
129C2×#106—32no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×C22⋊C8—32no—
135#389—31no—
136#389—31no—
137#389—31no—
138#389—31no—
139#385—61no—
140#385—61no—
141C4×C4⋊C4—61no—
142C4×C4⋊C4—61no—
143#340—32no—
144#340—32no—
145#336—31no—
146#336—31no—
147#336—32no—
148C24×C3—22no—
149C24×C3C2×D411yescharacteristic
150C3×#41—432no—
151C2×C3×Q8—24no—
152C2×C3×Q8—42no—
153C2×C3×Q8—42no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—28no—
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—44no—
165C2×C3×D4—44no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4C2×D414yes—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—28no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—41no—
180C22×C3×C4—44no—
181C22×C3×C4—44no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4C2×D414yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—22no—
202#217—24no—
203#217—42no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3#4112yescomplemented
209C4⋊Dic3#4112yescomplemented
210Dic3⋊C4—42no—
211Dic3⋊C4—42no—
212Dic3⋊C4—42no—
213Dic3⋊C4—42no—
214C4×Dic3—24no—
215C4×Dic3—42no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×#41—28no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C22×D312yes—
230C23×C4—22no—
231C23×C4#6711yescharacteristic
232C23×C4#6711yescharacteristic
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—32no—
236C4⋊Q8—24no—
237#127—24no—
238#124—28no—
239#121—28no—
240#121—24no—
241#121#6718yes—
242C4×D4—28no—
243C4×D4—24no—
244C4×D4—24no—
245#117—62no—
246#117—62no—
247#117—64no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—61no—
262C2×C42—24no—
263C2×C42—61no—
264C2×C42—61no—
265C2×C42#6711yescharacteristic
266C2×C42#6711yescharacteristic
267C2×C42C22×D311yescharacteristic
268C2×C42C22×D311yescharacteristic
269#107—64no—
270#106—64no—
271#102—64no—
272#102—64no—
273#102—64no—
274#102—64no—
275C22⋊C8—64no—
276#95—61no—
277#95—61no—
278#95—61no—
279#95—61no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—24no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3#12012yes—
294C23×C3C22×D411yescharacteristic
295C3×Q8—48no—
296C3×D4—24no—
297C3×D4—24no—
298C3×D4—48no—
299C3×D4—48no—
300C3×D4—416no—
301C3×D4—42no—
302C3×D4—42no—
303C2×C3×C4—28no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—44no—
319C2×C3×C4—416no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4#12012yes—
323C2×C3×C4#13512yes—
324C2×C3×C4AGL(1,ℤ/8)12yes—
325C2×C3×C4AGL(1,ℤ/8)12yes—
326C2×C3×C4C22×D412yes—
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C42−1+412yes—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C3⋊C8—42no—
343C3⋊C8—42no—
344C24—22no—
345C24C2×#6711yescharacteristic
346#41—432no—
347C2×Q8—24no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—28no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4C2×#6714yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4C2×#6711yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6714yes—
388C22×C4C23×D311yescharacteristic
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—32no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—122no—
404C4⋊C4—122no—
405C4⋊C4—122no—
406C4⋊C4—122no—
407C4⋊C4C2×#6714yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—64no—
411C22⋊C4—122no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—64no—
417C42—122no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#52511yescharacteristic
434C22×C3#53611yescharacteristic
435C22×C3C2×#13511yescharacteristic
436C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3#57511yescharacteristic
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—41no—
444C3×C4—44no—
445C3×C4#54712yes—
446C3×C4#54712yes—
447Dic3—42no—
448Dic3—42no—
449Dic3—81no—
450Dic3—81no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23#95312yes—
465C23C22×#6711yescharacteristic
466Q8—48no—
467D4—24no—
468D4—24no—
469D4—48no—
470D4—48no—
471D4—416no—
472D4—42no—
473D4—42no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—416no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4#93212yes—
506C2×C4#93212yes—
507C2×C4#95012yes—
508C2×C4#95312yes—
509C2×C4D3×#4111yescharacteristic
510C2×C4D3×#4111yescharacteristic
511C2×C4#101012yes—
512C2×C4C22×#6712yes—
513C8—122no—
514C8—122no—
515C2×C3—42no—
516C2×C3—84no—
517C2×C3#240611yescharacteristic
518C2×C3#300211yescharacteristic
519C2×C3#300311yescharacteristic
520C2×C3#310611yescharacteristic
521C2×C3#311511yescharacteristic
522C2×C3#322811yescharacteristic
523C2×C3#323511yescharacteristic
524C22—22no—
525C22—22no—
526C22—22no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22C2×#93211yescharacteristic
539C22C2×#93211yescharacteristic
540C22C2×#95011yescharacteristic
541C22#582211yescharacteristic
542C22#582511yescharacteristic
543C22#582811yescharacteristic
544C22C2×#95311yescharacteristic
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—44no—
550C4—41no—
551C4—122no—
552C4—122no—
553C4—241no—
554C4—241no—
555C4#582412yes—
556C4#582412yes—
557C3#2406211yescharacteristic, complemented, Sylow
558C2—42no—
559C2—84no—
560C2#8327211yescharacteristic
561C2#8327411yescharacteristic
562C2#8329911yescharacteristic
563C2#8330111yescharacteristic
564C2#8403811yescharacteristic
565C2#8408011yescharacteristic
566C2#8409011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC23×C3C22×D4
hypercenterC2×#532D3
hypocenterC3#24062
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5035C2semidirect
3. #82069C2semidirect
4. #71783C2semidirect
5. #71782C2semidirect
6. #71762C2semidirect
7. #71762C2semidirect
8. #71678C2semidirect
9. #71607C2semidirect
10. #71606C2semidirect
11. #71531C2semidirect
208. C4⋊Dic3#41semidirect
209. C4⋊Dic3#41semidirect
557. C3#24062semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#50352
#820692
#717832
#717822
#717622
#716782
#716072
#716062
#715312
#240623