Finite Group Databasebeta

← #10824588

Subgroups — #10824588

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82328C211yesmaximal, characteristic
2C2×#5018C211yesmaximal, characteristic
3C2×C4×Dic12C211yesmaximal, characteristic, complemented
4#73625C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic
6#70217C211yesmaximal, characteristic
7#70216C211yesmaximal, characteristic
8#70215C211yesmaximal, characteristic
9#70180C211yesmaximal, characteristic
10#69755C211yesmaximal, characteristic, complemented
11#69475C211yesmaximal, characteristic, complemented
12#68914C211yesmaximal, characteristic, complemented
13#68913C211yesmaximal, characteristic, complemented
14#68864C211yesmaximal, characteristic, complemented
15#68834C211yesmaximal, characteristic, complemented
16#23695—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C22×Dic12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29#5210—21no—
30#5210—21no—
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#5018—24no—
37#4975C2211yescharacteristic
38#4973C2211yescharacteristic
39#4973C2211yescharacteristic
40#4973C2211yescharacteristic
41#4972C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4944—21no—
48#4944—21no—
49#4933—21no—
50#4933—21no—
51#4932C2211yescharacteristic
52C2×#803C2211yescharacteristic
53C4×Dic12—24no—
54#4651—21no—
55#4650—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#440—31no—
63C2×C4×Q16—31no—
64#2326—31no—
65#2320—31no—
66#2083—31no—
67#2072—31no—
68#2068—31no—
69#2067—31no—
70#2056—31no—
71#2055—31no—
72#2042—31no—
73#2015—31no—
74#1768—31no—
75#1763—31no—
76C4×#102D311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95#936—22no—
96#936—22no—
97C2×#214—22no—
98C2×#214—22no—
99C2×#214—22no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×Dic12—22no—
115C2×Dic12—22no—
116C2×Dic12—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C4×Dic6D414yes—
121C3×#102—22no—
122C3×#102—22no—
123C3×C4×C8—22no—
124C3×#95—21no—
125#832—22no—
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⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#818—22no—
139#816—22no—
140#816—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#810—22no—
144#810—22no—
145#810—22no—
146#807—22no—
147#803—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4C2×D311yescharacteristic
154#440—64no—
155C4×Q16—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×C4×C8C2×D311yescharacteristic
170#391—31no—
171#391—31no—
172#391—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#383—61no—
184#382—61no—
185C4×C4⋊C4C2×D311yescharacteristic
186#376—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C2×D412yes—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8#4112yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C22⋊C4—42no—
242C3×C42—81no—
243C3×C42C2×D412yes—
244C3×C42#4112yes—
245#217—24no—
246#217—24no—
247#217—81no—
248#217—81no—
249#214—28no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3C2×D412yes—
254Dic3⋊C4—24no—
255Dic3⋊C4—42no—
256Dic3⋊C4—81no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—24no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8#4112yes—
272Dic12—28no—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16—62no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287#124—62no—
288#124—62no—
289C4×Q8—34no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×D4—22no—
294C4×D4—22no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107—62no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#102—22no—
327#102—22no—
328#102—62no—
329#97—62no—
330C4×C8—22no—
331#95—21no—
332#95—31no—
333#95—31no—
334#95—31no—
335#95—31no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—41no—
355C3×D4—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—84no—
370C2×C3×C4#12112yes—
371C2×C3×C4#12112yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—22no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C24—21no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4#23512yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—42no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—68no—
451SD16—68no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8#23512yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—64no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C22⋊C4—42no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—64no—
486C42—61no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—81no—
491C42—122no—
492C42#23512yes—
493C42D3×D412yes—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12111yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#53211yescharacteristic
504C22×C3#57611yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57811yescharacteristic
507C22×C3#57811yescharacteristic
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52511yescharacteristic
514C3×C4#52511yescharacteristic
515C3×C4#52911yescharacteristic
516C3×C4#52911yescharacteristic
517Dic3—41no—
518Dic3—41no—
519Dic3—42no—
520Dic3—42no—
521Dic3—41no—
522Dic3—41no—
523Dic3—42no—
524Dic3—81no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533Q8—62no—
534Q8—62no—
535Q8—62no—
536Q8—62no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540Q8—121no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—84no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4#88412yes—
584C2×C4#88412yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4C2×#23511yescharacteristic
588C2×C4C2×D3×D411yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—121no—
595C2×C3—82no—
596C2×C3#239211yescharacteristic
597C2×C3#304411yescharacteristic
598C2×C3#304511yescharacteristic
599C2×C3#312011yescharacteristic
600C2×C3#312411yescharacteristic
601C2×C3#317511yescharacteristic
602C2×C3#317811yescharacteristic
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22C2×#88411yescharacteristic
611C22#554711yescharacteristic
612C22#554811yescharacteristic
613C22#575711yescharacteristic
614C22#576211yescharacteristic
615C22#577111yescharacteristic
616C22#577111yescharacteristic
617C4—22no—
618C4—22no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—121no—
623C4—121no—
624C4—122no—
625C4—122no—
626C4—122no—
627C4—121no—
628C4—121no—
629C4—241no—
630C4#547911yescharacteristic
631C4#552811yescharacteristic
632C4#554711yescharacteristic
633C4#554711yescharacteristic
634C3#2369511yescharacteristic, complemented, Sylow
635C2—82no—
636C2#7358111yescharacteristic
637C2#7876711yescharacteristic
638C2#7886511yescharacteristic
639C2#7959711yescharacteristic
640C2#7960011yescharacteristic
641C2#7965711yescharacteristic
642C2#7966811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC4×#102D3
hypocenterC3#23695
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic12C2semidirect
4. #73625C2semidirect
10. #69755C2semidirect
11. #69475C2semidirect
12. #68914C2semidirect
13. #68913C2semidirect
14. #68864C2semidirect
15. #68834C2semidirect
634. C3#23695semidirect

Maximal subgroups

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

subgroupindex
#823282
C2×#50182
C2×C4×Dic122
#736252
C3×C4×#1022
#702172
#702162
#702152
#701802
#697552
#694752
#689142
#689132
#688642
#688342
#236953