Finite Group Databasebeta

← #10900874

Subgroups — #10900874

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×#5805C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic, complemented
3C2×#4733C211yesmaximal, characteristic, complemented
4#73951C211yesmaximal, characteristic, complemented
5C3×#1786C211yesmaximal, characteristic, complemented
6#70215C212yesmaximal, complemented
7#70210C212yesmaximal, complemented
8#70178C212yesmaximal, complemented
9#70177C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic, complemented
12#29632—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C2×#953—22no—
16#5805—24no—
17#5805—24no—
18C2×#940C2212yes—
19C2×#938—21no—
20C2×#938—21no—
21C2×#936C2212yes—
22C22×#214C2212yes—
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C22×Dic12C2211yescharacteristic, complemented
26C22×#204C2211yescharacteristic, complemented
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#380C2211yescharacteristic
31#5218—24no—
32C2×#832C2212yes—
33C2×#830C2211yescharacteristic
34C2×#821C2211yescharacteristic, complemented
35C2×#816C2211yescharacteristic, complemented
36#4984C2212yescomplemented
37#4973C2212yescomplemented
38C2×#810C2212yescomplemented
39C2×#809C2212yescomplemented
40#4953—22no—
41#4733—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#1788—31no—
56#1786D311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4—42no—
61C22×C3×D4—42no—
62C22×C3×D4D414yes—
63C22×C3×D4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C22×#67—21no—
66C22×#67—22no—
67C22×#67—22no—
68C2×#235—22no—
69C2×#235—42no—
70C2×#235—42no—
71C2×#235D414yes—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×Dic6C2311yescharacteristic
75C22×Dic6C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80#953—216no—
81C2×#217—22no—
82C2×#217—22no—
83C2×#217—21no—
84C2×#217D412yes—
85#940—24no—
86#940—24no—
87#938—28no—
88#938—28no—
89#936—24no—
90#936—24no—
91C2×#214—24no—
92C2×#214—42no—
93C2×#214—42no—
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⋊C8C2312yes—
105C2×Dic12—22no—
106C2×Dic12—42no—
107C2×#204—22no—
108C2×#204—42no—
109C3×#102—42no—
110C3×#102—42no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113#832—24no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117#830—22no—
118#821—22no—
119#816—22no—
120#810—42no—
121#809—42no—
122C22×#41—31no—
123C23×D4C2×D311yescharacteristic
124C22×Q16—31no—
125C22×SD16—32no—
126C22×SD16—31no—
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×#103—32no—
140C2×#102—32no—
141C2×#102—31no—
142C2×#102—32no—
143C2×#102C2×D312yes—
144C2×#102C2×D312yes—
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—22no—
156C24×C3—24no—
157C24×C3—44no—
158C24×C3—81no—
159C24×C3—81no—
160C24×C3—82no—
161C24×C3C2×D411yescharacteristic
162C23×D3—21no—
163C2×C3×D4—24no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—48no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4C2×D414yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C2×#67—24no—
182C2×#67—28no—
183C2×#67—28no—
184C2×#67—44no—
185C2×#67—82no—
186C2×#67—82no—
187C2×#67—82no—
188C2×#67—82no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D412yes—
194#235—44no—
195#235—44no—
196#235—48no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12C2×D412yes—
201C2×C4×D3—24no—
202C2×C4×D3—44no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—22no—
209C2×Dic6—22no—
210C2×Dic6C2×D412yes—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C22⋊C4—81no—
218#217—28no—
219#217—28no—
220#217—24no—
221#217—42no—
222#214—48no—
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—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234Dic12—44no—
235#204—44no—
236C25—21no—
237C2×#41—34no—
238C2×#41—62no—
239C2×#41—122no—
240C2×#41—122no—
241C22×Q8—31no—
242C22×Q8—31no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—31no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4—62no—
249C22×D4—62no—
250C22×D4—61no—
251C22×D4D1214yes—
252C22×D4C22×D312yes—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—62no—
256C2×Q16—122no—
257C2×SD16—62no—
258C2×SD16—64no—
259C2×SD16—122no—
260C2×SD16—122no—
261C2×SD16—122no—
262C22×C8—32no—
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
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—62no—
289#103—122no—
290#102—42no—
291#102—42no—
292#102—62no—
293#102—64no—
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—21no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—48no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—84no—
314C23×C3—84no—
315C23×C3—88no—
316C23×C3—88no—
317C23×C3#12012yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—42no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—42no—
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—48no—
327C3×D4—42no—
328C3×D4—42no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4#12012yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4H(5,𝔽2)12yes—
340C2×C3×C4H(5,𝔽2)12yes—
341#67—48no—
342#67—88no—
343#67—88no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357D12—22no—
358D12—22no—
359D12—42no—
360C4×D3—48no—
361Dic6—22no—
362Dic6—22no—
363Dic6—42no—
364Dic6—41no—
365Dic6—41no—
366Dic6—42no—
367C3×C8—41no—
368C3×C8—41no—
369C3⋊C8—42no—
370C24—22no—
371C24—24no—
372C24—44no—
373C24—61no—
374C24—81no—
375C24—81no—
376C24—82no—
377C24D3×D411yescharacteristic
378#41—124no—
379#41—124no—
380#41—128no—
381C2×Q8—32no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—32no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—64no—
401C2×D4—68no—
402C2×D4—68no—
403C2×D4—61no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—124no—
407C2×D4—242no—
408C2×D4—242no—
409C2×D4—242no—
410C2×D4—242no—
411C2×D4C2×D1214yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—61no—
418C22×C4—64no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—124no—
422C22×C4C2×D1212yes—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D412yes—
425C22×C4C23×D311yescharacteristic
426Q16—124no—
427SD16—124no—
428SD16—128no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—62no—
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—44no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—816no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#53512yes—
466C22×C3#57611yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57712yes—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—82no—
473C2×D3—82no—
474C3×C4—22no—
475C3×C4—82no—
476C3×C4#53412yes—
477C3×C4#53411yescharacteristic
478C3×C4#53411yescharacteristic
479Dic3—42no—
480Dic3—42no—
481Dic3—42no—
482Dic3—81no—
483Dic3—81no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—62no—
497C23—84no—
498C23—84no—
499C23—88no—
500C23—88no—
501C23—121no—
502C23—121no—
503C23—122no—
504C23—122no—
505C23#88212yes—
506C23C2×D3×D411yescharacteristic
507Q8—62no—
508Q8—62no—
509Q8—122no—
510Q8—122no—
511Q8—121no—
512Q8—121no—
513D4—24no—
514D4—24no—
515D4—48no—
516D4—42no—
517D4—42no—
518D4—62no—
519D4—62no—
520D4—128no—
521D4—122no—
522D4—248no—
523D4—248no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—128no—
539C2×C4—122no—
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—41no—
551C8—41no—
552C8—122no—
553C2×C3—42no—
554C2×C3—84no—
555C2×C3#261311yescharacteristic
556C2×C3#300911yescharacteristic
557C2×C3#301211yescharacteristic
558C2×C3#322212yes—
559C2×C3#322412yes—
560D3—82no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—816no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—122no—
575C22—122no—
576C22—122no—
577C22—242no—
578C22—242no—
579C22C2×#88211yescharacteristic
580C22#559212yes—
581C22#575711yescharacteristic
582C22#576211yescharacteristic
583C22#577512yes—
584C4—22no—
585C4—82no—
586C4—122no—
587C4—122no—
588C4—122no—
589C4—241no—
590C4—241no—
591C4#549312yes—
592C4#558911yescharacteristic
593C4#558911yescharacteristic
594C3#2963211yescharacteristic, complemented, Sylow
595C2—42no—
596C2—84no—
597C2—242no—
598C2#7385511yescharacteristic
599C2#7874711yescharacteristic
600C2#7892911yescharacteristic
601C2#7999912yes—
602C2#8014212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC2×#5805C2
hypercenter#1786D3
hypocenterC3#29632
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#5218C2semidirect
3. C2×#4733C2semidirect
4. #73951C2semidirect
5. C3×#1786C2semidirect
6. #70215C2semidirect
7. #70210C2semidirect
8. #70178C2semidirect
9. #70177C2semidirect
10. #69911C2semidirect
11. #69522C2semidirect
25. C22×Dic12C22semidirect
26. C22×#204C22semidirect
34. C2×#821C22semidirect
35. C2×#816C22semidirect
36. #4984C22semidirect
37. #4973C22semidirect
38. C2×#810C22semidirect
39. C2×#809C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
56. #1786D3semidirect
594. C3#29632semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C2×#52182
C2×#47332
#739512
C3×#17862
#702152
#702102
#701782
#701772
#699112
#695222
#296323