Finite Group Databasebeta

← #10798399

Subgroups — #10798399

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4758C211yesmaximal, characteristic
2#76295C211yesmaximal, characteristic
3#73616C211yesmaximal, characteristic, complemented
4C3×#1925C211yesmaximal, characteristic
5#70642C211yesmaximal, characteristic, complemented
6Dic3×#107C211yesmaximal, characteristic, complemented
7#70213C211yesmaximal, characteristic
8#70155C211yesmaximal, characteristic
9#70152C211yesmaximal, characteristic
10#69660C211yesmaximal, characteristic
11#69459C211yesmaximal, characteristic, complemented
12#68788C211yesmaximal, characteristic, complemented
13#68783C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68598C211yesmaximal, characteristic, complemented
16#21136—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4945—21no—
41#4944—21no—
42#4943—21no—
43#4942—21no—
44#4932C2211yescharacteristic
45C4×Dic3⋊C4—21no—
46C4×Dic3⋊C4—21no—
47#4928C2211yescharacteristic
48#4758C418yescomplemented
49#4649—21no—
50#4648—21no—
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×SD16—31no—
62#2319—31no—
63#2276—31no—
64#1925—31no—
65#1925D311yescharacteristic
66#1919—31no—
67#1919—31no—
68#1880—31no—
69#1880—31no—
70#1863—31no—
71#1863—31no—
72C4×#107—31no—
73#1763—31no—
74#1762—31no—
75#1746—31no—
76C22×C3×D4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91D4×Dic3—22no—
92D4×Dic3C2×C414yes—
93C2×#214—22no—
94C2×#214—22no—
95C2×#214C2×C414yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#887—22no—
115#887—22no—
116#887—22no—
117#887C2×C414yes—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×#107—22no—
121C3×#102—22no—
122C3×#102C2×C414yes—
123C3×#95—21no—
124#832—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#818—22no—
135#816—22no—
136#816C2×C414yes—
137#815—22no—
138C8×Dic3C2×C414yes—
139#810—22no—
140#808C2×C414yes—
141C22×SD16—31no—
142C2×C4×Q8—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×D4—31no—
146C4×SD16—38no—
147C2×#107—31no—
148C2×#107C2×D311yescharacteristic
149C2×#106—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#102—31no—
154C2×#102C2×D311yescharacteristic
155C2×#102C2×D311yescharacteristic
156C2×#97—31no—
157C2×C4×C8—31no—
158#390—31no—
159#390C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#388—61no—
163#388—61no—
164#381—61no—
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4C2×D311yescharacteristic
173#376—31no—
174#376—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C24×C3—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4C22×C412yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—24no—
202C22×Dic3—21no—
203C22×Dic3—81no—
204C22×Dic3—81no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C22×C412yes—
213C2×Dic6#4112yes—
214C2×Dic6#4112yes—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8C22×C412yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4C22×C412yes—
226C3×C4⋊C4#4112yes—
227C3×C22⋊C4—81no—
228C3×C22⋊C4—81no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—81no—
232#217—24no—
233#217—81no—
234#217—81no—
235#214—28no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3C22×C412yes—
241C4⋊Dic3#4112yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—24no—
244Dic3⋊C4—24no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247C4×Dic3—24no—
248C4×Dic3—24no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—24no—
252C4×Dic3—81no—
253C4×Dic3—81no—
254C4×Dic3—81no—
255C4×Dic3C22×C412yes—
256C4×Dic3#4112yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8C22×C412yes—
263C22×Q8—31no—
264C22×D4C22×D311yescharacteristic
265C23×C4—61no—
266C2×SD16—34no—
267C2×SD16—62no—
268C2×SD16—62no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4×Q8—34no—
274C4×Q8—62no—
275C4×Q8—62no—
276C4×Q8—62no—
277C4×Q8—62no—
278C4×Q8—62no—
279C4×D4—34no—
280C4×D4—62no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#107—22no—
308#107—62no—
309#106—34no—
310#103—34no—
311#103—62no—
312#103—62no—
313#102—22no—
314#102—62no—
315#102C4×D314yes—
316#97—62no—
317C4×C8—34no—
318#95—21no—
319#95—31no—
320#95—31no—
321#95—31no—
322#95—31no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3—22no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C22×D411yescharacteristic
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4C23×C411yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—84no—
384C2×Dic3C4×D412yes—
385C2×Dic3C4×D412yes—
386Dic6—22no—
387Dic6—22no—
388Dic6—22no—
389Dic6#11714yes—
390C3×C8—22no—
391C3×C8—41no—
392C3⋊C8—22no—
393C3⋊C8—41no—
394C24—21no—
395C2×Q8—32no—
396C2×Q8—32no—
397C2×Q8—32no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C2×C4×D312yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—64no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4C23×D311yescharacteristic
424SD16—68no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8C2×C4×D312yes—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4C2×C4×D312yes—
453C4⋊C4#23712yes—
454C22⋊C4—64no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—21no—
460C42—21no—
461C42—32no—
462C42—32no—
463C42—64no—
464C42—64no—
465C42—64no—
466C42—61no—
467C42—61no—
468C42—81no—
469C42—241no—
470C42—241no—
471C42—241no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3C2×C4×D411yescharacteristic
480C22×C3#52511yescharacteristic
481C22×C3#52511yescharacteristic
482C22×C3#57611yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57711yescharacteristic
485C22×C3#57811yescharacteristic
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—81no—
491C3×C4C2×#11711yescharacteristic
492C3×C4C4×#4111yescharacteristic
493C3×C4#52911yescharacteristic
494C3×C4#52911yescharacteristic
495Dic3—22no—
496Dic3—22no—
497Dic3—24no—
498Dic3—41no—
499Dic3—42no—
500Dic3—41no—
501Dic3—42no—
502Dic3—81no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D3×D411yescharacteristic
511Q8—34no—
512Q8—62no—
513Q8—62no—
514Q8—62no—
515D4—22no—
516D4—22no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—244no—
559C2×C4C22×C4×D311yescharacteristic
560C2×C4C2×#23711yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C8—22no—
566C8—41no—
567C8—62no—
568C8—121no—
569C2×C3—82no—
570C2×C3#229011yescharacteristic
571C2×C3#294911yescharacteristic
572C2×C3#295111yescharacteristic
573C2×C3#311911yescharacteristic
574C2×C3#311911yescharacteristic
575C2×C3#312111yescharacteristic
576C2×C3#312111yescharacteristic
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C4×D3×D411yescharacteristic
585C22#560611yescharacteristic
586C22#565611yescharacteristic
587C22#575711yescharacteristic
588C22#577111yescharacteristic
589C22#577511yescharacteristic
590C22#577811yescharacteristic
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—62no—
596C4—64no—
597C4—62no—
598C4—81no—
599C4—122no—
600C4—122no—
601C4—121no—
602C4—121no—
603C4—241no—
604C4D3×#11711yescharacteristic
605C4C4×#23711yescharacteristic
606C4#568011yescharacteristic
607C4#570611yescharacteristic
608C3#2113611yescharacteristic, complemented, Sylow
609C2—82no—
610C2#7544911yescharacteristic
611C2#7958911yescharacteristic
612C2#7977711yescharacteristic
613C2#8031611yescharacteristic
614C2#8057311yescharacteristic
615C2#8127211yescharacteristic
616C2#8135311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1925C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1925D3
hypocenterC3#21136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
3. #73616C2semidirect
5. #70642C2semidirect
6. Dic3×#107C2semidirect
11. #69459C2semidirect
12. #68788C2semidirect
13. #68783C2semidirect
14. #68683C2semidirect
15. #68598C2semidirect
48. #4758C4semidirect
608. C3#21136semidirect

Maximal subgroups

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

subgroupindex
C2×#47582
#762952
#736162
C3×#19252
#706422
Dic3×#1072
#702132
#701552
#701522
#696602
#694592
#687882
#687832
#686832
#685982
#211363