Finite Group Databasebeta

← #10763522

Subgroups — #10763522

Order 768.

1298 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×#5725C211yesmaximal, characteristic
2D3×#431C211yesmaximal, characteristic, complemented
3D3×#423C211yesmaximal, characteristic, complemented
4C3×#1546C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#70774C211yesmaximal, characteristic, complemented
7#70502C211yesmaximal, characteristic, complemented
8#68421C214yesmaximal, complemented
9#68417C214yesmaximal, complemented
10#17704—31nomaximal, Sylow
11C22×#239—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×#947—21no—
14C2×D3×M16—21no—
15#5725—21no—
16#5725—21no—
17#5725—21no—
18#5725—21no—
19#5725C412yes—
20#5725C412yes—
21#5725C2212yes—
22#5725C2212yes—
23C2×#895—21no—
24C2×#895C412yes—
25C2×#893—21no—
26C2×#893C412yes—
27C2×#874C2211yescharacteristic
28C2×C42×D3C2211yescharacteristic
29C3×#431C2211yescharacteristic
30C3×#423C2211yescharacteristic
31#5118—21no—
32#4925C2211yescharacteristic
33#4924C2211yescharacteristic
34#4837—21no—
35#4837C2212yescomplemented
36#4833C2212yescomplemented
37D3×C4⋊C8—21no—
38D3×C4⋊C8C2212yescomplemented
39D3×C4⋊C8C2212yescomplemented
40D3×C22⋊C8—21no—
41#4718—21no—
42#4685—21no—
43C4×D3×C8—21no—
44C3×#332C2214yescomplemented
45#4547C2214yescomplemented
46#4485C2214yescomplemented
47#4485C2214yescomplemented
48C2×#532—31no—
49C2×#431—31no—
50C2×#423—31no—
51#1547—34no—
52#1546—31no—
53#1546D311yescharacteristic, complemented
54#1545—31no—
55#1545—31no—
56#1540—34no—
57C22×C3×Q8—21no—
58C2×#239—22no—
59C2×#239—22no—
60C2×#239—22no—
61C2×#239—22no—
62C2×#239—44no—
63C2×#239—44no—
64C22×D12—22no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2×C412yes—
68C22×C4×D3—22no—
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3D411yescharacteristic
71C22×C4×D3D411yescharacteristic
72C2×C3×M16—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8C2×C412yes—
76C3×C4⋊Q8C2312yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2×C412yes—
79C2×C3×C42C2311yescharacteristic
80#947—22no—
81#947—22no—
82#947—22no—
83#947—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4×Dic3C2×C411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3D411yescharacteristic
89C2×#208—21no—
90D3×M16—44no—
91C2×ΓL(1,𝔽25)—22no—
92C2×D3×C8—22no—
93C2×D3×C8—22no—
94#895—22no—
95#895—22no—
96#895—22no—
97#895—22no—
98#895—22no—
99#895—22no—
100#895C2×C414yes—
101#895C2×C414yes—
102#893—22no—
103#893—22no—
104#893—22no—
105#893—22no—
106#893—22no—
107#893—22no—
108#893C2×C414yes—
109#893C2×C414yes—
110#874—21no—
111#874—21no—
112#874C2×C412yes—
113#874C2312yes—
114C42×D3C2×C411yescharacteristic
115C42×D3C2×C411yescharacteristic
116C42×D3D411yescharacteristic
117C42×D3D411yescharacteristic
118C42×D3D411yescharacteristic
119C42×D3D411yescharacteristic
120C42×D3C2311yescharacteristic
121C42×D3C2311yescharacteristic
122C3×C4⋊C8—21no—
123C3×C4⋊C8C2312yes—
124C3×C4⋊C8C2312yes—
125C3×C22⋊C8—21no—
126C3×C4×C8—21no—
127#830—21no—
128(C2×D3)⋊C8—22no—
129Dic3⋊C8—22no—
130Dic3⋊C8—22no—
131Dic3⋊C8—22no—
132C8×Dic3—22no—
133C4⋊(C3⋊C8)—21no—
134C4⋊(C3⋊C8)C2312yes—
135C4⋊(C3⋊C8)C2312yes—
136C4×C3⋊C8—21no—
137C22×#41—61no—
138C22×M16—61no—
139#532—32no—
140#532—32no—
141#532—32no—
142#532—32no—
143#532—61no—
144#532—61no—
145#532—61no—
146#532—61no—
147C2×C4⋊Q8C2×D311yescharacteristic
148C2×#127—31no—
149C2×#124—61no—
150C2×#121—32no—
151C2×#121—61no—
152C2×C4×D4—32no—
153C2×C4×D4—61no—
154C22×C42—31no—
155#431—31no—
156#431—32no—
157#431—61no—
158#431—61no—
159#431C2×D311yescharacteristic
160#423—31no—
161#423—32no—
162#423—61no—
163#423—61no—
164#423C2×D311yescharacteristic
165C2×C4⋊C8—32no—
166C2×C4⋊C8—32no—
167C2×C4⋊C8—61no—
168C2×C22⋊C8—61no—
169C2×C4×C8—61no—
170#332—34no—
171#332C2×D314yescomplemented
172#331—34no—
173#331—34no—
174C23×D3—22no—
175C23×D3C22⋊C411yescharacteristic
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C22×C3×C4—22no—
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×Dic3C22⋊C411yescharacteristic
185C22×Dic3C22⋊C411yescharacteristic
186C22×Dic3C22⋊C411yescharacteristic
187#239—48no—
188#239—48no—
189#239—44no—
190#239—44no—
191#239—44no—
192#239—44no—
193C2×D12—24no—
194C2×D12—24no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—44no—
201C2×D12—48no—
202C2×D12—42no—
203C2×D12—42no—
204C2×D12—44no—
205C2×D12C22×C414yes—
206C2×C4×D3—24no—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—48no—
211C2×C4×D3C22⋊C411yescharacteristic
212C2×C4×D3C22⋊C411yescharacteristic
213C2×C4×D3C22⋊C411yescharacteristic
214C2×C4×D3C22⋊C411yescharacteristic
215C2×C4×D3C22⋊C411yescharacteristic
216C2×C4×D3C22⋊C411yescharacteristic
217C2×C4×D3D814yes—
218C2×C4×D3SD1614yes—
219C2×C4×D3C22×C411yescharacteristic
220C2×C4×D3C22×C411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3C2×D411yescharacteristic
223C2×C4×D3C2×D411yescharacteristic
224C2×C4×D3C2×D411yescharacteristic
225C3×M16—42no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4C22×C414yes—
232C3×C42C22×C411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2411yescharacteristic
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240C4×Dic3C22⋊C411yescharacteristic
241C4×Dic3C22⋊C411yescharacteristic
242C4×Dic3C22⋊C411yescharacteristic
243C4×Dic3C22⋊C411yescharacteristic
244C4×Dic3C22⋊C411yescharacteristic
245C4×Dic3C22⋊C411yescharacteristic
246C4×Dic3C22×C411yescharacteristic
247C4×Dic3C22×C411yescharacteristic
248C4×Dic3C2×D411yescharacteristic
249C4×Dic3C2×D411yescharacteristic
250C4×Dic3C2×D411yescharacteristic
251C4×Dic3C2×D411yescharacteristic
252#208—42no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255ΓL(1,𝔽25)—44no—
256D3×C8—44no—
257D3×C8—44no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41—124no—
263C2×#41—124no—
264C22×Q8—21no—
265C22×D4—32no—
266C22×D4—62no—
267C22×D4—61no—
268C22×D4—61no—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—31no—
272C23×C4—62no—
273C2×M16—21no—
274C2×M16—62no—
275C2×M16—61no—
276C2×M16—124no—
277C22×C8—62no—
278C22×C8—62no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8C4×D312yes—
282C4⋊Q8C22×D312yes—
283#127—32no—
284#127—32no—
285#127—61no—
286#127—61no—
287#124—62no—
288#124—62no—
289#124—62no—
290#124—62no—
291#121—34no—
292#121—34no—
293#121—62no—
294#121—62no—
295#121—62no—
296#121—62no—
297#121—62no—
298#121—62no—
299C4×D4—34no—
300C4×D4—34no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C4×D312yes—
309C2×C22⋊C4—62no—
310C2×C22⋊C4—62no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42C22×D311yescharacteristic
323C4⋊C8—21no—
324C4⋊C8—32no—
325C4⋊C8—32no—
326C4⋊C8—62no—
327C4⋊C8—62no—
328C4⋊C8—62no—
329C4⋊C8—61no—
330C4⋊C8C22×D312yes—
331C4⋊C8C22×D312yes—
332C22⋊C8—21no—
333C22⋊C8—62no—
334C22⋊C8—61no—
335C4×C8—21no—
336C4×C8—62no—
337C4×C8—61no—
338C23×C3C2×C22⋊C411yescharacteristic
339C22×D3—24no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—44no—
343C22×D3—44no—
344C22×D3—44no—
345C22×D3#10214yes—
346C22×D3C2×C22⋊C411yescharacteristic
347C22×D3C2×C22⋊C411yescharacteristic
348C3×Q8—42no—
349C3×Q8—42no—
350C3×Q8—44no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—44no—
355C2×C3×C4C2×C22⋊C411yescharacteristic
356C2×C3×C4C2×C22⋊C411yescharacteristic
357C2×C3×C4C2×C22⋊C411yescharacteristic
358C2×C3×C4C2×D812yes—
359C2×C3×C4C2×SD1612yes—
360C2×C3×C4C23×C411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3#10214yes—
366C2×Dic3C2×C22⋊C411yescharacteristic
367C2×Dic3C2×C22⋊C411yescharacteristic
368C2×Dic3C2×C22⋊C411yescharacteristic
369C2×Dic3C2×C22⋊C411yescharacteristic
370C2×Dic3C2×C22⋊C411yescharacteristic
371C2×Dic3C2×C22⋊C411yescharacteristic
372D12—44no—
373D12—44no—
374D12—48no—
375D12—48no—
376D12—42no—
377D12—42no—
378D12—44no—
379D12—44no—
380D12—44no—
381C4×D3—22no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—41no—
386C4×D3—41no—
387C4×D3—41no—
388C4×D3—41no—
389C4×D3—48no—
390C4×D3—48no—
391C4×D3#10214yes—
392C4×D3#10214yes—
393C4×D3C2×D812yes—
394C4×D3C2×D812yes—
395C4×D3C2×SD1612yes—
396C4×D3C2×SD1612yes—
397C3×C8—42no—
398C3×C8—42no—
399C3⋊C8—42no—
400C3⋊C8—42no—
401C24—31no—
402C24—62no—
403#41—124no—
404#41—124no—
405#41—124no—
406#41—124no—
407#41—128no—
408#41—128no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×D4—34no—
414C2×D4—64no—
415C2×D4—64no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—124no—
422C2×D4—124no—
423C2×D4—128no—
424C2×D4—122no—
425C2×D4—122no—
426C22×C4—22no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—34no—
431C22×C4—34no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—128no—
449C22×C4C2×C4×D311yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4D3×D411yescharacteristic
452M16—42no—
453M16—124no—
454M16—122no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—124no—
460C2×C8—124no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4C2×C4×D314yes—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—31no—
473C42—31no—
474C42—31no—
475C42—31no—
476C42—31no—
477C42—31no—
478C42—31no—
479C42—31no—
480C42—31no—
481C42C2×C4×D311yescharacteristic
482C42D3×D411yescharacteristic
483C42D3×D411yescharacteristic
484C42C23×D311yescharacteristic
485C22×C3—21no—
486C22×C3—21no—
487C22×C3C2×#10212yes—
488C22×C3C22×C22⋊C411yescharacteristic
489C2×D3—21no—
490C2×D3—21no—
491C2×D3—21no—
492C2×D3—21no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C2×D3—88no—
498C2×D3C2×#10212yes—
499C2×D3C2×#10212yes—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—44no—
503C3×C4#41312yes—
504C3×C4C2×#10212yes—
505C3×C4C22×D811yescharacteristic
506C3×C4C22×SD1611yescharacteristic
507C3×C4#57512yes—
508Dic3—21no—
509Dic3—21no—
510Dic3—21no—
511Dic3—21no—
512Dic3C2×#10212yes—
513Dic3C2×#10212yes—
514C23—31no—
515C23—31no—
516C23—34no—
517C23—64no—
518C23—121no—
519C23—121no—
520C23—124no—
521C23—124no—
522C23—124no—
523C23D3×C22⋊C411yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—44no—
527D4—124no—
528D4—124no—
529D4—128no—
530D4—128no—
531D4—122no—
532D4—122no—
533D4—124no—
534D4—124no—
535D4—124no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—34no—
542C2×C4—32no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—32no—
546C2×C4—34no—
547C2×C4—34no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—31no—
551C2×C4—31no—
552C2×C4—44no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—128no—
562C2×C4—128no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4D3×C22⋊C411yescharacteristic
566C2×C4D3×C22⋊C411yescharacteristic
567C2×C4D3×C22⋊C411yescharacteristic
568C2×C4D3×D812yes—
569C2×C4D3×SD1612yes—
570C2×C4C22×C4×D311yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C8—42no—
574C8—42no—
575C8—122no—
576C8—122no—
577C2×C3—22no—
578C2×C3C2×#41311yescharacteristic
579C2×C3C22×#10211yescharacteristic
580C2×C3#289611yescharacteristic
581D3—22no—
582D3—22no—
583D3—84no—
584C22—21no—
585C22—21no—
586C22—32no—
587C22—32no—
588C22—61no—
589C22—61no—
590C22—61no—
591C22—61no—
592C22—124no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—248no—
597C22D3×#10212yes—
598C22C2×D3×C22⋊C411yescharacteristic
599C4—21no—
600C4—21no—
601C4—32no—
602C4—32no—
603C4—44no—
604C4—61no—
605C4—61no—
606C4—61no—
607C4—61no—
608C4#475112yes—
609C4D3×#10212yes—
610C4C2×D3×D811yescharacteristic
611C4C2×D3×SD1611yescharacteristic
612C4#577712yes—
613C3#1770411yescharacteristic, complemented, Sylow
614C2—22no—
615C2—62no—
616C2—62no—
617C2—244no—
618C2C2×#475111yescharacteristic
619C2C2×D3×#10211yescharacteristic
620C2#7895611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1546C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5725C2
cosocleC3×C42C24
hypercenter#1546D3
hypocenterC3#17704
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
2. D3×#431C2semidirect
3. D3×#423C2semidirect
4. C3×#1546C2semidirect
5. #71773C2semidirect
6. #70774C2semidirect
7. #70502C2semidirect
8. #68421C2semidirect
9. #68417C2semidirect
35. #4837C22semidirect
36. #4833C22semidirect
38. D3×C4⋊C8C22semidirect
39. D3×C4⋊C8C22semidirect
44. C3×#332C22semidirect
45. #4547C22semidirect
46. #4485C22semidirect
47. #4485C22semidirect
53. #1546D3semidirect
171. #332C2×D3semidirect
613. C3#17704semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
D3×#4312
D3×#4232
C3×#15462
#717732
#707742
#705022
#684212
#684172
#177043