Finite Group Databasebeta

← #10830833

Subgroups — #10830833

Order 768.

1546 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85786C211yesmaximal, characteristic
2C3×C4×#123C211yesmaximal, characteristic, complemented
3C2×#5053C212yesmaximal, complemented
4#82080C212yesmaximal, complemented
5#71619C211yesmaximal, characteristic, complemented
6#69598C212yesmaximal, complemented
7#69596C212yesmaximal, complemented
8#69539C212yesmaximal, complemented
9#69456C212yesmaximal, complemented
10#24261—31nomaximal, Sylow
11C2×#1004C2212yes—
12C2×C3×#123C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic, complemented
15C2×#930—21no—
16C2×#930C2212yes—
17C2×#929C2214yescomplemented
18#5555—22no—
19#5554—24no—
20D4×D12—22no—
21#5510—21no—
22#5493—21no—
23#5493C2212yes—
24#5477—21no—
25C3×#388—21no—
26C3×#387—21no—
27C3×#386C2211yescharacteristic
28C3×#385C2211yescharacteristic, complemented
29C3×#381C2211yescharacteristic, complemented
30C3×C4×C4⋊C4—21no—
31C3×C4×C22⋊C4—21no—
32C3×C4×C22⋊C4C2211yescharacteristic, complemented
33C2×#834C2212yes—
34#5213C2214yescomplemented
35#5212C2212yescomplemented
36#5053—28no—
37C2×#806C2214yescomplemented
38C2×#806C2214yescomplemented
39#4940—24no—
40#4939—22no—
41#4928—22no—
42#4526C2212yescomplemented
43#3496—31no—
44C2×#458—32no—
45#2854—32no—
46C4×#123D311yescharacteristic, complemented
47#2044—32no—
48#2022—32no—
49#1964—32no—
50#1799—31no—
51#1790—32no—
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C22×#67—22no—
58C22×#67C2314yes—
59#1004—24no—
60#1004—24no—
61#1004—44no—
62C2×#237—24no—
63C2×D3×D4—24no—
64C2×#235—22no—
65C22×D12—21no—
66C22×D12D412yes—
67C3×#123—22no—
68C3×#123D414yescomplemented
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4D411yescharacteristic
77C2×C3×C22⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4C2311yescharacteristic
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—21no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42D411yescharacteristic
86C2×#217C2314yes—
87#930—22no—
88#930—22no—
89#930—22no—
90#930—22no—
91#930—22no—
92#930—22no—
93#930—22no—
94#930—22no—
95#929—28no—
96#929—44no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—24no—
99C2×(C2×D3)⋊C4—24no—
100C2×(C2×D3)⋊C4D414yes—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3D412yes—
104C2×Dic3⋊C4—24no—
105#897—24no—
106#895—24no—
107#886—24no—
108#886—24no—
109#882—24no—
110#882—24no—
111#875—22no—
112#874—21no—
113C4×D12—22no—
114C4×D12—22no—
115#869—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121#834—24no—
122C3⋊#95—22no—
123C3⋊#95—24no—
124#806—28no—
125#806—44no—
126C2×H(5,𝔽2)—32no—
127#547—62no—
128#546—64no—
129D42—62no—
130#537—61no—
131#534—32no—
132#534—61no—
133#532—61no—
134C2×#123—34no—
135C2×#123C2×D311yescharacteristic
136C2×#121—32no—
137C2×#121—61no—
138C2×C4×D4C2×D311yescharacteristic
139C22×C42C2×D311yescharacteristic, complemented
140#458—68no—
141C2×#99—32no—
142C2×#99—34no—
143C2×#99—34no—
144#390—62no—
145#389—62no—
146#388—21no—
147#388—64no—
148#387—21no—
149#386—34no—
150#386C2×D311yescharacteristic
151#385C2×D311yescharacteristic, complemented
152#381C2×D311yescharacteristic, complemented
153#380—32no—
154C4×C4⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4C2×D311yescharacteristic, complemented
157#342—32no—
158C24×C3C2×D411yescharacteristic
159C24×C3C2411yescharacteristic
160C23×D3—22no—
161C23×D3C2×D414yes—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4C2×D412yes—
165C2×C3×D4#4112yes—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yescomplemented
187C2×#67—24no—
188C2×#67—24no—
189C2×#67—24no—
190C2×#67—24no—
191C2×#67—24no—
192C2×#67—24no—
193C2×#67—24no—
194C2×#67—44no—
195C2×#67—44no—
196C22×Dic3—22no—
197C22×Dic3—24no—
198C22×Dic3C2×D414yes—
199#237—48no—
200#237—48no—
201D3×D4—48no—
202D3×D4—48no—
203#235—44no—
204#235—44no—
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—22no—
210C2×D12—44no—
211C2×D12—82no—
212C2×C4×D3—24no—
213C2×Dic6—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—42no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4C2×D412yescomplemented
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—42no—
227C3×C42—42no—
228C3×C42—41no—
229C3×C42—41no—
230#217—24no—
231#217—24no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—48no—
237(C2×D3)⋊C4—48no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3—44no—
243Dic3⋊C4—48no—
244H(5,𝔽2)—64no—
245H(5,𝔽2)—64no—
246H(5,𝔽2)—124no—
247C2×#41—64no—
248C2×#41—62no—
249C22×D4—34no—
250C22×D4—32no—
251C22×D4—61no—
252C22×D4—64no—
253C22×D4—62no—
254C22×D4C22×D311yescharacteristic
255C23×C4#6711yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C4⋊Q8—61no—
260#127—61no—
261#124—62no—
262#123—22no—
263#123—64no—
264#123—64no—
265#123—68no—
266#123—124no—
267#123D1214yescomplemented
268C22⋊Q8—64no—
269#121—64no—
270#121—62no—
271#121—62no—
272#121—62no—
273#121—62no—
274#121—62no—
275#121—62no—
276#121—62no—
277#121—62no—
278#120—64no—
279#120—64no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—62no—
283C4×D4—62no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4#6711yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—34no—
294C2×C22⋊C4—34no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—64no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4C22×D311yescharacteristic
300C2×C22⋊C4C22×D311yescharacteristic
301C2×C22⋊C4C22×D311yescharacteristic
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42#6711yescharacteristic
308#99—64no—
309#99—68no—
310#99—124no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—64no—
317#95—62no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3#12112yes—
325C23×C3C22×D411yescharacteristic
326C23×C3C22×D412yes—
327C23×C3C22×D411yescharacteristic
328C23×C3C2×#4111yescharacteristic
329C23×C3H(5,𝔽2)12yes—
330C22×D3—24no—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—44no—
334C22×D3—44no—
335C22×D3—44no—
336C22×D3—82no—
337C22×D3—82no—
338C3×D4—24no—
339C3×D4—24no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—81no—
368C2×C3×C4#12012yes—
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4#12011yescharacteristic
371C2×C3×C4#12112yes—
372#67—48no—
373#67—44no—
374#67—44no—
375#67—44no—
376#67—44no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—44no—
381C2×Dic3—44no—
382C2×Dic3—44no—
383C2×Dic3—44no—
384D12—44no—
385D12—44no—
386D12—82no—
387D12—82no—
388D12—82no—
389C4×D3—48no—
390Dic6—44no—
391C24—34no—
392C24—62no—
393C24C2×#6711yescharacteristic
394C24C23×D311yescharacteristic
395#41—124no—
396#41—128no—
397#41—128no—
398#41—124no—
399C2×Q8—62no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—62no—
405C2×D4—64no—
406C2×D4—64no—
407C2×D4—64no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—124no—
415C2×D4—128no—
416C2×D4—128no—
417C2×D4—124no—
418C2×D4—124no—
419C2×D4—242no—
420C2×D4C2×D1212yes—
421C2×D4#23512yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—34no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—64no—
437C22×C4—62no—
438C22×C4—64no—
439C22×C4C2×D1212yescomplemented
440C22×C4D3×D412yes—
441C22×C4D3×D412yes—
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—128no—
455C4⋊C4—124no—
456C22⋊C4—24no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—42no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—128no—
469C22⋊C4—128no—
470C22⋊C4C2×D1212yescomplemented
471C42—21no—
472C42—21no—
473C42—42no—
474C42—41no—
475C42—42no—
476C42—41no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—22no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—81no—
489C22×C3C2×#12011yescharacteristic
490C22×C3C2×#12111yescharacteristic
491C22×C3#53511yescharacteristic
492C22×C3D4212yes—
493C22×C3#54612yes—
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C2×D3—84no—
498C2×D3—84no—
499C2×D3—164no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—81no—
509C3×C4#45812yescomplemented
510Dic3—44no—
511Dic3—84no—
512C23—21no—
513C23—21no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—64no—
519C23—64no—
520C23—64no—
521C23—124no—
522C23—124no—
523C23—124no—
524C23—242no—
525C23—242no—
526C23#93012yes—
527C23C22×D1211yescharacteristic
528C23C2×#23511yescharacteristic
529C23C2×D3×D412yes—
530C23#100912yes—
531C23C22×#6711yescharacteristic
532Q8—124no—
533D4—24no—
534D4—24no—
535D4—124no—
536D4—128no—
537D4—124no—
538D4—124no—
539D4—124no—
540D4—124no—
541D4—124no—
542D4—242no—
543D4—242no—
544D4—242no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—81no—
576C2×C4—128no—
577C2×C4—124no—
578C2×C4—124no—
579C2×C4—124no—
580C2×C4—124no—
581C2×C4#93012yes—
582C2×C4#93711yescharacteristic
583C2×C4#93712yes—
584C2×C4#93711yescharacteristic
585C2×C3—22no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—22no—
589C2×C3—42no—
590C2×C3#240811yescharacteristic
591C2×C3C2×#45811yescharacteristic
592C2×C3#302511yescharacteristic
593D3—84no—
594C22—21no—
595C22—22no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—81no—
606C22—124no—
607C22—124no—
608C22—124no—
609C22—244no—
610C22—244no—
611C22—484no—
612C22D4×D1212yes—
613C22#555012yes—
614C22C2×#93011yescharacteristic
615C22C2×#93711yescharacteristic
616C22#583011yescharacteristic
617C4—21no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—42no—
624C4—41no—
625C4—81no—
626C4—124no—
627C4—244no—
628C4#505312yescomplemented
629C3#2426111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—22no—
633C2—22no—
634C2—42no—
635C2—244no—
636C2#8240311yescharacteristic
637C2C2×#505311yescharacteristic
638C2#8293611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×C4×#123C2
socleC22×C3C2×#120
prime radical#85786C2
hypercenterC4×#123D3
hypocenterC3#24261
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
2. C3×C4×#123C2semidirect
3. C2×#5053C2semidirect
4. #82080C2semidirect
5. #71619C2semidirect
6. #69598C2semidirect
7. #69596C2semidirect
8. #69539C2semidirect
9. #69456C2semidirect
14. C22×C3×C42C22semidirect
17. C2×#929C22semidirect
28. C3×#385C22semidirect
29. C3×#381C22semidirect
32. C3×C4×C22⋊C4C22semidirect
34. #5213C22semidirect
35. #5212C22semidirect
37. C2×#806C22semidirect
38. C2×#806C22semidirect
42. #4526C22semidirect
46. C4×#123D3semidirect
68. C3×#123D4semidirect
139. C22×C42C2×D3semidirect
151. #385C2×D3semidirect
152. #381C2×D3semidirect
156. C4×C22⋊C4C2×D3semidirect
186. C22×C3×C4C2×D4semidirect
223. C3×C22⋊C4C2×D4semidirect
267. #123D12semidirect
439. C22×C4C2×D12semidirect
470. C22⋊C4C2×D12semidirect
509. C3×C4#458semidirect
628. C4#5053semidirect
629. C3#24261semidirect

Maximal subgroups

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

subgroupindex
#857862
C3×C4×#1232
C2×#50532
#820802
#716192
#695982
#695962
#695392
#694562
#242613