Finite Group Databasebeta

← #10875980

Subgroups — #10875980

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76341C211yesmaximal, characteristic
2#75981C211yesmaximal, characteristic, complemented
3C3×#1936C211yesmaximal, characteristic
4#70213C211yesmaximal, characteristic
5#70213C211yesmaximal, characteristic
6#70194C211yesmaximal, characteristic
7#70165C211yesmaximal, characteristic
8#69834C211yesmaximal, characteristic, complemented
9#69813C211yesmaximal, characteristic, complemented
10#69660C211yesmaximal, characteristic
11#69600C211yesmaximal, characteristic, complemented
12#68906C211yesmaximal, characteristic, complemented
13#68905C211yesmaximal, characteristic, complemented
14#68845C211yesmaximal, characteristic, complemented
15#68843C211yesmaximal, characteristic, complemented
16#27572—31nomaximal, Sylow
17C22×#214C2211yescharacteristic
18C2×#887C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#389C2211yescharacteristic
26#5221C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4984C2211yescharacteristic
33#4984C2211yescharacteristic
34#4977C2211yescharacteristic
35#4976—21no—
36#4975—21no—
37#4975C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4953—21no—
41#4952—21no—
42#4951—21no—
43#4945—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4649—21no—
47#4647—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61Q8×C4⋊C4—31no—
62#2101—31no—
63#2100—31no—
64#2071—31no—
65#2067—31no—
66#1936D311yescharacteristic
67#1918—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1863—31no—
73#1805—31no—
74#1763—31no—
75C22×C3×D4C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90C2×#214—22no—
91C2×#214—22no—
92C2×#214—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4D411yescharacteristic
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#887—22no—
114#887—22no—
115#887—22no—
116#887—41no—
117C4×Dic6—22no—
118C4×Dic6—41no—
119C3×#102—22no—
120C3×#102—41no—
121C3×#97—41no—
122C3×#95—21no—
123#832—41no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131#816—22no—
132#816—22no—
133Dic3⋊C8—41no—
134#810—22no—
135#807—22no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×Q8—31no—
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
148C2×#97C2×D311yescharacteristic
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389—61no—
157#389—61no—
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#384—31no—
162#384—61no—
163#382—31no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6#4112yes—
204C2×Dic6#4112yes—
205C2×Dic6#4112yes—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—81no—
218C3×C4⋊C4#4112yes—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—81no—
221C3×C42—21no—
222C3×C42—21no—
223#217—42no—
224#217—81no—
225#217—81no—
226#217—81no—
227#214—44no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—81no—
235C4⋊Dic3—81no—
236Dic3⋊C4—24no—
237Dic3⋊C4—22no—
238Dic3⋊C4—41no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—81no—
243C4×Dic3—24no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C4×Dic3—42no—
250C4×Dic3—81no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C22×Q8—31no—
258C22×D4C22×D311yescharacteristic
259C2×SD16—62no—
260C2×SD16—62no—
261C2×SD16—62no—
262C22×C8—31no—
263C22×C8—31no—
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—121no—
271C4×Q8—121no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#107—62no—
302C4⋊C8—121no—
303#103—62no—
304#103—62no—
305#103—62no—
306#102—22no—
307#102—41no—
308#102—121no—
309#97—41no—
310#95—21no—
311#95—31no—
312#95—31no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3C22×D411yescharacteristic
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—41no—
329C3×D4—41no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C2×C3×C42−1+411yescharacteristic
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3#13712yes—
375Dic6—22no—
376Dic6—22no—
377Dic6—22no—
378Dic6#12614yes—
379C3×C8—41no—
380C3×C8—41no—
381C3⋊C8—41no—
382C3⋊C8—41no—
383C24—21no—
384C2×Q8—32no—
385C2×Q8—32no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4D3×D411yescharacteristic
408C22×C4D3×D411yescharacteristic
409C22×C4C23×D311yescharacteristic
410SD16—124no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—62no—
429C4⋊C4—64no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—81no—
437C4⋊C4—121no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—241no—
442C4⋊C4—241no—
443C4⋊C4—241no—
444C4⋊C4#23512yes—
445C22⋊C4—42no—
446C22⋊C4—81no—
447C22⋊C4—122no—
448C22⋊C4—241no—
449C22⋊C4—241no—
450C22⋊C4—241no—
451C42—21no—
452C42—21no—
453C42—64no—
454C42—61no—
455C42—61no—
456C42—61no—
457C42—61no—
458C42—122no—
459C42—122no—
460C42—241no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3#52511yescharacteristic
469C22×C3#54611yescharacteristic
470C22×C3#54811yescharacteristic
471C22×C3C2×#13711yescharacteristic
472C22×C3#57511yescharacteristic
473C22×C3#57611yescharacteristic
474C22×C3#57711yescharacteristic
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4C2×#12611yescharacteristic
481C3×C4#53911yescharacteristic
482C3×C4#55211yescharacteristic
483C3×C4#55311yescharacteristic
484Dic3—22no—
485Dic3—24no—
486Dic3—41no—
487Dic3—41no—
488Dic3—41no—
489Dic3—81no—
490Dic3—81no—
491Dic3—81no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×D3×D411yescharacteristic
500Q8—34no—
501Q8—62no—
502Q8—62no—
503Q8—62no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—32no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—62no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4C2×#23511yescharacteristic
548C2×C4#100411yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4#101011yescharacteristic
553C8—41no—
554C8—41no—
555C8—121no—
556C8—121no—
557C2×C3—161no—
558C2×C3#248111yescharacteristic
559C2×C3#311711yescharacteristic
560C2×C3#311911yescharacteristic
561C2×C3#330911yescharacteristic
562C2×C3#332911yescharacteristic
563C2×C3#337111yescharacteristic
564C2×C3#337511yescharacteristic
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22#554711yescharacteristic
573C22#561011yescharacteristic
574C22#567711yescharacteristic
575C22#577311yescharacteristic
576C22#577411yescharacteristic
577C22#577511yescharacteristic
578C22D3×#13711yescharacteristic
579C4—42no—
580C4—41no—
581C4—41no—
582C4—64no—
583C4—62no—
584C4—81no—
585C4—81no—
586C4—121no—
587C4—121no—
588C4—121no—
589C4—241no—
590C4—241no—
591C4—241no—
592C4#552911yescharacteristic
593C4#555811yescharacteristic
594C4#566011yescharacteristic
595C4D3×#12611yescharacteristic
596C3#2757211yescharacteristic, complemented, Sylow
597C2—161no—
598C2#7554211yescharacteristic
599C2#7978411yescharacteristic
600C2#7987311yescharacteristic
601C2#8084611yescharacteristic
602C2#8087511yescharacteristic
603C2#8167511yescharacteristic
604C2#8171111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1936C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1936D3
hypocenterC3#27572
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #75981C2semidirect
8. #69834C2semidirect
9. #69813C2semidirect
11. #69600C2semidirect
12. #68906C2semidirect
13. #68905C2semidirect
14. #68845C2semidirect
15. #68843C2semidirect
596. C3#27572semidirect

Maximal subgroups

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

subgroupindex
#763412
#759812
C3×#19362
#702132
#701942
#701652
#698342
#698132
#696602
#696002
#689062
#689052
#688452
#688432
#275723