Finite Group Databasebeta

← #10827614

Subgroups — #10827614

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5529C211yesmaximal, characteristic, complemented
2C2×#5001C211yesmaximal, characteristic, complemented
3C2×#4731C211yesmaximal, characteristic, complemented
4#74016C211yesmaximal, characteristic
5C3×#1825C211yesmaximal, characteristic, complemented
6#69953C211yesmaximal, characteristic, complemented
7#69516C211yesmaximal, characteristic, complemented
8#68909C212yesmaximal, complemented
9#68899C212yesmaximal, complemented
10#68887C212yesmaximal, complemented
11#68880C212yesmaximal, complemented
12#23982—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C4×#67C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×#208C2211yescharacteristic
18C22×#204C2212yes—
19#5529—24no—
20#5529—24no—
21C2×#897C2211yescharacteristic, complemented
22C2×#895C2212yes—
23C2×D3×C4⋊C4C2211yescharacteristic, complemented
24C2×#890C2211yescharacteristic, complemented
25C2×#887C2211yescharacteristic, complemented
26C2×#883—21no—
27C2×#882—21no—
28C2×#879—21no—
29C2×#872—21no—
30C2×#870—21no—
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#380C2211yescharacteristic
33C2×#821C2212yes—
34C2×(C2×D3)⋊C8C2212yescomplemented
35#5001—24no—
36#4988C2212yes—
37C2×#810C2212yescomplemented
38C2×#809C2212yes—
39C2×#809C2212yes—
40#4955—21no—
41#4731—24no—
42#4672—21no—
43#4671—21no—
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46#4550C2212yescomplemented
47#4529C2212yescomplemented
48C2×#552—31no—
49C2×#450—31no—
50C2×#418—31no—
51#2408—31no—
52#2025—32no—
53#2000—32no—
54#1894—31no—
55#1880—32no—
56#1874—32no—
57#1870—31no—
58#1825D311yescharacteristic, complemented
59C24×D3—21no—
60C23×C3×C4C2311yescharacteristic
61C22×#67D411yescharacteristic
62C22×D12—22no—
63C22×D12—24no—
64C22×D12—22no—
65C22×D12—21no—
66C22×D12—22no—
67C22×D12—42no—
68C22×D12C2312yes—
69C22×C4×D3—21no—
70C22×C4×D3C2311yescharacteristic
71C22×Dic6C2311yescharacteristic
72C22×C3×C8C2312yes—
73C3×#117—22no—
74C3×#117D414yescomplemented
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—21no—
81C2×#217—21no—
82C2×#217D411yescharacteristic
83C4×#67—22no—
84C4×#67—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4D411yescharacteristic
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93#924—22no—
94#924D414yescomplemented
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2311yescharacteristic
102C2×#208—22no—
103C2×#208—42no—
104C22×C3⋊C8C2312yes—
105C2×#204—24no—
106C2×#204—24no—
107C2×#204—42no—
108#897—22no—
109#897—22no—
110#895—24no—
111#895—42no—
112D3×C4⋊C4—22no—
113D3×C4⋊C4—22no—
114#890—22no—
115#890—22no—
116#887—22no—
117#887—22no—
118#883—28no—
119#882—44no—
120#879—28no—
121#872—28no—
122#870—28no—
123C3×C22⋊C8—22no—
124C3×#95—21no—
125#821—24no—
126(C2×D3)⋊C8—42no—
127#810—42no—
128#809—24no—
129#809—24no—
130C23×D4—31no—
131C22×SD16—32no—
132C22×M16—31no—
133#552—64no—
134#552—64no—
135C2×#126—61no—
136C2×#125—31no—
137C2×#125—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—31no—
141C2×#121—32no—
142C2×#120—61no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—61no—
146C2×#117C2×D311yescharacteristic
147C22×C4⋊C4—31no—
148#450—64no—
149#418—64no—
150C2×#103—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×C22⋊C8—32no—
155C2×C22⋊C8C2×D311yescharacteristic
156#394—61no—
157#392—61no—
158#390—32no—
159#380C2×D311yescharacteristic
160#379—61no—
161#340—32no—
162#340C2×D312yescomplemented
163#340C2×D312yescomplemented
164#336—32no—
165C24×C3C2×D411yescharacteristic
166C23×D3—22no—
167C23×D3—44no—
168C23×D3—42no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—82no—
172C23×D3C2×D411yescharacteristic
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—41no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3C2×D411yescharacteristic
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C2×D12—28no—
191C2×D12—22no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×D12—48no—
197C2×D12—44no—
198C2×D12—44no—
199C2×D12—42no—
200C2×D12—44no—
201C2×D12—44no—
202C2×D12—82no—
203C2×D12#4114yes—
204C2×C4×D3—24no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—21no—
214C2×Dic6C2×D412yes—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D412yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—42no—
226C3×C42—24no—
227#217—22no—
228#217—24no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—44no—
233(C2×D3)⋊C4—82no—
234C4⋊Dic3—24no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3C2×D412yes—
238Dic3⋊C4—24no—
239Dic3⋊C4—24no—
240Dic3⋊C4—24no—
241Dic3⋊C4—22no—
242Dic3⋊C4—24no—
243C4×Dic3—24no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246#208—44no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250#204—48no—
251C25—61no—
252C22×Q8—31no—
253C22×D4—31no—
254C22×D4—32no—
255C22×D4—64no—
256C22×D4—62no—
257C22×D4—62no—
258C22×D4—62no—
259C22×D4—61no—
260C22×D4—122no—
261C23×C4—31no—
262C23×C4—61no—
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×M16—62no—
268C2×M16—122no—
269C22×C8—32no—
270C22×C8C22×D312yes—
271#126—68no—
272#125—68no—
273#125—62no—
274#125—62no—
275#123—68no—
276C22⋊Q8—34no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—62no—
280#121—64no—
281#121—122no—
282#120—124no—
283C4×Q8—62no—
284C4×Q8—62no—
285C4×D4—62no—
286C4×D4—62no—
287#117—22no—
288#117—68no—
289#117D1214yescomplemented
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4C22×D312yes—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C22⋊C4—62no—
312C2×C42—21no—
313C2×C42—31no—
314C2×C42—61no—
315#103—122no—
316#102—64no—
317#102—64no—
318#102—64no—
319C22⋊C8—22no—
320C22⋊C8—122no—
321#95—21no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3#12012yes—
327C23×C3C22×D411yescharacteristic
328C22×D3—21no—
329C22×D3—22no—
330C22×D3—21no—
331C22×D3—22no—
332C22×D3—24no—
333C22×D3—48no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—88no—
341C22×D3—88no—
342C22×D3—84no—
343C22×D3—84no—
344C22×D3AGL(1,ℤ/8)12yes—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4#12012yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C2×#4112yes—
361C2×C3×C4H(5,𝔽2)12yes—
362#67—44no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375D12—24no—
376D12—24no—
377D12—48no—
378D12—42no—
379D12—42no—
380D12—88no—
381C4×D3—48no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—42no—
385Dic6—24no—
386Dic6—41no—
387Dic6—41no—
388C3×C8—42no—
389C3⋊C8—42no—
390C24—31no—
391C24—62no—
392C24—124no—
393C24—122no—
394C24—122no—
395C24—122no—
396C24—242no—
397C24D3×D411yescharacteristic
398C2×Q8—32no—
399C2×Q8—61no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×D4—34no—
403C2×D4—68no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—128no—
412C2×D4—124no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—121no—
417C2×D4—122no—
418C2×D4—242no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—64no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4C2×D1212yes—
437C22×C4D3×D411yescharacteristic
438C22×C4D3×D412yes—
439C22×C4C23×D311yescharacteristic
440SD16—128no—
441M16—124no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—32no—
453C4⋊C4—64no—
454C4⋊C4—64no—
455C4⋊C4—64no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—62no—
459C4⋊C4—64no—
460C4⋊C4—61no—
461C4⋊C4C2×D1212yes—
462C4⋊C4C2×D1212yes—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—64no—
466C22⋊C4—62no—
467C22⋊C4—64no—
468C22⋊C4—62no—
469C22⋊C4—124no—
470C22⋊C4—124no—
471C22⋊C4—242no—
472C42—24no—
473C42—64no—
474C42—61no—
475C42—61no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3C2×#12011yescharacteristic
482C22×C3#54612yes—
483C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
484C22×C3#57511yescharacteristic
485C22×C3#57712yes—
486C2×D3—22no—
487C2×D3—24no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—42no—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—816no—
495C2×D3—82no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—22no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—82no—
503C3×C4#52511yescharacteristic
504C3×C4#53511yescharacteristic
505C3×C4#54612yes—
506Dic3—42no—
507Dic3—41no—
508Dic3—42no—
509Dic3—41no—
510Dic3—41no—
511C23—22no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—32no—
516C23—62no—
517C23—61no—
518C23—62no—
519C23—61no—
520C23—64no—
521C23—128no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—244no—
529C23—244no—
530C23—248no—
531C23—248no—
532C23#88212yes—
533C23C2×D3×D411yescharacteristic
534Q8—64no—
535Q8—121no—
536Q8—121no—
537D4—64no—
538D4—64no—
539D4—128no—
540D4—124no—
541D4—122no—
542D4—122no—
543D4—248no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—128no—
566C2×C4—122no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—122no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4#88212yes—
573C2×C4C22×D1211yescharacteristic
574C2×C4C2×D3×D411yescharacteristic
575C2×C4D3×#4112yes—
576C2×C4#100912yes—
577C8—42no—
578C8—122no—
579C2×C3—42no—
580C2×C3#240411yescharacteristic
581C2×C3#300611yescharacteristic
582C2×C3#301011yescharacteristic
583C2×C3#331012yes—
584C2×C3#332312yes—
585D3—42no—
586D3—84no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—42no—
592C22—62no—
593C22—64no—
594C22—121no—
595C22—122no—
596C22—121no—
597C22—124no—
598C22—124no—
599C22—124no—
600C22—2416no—
601C22—242no—
602C22—242no—
603C22—242no—
604C22—242no—
605C22C2×#88211yescharacteristic
606C22#565412yes—
607C22#577012yes—
608C22D3×AGL(1,ℤ/8)11yescharacteristic
609C22#577711yescharacteristic
610C4—22no—
611C4—42no—
612C4—42no—
613C4—82no—
614C4—121no—
615C4—122no—
616C4—121no—
617C4—122no—
618C4—121no—
619C4#552411yescharacteristic
620C4#552511yescharacteristic
621C4#565412yes—
622C3#2398211yescharacteristic, complemented, Sylow
623C2—42no—
624C2—122no—
625C2—244no—
626C2#7412511yescharacteristic
627C2#7883611yescharacteristic
628C2#7888411yescharacteristic
629C2#8138312yes—
630C2#8154912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1825C2
socleC23×C3C22×D4
prime radical#74016C2
hypercenter#1825D3
hypocenterC3#23982
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
1. C2×#5529C2semidirect
2. C2×#5001C2semidirect
3. C2×#4731C2semidirect
5. C3×#1825C2semidirect
6. #69953C2semidirect
7. #69516C2semidirect
8. #68909C2semidirect
9. #68899C2semidirect
10. #68887C2semidirect
11. #68880C2semidirect
21. C2×#897C22semidirect
23. C2×D3×C4⋊C4C22semidirect
24. C2×#890C22semidirect
25. C2×#887C22semidirect
34. C2×(C2×D3)⋊C8C22semidirect
37. C2×#810C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
46. #4550C22semidirect
47. #4529C22semidirect
58. #1825D3semidirect
74. C3×#117D4semidirect
94. #924D4semidirect
162. #340C2×D3semidirect
163. #340C2×D3semidirect
289. #117D12semidirect
622. C3#23982semidirect

Maximal subgroups

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

subgroupindex
C2×#55292
C2×#50012
C2×#47312
#740162
C3×#18252
#699532
#695162
#689092
#688992
#688872
#688802
#239823