Finite Group Databasebeta

← #10824493

Subgroups — #10824493

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
1C2×Dic3×SD16C211yesmaximal, characteristic
2C2×#4757C211yesmaximal, characteristic
3#76485C211yesmaximal, characteristic, complemented
4#76295C211yesmaximal, characteristic
5C3×#2055C211yesmaximal, characteristic
6#72057C211yesmaximal, characteristic
7#70488C211yesmaximal, characteristic, complemented
8#70442C211yesmaximal, characteristic, complemented
9#70284C211yesmaximal, characteristic, complemented
10#70220C211yesmaximal, characteristic, complemented
11#70166C211yesmaximal, characteristic
12#69683C211yesmaximal, characteristic, complemented
13#69651C211yesmaximal, characteristic
14#68836C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#23689—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32Dic3×SD16—24no—
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4945—21no—
44#4944—21no—
45#4943—21no—
46#4942—21no—
47#4932C2211yescharacteristic
48#4929C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4757—24no—
52#4655—21no—
53#4655—21no—
54#4652—21no—
55#4652—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#440—31no—
63C2×C4×SD16—31no—
64#2327—31no—
65#2319—31no—
66#2085—31no—
67#2073—31no—
68#2068—31no—
69#2066—31no—
70#2055—31no—
71#2055D311yescharacteristic
72#2043—31no—
73#2013—31no—
74#1768—31no—
75#1763—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96#944—22no—
97#944—22no—
98D4×Dic3—22no—
99D4×Dic3D414yes—
100C2×#214—22no—
101C2×#214—22no—
102C2×#214—22no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#887—22no—
120#887—22no—
121C3×C4⋊C8—22no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#95—21no—
125C3×#95—21no—
126#835—22no—
127#835—22no—
128#832—22no—
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
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#817—22no—
141#816—22no—
142#816—22no—
143#815—22no—
144C8×Dic3—22no—
145#810—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C22×SD16—31no—
149C22×SD16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391C2×D311yescharacteristic
172#390—31no—
173#390C2×D311yescharacteristic
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#387—61no—
181#387—61no—
182#382—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—24no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—81no—
218C22×Dic3—81no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C3×SD16—28no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8#4112yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C22⋊C4—81no—
241C3×C22⋊C4—81no—
242C3×C42—21no—
243C3×C42—21no—
244#217—24no—
245#217—81no—
246#217—81no—
247#214—28no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—42no—
254C4⋊Dic3C2×D412yes—
255Dic3⋊C4—24no—
256Dic3⋊C4—24no—
257Dic3⋊C4—42no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C4×Dic3—81no—
265C4×Dic3C2×D412yes—
266C4×Dic3#4112yes—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×D4C22×D311yescharacteristic
276C23×C4—61no—
277C2×SD16—22no—
278C2×SD16—22no—
279C2×SD16—22no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C4⋊Q8—62no—
288C4⋊Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×D4—34no—
294C4×D4—62no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#106—62no—
319C4⋊C8—22no—
320C4⋊C8—62no—
321#103—22no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#102—22no—
328#102—62no—
329#97—62no—
330C4×C8—62no—
331#95—21no—
332#95—21no—
333#95—31no—
334#95—31no—
335#95—31no—
336#95—31no—
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×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—41no—
359C3×D4—41no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
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—
402C2×Dic3—41no—
403C2×Dic3—84no—
404C2×Dic3#12112yes—
405C2×Dic3#12112yes—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C24—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8#23712yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4D3×D412yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—64no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—28no—
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#23712yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C22⋊C4—64no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—21no—
486C42—21no—
487C42—32no—
488C42—32no—
489C42—61no—
490C42—61no—
491C42—122no—
492C42—122no—
493C42—241no—
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#57711yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57711yescharacteristic
507C22×C3#57811yescharacteristic
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52511yescharacteristic
514C3×C4#52511yescharacteristic
515C3×C4#52911yescharacteristic
516C3×C4#52911yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—41no—
520Dic3—41no—
521Dic3—42no—
522Dic3—42no—
523Dic3—81no—
524Dic3—81no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—62no—
538Q8—62no—
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—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
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—62no—
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—121no—
584C2×C4—244no—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4C2×#23711yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—121no—
595C2×C3—82no—
596C2×C3#239211yescharacteristic
597C2×C3#304211yescharacteristic
598C2×C3#304611yescharacteristic
599C2×C3#312111yescharacteristic
600C2×C3#312611yescharacteristic
601C2×C3#317411yescharacteristic
602C2×C3#317811yescharacteristic
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#559611yescharacteristic
611C22D3×#12111yescharacteristic
612C22#560611yescharacteristic
613C22#576211yescharacteristic
614C22#577011yescharacteristic
615C22#577511yescharacteristic
616C22#577911yescharacteristic
617C4—42no—
618C4—41no—
619C4—41no—
620C4—62no—
621C4—62no—
622C4—81no—
623C4—81no—
624C4—122no—
625C4—122no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4#558611yescharacteristic
631C4#564911yescharacteristic
632C4#565611yescharacteristic
633C4#568011yescharacteristic
634C3#2368911yescharacteristic, complemented, Sylow
635C2—82no—
636C2#7689011yescharacteristic
637C2#8036511yescharacteristic
638C2#8043111yescharacteristic
639C2#8053711yescharacteristic
640C2#8057211yescharacteristic
641C2#8063511yescharacteristic
642C2#8078811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2055C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2055D3
hypocenterC3#23689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
3. #76485C2semidirect
7. #70488C2semidirect
8. #70442C2semidirect
9. #70284C2semidirect
10. #70220C2semidirect
12. #69683C2semidirect
14. #68836C2semidirect
15. #68781C2semidirect
634. C3#23689semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
C2×#47572
#764852
#762952
C3×#20552
#720572
#704882
#704422
#702842
#702202
#701662
#696832
#696512
#688362
#687812
#236893