Finite Group Databasebeta

← #10824583

Subgroups — #10824583

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×Q16C211yesmaximal, characteristic, complemented
2C2×#4817C211yesmaximal, characteristic
3#76484C211yesmaximal, characteristic
4#76483C211yesmaximal, characteristic, complemented
5C3×#2056C211yesmaximal, characteristic, complemented
6#72072C211yesmaximal, characteristic
7#70484C211yesmaximal, characteristic
8#70442C211yesmaximal, characteristic, complemented
9#70279C211yesmaximal, characteristic
10#70233C211yesmaximal, characteristic, complemented
11#70220C211yesmaximal, characteristic, complemented
12#69683C211yesmaximal, characteristic, complemented
13#69680C211yesmaximal, characteristic
14#68834C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic
16#23695—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32Dic3×Q16—24no—
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4975C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4939C2211yescharacteristic
44#4932C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47#4817—24no—
48#4669—21no—
49#4669—21no—
50#4668—21no—
51#4668—21no—
52#4654—21no—
53#4653—21no—
54#4651—21no—
55#4650—21no—
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
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#2056D311yescharacteristic, complemented
71#2055—31no—
72#2042—31no—
73#2015—31no—
74#1768—31no—
75#1763—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#947—22no—
90#947—22no—
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93Q8×Dic3—22no—
94Q8×Dic3D414yes—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
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#893—22no—
120#893—22no—
121C3×C4⋊C8—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#821—22no—
142#818—22no—
143#815—22no—
144C8×Dic3—22no—
145#809—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×Q16—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170#391—31no—
171#391C2×D311yescharacteristic
172#391C2×D311yescharacteristic
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⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—81no—
228C2×C4×D3—81no—
229C3×Q16—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×C42—21no—
241C3×C42—21no—
242#215—28no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—81no—
247(C2×D3)⋊C4—81no—
248C4⋊Dic3—21no—
249C4⋊Dic3—24no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—81no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—24no—
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×Q8C22×D311yescharacteristic
274C22×Q8C22×D311yescharacteristic
275C22×D4—31no—
276C23×C4—61no—
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—22no—
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—62no—
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—61no—
307C2×C22⋊C4—61no—
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#107—62no—
319C4⋊C8—22no—
320C4⋊C8—62no—
321#103—22no—
322#103—22no—
323#103—62no—
324#103—62no—
325#103—62no—
326#102—62no—
327#102—62no—
328#102—62no—
329#97—62no—
330C4×C8—62no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—31no—
336#95—31no—
337#95—31no—
338#95—31no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
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×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
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#12112yes—
404C2×Dic3#12112yes—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—84no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C24—61no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8D3×D412yes—
426C2×Q8#23712yes—
427C2×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—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—61no—
471C4⋊C4—61no—
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—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—21no—
486C42—21no—
487C42—32no—
488C42—32no—
489C42—64no—
490C42—61no—
491C42—61no—
492C42—122no—
493C42—241no—
494C22×C3C2×#12111yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53211yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
499C22×C3#57811yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52511yescharacteristic
515C3×C4#52511yescharacteristic
516C3×C4#52911yescharacteristic
517C3×C4#52911yescharacteristic
518Dic3—22no—
519Dic3—22no—
520Dic3—41no—
521Dic3—41no—
522Dic3—42no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—61no—
575C2×C4—61no—
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#239211yescharacteristic
596C2×C3#304411yescharacteristic
597C2×C3#304511yescharacteristic
598C2×C3#312011yescharacteristic
599C2×C3#312411yescharacteristic
600C2×C3#317511yescharacteristic
601C2×C3#317811yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#559611yescharacteristic
611C22D3×#12111yescharacteristic
612C22#560611yescharacteristic
613C22#576611yescharacteristic
614C22#576911yescharacteristic
615C22#577811yescharacteristic
616C22#577911yescharacteristic
617C4—42no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—62no—
622C4—62no—
623C4—81no—
624C4—81no—
625C4—121no—
626C4—121no—
627C4—122no—
628C4—241no—
629C4—241no—
630C4#558611yescharacteristic
631C4#564911yescharacteristic
632C4#565611yescharacteristic
633C4#568011yescharacteristic
634C3#2369511yescharacteristic, complemented, Sylow
635C2—242no—
636C2#7689011yescharacteristic
637C2#8046011yescharacteristic
638C2#8047811yescharacteristic
639C2#8057411yescharacteristic
640C2#8060211yescharacteristic
641C2#8074211yescharacteristic
642C2#8089111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2056D3
hypocenterC3#23695
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×Q16C2semidirect
4. #76483C2semidirect
5. C3×#2056C2semidirect
8. #70442C2semidirect
10. #70233C2semidirect
11. #70220C2semidirect
12. #69683C2semidirect
14. #68834C2semidirect
70. #2056D3semidirect
634. C3#23695semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×Q162
C2×#48172
#764842
#764832
C3×#20562
#720722
#704842
#704422
#702792
#702332
#702202
#696832
#696802
#688342
#687802
#236953