Finite Group Databasebeta

← #10868439

Subgroups — #10868439

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#82955C211yesmaximal, characteristic
2#76526C211yesmaximal, characteristic, complemented
3C3×#2015C211yesmaximal, characteristic
4#72066C211yesmaximal, characteristic, complemented
5#72065C211yesmaximal, characteristic, complemented
6#72051C211yesmaximal, characteristic
7#72047C211yesmaximal, characteristic
8#71722C211yesmaximal, characteristic
9#70241C211yesmaximal, characteristic, complemented
10#70222C211yesmaximal, characteristic, complemented
11#70181C211yesmaximal, characteristic
12Dic3×#102C211yesmaximal, characteristic
13#69666C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#27070—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#391C2211yescharacteristic
25#5229C2211yescharacteristic
26#5229C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5221C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C2×#808C2211yescharacteristic
41#4951—21no—
42#4951—21no—
43#4950—21no—
44Dic3×C22⋊C4—21no—
45C4×C4⋊Dic3C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4928C2211yescharacteristic
48C3⋊#401—21no—
49#4659—21no—
50#4655—21no—
51#4651—21no—
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2514—31no—
61#2367—31no—
62#2092—31no—
63#2055—31no—
64#2042—31no—
65#2015D311yescharacteristic
66#1969—31no—
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×C3×D4C2311yescharacteristic
78C23×Dic3—21no—
79C2×C3×Q16—22no—
80C2×C3×Q16—22no—
81C2×C3×Q16—41no—
82C2×C3×SD16—22no—
83C2×C3×SD16—41no—
84C2×C3×SD16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#124—22no—
88C3×#124—22no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96Q8×Dic3—22no—
97Q8×Dic3—41no—
98D4×Dic3—22no—
99D4×Dic3—41no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3D411yescharacteristic
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C3×#103—41no—
116C3×#102—22no—
117C3×#95—21no—
118C3×#95—21no—
119#835—22no—
120#835—22no—
121#835—41no—
122#835—41no—
123#832—22no—
124#832—41no—
125C3⋊#95—21no—
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⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#818—22no—
139Dic3⋊C8—41no—
140C8×Dic3—41no—
141#808—41no—
142C22×Q16C2×D311yescharacteristic
143C22×SD16C2×D311yescharacteristic
144C2×#124C2×D311yescharacteristic
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×#107—31no—
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102C2×D311yescharacteristic
158C2×C4×C8—31no—
159#401—61no—
160#400—61no—
161#391—31no—
162#391—31no—
163#391—31no—
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#336C2×D311yescharacteristic
185C24×C3—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—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4#4112yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—42no—
215C22×Dic3—42no—
216C22×Dic3—81no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C3×Q16—44no—
222C3×SD16—44no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8#4112yes—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C42—21no—
234C3×C42—21no—
235#217—42no—
236#217—42no—
237#217—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—22no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—41no—
247C4⋊Dic3—42no—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—42no—
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×D4C22×D311yescharacteristic
266C23×C4—61no—
267C2×Q16—22no—
268C2×Q16—22no—
269C2×Q16—41no—
270C2×SD16—22no—
271C2×SD16—41no—
272C2×SD16—41no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#124—22no—
278#124—22no—
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—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
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—62no—
304#106—121no—
305C4⋊C8—121no—
306#103—41no—
307#103—62no—
308#103—62no—
309#103—121no—
310#103—121no—
311#102—22no—
312#102—62no—
313#102—121no—
314C4×C8—121no—
315#95—21no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
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×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
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—41no—
364C2×C3×C4—41no—
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
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—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—84no—
395C2×Dic3#13512yes—
396C3×C8—22no—
397C3×C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C24—21no—
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#23712yes—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4#23712yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—61no—
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—44no—
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—32no—
453C4⋊C4—62no—
454C4⋊C4—61no—
455C4⋊C4—61no—
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—24no—
465C22⋊C4—24no—
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—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3#52511yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#55011yescharacteristic
488C22×C3C2×#13511yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—42no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#52911yescharacteristic
498C3×C4#53811yescharacteristic
499C3×C4#55111yescharacteristic
500C3×C4#55311yescharacteristic
501Dic3—22no—
502Dic3—41no—
503Dic3—42no—
504Dic3—41no—
505Dic3—41no—
506Dic3—81no—
507Dic3—81no—
508Dic3—81no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×D3×D411yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—32no—
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—61no—
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—62no—
557C2×C4—121no—
558C2×C4—121no—
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—121no—
568C2×C4—244no—
569C2×C4C2×#23711yescharacteristic
570C2×C4C2×#23711yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4#100411yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C8—22no—
576C8—41no—
577C8—121no—
578C8—121no—
579C2×C3—82no—
580C2×C3#246711yescharacteristic
581C2×C3#310211yescharacteristic
582C2×C3#312011yescharacteristic
583C2×C3#329611yescharacteristic
584C2×C3#331311yescharacteristic
585C2×C3#339311yescharacteristic
586C2×C3#340111yescharacteristic
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22#558611yescharacteristic
595C22#561211yescharacteristic
596C22#571811yescharacteristic
597C22D3×#13511yescharacteristic
598C22#577111yescharacteristic
599C22#577711yescharacteristic
600C22#577811yescharacteristic
601C4—41no—
602C4—41no—
603C4—42no—
604C4—62no—
605C4—81no—
606C4—81no—
607C4—121no—
608C4—122no—
609C4—121no—
610C4—121no—
611C4—241no—
612C4—241no—
613C4—241no—
614C4#558711yescharacteristic
615C4#559311yescharacteristic
616C4#560011yescharacteristic
617C4#560111yescharacteristic
618C3#2707011yescharacteristic, complemented, Sylow
619C2—82no—
620C2#7668411yescharacteristic
621C2#8021211yescharacteristic
622C2#8021711yescharacteristic
623C2#8061511yescharacteristic
624C2#8061811yescharacteristic
625C2#8181311yescharacteristic
626C2#8186611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2015C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2015D3
hypocenterC3#27070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2015

Complements

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

NH ≅ G/Ntype
2. #76526C2semidirect
4. #72066C2semidirect
5. #72065C2semidirect
9. #70241C2semidirect
10. #70222C2semidirect
13. #69666C2semidirect
14. #68624C2semidirect
15. #68610C2semidirect
618. C3#27070semidirect

Maximal subgroups

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

subgroupindex
#829552
#765262
C3×#20152
#720662
#720652
#720512
#720472
#717222
#702412
#702222
#701812
Dic3×#1022
#696662
#686242
#686102
#270703