Finite Group Databasebeta

← #10798342

Subgroups — #10798342

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×Dic3×SD16C211yesmaximal, characteristic
2#82450C211yesmaximal, characteristic, complemented
3#76295C211yesmaximal, characteristic
4C3×#1880C211yesmaximal, characteristic
5#72048C211yesmaximal, characteristic
6#71629C211yesmaximal, characteristic, complemented
7#70237C211yesmaximal, characteristic, complemented
8#70220C211yesmaximal, characteristic, complemented
9#70174C211yesmaximal, characteristic
10#70152C211yesmaximal, characteristic
11#69637C211yesmaximal, characteristic
12#68818C211yesmaximal, characteristic, complemented
13#68784C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68598C211yesmaximal, characteristic, complemented
16#21131—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×#835C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30Dic3×SD16C418yescomplemented
31C2×#817C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34Dic3×C4⋊C4—21no—
35Dic3×C4⋊C4—21no—
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4945—21no—
43#4944—21no—
44#4943—21no—
45#4942—21no—
46C4×C4⋊Dic3—21no—
47C4×C4⋊Dic3C2211yescharacteristic
48#4932—21no—
49#4932C2211yescharacteristic
50#4928C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61C2×C4×SD16—31no—
62#2319—31no—
63#2276—31no—
64#1926—31no—
65#1926—31no—
66#1920—31no—
67#1920—31no—
68#1880—31no—
69#1880D311yescharacteristic
70#1865—31no—
71#1865—31no—
72C4×#106—31no—
73#1763—31no—
74#1762—31no—
75#1746—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×Dic3—21no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16C2×C414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99Q8×Dic3C2×C414yes—
100D4×Dic3—22no—
101D4×Dic3C2×C414yes—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C3×#103—22no—
121C3×#102—22no—
122#835—22no—
123#835C2×C414yes—
124#832—22no—
125#832C2×C414yes—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136#817C2×C414yes—
137#815—22no—
138C8×Dic3C2×C414yes—
139#808—22no—
140#808—22no—
141C22×SD16C2×D311yescharacteristic
142C2×C4×Q8—31no—
143C2×C4×Q8—31no—
144C2×C4×Q8C2×D311yescharacteristic
145C2×C4×D4—31no—
146C4×SD16—38no—
147C2×#106—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#102—31no—
154C2×#102—31no—
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—31no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4—61no—
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×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C22×C412yes—
190C2×C3×Q8#4112yes—
191C2×C3×Q8#4112yes—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4C22×C412yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—24no—
208C22×Dic3—21no—
209C22×Dic3—81no—
210C22×Dic3—81no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C3×SD16—28no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C22×C412yes—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4#4112yes—
227C3×C22⋊C4—81no—
228C3×C22⋊C4—81no—
229C3×C42—24no—
230C3×C42—21no—
231C3×C42—21no—
232#217—24no—
233#217—81no—
234#217—81no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—24no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3C22×C412yes—
243C4⋊Dic3C22×C412yes—
244C4⋊Dic3#4112yes—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247C4×Dic3—24no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—24no—
251C4×Dic3—81no—
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×Q8C22×D311yescharacteristic
264C22×D4C22×D311yescharacteristic
265C23×C4—61no—
266C2×SD16—22no—
267C2×SD16—22no—
268C2×SD16C4×D314yes—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—34no—
276C4×Q8—62no—
277C4×Q8—62no—
278C4×Q8—62no—
279C4×D4—34no—
280C4×D4—62no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
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#106—34no—
308#106—62no—
309#106—62no—
310#103—22no—
311#103—34no—
312#103—62no—
313#102—22no—
314#102—34no—
315#102—62no—
316#97—62no—
317C4×C8—34no—
318#95—31no—
319#95—31no—
320#95—31no—
321#95—31no—
322#95—61no—
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×Q8—22no—
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8#11714yes—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—41no—
343C3×D4—41no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4C23×C411yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—84no—
388C2×Dic3C4×D412yes—
389C2×Dic3C4×D412yes—
390C3×C8—22no—
391C3×C8—41no—
392C3⋊C8—22no—
393C3⋊C8—41no—
394C24—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C2×C4×D312yes—
399C2×Q8#23712yes—
400C2×Q8#23712yes—
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—28no—
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—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—32no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23712yes—
454C22⋊C4—64no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—24no—
460C42—21no—
461C42—21no—
462C42—32no—
463C42—32no—
464C42—64no—
465C42—64no—
466C42—61no—
467C42—61no—
468C42—241no—
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#57711yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57711yescharacteristic
485C22×C3#57711yescharacteristic
486C3×C4—24no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4C2×#11711yescharacteristic
492C3×C4C4×#4111yescharacteristic
493C3×C4#52911yescharacteristic
494C3×C4#52911yescharacteristic
495Dic3—22no—
496Dic3—22no—
497Dic3—41no—
498Dic3—42no—
499Dic3—41no—
500Dic3—42no—
501Dic3—42no—
502Dic3—81no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D3×D411yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—22no—
514Q8#87714yes—
515D4—22no—
516D4—22no—
517D4—41no—
518D4—41no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—61no—
549C2×C4—121no—
550C2×C4—121no—
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×C4C2×#23711yescharacteristic
562C2×C4C2×#23711yescharacteristic
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#295211yescharacteristic
573C2×C3#312511yescharacteristic
574C2×C3#312511yescharacteristic
575C2×C3#312611yescharacteristic
576C2×C3#312611yescharacteristic
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C4×D3×D411yescharacteristic
585C22#558611yescharacteristic
586C22#560611yescharacteristic
587C22#576211yescharacteristic
588C22#577011yescharacteristic
589C22#577511yescharacteristic
590C22#577811yescharacteristic
591C4—24no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—62no—
596C4—62no—
597C4—81no—
598C4—122no—
599C4—122no—
600C4—121no—
601C4—122no—
602C4—121no—
603C4—241no—
604C4C2×#87711yescharacteristic
605C4C4×#23711yescharacteristic
606C4#560011yescharacteristic
607C4#564911yescharacteristic
608C3#2113111yescharacteristic, complemented, Sylow
609C2—82no—
610C2#7465711yescharacteristic
611C2#7965011yescharacteristic
612C2#7977711yescharacteristic
613C2#8003311yescharacteristic
614C2#8007511yescharacteristic
615C2#8036511yescharacteristic
616C2#8057511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1880D3
hypocenterC3#21131
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. #82450C2semidirect
6. #71629C2semidirect
7. #70237C2semidirect
8. #70220C2semidirect
12. #68818C2semidirect
13. #68784C2semidirect
14. #68623C2semidirect
15. #68598C2semidirect
30. Dic3×SD16C4semidirect
608. C3#21131semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
#824502
#762952
C3×#18802
#720482
#716292
#702372
#702202
#701742
#701522
#696372
#688182
#687842
#686232
#685982
#211313