Finite Group Databasebeta

← #10827391

Subgroups — #10827391

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×#5565C211yesmaximal, characteristic
2C22×#816C211yesmaximal, characteristic, complemented
3#82349C211yesmaximal, characteristic, complemented
4C2×#4738C211yesmaximal, characteristic, complemented
5C3×#1869C211yesmaximal, characteristic, complemented
6#71973C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69931C212yesmaximal, complemented
9#69925C211yesmaximal, characteristic, complemented
10#68690C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#23964—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#939—21no—
17C2×#930C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#924C2212yes—
22C22×#204C2212yes—
23#5565—24no—
24#5565—24no—
25C2×#898—21no—
26C2×#883—21no—
27C2×#67⋊C4—21no—
28C2×C4×D12C2211yescharacteristic
29C2×#870C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2212yes—
32#5220—21no—
33#5212C2211yescharacteristic
34C2×#821C2212yes—
35C2×#816—24no—
36C2×#816C2212yes—
37C2×#816C2212yes—
38C2×#816C2212yes—
39#4979—21no—
40#4947—21no—
41#4946—21no—
42#4929C2212yes—
43#4738—24no—
44C3×#336C2212yes—
45#4551C2212yes—
46#4551C2212yes—
47#4551C2212yes—
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#1869D311yescharacteristic, complemented
58#1824—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—21no—
64C23×Dic3—21no—
65C23×Dic3—21no—
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C22×Dic6—22no—
69C22×Dic6—24no—
70C22×Dic6—22no—
71C22×Dic6—21no—
72C22×Dic6—42no—
73C22×Dic6C2312yes—
74C22×C3×C8—22no—
75C22×C3×C8—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#217—21no—
85C2×#217—21no—
86C2×#217—22no—
87#939—28no—
88#930—22no—
89#930—22no—
90#930—22no—
91#930D414yes—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95#926—22no—
96#926—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3D414yes—
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104#924—24no—
105#924—42no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—22no—
109C2×C4×Dic3—21no—
110C2×#204—24no—
111C2×#204—24no—
112C2×#204—42no—
113#898—28no—
114#883—28no—
115#67⋊C4—28no—
116C4×D12—22no—
117C4×D12—22no—
118#870—22no—
119#870—22no—
120C3×C22⋊C8—22no—
121C3×C22⋊C8—42no—
122C3⋊#95—22no—
123C3⋊#95—22no—
124C3⋊#95—21no—
125#821—42no—
126#816—24no—
127#816—24no—
128#816—24no—
129C23×Q8—31no—
130C22×SD16—32no—
131C23×C8C2×D311yescharacteristic
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—61no—
143C2×C4×D4C2×D311yescharacteristic
144C2×#117—31no—
145C22×C4⋊C4—31no—
146#448—64no—
147C2×#103—32no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—64no—
151C2×#102—32no—
152C2×C22⋊C8C2×D311yescharacteristic
153C2×C22⋊C8C2×D312yes—
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336C2×D312yes—
164C24×C3C2×D411yescharacteristic
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C2×#67—24no—
184C2×#67—24no—
185C2×#67—42no—
186C2×#67—42no—
187C2×#67—82no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—24no—
193C22×Dic3—22no—
194C22×Dic3—44no—
195C22×Dic3—42no—
196C22×Dic3—82no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12C2×D412yes—
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—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—44no—
221C3×C4⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C42—21no—
225C3×C42—21no—
226#217—24no—
227#217—44no—
228#217—42no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—24no—
232C4⋊Dic3—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—22no—
237C4⋊Dic3—22no—
238C4⋊Dic3C2×D412yes—
239C4⋊Dic3C2×D412yes—
240C4⋊Dic3C2×D412yes—
241Dic3⋊C4—24no—
242Dic3⋊C4—24no—
243Dic3⋊C4—44no—
244C4×Dic3—24no—
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—61no—
254C22×D4—61no—
255C22×D4D1211yescharacteristic
256C23×C4—61no—
257C23×C4—61no—
258C23×C4—61no—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C22×C8—22no—
265C22×C8—42no—
266C22×C8C22×D312yes—
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—62no—
277#121—68no—
278#121—62no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—68no—
284#117—62no—
285#117—62no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4D1211yescharacteristic
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4D1211yescharacteristic
304C2×C22⋊C4D1211yescharacteristic
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#103—64no—
308#103—64no—
309#103—64no—
310#102—122no—
311C22⋊C8—22no—
312C22⋊C8—42no—
313#95—61no—
314#95—62no—
315#95—62no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3#12012yes—
324C23×C3#13512yes—
325C23×C3#13712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—42no—
332C3×D4—44no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4#12012yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×C3×C4H(5,𝔽2)12yes—
352#67—48no—
353#67—84no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—48no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—84no—
369D12—24no—
370D12—41no—
371D12—41no—
372C4×D3—84no—
373Dic6—24no—
374Dic6—24no—
375Dic6—48no—
376Dic6—42no—
377Dic6—42no—
378C3×C8—42no—
379C3×C8—42no—
380C24—61no—
381C24C2×D1211yescharacteristic
382C24C2×D1211yescharacteristic
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—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—32no—
398C2×D4—64no—
399C2×D4—64no—
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—22no—
411C22×C4—22no—
412C22×C4—64no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—64no—
418C22×C4—62no—
419C22×C4—124no—
420C22×C4—122no—
421C22×C4—242no—
422C22×C4—242no—
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
425C22×C4C2×D1211yescharacteristic
426C22×C4C2×D1211yescharacteristic
427C22×C4D3×D412yes—
428C22×C4D3×D412yes—
429C22×C4C23×D311yescharacteristic
430SD16—128no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—44no—
438C4⋊C4—22no—
439C4⋊C4—32no—
440C4⋊C4—32no—
441C4⋊C4—32no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—124no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—124no—
458C22⋊C4—122no—
459C42—21no—
460C42—21no—
461C42—64no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3#45214yes—
472C22×C3C2×#12011yescharacteristic
473C22×C3#54612yes—
474C22×C3C2×#13511yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57712yes—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—82no—
481C2×D3—82no—
482C3×C4—22no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4#52511yescharacteristic
487C3×C4#53511yescharacteristic
488C3×C4#54612yes—
489Dic3—44no—
490Dic3—42no—
491Dic3—42no—
492Dic3—42no—
493Dic3—82no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—62no—
502C23—121no—
503C23—121no—
504C23—122no—
505C23—122no—
506C23#88212yes—
507C23#90412yes—
508C23#90912yes—
509C23C22×D1211yescharacteristic
510Q8—64no—
511Q8—64no—
512Q8—128no—
513Q8—122no—
514Q8—122no—
515D4—44no—
516D4—64no—
517D4—128no—
518D4—121no—
519D4—121no—
520D4—244no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—64no—
539C2×C4—128no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—121no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—244no—
550C2×C4—244no—
551C2×C4#88212yes—
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4#100412yes—
555C2×C4D3×#4112yes—
556C8—42no—
557C8—42no—
558C2×C3—24no—
559C2×C3—81no—
560C2×C3#240411yescharacteristic
561C2×C3C2×#45211yescharacteristic
562C2×C3#300911yescharacteristic
563C2×C3#330112yes—
564C2×C3#332012yes—
565D3—82no—
566C22—22no—
567C22—24no—
568C22—22no—
569C22—24no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—41no—
574C22—42no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—242no—
579C22—242no—
580C22#473314yes—
581C22C2×#88211yescharacteristic
582C22#555412yes—
583C22C2×#90411yescharacteristic
584C22C2×#90911yescharacteristic
585C22#575112yes—
586C4—22no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—122no—
591C4—122no—
592C4—122no—
593C4—124no—
594C4—242no—
595C4#559211yescharacteristic
596C4#561012yes—
597C4#565611yescharacteristic
598C3#2396411yescharacteristic, complemented, Sylow
599C2—24no—
600C2—81no—
601C2—242no—
602C2#7488511yescharacteristic
603C2C2×#473311yescharacteristic
604C2#7867111yescharacteristic
605C2#7945012yes—
606C2#7951512yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1869C2
socleC23×C3C22×D4
prime radicalC2×#5565C2
hypercenter#1869D3
hypocenterC3#23964
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#816C2semidirect
3. #82349C2semidirect
4. C2×#4738C2semidirect
5. C3×#1869C2semidirect
6. #71973C2semidirect
7. #69962C2semidirect
8. #69931C2semidirect
9. #69925C2semidirect
10. #68690C2semidirect
11. #68672C2semidirect
57. #1869D3semidirect
598. C3#23964semidirect

Maximal subgroups

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

subgroupindex
C2×#55652
C22×#8162
#823492
C2×#47382
C3×#18692
#719732
#699622
#699312
#699252
#686902
#686722
#239643