Finite Group Databasebeta

← #10846008

Subgroups — #10846008

Order 768.

1206 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic, complemented
2C2×#5159C212yesmaximal, complemented
3#83188C211yesmaximal, characteristic, complemented
4#82417C211yesmaximal, characteristic, complemented
5C2×#4838C211yesmaximal, characteristic, complemented
6#72043C212yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic, complemented
9#70454C211yesmaximal, characteristic, complemented
10#70203C212yesmaximal, complemented
11#70176C212yesmaximal, complemented
12#25392—31nomaximal, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#940—21no—
17C2×#938C2212yes—
18C2×#938C2212yes—
19C2×#938C2212yes—
20C2×#930—21no—
21C22×ΓL(1,𝔽25)C2212yes—
22C2×C4×D12C2211yescharacteristic
23C3×#459—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2212yes—
26#5223—22no—
27#5221—21no—
28C2×#832C2212yes—
29C2×#832C2211yescharacteristic
30C2×#832C2212yes—
31C2×#832C2211yescharacteristic
32#5212—21no—
33#5159—28no—
34C2×(C2×D3)⋊C8C2212yes—
35C2×#817C2212yes—
36#4987C2212yes—
37#4961—22no—
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40C2×C4×C3⋊C8C2211yescharacteristic
41#4838—24no—
42#4551C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#469—32no—
46C2×#459D311yescharacteristic, complemented
47C2×#436—31no—
48#2541—31no—
49#2537—31no—
50#2034—32no—
51#1999—32no—
52#1957—31no—
53#1948—31no—
54#1895—32no—
55C4×#102—31no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×#67—22no—
62C22×#67—22no—
63C22×#67—22no—
64C22×D12D411yescharacteristic
65C22×C4×D3C2312yes—
66C2×C3×D8—24no—
67C2×C3×D8—24no—
68C2×C3×D8—42no—
69C22×C3×C8C2312yes—
70C3×#127—22no—
71C3×#127—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80#940—44no—
81#938—24no—
82#938—24no—
83#938—24no—
84#938—42no—
85#938—42no—
86#938—42no—
87#930—44no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4D412yes—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4×Dic3—21no—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C2×ΓL(1,𝔽25)—24no—
100C2×ΓL(1,𝔽25)—24no—
101C2×ΓL(1,𝔽25)—42no—
102C4×D12—22no—
103C4×D12—22no—
104C3×C4⋊C8D414yescomplemented
105C3×#102—24no—
106#832—22no—
107#832—24no—
108#832—42no—
109#832D414yescomplemented
110C3⋊#95—21no—
111C3⋊#95—22no—
112(C2×D3)⋊C8—24no—
113#817—24no—
114C4×C3⋊C8—22no—
115C22×D8C2×D312yes—
116C22×M16—32no—
117C2×#127—61no—
118C2×#127C2×D311yescharacteristic
119C2×#121—32no—
120C2×#121—32no—
121C2×#121—32no—
122C2×#121—61no—
123C2×C4×D4—31no—
124C2×C4×D4C2×D311yescharacteristic
125#469—68no—
126#459—24no—
127#436—64no—
128C2×#106—32no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—31no—
131C2×#102—32no—
132C2×#102—31no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×C22⋊C8—32no—
136C2×C4×C8—31no—
137#394—62no—
138#392—62no—
139#390—61no—
140#384—31no—
141#380—32no—
142#380—61no—
143C4×C4⋊C4—31no—
144#340—32no—
145#340—31no—
146#340—31no—
147C24×C3—21no—
148C24×C3—22no—
149C23×D3C2×D412yes—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—42no—
162C2×C3×D4—42no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—81no—
166C2×C3×D4—81no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—21no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—44no—
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—82no—
182C2×#67—82no—
183C2×#67—82no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3C2×D412yes—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C3×D8—48no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8C2×D414yes—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C22⋊C4—42no—
204C3×C42C2×D412yes—
205C3×C42#4112yes—
206#217—44no—
207#217—44no—
208#217—44no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—44no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4⋊Dic3—81no—
218C4⋊Dic3C2×D412yes—
219C4⋊Dic3#4112yes—
220C4×Dic3—42no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8C2×D412yes—
227ΓL(1,𝔽25)—48no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—31no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4C22×D312yes—
235C22×D4C22×D311yescharacteristic
236C23×C4—21no—
237C23×C4—32no—
238C2×D8—24no—
239C2×D8—24no—
240C2×D8—42no—
241C2×M16—64no—
242C2×M16—64no—
243C2×M16—122no—
244C22×C8—31no—
245C22×C8—31no—
246C22×C8C22×D312yes—
247#127—22no—
248#127—22no—
249#127—124no—
250#121—64no—
251#121—64no—
252#121—64no—
253#121—122no—
254#121—122no—
255#121—122no—
256#121—124no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—62no—
260C4×D4—62no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—32no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#106—64no—
277C4⋊C8#6714yescomplemented
278#102—24no—
279#102—34no—
280#102—62no—
281#102—64no—
282#102—122no—
283C22⋊C8—64no—
284C4×C8—62no—
285#95—62no—
286#95—61no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3C22×D411yescharacteristic
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—42no—
312C3×D4—41no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—42no—
317C3×D4—84no—
318C3×D4—84no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—84no—
325C2×C3×C4#12112yes—
326C2×C3×C4#12112yes—
327C2×C3×C4AGL(1,ℤ/8)12yes—
328C2×C3×C4AGL(1,ℤ/8)12yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335#67—88no—
336#67—88no—
337#67—88no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350D12—44no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355C3×C8—24no—
356C3⋊C8—22no—
357C3⋊C8—41no—
358C24—21no—
359C24—22no—
360C24—32no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4—124no—
386C2×D4—124no—
387C2×D4—242no—
388C2×D4—242no—
389C2×D4—242no—
390C2×D4D3×D412yes—
391C22×C4—21no—
392C22×C4—32no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4D3×D411yescharacteristic
406C22×C4D3×D411yescharacteristic
407C22×C4C23×D311yescharacteristic
408D8—48no—
409M16—128no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—32no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8C2×#6714yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—32no—
422C4⋊C4—32no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—122no—
429C4⋊C4—241no—
430C22⋊C4—42no—
431C22⋊C4—64no—
432C22⋊C4—124no—
433C22⋊C4—124no—
434C22⋊C4—124no—
435C22⋊C4—124no—
436C42—122no—
437C42#23712yes—
438C42C2×#6712yes—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3C2×#12111yescharacteristic
454C22×C3D4211yescharacteristic
455C22×C3#54611yescharacteristic
456C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
457C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
458C22×C3#57611yescharacteristic
459C22×C3#57711yescharacteristic
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—42no—
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—42no—
467C3×C4—22no—
468C3×C4—81no—
469C3×C4#46914yescomplemented
470C3×C4#53211yescharacteristic
471C3×C4#54011yescharacteristic
472C3×C4D4211yescharacteristic
473C3×C4#55011yescharacteristic
474Dic3—42no—
475Dic3—42no—
476Dic3—42no—
477Dic3—82no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—62no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23C2×D3×D411yescharacteristic
500D4—22no—
501D4—22no—
502D4—42no—
503D4—41no—
504D4—42no—
505D4—42no—
506D4—41no—
507D4—42no—
508D4—84no—
509D4—84no—
510D4—124no—
511D4—248no—
512D4—248no—
513D4—248no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—84no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4#91112yes—
537C2×C4#91112yes—
538C2×C4#93812yes—
539C2×C4#93812yes—
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×#23711yescharacteristic
542C2×C4#100411yescharacteristic
543C2×C4D3×#4111yescharacteristic
544C2×C4#100911yescharacteristic
545C2×C4C22×#6711yescharacteristic
546C8—24no—
547C8—62no—
548C8—121no—
549C2×C3—82no—
550C2×C3—162no—
551C2×C3#242811yescharacteristic
552C2×C3C2×#46911yescharacteristic
553C2×C3#306211yescharacteristic
554C2×C3#327211yescharacteristic
555C2×C3#327411yescharacteristic
556C2×C3#332511yescharacteristic
557C2×C3#333211yescharacteristic
558D3—82no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—122no—
579C22—122no—
580C22#560911yescharacteristic
581C22#561211yescharacteristic
582C22C2×#91111yescharacteristic
583C22C2×#91111yescharacteristic
584C22#577411yescharacteristic
585C22#577511yescharacteristic
586C22C2×#93811yescharacteristic
587C4—22no—
588C4—81no—
589C4—122no—
590C4—122no—
591C4—122no—
592C4—242no—
593C4#515914yescomplemented
594C4#559611yescharacteristic
595C4#561911yescharacteristic
596C4D4×#6711yescharacteristic
597C4#583411yescharacteristic
598C3#2539211yescharacteristic, complemented, Sylow
599C2—82no—
600C2—162no—
601C2—242no—
602C2#8034911yescharacteristic
603C2#8035611yescharacteristic
604C2#8035711yescharacteristic
605C2#8035911yescharacteristic
606C2#8264811yescharacteristic
607C2C2×#515911yescharacteristic
608C2#8395111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#25392
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×C3×#459C2semidirect
2. C2×#5159C2semidirect
3. #83188C2semidirect
4. #82417C2semidirect
5. C2×#4838C2semidirect
6. #72043C2semidirect
7. #71764C2semidirect
8. C4×#832C2semidirect
9. #70454C2semidirect
10. #70203C2semidirect
11. #70176C2semidirect
46. C2×#459D3semidirect
104. C3×C4⋊C8D4semidirect
109. #832D4semidirect
277. C4⋊C8#67semidirect
469. C3×C4#469semidirect
593. C4#5159semidirect
598. C3#25392semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×#51592
#831882
#824172
C2×#48382
#720432
#717642
C4×#8322
#704542
#702032
#701762
#253923