Finite Group Databasebeta

← #10900871

Subgroups — #10900871

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5830C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic, complemented
3C2×#4731C211yesmaximal, characteristic, complemented
4#73951C211yesmaximal, characteristic, complemented
5C3×#1788C211yesmaximal, characteristic, complemented
6#70278C212yesmaximal, complemented
7#70245C212yesmaximal, complemented
8#70210C212yesmaximal, complemented
9#70177C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic, complemented
11#69521C211yesmaximal, characteristic, complemented
12#29632—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15#5830—24no—
16#5830—24no—
17C2×#947C2212yes—
18C22×#216C2211yescharacteristic, complemented
19C2×#940C2212yes—
20C2×#937—22no—
21C22×#214C2211yescharacteristic, complemented
22C2×#930—21no—
23C2×#930—21no—
24C2×#926C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C22×#204C2212yes—
27C2×C3×#103C2212yescomplemented
28C2×C3×#102C2212yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#380C2211yescharacteristic
31#5239—24no—
32C2×#835C2211yescharacteristic, complemented
33C2×#832C2211yescharacteristic, complemented
34C2×#830C2211yescharacteristic
35C2×#821C2212yes—
36#4984C2212yescomplemented
37#4973C2212yescomplemented
38C2×#809C2212yes—
39C2×#809C2212yes—
40#4953—22no—
41#4731—24no—
42#4661—21no—
43#4659—21no—
44#4550C2212yescomplemented
45#4550C2212yescomplemented
46C2×#535—31no—
47C2×#452—31no—
48C2×#450—31no—
49#2625—31no—
50#2042—32no—
51#2041—32no—
52#2015—32no—
53#2014—32no—
54#1818—31no—
55#1788D311yescharacteristic, complemented
56#1786—31no—
57C24×D3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67—22no—
68C22×#67—22no—
69C22×D12—21no—
70C22×D12—22no—
71C22×D12—42no—
72C22×D12—42no—
73C22×D12D414yes—
74C22×D12C2312yes—
75C22×Dic6C2311yescharacteristic
76C22×C3×C8C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×#217—21no—
80C2×#217D412yes—
81#947—24no—
82#947—24no—
83C2×#216—22no—
84C2×#216—42no—
85#940—24no—
86#940—24no—
87#937—216no—
88C2×#214—22no—
89C2×#214—42no—
90#930—28no—
91#930—28no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94#926—22no—
95#926—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3D411yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100#924—22no—
101#924—22no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3C2312yes—
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C2×#204—24no—
107C2×#204—42no—
108C2×#204—42no—
109C3×#103—42no—
110C3×#102—42no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113#835—22no—
114#832—22no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#821—24no—
120#809—42no—
121#809—42no—
122C22×#41C2×D311yescharacteristic
123C23×D4—31no—
124C22×Q16—31no—
125C22×SD16—31no—
126C22×SD16—32no—
127#535—64no—
128#535—64no—
129C2×#127—32no—
130C2×#124—32no—
131C2×C22⋊Q8—31no—
132C2×#121—61no—
133C2×#121—61no—
134C2×#120—62no—
135C2×#117—31no—
136#452—64no—
137#450—64no—
138C2×#103—31no—
139C2×#103C2×D312yescomplemented
140C2×#102—31no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102C2×D312yescomplemented
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#398—61no—
149#397—62no—
150#391—32no—
151#389—32no—
152#380C2×D311yescharacteristic
153#336—32no—
154#336—32no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C23×D3—22no—
158C23×D3—24no—
159C23×D3—44no—
160C23×D3—81no—
161C23×D3—82no—
162C23×D3—81no—
163C3×#41—44no—
164C3×#41—44no—
165C3×#41—48no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—44no—
175C2×C3×D4C2×D412yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—44no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—28no—
187C2×#67—28no—
188C2×#67—82no—
189C2×#67—82no—
190C2×#67—82no—
191C2×#67—82no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—24no—
201C2×D12—22no—
202C2×D12—48no—
203C2×D12—44no—
204C2×D12—44no—
205C2×D12—44no—
206C2×D12C2×D414yes—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C22⋊C4—81no—
217#217—24no—
218#217—42no—
219#216—44no—
220#214—44no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—28no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—82no—
228Dic3⋊C4—42no—
229C4×Dic3—22no—
230C4×Dic3—22no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235#204—48no—
236C25—61no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C2×#41D1214yes—
241C22×Q8—31no—
242C22×Q8C22×D311yescharacteristic
243C22×D4—21no—
244C22×D4—34no—
245C22×D4—32no—
246C22×D4—62no—
247C22×D4—62no—
248C22×D4—62no—
249C22×D4—61no—
250C22×D4—122no—
251C22×D4—122no—
252C22×D4C22×D311yescharacteristic
253C23×C4—21no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—62no—
256C2×Q16—122no—
257C2×SD16—64no—
258C2×SD16—62no—
259C2×SD16—122no—
260C2×SD16—122no—
261C2×SD16—122no—
262C22×C8—31no—
263C22×C8—31no—
264C22×C8C22×D312yes—
265#127—64no—
266#127—64no—
267#124—64no—
268#124—64no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271#121—68no—
272#121—68no—
273#120—616no—
274#117—62no—
275#117—62no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4C22×D312yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—32no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—61no—
287C2×C42—32no—
288#103—42no—
289#103—62no—
290#102—42no—
291#102—64no—
292#102—62no—
293#102—122no—
294#102—122no—
295C22⋊C8—22no—
296C22⋊C8—62no—
297#95—21no—
298#95—62no—
299#95—62no—
300#95—62no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3—41no—
310C23×C3#12012yes—
311C23×C3C22×D411yescharacteristic
312C22×D3—24no—
313C22×D3—48no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—84no—
321C22×D3—84no—
322C22×D3—88no—
323C22×D3—88no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—42no—
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—48no—
330C3×D4—42no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4#12012yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4H(5,𝔽2)12yes—
345C2×C3×C4H(5,𝔽2)12yes—
346#67—88no—
347#67—88no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359D12—24no—
360D12—24no—
361D12—48no—
362D12—42no—
363D12—42no—
364Dic6—42no—
365Dic6—41no—
366Dic6—41no—
367C3×C8—42no—
368C3⋊C8—41no—
369C3⋊C8—41no—
370C24—21no—
371C24—62no—
372C24—64no—
373C24—124no—
374C24—242no—
375C24—241no—
376C24—241no—
377C24D3×D411yescharacteristic
378#41—44no—
379#41—44no—
380#41—48no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8C2×D1212yes—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—34no—
394C2×D4—44no—
395C2×D4—68no—
396C2×D4—68no—
397C2×D4—64no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—62no—
402C2×D4—62no—
403C2×D4—128no—
404C2×D4—124no—
405C2×D4—124no—
406C2×D4—124no—
407C2×D4—242no—
408C2×D4—242no—
409C2×D4—242no—
410C2×D4—242no—
411C2×D4C2×D1212yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—44no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4C2×D1212yes—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D412yes—
425C22×C4C23×D311yescharacteristic
426Q16—124no—
427SD16—128no—
428SD16—124no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—62no—
435C2×C8—62no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—122no—
443C4⋊C4—242no—
444C22⋊C4—68no—
445C22⋊C4—68no—
446C22⋊C4—64no—
447C22⋊C4—81no—
448C22⋊C4—122no—
449C42—62no—
450C42—62no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#53512yes—
463C22×C3#57611yescharacteristic
464C22×C3#57712yes—
465C22×C3#57711yescharacteristic
466C2×D3—44no—
467C2×D3—44no—
468C2×D3—44no—
469C2×D3—816no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—82no—
477C3×C4#53411yescharacteristic
478C3×C4#53411yescharacteristic
479C3×C4#53412yes—
480Dic3—42no—
481Dic3—42no—
482Dic3—81no—
483Dic3—81no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—64no—
494C23—128no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23—244no—
502C23—244no—
503C23—248no—
504C23—248no—
505C23#88212yes—
506C23C2×D3×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—42no—
510Q8—121no—
511Q8—121no—
512Q8—122no—
513D4—22no—
514D4—22no—
515D4—48no—
516D4—42no—
517D4—64no—
518D4—64no—
519D4—128no—
520D4—122no—
521D4—122no—
522D4—248no—
523D4—248no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4#88212yes—
546C2×C4C22×D1211yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4#100412yes—
549C2×C4#100412yes—
550C8—42no—
551C8—121no—
552C8—121no—
553C2×C3—42no—
554C2×C3—82no—
555C2×C3#261311yescharacteristic
556C2×C3#300911yescharacteristic
557C2×C3#301211yescharacteristic
558C2×C3#322212yes—
559C2×C3#322412yes—
560D3—84no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—82no—
570C22—82no—
571C22—124no—
572C22—124no—
573C22—124no—
574C22—2416no—
575C22—242no—
576C22—242no—
577C22—242no—
578C22—242no—
579C22C2×#88211yescharacteristic
580C22#559212yes—
581C22#577012yes—
582C22#577411yescharacteristic
583C22#577811yescharacteristic
584C4—22no—
585C4—42no—
586C4—82no—
587C4—122no—
588C4—122no—
589C4—241no—
590C4—241no—
591C4#549311yescharacteristic
592C4#549311yescharacteristic
593C4#558912yes—
594C3#2963211yescharacteristic, complemented, Sylow
595C2—42no—
596C2—82no—
597C2—244no—
598C2#7385511yescharacteristic
599C2#7883511yescharacteristic
600C2#7888611yescharacteristic
601C2#8011912yes—
602C2#8026212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC2×#5830C2
hypercenter#1788D3
hypocenterC3#29632
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. C2×#5239C2semidirect
3. C2×#4731C2semidirect
4. #73951C2semidirect
5. C3×#1788C2semidirect
6. #70278C2semidirect
7. #70245C2semidirect
8. #70210C2semidirect
9. #70177C2semidirect
10. #69911C2semidirect
11. #69521C2semidirect
18. C22×#216C22semidirect
21. C22×#214C22semidirect
27. C2×C3×#103C22semidirect
28. C2×C3×#102C22semidirect
32. C2×#835C22semidirect
33. C2×#832C22semidirect
36. #4984C22semidirect
37. #4973C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
55. #1788D3semidirect
139. C2×#103C2×D3semidirect
144. C2×#102C2×D3semidirect
594. C3#29632semidirect

Maximal subgroups

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

subgroupindex
C2×#58302
C2×#52392
C2×#47312
#739512
C3×#17882
#702782
#702452
#702102
#701772
#699112
#695212
#296323