Finite Group Databasebeta

← #10823412

Subgroups — #10823412

Order 768.

1508 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic, complemented
2C3×C4×#121C211yesmaximal, characteristic, complemented
3C2×#5014C212yesmaximal, complemented
4#71684C212yesmaximal, complemented
5#71529C211yesmaximal, characteristic, complemented
6#69658C211yesmaximal, characteristic, complemented
7#69638C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69605C211yesmaximal, characteristic, complemented
10#69524C212yesmaximal, complemented
11#69474C212yesmaximal, complemented
12#23573—31nomaximal, Sylow
13C22×#235—21no—
14C2×C3×#121C2211yescharacteristic
15C2×C3×C4×D4—21no—
16C2×C3×C4×D4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#213C2212yes—
19C2×#930—21no—
20C2×#930C2212yes—
21#5477—28no—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×D12C2212yes—
26C2×#869C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#388—21no—
30C3×#386—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C2×#832C2212yes—
35C2×#830C2212yes—
36#5014—28no—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2212yes—
41C2×#803C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#442—32no—
48C4×#121D311yescharacteristic, complemented
49#2058—31no—
50#2041—31no—
51#2034—32no—
52#2023—31no—
53#2011—31no—
54#1999—32no—
55#1762—32no—
56#1749—31no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—22no—
64C2×#235—28no—
65C2×#235—44no—
66C2×#235—44no—
67C22×D12—21no—
68C22×D12—21no—
69C22×D12C2312yes—
70C22×C4×D3—22no—
71C22×Dic6—21no—
72C3×#121—22no—
73C3×#121—22no—
74C3×C4×D4—24no—
75C3×C4×D4—44no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—24no—
86C2×C3×C42—21no—
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#213—24no—
92C2×#213—24no—
93C2×#213—24no—
94#930—28no—
95#930—24no—
96#930—24no—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2312yes—
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2312yes—
103#875—28no—
104#874—24no—
105C4×D12—28no—
106C4×D12—24no—
107C4×D12D418yescomplemented
108#869—24no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112#832—24no—
113#830—24no—
114#809—24no—
115#809—24no—
116#809—24no—
117#808—24no—
118#803—24no—
119C22×#41—61no—
120C22×D8—32no—
121#532—68no—
122C2×C4⋊Q8—31no—
123C2×#127—31no—
124C2×#124—61no—
125C2×#121—32no—
126C2×#121—61no—
127C2×#121C2×D311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C2×C4×D4C2×D311yescharacteristic
132C22×C42C2×D311yescharacteristic
133#442—68no—
134C2×#106—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×#97—32no—
141#390C2×D311yescharacteristic
142#389C2×D311yescharacteristic
143#388—21no—
144#386—21no—
145#380C2×D311yescharacteristic
146C4×C4⋊C4C2×D311yescharacteristic
147C4×C22⋊C4—21no—
148#340—32no—
149#340—32no—
150#336—31no—
151#336—31no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—22no—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4#4112yescomplemented
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—41no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C2×#67—28no—
185C2×#67—44no—
186C2×#67—44no—
187C22×Dic3—22no—
188#235—432no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—24no—
194C2×D12—24no—
195C2×D12—22no—
196C2×D12—42no—
197C2×D12—42no—
198C2×D12—42no—
199C2×D12—42no—
200C2×D12C2×D414yes—
201C2×C4×D3—28no—
202C2×C4×D3—44no—
203C2×C4×D3—44no—
204C2×Dic6—24no—
205C2×Dic6—42no—
206C2×Dic6—42no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4#4112yescomplemented
212C3×C22⋊C4—24no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—42no—
217C3×C42—22no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42—42no—
221C3×C42C2×D414yes—
222#213—216no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—28no—
225C4⋊Dic3—24no—
226C4⋊Dic3—22no—
227C4⋊Dic3C2×D414yes—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8#4114yescomplemented
234C2×#41—68no—
235C2×#41—124no—
236C2×#41—124no—
237C22×Q8—61no—
238C22×D4—21no—
239C22×D4—32no—
240C22×D4—62no—
241C22×D4—61no—
242C22×D4—61no—
243C22×D4C22×D311yescharacteristic
244C23×C4—21no—
245C23×C4—62no—
246C23×C4#6711yescharacteristic
247C23×C4#6711yescharacteristic
248C23×C4C22×D311yescharacteristic
249C2×D8—64no—
250C2×D8—64no—
251C2×D8—64no—
252C22×C8—32no—
253C22×C8—32no—
254C4⋊Q8—64no—
255#127—64no—
256#124—68no—
257#121—22no—
258#121—22no—
259#121—64no—
260#121—64no—
261#121—68no—
262C4×D4—24no—
263C4×D4—38no—
264C4×D4—44no—
265C4×D4—68no—
266C4×D4—64no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42#6711yescharacteristic
283C2×C42#6711yescharacteristic
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D311yescharacteristic
286#106—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290#102—64no—
291C22⋊C8—64no—
292#97—64no—
293#95—21no—
294#95—21no—
295#95—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3AGL(1,ℤ/8)12yes—
310C23×C3C22×D411yescharacteristic
311C22×D3—24no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C3×D4—24no—
319C3×D4—48no—
320C3×D4—42no—
321C2×C3×C4—28no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—44no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—84no—
346C2×C3×C4#12012yes—
347C2×C3×C4#12012yes—
348C2×C3×C4AGL(1,ℤ/8)12yes—
349C2×C3×C4AGL(1,ℤ/8)12yes—
350C2×C3×C4C22×D412yes—
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4112yes—
354#67—416no—
355C2×Dic3—24no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358D12—24no—
359D12—24no—
360D12—48no—
361D12—48no—
362D12—42no—
363D12—42no—
364C4×D3—416no—
365Dic6—48no—
366C3⋊C8—24no—
367C3⋊C8—42no—
368C24—21no—
369C24—62no—
370C24C2×#6711yescharacteristic
371#41—1232no—
372C2×Q8—64no—
373C2×Q8—122no—
374C2×Q8—122no—
375C2×D4—24no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—34no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—64no—
383C2×D4—64no—
384C2×D4—68no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—124no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4#23512yescomplemented
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—62no—
414C22×C4—68no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C23×D311yescharacteristic
423D8—616no—
424C2×C8—34no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—62no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—34no—
434C4⋊C4—42no—
435C4⋊C4—64no—
436C4⋊C4—62no—
437C4⋊C4#23512yescomplemented
438C22⋊C4—24no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—68no—
444C22⋊C4—68no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42D3×D414yes—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#52511yescharacteristic
466C22×C3#53211yescharacteristic
467C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
468C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
469C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
470C22×C3#57511yescharacteristic
471C2×D3—44no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—82no—
475C2×D3—82no—
476C2×D3—82no—
477C2×D3—82no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#52512yes—
487C3×C4#53212yes—
488Dic3—44no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—64no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23#94912yes—
510C23C22×#6711yescharacteristic
511Q8—128no—
512D4—24no—
513D4—48no—
514D4—42no—
515D4—64no—
516D4—64no—
517D4—128no—
518D4—128no—
519D4—1216no—
520D4—122no—
521D4—122no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—64no—
547C2×C4—84no—
548C2×C4—1216no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4#93212yes—
552C2×C4#93712yes—
553C2×C4#93712yes—
554C2×C4#94912yes—
555C2×C4C2×#23511yescharacteristic
556C2×C4C2×#23511yescharacteristic
557C2×C4C2×D3×D412yes—
558C2×C4D3×#4112yes—
559C8—64no—
560C8—122no—
561C2×C3—42no—
562C2×C3—82no—
563C2×C3#238811yescharacteristic
564C2×C3#300311yescharacteristic
565C2×C3#300611yescharacteristic
566C2×C3#310911yescharacteristic
567C2×C3#311511yescharacteristic
568C2×C3#318011yescharacteristic
569C2×C3#318411yescharacteristic
570D3—84no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—124no—
586C22—124no—
587C22—124no—
588C22—242no—
589C22—242no—
590C22—242no—
591C22—242no—
592C22#579111yescharacteristic
593C22C2×#93211yescharacteristic
594C22C2×#93711yescharacteristic
595C22C2×#94911yescharacteristic
596C22C2×#94911yescharacteristic
597C22#582211yescharacteristic
598C22#583211yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—124no—
608C4#554712yes—
609C4#554812yes—
610C3#2357311yescharacteristic, complemented, Sylow
611C2—42no—
612C2—82no—
613C2—244no—
614C2#8231011yescharacteristic
615C2#8240011yescharacteristic
616C2#8253111yescharacteristic
617C2#8262411yescharacteristic
618C2#8292711yescharacteristic
619C2#8313711yescharacteristic
620C2#8329511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#121C2
socleC23×C3C22×D4
hypercenterC4×#121D3
hypocenterC3#23573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C3×C4×#121C2semidirect
3. C2×#5014C2semidirect
4. #71684C2semidirect
5. #71529C2semidirect
6. #69658C2semidirect
7. #69638C2semidirect
8. #69627C2semidirect
9. #69605C2semidirect
10. #69524C2semidirect
11. #69474C2semidirect
48. C4×#121D3semidirect
107. C4×D12D4semidirect
161. C2×C3×D4#41semidirect
211. C3×C4⋊C4#41semidirect
233. C2×C3⋊C8#41semidirect
396. C2×D4#235semidirect
437. C4⋊C4#235semidirect
610. C3#23573semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C3×C4×#1212
C2×#50142
#716842
#715292
#696582
#696382
#696272
#696052
#695242
#694742
#235733