Finite Group Databasebeta

← #34031039

Subgroups — #34031039

Order 18688.

2576 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626470C213yesmaximal, complemented
2#22626467C213yesmaximal, complemented
3#22626462C216yesmaximal, complemented
4#22626453C213yesmaximal, complemented
5#22626432C216yesmaximal, complemented
6#22626421C213yesmaximal, complemented
7#22626416C213yesmaximal, complemented
8#120×#64408C213yesmaximal, complemented
9H(3,𝔽4)×D73C211yesmaximal, characteristic
10C24×#64408C2211yescharacteristic, complemented
11C2×Q8×#64408C2213yescomplemented
12C2×D4×#64408C2213yes—
13C2×D4×#64408C2216yes—
14C22×C22⋊#64408C2213yescomplemented
15(C2×#64408)⋊Q8C2213yescomplemented
16(C2×#64408)⋊Q8C2216yescomplemented
17#17989539C2216yes—
18#17989539C2213yes—
19#17989537C2216yescomplemented
20#17989537C2216yescomplemented
21#17989537C2216yescomplemented
22#17989537C2216yescomplemented
23#17989537C2216yescomplemented
24#17989537C2216yescomplemented
25#17989535C2216yes—
26#17989535C2213yes—
27#17989535C2213yes—
28#17989535C2216yes—
29#17989534C2216yescomplemented
30#17989534C2213yescomplemented
31#17989534C2213yescomplemented
32#17989534C2216yescomplemented
33#17989533C2211yescharacteristic, complemented
34#17989532C2212yescomplemented
35#17989528C2213yescomplemented
36#17989528C2211yescharacteristic, complemented
37C4×C4⋊#64408C2216yescomplemented
38C22⋊C4×#64408C2213yescomplemented
39C22⋊C4×#64408C2216yescomplemented
40C22⋊C4×#64408C2213yescomplemented
41C22⋊C4×#64408C2216yescomplemented
42#17989523C2213yescomplemented
43#17989330C413yescomplemented
44#17989330C413yescomplemented
45#17989329C413yescomplemented
46#17989329C416yescomplemented
47#124×D73C2213yes—
48#124×D73C2216yes—
49#120×D73C2213yes—
50#120×D73C2213yes—
51H(3,𝔽4)×C73C411yescharacteristic, complemented
52C23×#64408—46no—
53C23×#64408C2313yes—
54C23×#64408C2313yes—
55C23×#64408C2313yes—
56C23×#64408C2313yes—
57Q8×#64408—46no—
58D4×#64408—46no—
59D4×#64408—412no—
60C2×C4⋊#64408C2316yes—
61C2×C4⋊#64408C2313yes—
62C2×C4⋊#64408C2313yes—
63C2×C4⋊#64408C2316yes—
64C2×C22⋊#64408—46no—
65C2×C22⋊#64408C2316yes—
66C2×C22⋊#64408C2316yes—
67C2×C22⋊#64408C2313yes—
68C2×C22⋊#64408C2316yes—
69C2×C22⋊#64408C2313yes—
70C2×C22⋊#64408C2313yes—
71C2×C22⋊#64408C2316yes—
72C2×C22⋊#64408C2313yes—
73C2×C4×#64408C2313yes—
74C2×C4×#64408C2316yes—
75C2×C4×#64408C2316yes—
76C2×C4×#64408C2313yes—
77(C2×#64408)⋊C4C2316yes—
78(C2×#64408)⋊C4C2316yes—
79(C2×#64408)⋊C4C2316yes—
80(C2×#64408)⋊C4C2316yes—
81(C2×#64408)⋊C4C2316yes—
82(C2×#64408)⋊C4C2316yes—
83C24×D73C2311yescharacteristic
84C24×D73C2311yescharacteristic
85C2×Q8×D73C2313yes—
86C2×D4×D73C2313yes—
87C2×D4×D73C2316yes—
88#14556580C2×C413yescomplemented
89#14556580C2×C416yescomplemented
90#14556579C2×C413yescomplemented
91#14556579C2×C416yescomplemented
92#14556578C2×C413yescomplemented
93#14556578C2×C416yescomplemented
94#14556577C2×C416yescomplemented
95#14556577C2×C416yescomplemented
96#14556577C2×C416yescomplemented
97#14556577C2×C416yescomplemented
98#14556577C2×C416yescomplemented
99#14556577C2×C416yescomplemented
100#14556561C2×C416yescomplemented
101#14556561C2×C413yescomplemented
102#14556561C2×C416yescomplemented
103#14556561C2×C413yescomplemented
104#14556560C2×C413yescomplemented
105#14556560C2×C413yescomplemented
106#14556559C2×C413yescomplemented
107#14556559C2×C416yescomplemented
108#14556559C2×C413yescomplemented
109#14556559C2×C416yescomplemented
110C22⋊C4×D73C2313yes—
111C22⋊C4×D73C2316yes—
112C22⋊C4×D73C2313yes—
113C22⋊C4×D73C2316yes—
114C42×D73C2311yescharacteristic
115C42×D73C2312yes—
116#124×C73C2×C413yescomplemented
117#124×C73C2×C416yescomplemented
118#120×C73C2×C413yescomplemented
119#120×C73C2×C413yescomplemented
120C22×#64408—212no—
121C22×#64408—26no—
122C22×#64408—26no—
123C22×#64408—43no—
124C22×#64408—46no—
125C22×#64408—46no—
126C22×#64408—43no—
127C22×#64408—46no—
128C22×#64408—416no—
129C22×#64408C2411yescharacteristic
130C22×#64408C2416yes—
131C22×#64408C2413yes—
132C22×#64408C2413yes—
133C22×#64408C2413yes—
134C4⋊#64408—212no—
135C4⋊#64408—43no—
136C4⋊#64408—46no—
137C4⋊#64408—43no—
138C22⋊#64408—212no—
139C22⋊#64408—46no—
140C22⋊#64408—46no—
141C22⋊#64408—43no—
142C22⋊#64408—43no—
143C22⋊#64408—43no—
144C22⋊#64408—43no—
145C22⋊#64408—46no—
146C4×#64408—26no—
147C4×#64408—46no—
148C4×#64408—43no—
149C4×#64408—46no—
150C23×D73—43no—
151C23×D73—43no—
152C23×D73—43no—
153C23×D73—43no—
154C23×D73C22×C413yes—
155C23×D73C22×C413yescomplemented
156C23×D73C2413yes—
157C23×D73C2413yes—
158C2×Dic146C22×C416yescomplemented
159C2×Dic146C22×C413yescomplemented
160Q8×D73—43no—
161Q8×D73—43no—
162C2×D292C22×C413yes—
163C2×D292C22×C416yes—
164C2×C22⋊D73C22×C413yes—
165C2×C22⋊D73C22×C416yes—
166C2×C22⋊D73C22×C413yes—
167C2×C22⋊D73C22×C416yes—
168D4×D73—43no—
169D4×D73—46no—
170D4×D73—43no—
171D4×D73—46no—
172C2×C4×D73C2413yes—
173C2×C4×D73C2416yes—
174C22⋊Dic73C22×C413yes—
175C22⋊Dic73C22×C416yes—
176C22⋊Dic73C22×C413yescomplemented
177C22⋊Dic73C22×C416yescomplemented
178#11924005C22×C416yes—
179#11924005C22×C416yes—
180#11924005C22×C416yes—
181#11924005C22×C416yescomplemented
182#11924005C22×C416yescomplemented
183#11924005C22×C416yescomplemented
184C4×Dic73C22×C413yescomplemented
185C4×Dic73C22×C416yescomplemented
186C24×C73C22×C411yescharacteristic
187C24×C73C22×C411yescharacteristic, complemented
188C2×Q8×C73C22×C413yescomplemented
189C2×D4×C73C22×C413yes—
190C2×D4×C73C22×C416yes—
191C22⋊C4×C73C22×C416yes—
192C22⋊C4×C73C22×C413yes—
193C22⋊C4×C73C22×C416yescomplemented
194C22⋊C4×C73C22×C413yescomplemented
195C42×C73C22×C411yescharacteristic, complemented
196C42×C73C22×C412yescomplemented
197C23×C73—43no—
198C23×C73—43no—
199C23×C73C23×C413yes—
200C23×C73C23×C413yes—
201C22×D73—26no—
202C22×D73—23no—
203C22×D73—26no—
204C22×D73—23no—
205C22×D73—23no—
206C22×D73—26no—
207C22×D73—23no—
208C22×D73—26no—
209C22×D73—46no—
210C22×D73—43no—
211C22×D73—43no—
212C22×D73—46no—
213C22×D73—412no—
214C22×D73—44no—
215C22×D73—82no—
216C22×D73—86no—
217C22×D73C23×C413yes—
218C22×D73C23×C413yes—
219C22×D73C2511yescharacteristic
220C2×#64408—26no—
221C2×#64408—23no—
222C2×#64408—26no—
223C2×#64408—26no—
224C2×#64408—23no—
225C2×#64408—23no—
226C2×#64408—26no—
227C2×#64408—26no—
228C2×#64408—26no—
229C2×#64408—412no—
230C2×#64408—412no—
231C2×#64408—86no—
232C2×#64408—86no—
233C2×#64408H(5,𝔽2)16yes—
234Q8×C73—43no—
235D4×C73—43no—
236D4×C73—46no—
237C2×C4×C73C23×C413yes—
238C2×C4×C73C23×C416yes—
239C22⋊D73—43no—
240C22⋊D73—43no—
241C22⋊D73—46no—
242C22⋊D73—46no—
243C2×Dic73C23×C413yes—
244C2×Dic73C23×C416yes—
245D292—43no—
246D292—46no—
247C4×D73—23no—
248C4×D73—26no—
249C4×D73—23no—
250C4×D73—26no—
251Dic146—46no—
252Dic146—43no—
253C22×C73—23no—
254C22×C73—26no—
255C22×C73—26no—
256C22×C73—23no—
257C22×C73—44no—
258C22×C73—82no—
259C22×C73C24×C411yescharacteristic
260C2×D73—23no—
261C2×D73—26no—
262C2×D73—26no—
263C2×D73—23no—
264C2×D73—412no—
265C2×D73—43no—
266C2×D73—43no—
267C2×D73—43no—
268C2×D73—43no—
269C2×D73—46no—
270C2×D73—46no—
271C2×D73—86no—
272C2×D73C2×H(5,𝔽2)13yes—
273C2×D73#58513yes—
274#64408—26no—
275#64408—41no—
276#64408—43no—
277#64408—43no—
278#64408—46no—
279C4×C73—23no—
280C4×C73—26no—
281Dic73—23no—
282Dic73—26no—
283#34512—731nomaximal, Sylow
284C2×C73—43no—
285C2×C73—43no—
286C2×C73C4×H(5,𝔽2)13yes—
287D73—43no—
288D73—43no—
289D73#353211yescharacteristic
290D73#354013yes—
291C2×H(3,𝔽4)—731no—
292#2375—733no—
293#2372—733no—
294#2367—736no—
295#2358—733no—
296#2337—736no—
297#2326—733no—
298#2321—733no—
299C4×#120—733no—
300C73#3451211yescharacteristic, complemented, Sylow
301C24×C4—731no—
302H(3,𝔽4)—736no—
303H(3,𝔽4)—733no—
304H(3,𝔽4)—733no—
305H(3,𝔽4)—733no—
306H(3,𝔽4)#6440811yescharacteristic, complemented
307C2×#124—733no—
308C2×#124—736no—
309C2×#120—733no—
310C2×#120—733no—
311C2×C4×Q8—733no—
312C2×C4×D4—733no—
313C2×C4×D4—736no—
314C22×C22⋊C4—733no—
315#391—733no—
316#391—736no—
317#390—736no—
318#390—733no—
319#388—736no—
320#388—736no—
321#388—736no—
322#388—736no—
323#388—736no—
324#388—736no—
325#386—736no—
326#386—733no—
327#386—736no—
328#386—733no—
329#385—733no—
330#385—736no—
331#385—736no—
332#385—733no—
333#384—731no—
334#383—732no—
335#379—731no—
336#379—733no—
337C4×C4⋊C4—736no—
338C4×C22⋊C4—733no—
339C4×C22⋊C4—733no—
340C4×C22⋊C4—736no—
341C4×C22⋊C4—736no—
342#376—733no—
343C25—731no—
344C25—731no—
345C22×Q8—733no—
346C22×D4—733no—
347C22×D4—736no—
348C23×C4—2926no—
349C23×C4—733no—
350C23×C4—733no—
351C23×C4—733no—
352C23×C4—733no—
353#124—733no—
354#124—736no—
355#124—733no—
356#124—736no—
357#124—736no—
358#124—736no—
359#124—736no—
360#124—736no—
361#124—736no—
362#124—736no—
363#124—733no—
364#124—736no—
365#124C2×#6440813yescomplemented
366#124C2×#6440816yescomplemented
367#120—733no—
368#120—733no—
369#120—736no—
370#120—733no—
371#120—736no—
372#120—733no—
373#120—733no—
374#120—736no—
375#120—733no—
376#120—736no—
377#120C2×#6440813yescomplemented
378#120C2×#6440813yescomplemented
379C4×Q8—2926no—
380C4×D4—29212no—
381C4×D4—2926no—
382C2×C4⋊C4—733no—
383C2×C4⋊C4—733no—
384C2×C4⋊C4—736no—
385C2×C4⋊C4—736no—
386C2×C22⋊C4—2926no—
387C2×C22⋊C4—736no—
388C2×C22⋊C4—733no—
389C2×C22⋊C4—736no—
390C2×C22⋊C4—733no—
391C2×C22⋊C4—733no—
392C2×C22⋊C4—736no—
393C2×C22⋊C4—733no—
394C2×C22⋊C4—736no—
395C2×C22⋊C4—733no—
396C2×C22⋊C4—736no—
397C2×C22⋊C4—733no—
398C2×C22⋊C4—736no—
399C2×C42—736no—
400C2×C42—736no—
401C2×C42—733no—
402C2×C42—731no—
403C2×C42—732no—
404C2×C42—733no—
405#95—736no—
406#95—736no—
407#95—736no—
408#95—736no—
409#95—736no—
410#95—736no—
411C24—2923no—
412C24—2923no—
413C24—2923no—
414C24—2923no—
415C24—733no—
416C24—733no—
417C24—733no—
418C24—733no—
419C24C22×#6440811yescharacteristic
420C24C22×#6440811yescharacteristic, complemented
421C2×Q8—2923no—
422C2×Q8—2923no—
423C2×Q8—733no—
424C2×Q8—736no—
425C2×Q8C22×#6440813yescomplemented
426C2×D4—2926no—
427C2×D4—2923no—
428C2×D4—2923no—
429C2×D4—2926no—
430C2×D4—733no—
431C2×D4—736no—
432C2×D4—733no—
433C2×D4—736no—
434C2×D4—733no—
435C2×D4—736no—
436C2×D4C22×#6440813yes—
437C2×D4C22×#6440816yes—
438C22×C4—29216no—
439C22×C4—2926no—
440C22×C4—2923no—
441C22×C4—2923no—
442C22×C4—2926no—
443C22×C4—2926no—
444C22×C4—733no—
445C22×C4—736no—
446C22×C4—733no—
447C22×C4—736no—
448C22×C4—731no—
449C22×C4—733no—
450C22×C4—733no—
451C22×C4—1466no—
452C22×C4—14612no—
453C22×C4—1466no—
454C4⋊C4—2926no—
455C4⋊C4—2923no—
456C4⋊C4—2923no—
457C4⋊C4—14612no—
458C22⋊C4—2926no—
459C22⋊C4—2926no—
460C22⋊C4—2923no—
461C22⋊C4—2923no—
462C22⋊C4—2923no—
463C22⋊C4—2923no—
464C22⋊C4—2926no—
465C22⋊C4—736no—
466C22⋊C4—733no—
467C22⋊C4—736no—
468C22⋊C4—733no—
469C22⋊C4—736no—
470C22⋊C4—736no—
471C22⋊C4—736no—
472C22⋊C4—736no—
473C22⋊C4—736no—
474C22⋊C4—736no—
475C22⋊C4—14612no—
476C22⋊C4C22×#6440816yes—
477C22⋊C4C22×#6440813yes—
478C22⋊C4C22×#6440816yescomplemented
479C22⋊C4C22×#6440813yescomplemented
480C42—2923no—
481C42—2926no—
482C42—2926no—
483C42—733no—
484C42—736no—
485C42—1466no—
486C42C22×#6440811yescharacteristic, complemented
487C42C22×#6440812yescomplemented
488C23—43no—
489C23—43no—
490C23—2926no—
491C23—2923no—
492C23—2926no—
493C23—2923no—
494C23—2924no—
495C23—29212no—
496C23—5842no—
497C23—5846no—
498C23—731no—
499C23—733no—
500C23—733no—
501C23—1466no—
502C23—1463no—
503C23—1466no—
504C23—1463no—
505C23—1466no—
506C23—1463no—
507C23—1463no—
508C23—1466no—
509C23C23×#6440813yes—
510C23C23×#6440813yes—
511Q8—43no—
512Q8—2926no—
513Q8—2923no—
514D4—46no—
515D4—43no—
516D4—2926no—
517D4—2923no—
518D4—2926no—
519D4—2926no—
520D4—2923no—
521D4—2923no—
522C2×C4—29212no—
523C2×C4—29212no—
524C2×C4—5846no—
525C2×C4—5846no—
526C2×C4—736no—
527C2×C4—736no—
528C2×C4—733no—
529C2×C4—1466no—
530C2×C4—1463no—
531C2×C4—1463no—
532C2×C4—1466no—
533C2×C4—1463no—
534C2×C4—1466no—
535C2×C4—1466no—
536C2×C4—1466no—
537C2×C4—1463no—
538C2×C4—1466no—
539C2×C4—1463no—
540C2×C4—1466no—
541C2×C4—1466no—
542C2×C4C23×#6440816yes—
543C2×C4C23×#6440813yes—
544C22—26no—
545C22—23no—
546C22—26no—
547C22—23no—
548C22—44no—
549C22—82no—
550C22—29212no—
551C22—2923no—
552C22—2923no—
553C22—2923no—
554C22—2923no—
555C22—2926no—
556C22—2926no—
557C22—5846no—
558C22—733no—
559C22—733no—
560C22—1466no—
561C22—1463no—
562C22—1466no—
563C22—1463no—
564C22C24×#6440811yescharacteristic
565C4—26no—
566C4—23no—
567C4—2921no—
568C4—2923no—
569C4—2926no—
570C4—2923no—
571C4—1466no—
572C4—1466no—
573C4—1463no—
574C2—43no—
575C2—43no—
576C2—2923no—
577C2—2923no—
578C2—731no—
579C2—733no—
580C2H(5,𝔽2)×#6440813yes—
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
FittingH(3,𝔽4)×C73C4
prime radicalH(3,𝔽4)×D73C2
cosocleC22×D73C25
hypercenterH(3,𝔽4)#64408
hypocenterC73#34512
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22626470C2semidirect
2. #22626467C2semidirect
3. #22626462C2semidirect
4. #22626453C2semidirect
5. #22626432C2semidirect
6. #22626421C2semidirect
7. #22626416C2semidirect
8. #120×#64408C2semidirect
10. C24×#64408C22semidirect
11. C2×Q8×#64408C22semidirect
14. C22×C22⋊#64408C22semidirect
15. (C2×#64408)⋊Q8C22semidirect
16. (C2×#64408)⋊Q8C22semidirect
19. #17989537C22semidirect
20. #17989537C22semidirect
21. #17989537C22semidirect
22. #17989537C22semidirect
23. #17989537C22semidirect
24. #17989537C22semidirect
29. #17989534C22semidirect
30. #17989534C22semidirect
31. #17989534C22semidirect
32. #17989534C22semidirect
33. #17989533C22semidirect
34. #17989532C22semidirect
35. #17989528C22semidirect
36. #17989528C22semidirect
37. C4×C4⋊#64408C22semidirect
38. C22⋊C4×#64408C22semidirect
39. C22⋊C4×#64408C22semidirect
40. C22⋊C4×#64408C22semidirect
41. C22⋊C4×#64408C22semidirect
42. #17989523C22semidirect
43. #17989330C4semidirect
44. #17989330C4semidirect
45. #17989329C4semidirect
46. #17989329C4semidirect
51. H(3,𝔽4)×C73C4semidirect
88. #14556580C2×C4semidirect
89. #14556580C2×C4semidirect
90. #14556579C2×C4semidirect
91. #14556579C2×C4semidirect
92. #14556578C2×C4semidirect
93. #14556578C2×C4semidirect
94. #14556577C2×C4semidirect
95. #14556577C2×C4semidirect
96. #14556577C2×C4semidirect
97. #14556577C2×C4semidirect
98. #14556577C2×C4semidirect
99. #14556577C2×C4semidirect
100. #14556561C2×C4semidirect
101. #14556561C2×C4semidirect
102. #14556561C2×C4semidirect
103. #14556561C2×C4semidirect
104. #14556560C2×C4semidirect
105. #14556560C2×C4semidirect
106. #14556559C2×C4semidirect
107. #14556559C2×C4semidirect
108. #14556559C2×C4semidirect
109. #14556559C2×C4semidirect
116. #124×C73C2×C4semidirect
117. #124×C73C2×C4semidirect
118. #120×C73C2×C4semidirect
119. #120×C73C2×C4semidirect
155. C23×D73C22×C4semidirect
158. C2×Dic146C22×C4semidirect
159. C2×Dic146C22×C4semidirect
176. C22⋊Dic73C22×C4semidirect
177. C22⋊Dic73C22×C4semidirect
181. #11924005C22×C4semidirect
182. #11924005C22×C4semidirect
183. #11924005C22×C4semidirect
184. C4×Dic73C22×C4semidirect
185. C4×Dic73C22×C4semidirect
187. C24×C73C22×C4semidirect
188. C2×Q8×C73C22×C4semidirect
193. C22⋊C4×C73C22×C4semidirect
194. C22⋊C4×C73C22×C4semidirect
195. C42×C73C22×C4semidirect
196. C42×C73C22×C4semidirect
300. C73#34512semidirect
306. H(3,𝔽4)#64408semidirect
365. #124C2×#64408semidirect
366. #124C2×#64408semidirect
377. #120C2×#64408semidirect
378. #120C2×#64408semidirect
420. C24C22×#64408semidirect
425. C2×Q8C22×#64408semidirect
478. C22⋊C4C22×#64408semidirect
479. C22⋊C4C22×#64408semidirect
486. C42C22×#64408semidirect
487. C42C22×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264702
#226264672
#226264622
#226264532
#226264322
#226264212
#226264162
#120×#644082
H(3,𝔽4)×D732
#3451273