Finite Group Databasebeta

← #10828753

Subgroups — #10828753

Order 768.

1354 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5555C211yesmaximal, characteristic
2C3×#2486C211yesmaximal, characteristic, complemented
3C2×#4967C211yesmaximal, characteristic, complemented
4C22×#807C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic, complemented
6#69625C212yesmaximal, complemented
7#69604C212yesmaximal, complemented
8#69523C212yesmaximal, complemented
9#69515C211yesmaximal, characteristic, complemented
10#69504C212yesmaximal, complemented
11#69501C211yesmaximal, characteristic, complemented
12#24088—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#123—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#930C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22#5555—24no—
23#5555—24no—
24C2×#897—21no—
25C2×#886—21no—
26C2×#886—21no—
27C2×C4×D12C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C3×#398—21no—
30C3×#397—21no—
31C3×#389C2212yes—
32C3×#381—21no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C2×#830C2211yescharacteristic
36C2×#830C2212yes—
37#4967—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807—24no—
41C2×#807C2212yes—
42C2×#807C2212yes—
43C2×#807C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#547—31no—
49C2×#480—31no—
50C22×#107—31no—
51#2486D311yescharacteristic, complemented
52#2057—32no—
53#2033—32no—
54#1893—31no—
55#1869—31no—
56#1862—32no—
57#1852—32no—
58#1823—31no—
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C22×#67—21no—
65C23×Dic3—21no—
66C23×Dic3—21no—
67C2×#237—28no—
68C2×#237—44no—
69C2×#237—44no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×Dic6—21no—
73C3×#123—44no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
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—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3D414yes—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C22×C3⋊C8—22no—
106C22×C3⋊C8—24no—
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2312yes—
109#897—28no—
110#886—28no—
111#886—28no—
112C4×D12—22no—
113C4×D12—22no—
114#869—22no—
115#869—22no—
116C3×#95—22no—
117C3×#95—21no—
118C3×#95—22no—
119#830—22no—
120#830—24no—
121#809—24no—
122#809—24no—
123#807—24no—
124#807—24no—
125#807—24no—
126C22×#41—61no—
127C23×C8—31no—
128#547—64no—
129#547—64no—
130C2×C4⋊Q8—31no—
131C2×#123—21no—
132C2×#123—61no—
133C2×#123—61no—
134C2×C22⋊Q8—61no—
135C2×#121—31no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×C4×D4C2×D311yescharacteristic
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4C2×D311yescharacteristic
142#480—64no—
143C2×#107—32no—
144C2×#107—32no—
145C2×#107—32no—
146C2×#107—64no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8—31no—
150C2×C22⋊C8—32no—
151#398—21no—
152#397—21no—
153#389C2×D312yes—
154#381—21no—
155#380C2×D311yescharacteristic
156#380C2×D312yes—
157#340—32no—
158#340—32no—
159#340—32no—
160#336—32no—
161C24×C3C2×D411yescharacteristic
162C24×C3C2×D411yescharacteristic
163C23×D3—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4D814yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—24no—
187C2×#67—42no—
188C2×#67—42no—
189C2×#67—42no—
190C2×#67—42no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—42no—
197C22×Dic3—42no—
198C22×Dic3—42no—
199C22×Dic3—42no—
200#237—432no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—21no—
204C2×D12C2×D412yes—
205C2×C4×D3—24no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×Dic6—24no—
209C2×Dic6—42no—
210C2×Dic6—42no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—44no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—44no—
222C3×C42—21no—
223C3×C42—21no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227C4⋊Dic3—24no—
228C4⋊Dic3—24no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3C2×D412yes—
234C4⋊Dic3C2×D412yes—
235C4⋊Dic3C2×D412yes—
236C2×C3⋊C8—28no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8#4114yescomplemented
243C2×#41—68no—
244C2×#41—124no—
245C2×#41—124no—
246C22×Q8—61no—
247C22×D4—31no—
248C22×D4—61no—
249C22×D4—61no—
250C22×D4#6711yescharacteristic
251C23×C4—21no—
252C23×C4—61no—
253C23×C4—61no—
254C23×C4—61no—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C22×C8—32no—
258C22×C8—32no—
259C22×C8—64no—
260C22×C8—62no—
261C4⋊Q8—62no—
262C4⋊Q8—62no—
263#123—44no—
264#123—68no—
265#123—68no—
266C22⋊Q8—68no—
267#121—34no—
268#121—62no—
269#121—62no—
270#121—62no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—62no—
274C4×D4—62no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4C22×D312yes—
291C2×C4⋊C4C22×D312yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4#6711yescharacteristic
298C2×C22⋊C4#6711yescharacteristic
299C2×C42C22×D311yescharacteristic
300#107—64no—
301#107—64no—
302#107—64no—
303#102—64no—
304#102—64no—
305C22⋊C8—64no—
306C22⋊C8—62no—
307#95—22no—
308#95—21no—
309#95—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—21no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3#12012yes—
317C23×C3C2×D812yes—
318C23×C3C2×D812yes—
319C23×C3AGL(1,ℤ/8)12yes—
320C23×C3C22×D411yescharacteristic
321C22×D3—22no—
322C22×D3—42no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—42no—
326C3×D4—28no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—84no—
347C2×C3×C4#12012yes—
348C2×C3×C4C2×D812yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4112yes—
352C2×C3×C42−1+412yes—
353#67—48no—
354#67—48no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—48no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—48no—
366D12—24no—
367D12—41no—
368D12—41no—
369C4×D3—48no—
370Dic6—48no—
371C3⋊C8—24no—
372C3⋊C8—42no—
373C24—61no—
374C24C2×#6711yescharacteristic
375C24C2×#6711yescharacteristic
376#41—1232no—
377C2×Q8—64no—
378C2×Q8—122no—
379C2×Q8—122no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—32no—
383C2×D4—64no—
384C2×D4—64no—
385C2×D4—61no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—122no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4#21314yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4D3×D412yes—
417C22×C4D3×D412yes—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C23×D311yescharacteristic
423C2×C8—34no—
424C2×C8—68no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—62no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—32no—
436C4⋊C4—32no—
437C4⋊C4—32no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—64no—
441C4⋊C4—64no—
442C4⋊C4—64no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—64no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C42—21no—
454C42—21no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—24no—
460C22×C3—22no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3#44714yes—
465C22×C3C2×#12011yescharacteristic
466C22×C3#54712yes—
467C22×C3C22×D811yescharacteristic
468C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
469C22×C3#57812yes—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—82no—
474C2×D3—82no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—82no—
480C3×C4—82no—
481C3×C4#52511yescharacteristic
482C3×C4#53611yescharacteristic
483C3×C4#54712yes—
484Dic3—42no—
485Dic3—42no—
486Dic3—42no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—62no—
494C23—121no—
495C23—121no—
496C23—122no—
497C23—122no—
498C23C2×#21312yes—
499C23C2×#21312yes—
500C23#93712yes—
501C23#94912yes—
502C23C22×#6711yescharacteristic
503Q8—128no—
504D4—28no—
505D4—64no—
506D4—128no—
507D4—128no—
508D4—121no—
509D4—121no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—84no—
533C2×C4—128no—
534C2×C4—128no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—128no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4C2×#21312yes—
543C2×C4#93712yes—
544C2×C4C2×D3×D411yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4D3×#4112yes—
547C2×C4#101012yes—
548C8—64no—
549C8—122no—
550C2×C3—24no—
551C2×C3—42no—
552C2×C3#240611yescharacteristic
553C2×C3C2×#44711yescharacteristic
554C2×C3#301111yescharacteristic
555C2×C3#333412yes—
556C2×C3#334312yes—
557D3—82no—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—22no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—242no—
571C22—242no—
572C22#503614yes—
573C22C22×#21311yescharacteristic
574C22C2×#93711yescharacteristic
575C22C2×#94911yescharacteristic
576C22#582412yes—
577C22#583812yes—
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—82no—
583C4—82no—
584C4—122no—
585C4—122no—
586C4—122no—
587C4#564811yescharacteristic
588C4#565512yes—
589C4#565611yescharacteristic
590C3#2408811yescharacteristic, complemented, Sylow
591C2—24no—
592C2—42no—
593C2—242no—
594C2C2×#503611yescharacteristic
595C2#8284111yescharacteristic
596C2#8341111yescharacteristic
597C2#8351612yes—
598C2#8359212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2486C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#2486D3
hypocenterC3#24088
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C3×#2486C2semidirect
3. C2×#4967C2semidirect
4. C22×#807C2semidirect
5. #71610C2semidirect
6. #69625C2semidirect
7. #69604C2semidirect
8. #69523C2semidirect
9. #69515C2semidirect
10. #69504C2semidirect
11. #69501C2semidirect
51. #2486D3semidirect
242. C2×C3⋊C8#41semidirect
590. C3#24088semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C3×#24862
C2×#49672
C22×#8072
#716102
#696252
#696042
#695232
#695152
#695042
#695012
#240883