Finite Group Databasebeta

← #10900872

Subgroups — #10900872

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×C3×#450C211yesmaximal, characteristic, complemented
3#82787C211yesmaximal, characteristic, complemented
4C2×#5047C211yesmaximal, characteristic, complemented
5#72128C211yesmaximal, characteristic, complemented
6#72058C212yesmaximal, complemented
7#72056C212yesmaximal, complemented
8#70216C212yesmaximal, complemented
9#70211C212yesmaximal, complemented
10#69960C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#29632—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×#953—22no—
18C22×#216C2211yescharacteristic, complemented
19C22×#215C2211yescharacteristic, complemented
20#5805—24no—
21#5805—24no—
22C2×#940C2212yes—
23C2×#938—21no—
24C2×#938—21no—
25C2×#936C2212yes—
26C2×#926C2211yescharacteristic
27C3×#450—24no—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30#5229C2212yescomplemented
31C2×#832C2212yes—
32C2×#832C2212yes—
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#821C2212yescomplemented
36C2×#816C2212yescomplemented
37#5047—24no—
38#4986—21no—
39#4981—21no—
40#4972C2212yescomplemented
41C2×#810C2211yescharacteristic, complemented
42C2×#809C2211yescharacteristic, complemented
43#4952—22no—
44#4550C2212yescomplemented
45#4550C2212yescomplemented
46C2×#535—31no—
47C2×#452—31no—
48C2×#450D311yescharacteristic, complemented
49#2625—31no—
50#2042—32no—
51#2041—32no—
52#2015—32no—
53#2014—32no—
54#1818—31no—
55#1788—31no—
56#1786—31no—
57C25×C3—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—22no—
61C22×C3×D4—42no—
62C22×C3×D4—42no—
63C22×C3×D4D414yes—
64C22×C3×D4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C22×#67—21no—
67C22×#67—22no—
68C22×#67—22no—
69C2×#235—22no—
70C2×#235—42no—
71C2×#235—42no—
72C2×#235D414yes—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×Dic6C2311yescharacteristic
76C2×C3×SD16—24no—
77C2×C3×SD16—42no—
78C2×C3×SD16—42no—
79C22×C3×C8C2312yes—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85#953—216no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—21no—
89C2×#217D412yes—
90C2×#216—22no—
91C2×#216—42no—
92C2×#215—22no—
93C2×#215—42no—
94#940—24no—
95#940—24no—
96#938—28no—
97#938—28no—
98#936—24no—
99#936—24no—
100#926—22no—
101#926—22no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2312yes—
105C2×Dic3⋊C4—22no—
106C2×C4×Dic3C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C3×#102—24no—
110C3×C22⋊C8—22no—
111#832—42no—
112#832—42no—
113C3⋊#95—21no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117#830—22no—
118#821—42no—
119#816—42no—
120#810—22no—
121#809—22no—
122C22×#41—31no—
123C23×D4C2×D311yescharacteristic
124C22×Q16—31no—
125C22×SD16—31no—
126C22×SD16C2×D312yes—
127#535—64no—
128#535—64no—
129C2×#127—32no—
130C2×#124—32no—
131C2×C22⋊Q8C2×D311yescharacteristic
132C2×#121—61no—
133C2×#121—61no—
134C2×#120—62no—
135C2×#117—31no—
136#452—64no—
137#450—24no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#102—31no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
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#380—31no—
153#336—32no—
154#336—32no—
155C24×C3—22no—
156C24×C3—24no—
157C24×C3—44no—
158C24×C3—81no—
159C24×C3—82no—
160C24×C3—81no—
161C24×C3C2×D411yescharacteristic
162C23×D3—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—48no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4C2×D414yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—24no—
187C2×#67—28no—
188C2×#67—28no—
189C2×#67—44no—
190C2×#67—82no—
191C2×#67—82no—
192C2×#67—82no—
193C2×#67—82no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3C2×D412yes—
197C22×Dic3C2×D412yes—
198#235—44no—
199#235—44no—
200#235—48no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—21no—
204C2×D12C2×D412yes—
205C2×C4×D3—24no—
206C2×C4×D3—44no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—21no—
210C2×Dic6C2×D412yes—
211C3×SD16—48no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—42no—
218C3×C22⋊C4—42no—
219#217—28no—
220#217—28no—
221#217—24no—
222#217—81no—
223#216—44no—
224#215—44no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229Dic3⋊C4—82no—
230C4×Dic3—22no—
231C4×Dic3—22no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C25—21no—
237C2×#41—34no—
238C2×#41—62no—
239C2×#41—122no—
240C2×#41—122no—
241C22×Q8—31no—
242C22×Q8C22×D311yescharacteristic
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×D4#6714yes—
252C22×D4C22×D312yes—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—62no—
256C2×Q16—122no—
257C2×SD16—24no—
258C2×SD16—42no—
259C2×SD16—42no—
260C2×SD16—62no—
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—22no—
270C22⋊Q8—22no—
271#121—68no—
272#121—68no—
273#120—616no—
274#117—62no—
275#117—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4C22×D311yescharacteristic
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—24no—
291#102—62no—
292#102—122no—
293#102—122no—
294#102—122no—
295C22⋊C8—22no—
296C22⋊C8—62no—
297#95—62no—
298#95—62no—
299#95—62no—
300#95—61no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
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—42no—
323C22×D3—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—42no—
327C3×D4—24no—
328C3×D4—24no—
329C3×D4—48no—
330C3×D4—42no—
331C3×D4—42no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
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—48no—
347#67—88no—
348#67—88no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360D12—22no—
361D12—22no—
362D12—42no—
363C4×D3—48no—
364Dic6—22no—
365Dic6—22no—
366Dic6—42no—
367C3×C8—42no—
368C3⋊C8—41no—
369C3⋊C8—41no—
370C24—22no—
371C24—24no—
372C24—44no—
373C24—61no—
374C24—82no—
375C24—81no—
376C24—81no—
377C24D3×D411yescharacteristic
378#41—124no—
379#41—124no—
380#41—128no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—32no—
386C2×Q8—61no—
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—62no—
404C2×D4—62no—
405C2×D4—61no—
406C2×D4—124no—
407C2×D4—242no—
408C2×D4—242no—
409C2×D4—242no—
410C2×D4—242no—
411C2×D4C2×#6714yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—61no—
419C22×C4—64no—
420C22×C4—62no—
421C22×C4—124no—
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D412yes—
424C22×C4C2×#6712yes—
425C22×C4C23×D311yescharacteristic
426Q16—124no—
427SD16—48no—
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—21no—
437C4⋊C4—21no—
438C4⋊C4—42no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—242no—
444C22⋊C4—42no—
445C22⋊C4—68no—
446C22⋊C4—68no—
447C22⋊C4—64no—
448C22⋊C4—241no—
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#57712yes—
468C22×C3#57711yescharacteristic
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—82no—
473C2×D3—82no—
474C3×C4—22no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4#53412yes—
478C3×C4#53411yescharacteristic
479C3×C4#53411yescharacteristic
480Dic3—42no—
481Dic3—42no—
482Dic3—42no—
483Dic3—82no—
484C23—22no—
485C23—21no—
486C23—21no—
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—122no—
502C23—121no—
503C23—121no—
504C23—122no—
505C23#93712yes—
506C23C2×D3×D411yescharacteristic
507Q8—42no—
508Q8—41no—
509Q8—41no—
510Q8—62no—
511Q8—62no—
512Q8—122no—
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—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—128no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4#93712yes—
546C2×C4C2×D3×D411yescharacteristic
547C2×C4#100412yes—
548C2×C4#100412yes—
549C2×C4C22×#6711yescharacteristic
550C8—42no—
551C8—121no—
552C8—121no—
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—
579C22#559212yes—
580C22#576212yes—
581C22#577411yescharacteristic
582C22#577511yescharacteristic
583C22C2×#93711yescharacteristic
584C4—22no—
585C4—81no—
586C4—81no—
587C4—122no—
588C4—122no—
589C4—122no—
590C4—242no—
591C4#558911yescharacteristic
592C4#558911yescharacteristic
593C4#580412yes—
594C3#2963211yescharacteristic, complemented, Sylow
595C2—42no—
596C2—84no—
597C2—242no—
598C2#8014212yes—
599C2#8014312yes—
600C2#8260711yescharacteristic
601C2#8385911yescharacteristic
602C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#450C2
socleC23×C3C22×D4
prime radicalC2×#5805C2
hypercenterC2×#450D3
hypocenterC3#29632
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
2. C2×C3×#450C2semidirect
3. #82787C2semidirect
4. C2×#5047C2semidirect
5. #72128C2semidirect
6. #72058C2semidirect
7. #72056C2semidirect
8. #70216C2semidirect
9. #70211C2semidirect
10. #69960C2semidirect
11. #69911C2semidirect
18. C22×#216C22semidirect
19. C22×#215C22semidirect
30. #5229C22semidirect
35. C2×#821C22semidirect
36. C2×#816C22semidirect
40. #4972C22semidirect
41. C2×#810C22semidirect
42. C2×#809C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
48. C2×#450D3semidirect
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×C3×#4502
#827872
C2×#50472
#721282
#720582
#720562
#702162
#702112
#699602
#699112
#296323