Finite Group Databasebeta

← #10848738

Subgroups — #10848738

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic, complemented
2#82077C211yesmaximal, characteristic, complemented
3C2×#4712C211yesmaximal, characteristic, complemented
4C2×#4714C211yesmaximal, characteristic, complemented
5C3×#1749C211yesmaximal, characteristic, complemented
6#69962C212yesmaximal, complemented
7#69961C212yesmaximal, complemented
8#69828C211yesmaximal, characteristic, complemented
9#69823C211yesmaximal, characteristic, complemented
10#68673C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#25602—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2212yes—
17C2×#930C2211yescharacteristic
18C2×#924C2211yescharacteristic
19C22×D24C2212yes—
20C22×#204C2212yes—
21#5477—24no—
22#5477—24no—
23C2×#875—21no—
24C2×#875C2211yescharacteristic
25C2×#874C2211yescharacteristic
26C2×C4×D12—21no—
27C2×C4×D12C2212yes—
28C2×#869C2211yescharacteristic
29C2×C3×C22⋊C8C2212yes—
30C2×C3×#97C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32#5213—21no—
33#5213—21no—
34C2×#821C2212yes—
35C2×#821C2212yes—
36C2×#821C2212yes—
37C2×#816C2212yes—
38#4940—21no—
39#4940—21no—
40#4939C2211yescharacteristic
41#4936C2211yescharacteristic
42#4929C2211yescharacteristic
43#4714—24no—
44#4712—24no—
45C3×#336C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#532—31no—
49C2×#496—31no—
50C2×#489—31no—
51#2850—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1961—31no—
56#1879—32no—
57#1877—32no—
58#1749D311yescharacteristic, complemented
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—22no—
63C22×#67—21no—
64C2×#235—28no—
65C2×#235—44no—
66C2×#235—44no—
67C22×D12—21no—
68C22×D12—21no—
69C22×D12C2311yescharacteristic
70C22×D12C2312yes—
71C22×C4×D3—22no—
72C22×Dic6—21no—
73C22×Dic6C2311yescharacteristic
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—21no—
83#930—28no—
84#930—24no—
85#930—24no—
86#930—42no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2312yes—
98#924—42no—
99C2×Dic3⋊C4—21no—
100C2×D24—24no—
101C2×D24—24no—
102C2×D24—42no—
103C2×#204—24no—
104C2×#204—24no—
105C2×#204—42no—
106#875—28no—
107#875—22no—
108#875—22no—
109#874—22no—
110#874D414yescomplemented
111C4×D12—28no—
112C4×D12—24no—
113C4×D12—24no—
114#869—22no—
115#869D414yescomplemented
116C3×C22⋊C8—24no—
117C3×#97—22no—
118C3×#97—22no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123#821—24no—
124#821—42no—
125#821—42no—
126#816—24no—
127C22×#41—61no—
128C22×SD16—32no—
129C22×D8—32no—
130#532—64no—
131#532—64no—
132C2×C4⋊Q8—31no—
133C2×#127—31no—
134C2×#124—31no—
135C2×#124—61no—
136C2×C22⋊Q8—31no—
137C2×#121—32no—
138C2×#121—31no—
139C2×#121—61no—
140C2×C4×D4—32no—
141C2×C4×D4—61no—
142C22×C42C2×D311yescharacteristic
143#496—64no—
144#489—64no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8C2×D312yes—
150C2×#97C2×D311yescharacteristic
151C2×#97C2×D311yescharacteristic
152#391—31no—
153#390—31no—
154#388—61no—
155#388—61no—
156#386—61no—
157#386—61no—
158#382—31no—
159#340—32no—
160#340—32no—
161#336C2×D312yes—
162C24×C3C2×D411yescharacteristic
163C23×D3—22no—
164C23×D3—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
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—41no—
174C22×C3×C4—41no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2411yescharacteristic
181C2×#67—28no—
182C2×#67—44no—
183C2×#67—44no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67—42no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191#235—432no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—24no—
197C2×D12—24no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—42no—
204C2×D12—42no—
205C2×D12—42no—
206C2×D12—42no—
207C2×D12C2×D414yes—
208C2×C4×D3—28no—
209C2×C4×D3—44no—
210C2×C4×D3—44no—
211C2×Dic6—24no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8#4114yescomplemented
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42C2×D412yes—
229C3×C42C2×D412yes—
230#217—42no—
231(C2×D3)⋊C4—28no—
232(C2×D3)⋊C4—28no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238C4⋊Dic3—24no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3C2×D414yes—
243Dic3⋊C4—42no—
244D24—48no—
245#204—48no—
246C2×#41—68no—
247C2×#41—124no—
248C2×#41—124no—
249C22×Q8—31no—
250C22×Q8—61no—
251C22×D4—32no—
252C22×D4—31no—
253C22×D4—62no—
254C22×D4—61no—
255C22×D4—61no—
256C22×D4—61no—
257C23×C4—62no—
258C23×C4D1211yescharacteristic
259C23×C4D1211yescharacteristic
260C23×C4C22×D311yescharacteristic
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—122no—
264C2×D8—64no—
265C2×D8—64no—
266C2×D8—122no—
267C22×C8C22×D312yes—
268C22×C8C22×D312yes—
269C4⋊Q8—34no—
270C4⋊Q8—62no—
271#127—34no—
272#127—62no—
273#124—68no—
274#124—62no—
275#124—62no—
276C22⋊Q8—122no—
277#121—64no—
278#121—64no—
279#121—68no—
280#121—122no—
281C4×D4—64no—
282C4×D4—64no—
283C4×D4—68no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42D1211yescharacteristic
300C2×C42D1211yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#103—64no—
304#102—64no—
305#102—122no—
306#102—122no—
307C22⋊C8—24no—
308#97—22no—
309#97—22no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3AGL(1,ℤ/8)12yes—
321C23×C3C22×D411yescharacteristic
322C22×D3—24no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C2×C3×C4—28no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—44no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4#12012yes—
353C2×C3×C4#12012yes—
354C2×C3×C4AGL(1,ℤ/8)12yes—
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4112yes—
358C2×C3×C4H(5,𝔽2)12yes—
359#67—416no—
360#67—84no—
361C2×Dic3—24no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—24no—
374D12—24no—
375D12—48no—
376D12—48no—
377D12—42no—
378D12—42no—
379D12—42no—
380D12—42no—
381C4×D3—416no—
382Dic6—48no—
383Dic6—42no—
384Dic6—42no—
385C3×C8—24no—
386C3×C8—42no—
387C24—62no—
388C24—61no—
389C24C2×D1211yescharacteristic
390#41—1232no—
391C2×Q8—64no—
392C2×Q8—62no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—62no—
396C2×Q8—122no—
397C2×Q8—122no—
398C2×D4—34no—
399C2×D4—68no—
400C2×D4—64no—
401C2×D4—64no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—122no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—122no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—122no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—68no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—124no—
436C22×C4—124no—
437C22×C4C2×D1211yescharacteristic
438C22×C4C2×D1211yescharacteristic
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1211yescharacteristic
441C22×C4C2×D1211yescharacteristic
442C22×C4C23×D311yescharacteristic
443SD16—128no—
444D8—128no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8#23514yescomplemented
451C4⋊C4—34no—
452C4⋊C4—64no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—62no—
456C4⋊C4—122no—
457C22⋊C4—68no—
458C22⋊C4—68no—
459C22⋊C4—122no—
460C22⋊C4—122no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42D3×D412yes—
470C42D3×D412yes—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3C2×#12011yescharacteristic
479C22×C3#53211yescharacteristic
480C22×C3#53511yescharacteristic
481C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
483C22×C3#57511yescharacteristic
484C22×C3#57511yescharacteristic
485C2×D3—44no—
486C2×D3—44no—
487C2×D3—44no—
488C2×D3—82no—
489C2×D3—82no—
490C2×D3—82no—
491C2×D3—82no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4#54612yes—
505C3×C4#55012yes—
506Dic3—44no—
507Dic3—81no—
508Dic3—81no—
509Dic3—81no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—64no—
517C23—122no—
518C23—122no—
519C23—122no—
520C23—122no—
521C23—122no—
522C23—122no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23#90812yes—
531C23C22×D1211yescharacteristic
532Q8—128no—
533Q8—122no—
534Q8—122no—
535D4—64no—
536D4—64no—
537D4—128no—
538D4—128no—
539D4—1216no—
540D4—122no—
541D4—122no—
542D4—122no—
543D4—122no—
544D4—244no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—64no—
562C2×C4—1216no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4#88212yes—
575C2×C4#88212yes—
576C2×C4#90812yes—
577C2×C4C2×#23512yes—
578C2×C4C2×D3×D411yescharacteristic
579C2×C4C2×D3×D411yescharacteristic
580C2×C4#100912yes—
581C8—24no—
582C8—42no—
583C2×C3—42no—
584C2×C3#243111yescharacteristic
585C2×C3#300311yescharacteristic
586C2×C3#300511yescharacteristic
587C2×C3#317111yescharacteristic
588C2×C3#318011yescharacteristic
589C2×C3#321311yescharacteristic
590C2×C3#321411yescharacteristic
591D3—84no—
592D3—161no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—124no—
601C22—124no—
602C22—124no—
603C22—242no—
604C22—242no—
605C22—242no—
606C22—242no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#547711yescharacteristic
615C22C2×#88211yescharacteristic
616C22#552511yescharacteristic
617C22C2×#90811yescharacteristic
618C22C2×#90811yescharacteristic
619C22#574811yescharacteristic
620C22#574811yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—124no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#555012yes—
631C4#555312yes—
632C3#2560211yescharacteristic, complemented, Sylow
633C2—42no—
634C2—244no—
635C2—481no—
636C2#7358611yescharacteristic
637C2#7850011yescharacteristic
638C2#7853211yescharacteristic
639C2#7863711yescharacteristic
640C2#7863811yescharacteristic
641C2#7934911yescharacteristic
642C2#7935011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
hypercenter#1749D3
hypocenterC3#25602
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. #82077C2semidirect
3. C2×#4712C2semidirect
4. C2×#4714C2semidirect
5. C3×#1749C2semidirect
6. #69962C2semidirect
7. #69961C2semidirect
8. #69828C2semidirect
9. #69823C2semidirect
10. #68673C2semidirect
11. #68672C2semidirect
58. #1749D3semidirect
110. #874D4semidirect
115. #869D4semidirect
223. C2×C3×C8#41semidirect
450. C2×C8#235semidirect
632. C3#25602semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
#820772
C2×#47122
C2×#47142
C3×#17492
#699622
#699612
#698282
#698232
#686732
#686722
#256023