Finite Group Databasebeta

← #10876039

Subgroups — #10876039

Order 768.

746 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82334C211yesmaximal, characteristic
2#76521C211yesmaximal, characteristic, complemented
3C3×#1877C211yesmaximal, characteristic
4#72047C211yesmaximal, characteristic
5#72047C211yesmaximal, characteristic
6#71604C211yesmaximal, characteristic
7#70256C211yesmaximal, characteristic, complemented
8#70234C211yesmaximal, characteristic, complemented
9#70167C211yesmaximal, characteristic
10#70152C211yesmaximal, characteristic
11#69683C211yesmaximal, characteristic, complemented
12#68844C211yesmaximal, characteristic, complemented
13#68787C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68614C211yesmaximal, characteristic, complemented
16#27576—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#391C2211yescharacteristic
24#5229C2211yescharacteristic
25#5229C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5210—21no—
31C2×C3⋊#95—21no—
32C2×#815C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4972C2211yescharacteristic
36#4972C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4946—21no—
39#4942—21no—
40#4935C2211yescharacteristic
41C4×Dic3⋊C4C2211yescharacteristic
42#4928C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4660—21no—
45#4659—21no—
46#4655—21no—
47#4653—21no—
48C3×#340C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2511—31no—
61#2349—31no—
62#2101—31no—
63#2100—31no—
64#2074—31no—
65#2068—31no—
66#1935—31no—
67#1918—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877D311yescharacteristic
72#1862—31no—
73#1802—31no—
74#1762—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×Dic3—21no—
79C23×Dic3—21no—
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×D4—22no—
86C3×C4×D4—41no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95D4×Dic3—22no—
96D4×Dic3—41no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4D411yescharacteristic
107C2×Dic3⋊C4D411yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C3×#103—41no—
114C3×#102—22no—
115C3×#95—21no—
116C3×#95—21no—
117#835—22no—
118#835—41no—
119#832—22no—
120#832—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#815—41no—
134#807—22no—
135C4⋊(C3⋊C8)—41no—
136C22×SD16C2×D311yescharacteristic
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4C2×D311yescharacteristic
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103C2×D311yescharacteristic
145C2×#102—31no—
146C2×#102—31no—
147C2×#102C2×D311yescharacteristic
148C2×#97—31no—
149#400—61no—
150#400—61no—
151#391—31no—
152#391—31no—
153#391C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389—31no—
157#387—61no—
158#387—61no—
159#385—61no—
160#385—61no—
161#384—31no—
162#382—31no—
163#380—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C2×#95—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336C2×D311yescharacteristic
179C24×C3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4#4112yes—
190C2×C3×D4#4112yes—
191C2×C3×D4#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—24no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3—81no—
209C22×Dic3—81no—
210C22×Dic3—81no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C3×SD16—44no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C22⋊C4—42no—
227C3×C42—21no—
228C3×C42—21no—
229#217—24no—
230#217—42no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—42no—
239C4⋊Dic3—81no—
240C4⋊Dic3#4112yes—
241Dic3⋊C4—22no—
242Dic3⋊C4—41no—
243Dic3⋊C4—81no—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
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×Q8C22×D311yescharacteristic
258C22×D4C22×D311yescharacteristic
259C23×C4—21no—
260C23×C4—61no—
261C23×C4—61no—
262C2×SD16—22no—
263C2×SD16—22no—
264C2×SD16—22no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C4×D4—22no—
270C4×D4—41no—
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—121no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296C4⋊C8—121no—
297#103—41no—
298#103—62no—
299#103—121no—
300#102—22no—
301#102—62no—
302#102—62no—
303#97—121no—
304#95—21no—
305#95—21no—
306#95—31no—
307#95—31no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3C22×D411yescharacteristic
326C3×Q8—41no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4#12614yes—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
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×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—84no—
349C2×C3×C4#13712yes—
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355C2×C3×C42−1+411yescharacteristic
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—22no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—84no—
381C2×Dic3—84no—
382C3×C8—41no—
383C3×C8—41no—
384C3⋊C8—41no—
385C3⋊C8—41no—
386C24—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4#23512yes—
397C2×D4#23712yes—
398C2×D4#23712yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—61no—
409C22×C4—64no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—81no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4C23×D311yescharacteristic
422SD16—44no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—32no—
440C4⋊C4—62no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—121no—
449C4⋊C4—122no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C22⋊C4—42no—
455C22⋊C4—64no—
456C22⋊C4—122no—
457C42—21no—
458C42—21no—
459C42—61no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—241no—
464C22×C3—24no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3#52511yescharacteristic
472C22×C3#54611yescharacteristic
473C22×C3#54811yescharacteristic
474C22×C3C2×#13711yescharacteristic
475C22×C3#57511yescharacteristic
476C22×C3#57611yescharacteristic
477C22×C3#57711yescharacteristic
478C3×C4—22no—
479C3×C4—41no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4C2×#12611yescharacteristic
484C3×C4#53911yescharacteristic
485C3×C4#55211yescharacteristic
486C3×C4#55311yescharacteristic
487Dic3—41no—
488Dic3—41no—
489Dic3—42no—
490Dic3—41no—
491Dic3—41no—
492Dic3—81no—
493Dic3—81no—
494Dic3—81no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C2×D3×D411yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507D4—22no—
508D4—22no—
509D4—22no—
510D4#87914yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
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—41no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
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—62no—
538C2×C4—84no—
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×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—244no—
551C2×C4—244no—
552C2×C4#91512yes—
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4C2×#23711yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4#101011yescharacteristic
559C8—41no—
560C8—41no—
561C8—121no—
562C8—121no—
563C2×C3—44no—
564C2×C3#248111yescharacteristic
565C2×C3#311611yescharacteristic
566C2×C3#311811yescharacteristic
567C2×C3#331411yescharacteristic
568C2×C3#332311yescharacteristic
569C2×C3#337011yescharacteristic
570C2×C3#337211yescharacteristic
571C22—24no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22#555211yescharacteristic
579C22#559111yescharacteristic
580C22#560611yescharacteristic
581C22C2×#91511yescharacteristic
582C22#577011yescharacteristic
583C22#577411yescharacteristic
584C22#577711yescharacteristic
585C4—22no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—121no—
591C4—121no—
592C4—121no—
593C4—122no—
594C4—121no—
595C4—241no—
596C4—241no—
597C4—241no—
598C4C2×#87911yescharacteristic
599C4#554011yescharacteristic
600C4#560211yescharacteristic
601C4#565011yescharacteristic
602C3#2757611yescharacteristic, complemented, Sylow
603C2—44no—
604C2#7473611yescharacteristic
605C2#7962811yescharacteristic
606C2#7975311yescharacteristic
607C2#8016311yescharacteristic
608C2#8025811yescharacteristic
609C2#8041111yescharacteristic
610C2#8050911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1877D3
hypocenterC3#27576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #76521C2semidirect
7. #70256C2semidirect
8. #70234C2semidirect
11. #69683C2semidirect
12. #68844C2semidirect
13. #68787C2semidirect
14. #68643C2semidirect
15. #68614C2semidirect
602. C3#27576semidirect

Maximal subgroups

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

subgroupindex
#823342
#765212
C3×#18772
#720472
#716042
#702562
#702342
#701672
#701522
#696832
#688442
#687872
#686432
#686142
#275763