Finite Group Databasebeta

← #10900870

Subgroups — #10900870

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×C3×#452C211yesmaximal, characteristic, complemented
3#82787C211yesmaximal, characteristic, complemented
4C2×#5046C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic, complemented
6#72072C212yesmaximal, complemented
7#72056C212yesmaximal, complemented
8#70279C212yesmaximal, complemented
9#70211C212yesmaximal, complemented
10#69959C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#29632—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C22×C3×Q16C2211yescharacteristic, complemented
16C22×C3×SD16C2211yescharacteristic, complemented
17C2×C3×C22⋊Q8C2211yescharacteristic
18#5830—24no—
19#5830—24no—
20C2×#947C2212yes—
21C22×#215C2212yes—
22C2×#940C2212yes—
23C2×#937—22no—
24C2×#930—21no—
25C2×#930—21no—
26C2×#926C2211yescharacteristic
27C3×#452—24no—
28C2×C3×#103C2211yescharacteristic, complemented
29C2×C3×#102C2211yescharacteristic, complemented
30C2×C3×C22⋊C8C2211yescharacteristic
31#5229C2212yescomplemented
32C2×#835C2212yescomplemented
33C2×#832C2212yescomplemented
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#821C2212yes—
37C2×#821C2212yes—
38#5046—24no—
39#4986—21no—
40#4981—21no—
41#4972C2212yescomplemented
42C2×#809C2212yes—
43#4952—22no—
44#4550C2212yescomplemented
45#4550C2212yescomplemented
46C2×#535—31no—
47C2×#452D311yescharacteristic, complemented
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#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×Q8C2311yescharacteristic
64C22×C3×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×#67—22no—
70C22×D12—21no—
71C22×D12—22no—
72C22×D12—42no—
73C22×D12—42no—
74C22×D12D414yes—
75C22×D12C2312yes—
76C2×C3×Q16—22no—
77C2×C3×Q16—42no—
78C2×C3×SD16—22no—
79C2×C3×SD16—42no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×#217—21no—
88C2×#217D412yes—
89#947—24no—
90#947—24no—
91C2×#215—24no—
92C2×#215—42no—
93C2×#215—42no—
94#940—24no—
95#940—24no—
96#937—216no—
97#930—28no—
98#930—28no—
99C2×(C2×D3)⋊C4—22no—
100C2×(C2×D3)⋊C4—22no—
101#926—22no—
102#926—22no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2312yes—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3C2312yes—
108C22×C3⋊C8C2312yes—
109C3×#103—22no—
110C3×#102—22no—
111C3×C22⋊C8—22no—
112#835—42no—
113#832—42no—
114C3⋊#95—21no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#830—22no—
119#821—42no—
120#821—42no—
121#809—24no—
122C22×#41C2×D311yescharacteristic
123C23×D4—31no—
124C22×Q16C2×D311yescharacteristic, complemented
125C22×SD16—32no—
126C22×SD16C2×D311yescharacteristic, complemented
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—24no—
137#450—64no—
138C2×#103—32no—
139C2×#103C2×D311yescharacteristic, complemented
140C2×#102—32no—
141C2×#102—32no—
142C2×#102—32no—
143C2×#102—32no—
144C2×#102C2×D311yescharacteristic, complemented
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—21no—
156C24×C3C2×D411yescharacteristic
157C23×D3—22no—
158C23×D3—24no—
159C23×D3—44no—
160C23×D3—81no—
161C23×D3—81no—
162C23×D3—82no—
163C3×#41—44no—
164C3×#41—44no—
165C3×#41—48no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—24no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—44no—
179C2×C3×D4C2×D412yes—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—24no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—44no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2411yescharacteristic
191C2×#67—28no—
192C2×#67—28no—
193C2×#67—82no—
194C2×#67—82no—
195C2×#67—82no—
196C2×#67—82no—
197C22×Dic3—22no—
198C22×Dic3C2×D412yes—
199C22×Dic3C2×D412yes—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—24no—
205C2×D12—22no—
206C2×D12—48no—
207C2×D12—44no—
208C2×D12—44no—
209C2×D12—44no—
210C2×D12C2×D414yes—
211C3×Q16—44no—
212C3×SD16—44no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—42no—
220C3×C22⋊C4—42no—
221#217—24no—
222#217—81no—
223#215—48no—
224(C2×D3)⋊C4—28no—
225(C2×D3)⋊C4—28no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230Dic3⋊C4—82no—
231C4×Dic3—22no—
232C4×Dic3—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C25—61no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C2×#41#6714yes—
241C22×Q8C22×D311yescharacteristic
242C22×Q8C22×D311yescharacteristic
243C22×D4—21no—
244C22×D4—34no—
245C22×D4—32no—
246C22×D4—62no—
247C22×D4—62no—
248C22×D4—61no—
249C22×D4—62no—
250C22×D4—122no—
251C22×D4—122no—
252C22×D4C22×D311yescharacteristic
253C23×C4—21no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—22no—
256C2×Q16—42no—
257C2×SD16—22no—
258C2×SD16—42no—
259C2×SD16—64no—
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—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—22no—
289#103—122no—
290#102—22no—
291#102—64no—
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—21no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
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×Q8—42no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—48no—
333C3×D4—42no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—48no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4#12012yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4H(5,𝔽2)12yes—
350C2×C3×C4H(5,𝔽2)12yes—
351#67—88no—
352#67—88no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362D12—24no—
363D12—24no—
364D12—48no—
365D12—42no—
366D12—42no—
367C3×C8—41no—
368C3×C8—41no—
369C3⋊C8—42no—
370C24—21no—
371C24—62no—
372C24—64no—
373C24—124no—
374C24—241no—
375C24—241no—
376C24—242no—
377C24D3×D411yescharacteristic
378#41—44no—
379#41—44no—
380#41—48no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8C2×#6712yes—
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×#6712yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—44no—
421C22×C4—62no—
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D412yes—
424C22×C4C2×#6712yes—
425C22×C4C23×D311yescharacteristic
426Q16—44no—
427SD16—44no—
428SD16—128no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—62no—
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—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#53512yes—
463C22×C3#57611yescharacteristic
464C22×C3#57711yescharacteristic
465C22×C3#57712yes—
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—81no—
477C3×C4—81no—
478C3×C4#53411yescharacteristic
479C3×C4#53411yescharacteristic
480C3×C4#53412yes—
481Dic3—42no—
482Dic3—42no—
483Dic3—82no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—42no—
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#93712yes—
506C23C2×D3×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—42no—
510Q8—41no—
511Q8—41no—
512Q8—42no—
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—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
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—41no—
551C8—41no—
552C8—122no—
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—
579C22#559212yes—
580C22#576911yescharacteristic
581C22#577011yescharacteristic
582C22#577812yes—
583C22C2×#93711yescharacteristic
584C4—22no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—122no—
589C4—122no—
590C4—242no—
591C4#558912yes—
592C4#580411yescharacteristic
593C4#580411yescharacteristic
594C3#2963211yescharacteristic, complemented, Sylow
595C2—42no—
596C2—82no—
597C2—244no—
598C2#8026212yes—
599C2#8026312yes—
600C2#8260711yescharacteristic
601C2#8391211yescharacteristic
602C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#452C2
socleC23×C3C22×D4
prime radicalC2×#5830C2
hypercenterC2×#452D3
hypocenterC3#29632
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#452

Complements

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

NH ≅ G/Ntype
2. C2×C3×#452C2semidirect
3. #82787C2semidirect
4. C2×#5046C2semidirect
5. #72135C2semidirect
6. #72072C2semidirect
7. #72056C2semidirect
8. #70279C2semidirect
9. #70211C2semidirect
10. #69959C2semidirect
11. #69911C2semidirect
15. C22×C3×Q16C22semidirect
16. C22×C3×SD16C22semidirect
28. C2×C3×#103C22semidirect
29. C2×C3×#102C22semidirect
31. #5229C22semidirect
32. C2×#835C22semidirect
33. C2×#832C22semidirect
41. #4972C22semidirect
44. #4550C22semidirect
45. #4550C22semidirect
47. C2×#452D3semidirect
124. C22×Q16C2×D3semidirect
126. C22×SD16C2×D3semidirect
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×C3×#4522
#827872
C2×#50462
#721352
#720722
#720562
#702792
#702112
#699592
#699112
#296323