Finite Group Databasebeta

← #10868450

Subgroups — #10868450

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76525C211yesmaximal, characteristic, complemented
2#76493C211yesmaximal, characteristic
3C3×#1969C211yesmaximal, characteristic, complemented
4#72072C211yesmaximal, characteristic
5#72065C211yesmaximal, characteristic, complemented
6#70640C211yesmaximal, characteristic
7#70610C211yesmaximal, characteristic, complemented
8#70280C211yesmaximal, characteristic
9#70245C211yesmaximal, characteristic
10#70241C211yesmaximal, characteristic, complemented
11#70222C211yesmaximal, characteristic, complemented
12#69784C211yesmaximal, characteristic, complemented
13C4×#821C211yesmaximal, characteristic
14#69687C211yesmaximal, characteristic
15#69666C211yesmaximal, characteristic, complemented
16#27070—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4988C2211yescharacteristic
37#4984C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43C4×Dic3⋊C4C2211yescharacteristic
44#4670—21no—
45#4669—21no—
46#4668—21no—
47#4666—21no—
48C3⋊#401—21no—
49#4660—21no—
50#4653—21no—
51#4651—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2514—31no—
61#2367—31no—
62#2092—31no—
63#2055—31no—
64#2042—31no—
65#2015—31no—
66#1969D311yescharacteristic, complemented
67#1927—31no—
68#1927—31no—
69#1921—31no—
70#1916—31no—
71#1878—31no—
72#1877—31no—
73#1805—31no—
74C4×#102—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C2×C3×Q16—22no—
80C2×C3×Q16—22no—
81C2×C3×Q16—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87#947—22no—
88#947—22no—
89Q8×Dic3—22no—
90Q8×Dic3—41no—
91C2×#215—22no—
92C2×#215—41no—
93C2×#215—41no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C4×D12—22no—
114C4×D12—41no—
115C3×#107—22no—
116C3×#103—41no—
117C3×#103—41no—
118C3×C4×C8—41no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123#835—22no—
124#835—22no—
125#835—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#821—22no—
139Dic3⋊C8—41no—
140#809—41no—
141#808—41no—
142C22×Q16C2×D311yescharacteristic
143C22×SD16—31no—
144C2×#124—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×#107C2×D311yescharacteristic
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C4×C8C2×D311yescharacteristic
159#401—61no—
160#400—61no—
161#391—31no—
162#391—31no—
163#391C2×D311yescharacteristic
164#391C2×D311yescharacteristic
165#390—31no—
166#389—31no—
167#388—61no—
168#388—61no—
169#387—61no—
170#385—61no—
171#383—61no—
172#382—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C22⋊C4—61no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C23×D3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8#4112yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12#4112yes—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3—81no—
221C3×Q16—44no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8#4112yes—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C42—21no—
233C3×C42—21no—
234#215—44no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—42no—
246C4⋊Dic3—42no—
247C4⋊Dic3#4112yes—
248Dic3⋊C4—22no—
249Dic3⋊C4—41no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—42no—
255C4×Dic3—42no—
256C4×Dic3—81no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C22×Q8C22×D311yescharacteristic
264C22×Q8C22×D311yescharacteristic
265C22×D4—31no—
266C23×C4—61no—
267C2×Q16—22no—
268C2×Q16—22no—
269C2×Q16—41no—
270C2×SD16—62no—
271C2×SD16—121no—
272C2×SD16—121no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#124—62no—
278#124—62no—
279C4×Q8—62no—
280C4×Q8—121no—
281C4×D4—62no—
282C4×D4—121no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—22no—
304#106—121no—
305C4⋊C8—121no—
306#103—41no—
307#103—41no—
308#103—62no—
309#103—62no—
310#103—121no—
311#102—62no—
312#102—62no—
313#102—121no—
314C4×C8—41no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4#13512yes—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391D12—22no—
392D12—22no—
393D12—41no—
394D12—41no—
395C4×D3—84no—
396C3×C8—22no—
397C3×C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C24—61no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8#23512yes—
413C2×D4—32no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—241no—
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4C23×D311yescharacteristic
436Q16—44no—
437SD16—124no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8#23712yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—32no—
455C4⋊C4—62no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—121no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C22⋊C4—64no—
465C22⋊C4—64no—
466C22⋊C4—122no—
467C22⋊C4—122no—
468C22⋊C4—241no—
469C42—21no—
470C42—21no—
471C42—61no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—122no—
476C42—122no—
477C42—241no—
478C22×C3#52511yescharacteristic
479C22×C3#54611yescharacteristic
480C22×C3#55011yescharacteristic
481C22×C3C2×#13511yescharacteristic
482C22×C3#57511yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57811yescharacteristic
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#52911yescharacteristic
499C3×C4#53811yescharacteristic
500C3×C4#55111yescharacteristic
501C3×C4#55311yescharacteristic
502Dic3—41no—
503Dic3—41no—
504Dic3—42no—
505Dic3—41no—
506Dic3—41no—
507Dic3—81no—
508Dic3—81no—
509C23—62no—
510C23—121no—
511C23—121no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23C2×D3×D411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525D4—62no—
526D4—62no—
527D4—121no—
528D4—121no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—244no—
568C2×C4#91912yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4#100411yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C8—22no—
576C8—41no—
577C8—121no—
578C8—121no—
579C2×C3#246711yescharacteristic
580C2×C3#310211yescharacteristic
581C2×C3#312011yescharacteristic
582C2×C3#329611yescharacteristic
583C2×C3#331311yescharacteristic
584C2×C3#339311yescharacteristic
585C2×C3#340111yescharacteristic
586D3—82no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—241no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22#554711yescharacteristic
595C22#561011yescharacteristic
596C22#571811yescharacteristic
597C22C2×#91911yescharacteristic
598C22#576611yescharacteristic
599C22#576811yescharacteristic
600C22#577011yescharacteristic
601C4—22no—
602C4—42no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—121no—
608C4—121no—
609C4—121no—
610C4—122no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4#549511yescharacteristic
615C4#554011yescharacteristic
616C4#561311yescharacteristic
617C4#568011yescharacteristic
618C3#2707011yescharacteristic, complemented, Sylow
619C2—242no—
620C2#7632811yescharacteristic
621C2#7974611yescharacteristic
622C2#7974811yescharacteristic
623C2#8077511yescharacteristic
624C2#8078311yescharacteristic
625C2#8179511yescharacteristic
626C2#8181611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1969C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1969D3
hypocenterC3#27070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1969

Complements

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

NH ≅ G/Ntype
1. #76525C2semidirect
3. C3×#1969C2semidirect
5. #72065C2semidirect
7. #70610C2semidirect
10. #70241C2semidirect
11. #70222C2semidirect
12. #69784C2semidirect
15. #69666C2semidirect
66. #1969D3semidirect
618. C3#27070semidirect

Maximal subgroups

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

subgroupindex
#765252
#764932
C3×#19692
#720722
#720652
#706402
#706102
#702802
#702452
#702412
#702222
#697842
C4×#8212
#696872
#696662
#270703