Finite Group Databasebeta

← #10827388

Subgroups — #10827388

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5514C211yesmaximal, characteristic
2C22×#810C211yesmaximal, characteristic, complemented
3C2×#4738C211yesmaximal, characteristic, complemented
4#74017C211yesmaximal, characteristic, complemented
5C3×#1824C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic, complemented
7#69515C211yesmaximal, characteristic, complemented
8#68911C212yesmaximal, complemented
9#68896C212yesmaximal, complemented
10#68887C212yesmaximal, complemented
11#68879C212yesmaximal, complemented
12#23964—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2211yescharacteristic
17C2×C4×#67—21no—
18C2×C4×#67C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×#204C2212yes—
21#5514—24no—
22#5514—24no—
23C2×#898—21no—
24C2×#897C2212yes—
25C2×#895C2211yescharacteristic
26C2×#894—21no—
27C2×#893C2211yescharacteristic
28C2×#892C2211yescharacteristic
29C2×#890C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33C2×#816C2212yes—
34#4973C2212yes—
35C2×#810—24no—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39#4958—21no—
40#4950—21no—
41#4738—24no—
42#4673—21no—
43#4656—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#553—31no—
49C2×#448—31no—
50C22×#103—31no—
51#2409—31no—
52#2026—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1873—32no—
57#1869—31no—
58#1824D311yescharacteristic, complemented
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C22×#67D411yescharacteristic
63C23×Dic3—21no—
64C22×D12—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—22no—
69C22×Dic6—24no—
70C22×Dic6—22no—
71C22×Dic6—21no—
72C22×Dic6—42no—
73C22×Dic6C2312yes—
74C22×C3×C8C2312yes—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4D414yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×#217D411yescharacteristic
83#930—28no—
84#930—22no—
85#930D414yes—
86C4×#67—28no—
87C4×#67—22no—
88C4×#67—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8—22no—
103C22×C3⋊C8—42no—
104C22×C3⋊C8C2312yes—
105C2×#204—24no—
106C2×#204—24no—
107C2×#204—42no—
108#898—28no—
109#897—24no—
110#897—42no—
111#895—22no—
112#895—22no—
113#894—28no—
114#893—22no—
115#893—22no—
116#892—22no—
117#892—22no—
118#890—22no—
119#890—22no—
120C3×C22⋊C8—22no—
121C3×#95—21no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124(C2×D3)⋊C8—42no—
125#816—24no—
126#810—24no—
127#810—24no—
128#809—42no—
129C23×Q8—31no—
130C22×SD16—32no—
131C23×C8—31no—
132#553—64no—
133#553—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#123—61no—
137C2×C22⋊Q8—32no—
138C2×#121—31no—
139C2×#121—31no—
140C2×#121—61no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—31no—
145C22×C4⋊C4C2×D311yescharacteristic
146#448—64no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—64no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×C22⋊C8C2×D311yescharacteristic
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380C2×D311yescharacteristic
160#340—32no—
161#340C2×D312yes—
162#340C2×D312yes—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C23×D3C2×D411yescharacteristic
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—82no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—22no—
179C2×#67—24no—
180C2×#67—22no—
181C2×#67—22no—
182C2×#67—42no—
183C2×#67—42no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—44no—
187C22×Dic3—42no—
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—21no—
194C2×D12—24no—
195C2×D12—82no—
196C2×D12C2×D412yes—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—24no—
202C2×C4×D3—82no—
203C2×Dic6—28no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—48no—
210C2×Dic6—44no—
211C2×Dic6—44no—
212C2×Dic6—42no—
213C2×Dic6#4114yes—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D412yes—
224C3×C22⋊C4—42no—
225#217—22no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—22no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—44no—
232C4⋊Dic3—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3—44no—
235C4⋊Dic3C2×D412yes—
236Dic3⋊C4—24no—
237Dic3⋊C4—22no—
238C4×Dic3—24no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—22no—
244C2×C3⋊C8—44no—
245#204—48no—
246C22×Q8—32no—
247C22×Q8—64no—
248C22×Q8—61no—
249C22×Q8—62no—
250C22×Q8—62no—
251C22×Q8—122no—
252C22×D4—31no—
253C22×D4—31no—
254C22×D4—61no—
255C22×D4—61no—
256C23×C4—21no—
257C23×C4—31no—
258C23×C4—61no—
259C23×C4—61no—
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C22×C8—32no—
265C22×C8—62no—
266C22×C8—122no—
267C22×C8C22×D312yes—
268#126—68no—
269#125—62no—
270#125—62no—
271#123—68no—
272C22⋊Q8—64no—
273C22⋊Q8—122no—
274#121—34no—
275#121—62no—
276#121—68no—
277#121—62no—
278#121—62no—
279C4×D4—62no—
280C4×D4—68no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—62no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4D1214yes—
294C2×C4⋊C4C22×D312yes—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—62no—
305C2×C42—31no—
306C2×C42—61no—
307#103—64no—
308#103—64no—
309#103—64no—
310#102—122no—
311C22⋊C8—22no—
312C22⋊C8—122no—
313#95—21no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3#12012yes—
319C23×C3#13512yes—
320C23×C3#13712yes—
321C23×C3C22×D411yescharacteristic
322C22×D3—21no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—22no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—42no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4#12012yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)12yes—
349#67—44no—
350#67—48no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—24no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—22no—
360C2×Dic3—48no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365D12—24no—
366D12—41no—
367D12—41no—
368D12—84no—
369C4×D3—41no—
370C4×D3—41no—
371C4×D3—42no—
372C4×D3—84no—
373Dic6—24no—
374Dic6—24no—
375Dic6—48no—
376Dic6—42no—
377Dic6—42no—
378C3×C8—42no—
379C3⋊C8—42no—
380C24—31no—
381C24—61no—
382C24D3×D411yescharacteristic
383C2×Q8—34no—
384C2×Q8—68no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×Q8—124no—
391C2×Q8—128no—
392C2×Q8—124no—
393C2×Q8—122no—
394C2×D4—32no—
395C2×D4—64no—
396C2×D4—64no—
397C2×D4—62no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—61no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—122no—
404C2×D4—122no—
405C2×D4—242no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—64no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—82no—
423C22×C4—124no—
424C22×C4—122no—
425C22×C4—242no—
426C22×C4C2×D1212yes—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D412yes—
429C22×C4C23×D311yescharacteristic
430SD16—128no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—124no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—32no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—62no—
447C4⋊C4—61no—
448C4⋊C4—124no—
449C4⋊C4C2×D1212yes—
450C4⋊C4C2×D1212yes—
451C22⋊C4—42no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—62no—
456C22⋊C4—62no—
457C22⋊C4—64no—
458C22⋊C4—124no—
459C42—64no—
460C42—61no—
461C42—61no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3#45214yes—
467C22×C3C2×#12011yescharacteristic
468C22×C3#54612yes—
469C22×C3C2×#13511yescharacteristic
470C22×C3C2×#13711yescharacteristic
471C22×C3#57712yes—
472C2×D3—41no—
473C2×D3—42no—
474C2×D3—41no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—41no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—82no—
481C2×D3—82no—
482C3×C4—22no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—82no—
486C3×C4#52511yescharacteristic
487C3×C4#53511yescharacteristic
488C3×C4#54612yes—
489Dic3—44no—
490Dic3—41no—
491Dic3—41no—
492Dic3—42no—
493Dic3—41no—
494C23—22no—
495C23—22no—
496C23—62no—
497C23—62no—
498C23—61no—
499C23—61no—
500C23—61no—
501C23—62no—
502C23—122no—
503C23—121no—
504C23—121no—
505C23—122no—
506C23#88212yes—
507C23#91612yes—
508C23#91812yes—
509C23C2×D3×D411yescharacteristic
510Q8—64no—
511Q8—64no—
512Q8—128no—
513Q8—122no—
514Q8—122no—
515D4—64no—
516D4—124no—
517D4—128no—
518D4—121no—
519D4—121no—
520D4—244no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—64no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—62no—
541C2×C4—84no—
542C2×C4—128no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—122no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—244no—
551C2×C4#88212yes—
552C2×C4C22×D1211yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4D3×#4112yes—
555C2×C4#100912yes—
556C8—42no—
557C8—122no—
558C2×C3—24no—
559C2×C3#240411yescharacteristic
560C2×C3C2×#45211yescharacteristic
561C2×C3#300911yescharacteristic
562C2×C3#330112yes—
563C2×C3#332012yes—
564D3—81no—
565D3—82no—
566C22—22no—
567C22—24no—
568C22—22no—
569C22—24no—
570C22—122no—
571C22—121no—
572C22—122no—
573C22—121no—
574C22—122no—
575C22—122no—
576C22—121no—
577C22—122no—
578C22—242no—
579C22—242no—
580C22#480814yes—
581C22C2×#88211yescharacteristic
582C22#565412yes—
583C22C2×#91611yescharacteristic
584C22C2×#91811yescharacteristic
585C22#577512yes—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—121no—
591C4—121no—
592C4—124no—
593C4—121no—
594C4—122no—
595C4#552411yescharacteristic
596C4#552511yescharacteristic
597C4#565412yes—
598C3#2396411yescharacteristic, complemented, Sylow
599C2—24no—
600C2—241no—
601C2—242no—
602C2#7412511yescharacteristic
603C2C2×#480811yescharacteristic
604C2#7892911yescharacteristic
605C2#8134512yes—
606C2#8149112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1824C2
socleC23×C3C22×D4
prime radicalC2×#5514C2
hypercenter#1824D3
hypocenterC3#23964
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. C22×#810C2semidirect
3. C2×#4738C2semidirect
4. #74017C2semidirect
5. C3×#1824C2semidirect
6. #69952C2semidirect
7. #69515C2semidirect
8. #68911C2semidirect
9. #68896C2semidirect
10. #68887C2semidirect
11. #68879C2semidirect
58. #1824D3semidirect
598. C3#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#55142
C22×#8102
C2×#47382
#740172
C3×#18242
#699522
#695152
#689112
#688962
#688872
#688792
#239643