Finite Group Databasebeta

← #10798333

Subgroups — #10798333

Order 768.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4788C211yesmaximal, characteristic
2#76483C211yesmaximal, characteristic, complemented
3#73751C211yesmaximal, characteristic
4C3×#1928C211yesmaximal, characteristic, complemented
5#70638C211yesmaximal, characteristic
6Dic3×#107C211yesmaximal, characteristic, complemented
7#70276C211yesmaximal, characteristic
8#70224C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#69687C211yesmaximal, characteristic
11#69460C211yesmaximal, characteristic, complemented
12#68789C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic
14#68670C211yesmaximal, characteristic
15#68598C211yesmaximal, characteristic, complemented
16#21130—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C4×(C2×D3)⋊C4—21no—
41C4×(C2×D3)⋊C4—21no—
42#4932C2211yescharacteristic
43#4928C2211yescharacteristic
44#4788C418yescomplemented
45#4666—21no—
46D3×#95—21no—
47#4654—21no—
48#4653—21no—
49#4651—21no—
50#4650—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#2320—31no—
63#2275—31no—
64#1928—31no—
65#1928D311yescharacteristic, complemented
66#1920—31no—
67#1920—31no—
68#1877—31no—
69#1877—31no—
70#1862—31no—
71#1862—31no—
72C4×#107—31no—
73#1763—31no—
74#1762—31no—
75#1746—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88Q8×Dic3—22no—
89Q8×Dic3C2×C414yescomplemented
90C2×#215—22no—
91C2×#215—22no—
92C2×#215C2×C414yes—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#893—22no—
111#893—22no—
112#893—22no—
113#893C2×C414yes—
114C4×D12—22no—
115C4×D12—22no—
116C3×#107—22no—
117C3×#103—22no—
118C3×#103C2×C414yescomplemented
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122#835—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#821—22no—
135#821C2×C414yes—
136#818—22no—
137#815—22no—
138C8×Dic3C2×C414yescomplemented
139#809—22no—
140#808C2×C414yescomplemented
141C22×SD16—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—31no—
146C4×SD16—38no—
147C2×#107—31no—
148C2×#107C2×D311yescharacteristic
149C2×#106—31no—
150C2×#103—31no—
151C2×#103C2×D311yescharacteristic
152C2×#103C2×D311yescharacteristic
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#97—31no—
157C2×C4×C8—31no—
158#391—31no—
159#391C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#387—61no—
163#387—61no—
164#383—61no—
165#382—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4C2×D311yescharacteristic
170C4×C22⋊C4—61no—
171C4×C22⋊C4—61no—
172C4×C22⋊C4—61no—
173#376—31no—
174C2×#95—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
185C23×D3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8C22×C412yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12C22×C412yes—
210C2×D12#4112yes—
211C2×D12#4112yes—
212C2×C4×D3—24no—
213C2×C4×D3—24no—
214C2×C4×D3—24no—
215C2×C4×D3—81no—
216C2×C4×D3—81no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C22×C412yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C22×C412yes—
232C3×C4⋊C4#4112yes—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—81no—
236#215—28no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—24no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3C22×C412yes—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
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×Q8C22×D311yescharacteristic
264C22×D4—31no—
265C23×C4—61no—
266C23×C4—61no—
267C23×C4—61no—
268C2×SD16—34no—
269C2×SD16—62no—
270C2×SD16—62no—
271C22×C8—31no—
272C22×C8—31no—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4×Q8—34no—
276C4×Q8—62no—
277C4×D4—34no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
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⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—21no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—22no—
304#107—62no—
305#106—34no—
306#103—22no—
307#103—62no—
308#103C4×D314yescomplemented
309#102—34no—
310#102—62no—
311#102—62no—
312#97—62no—
313C4×C8—34no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—31no—
320#95—31no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3C22×D411yescharacteristic
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×Q8—41no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
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×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4C23×C411yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
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×Dic3C4×D412yes—
384C2×Dic3C4×D412yes—
385D12—22no—
386D12—22no—
387D12—22no—
388D12#11714yes—
389C4×D3—84no—
390C4×D3—84no—
391C4×D3—84no—
392C3×C8—22no—
393C3×C8—41no—
394C3⋊C8—22no—
395C3⋊C8—41no—
396C24—61no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×C4×D312yes—
403C2×D4—32no—
404C2×D4—32no—
405C2×D4—32no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—64no—
418C22×C4—64no—
419C22×C4—64no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—241no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4—241no—
427C22×C4—241no—
428C22×C4—241no—
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4C23×D311yescharacteristic
432SD16—68no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—32no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8C2×C4×D312yes—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—32no—
451C4⋊C4—64no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—241no—
457C4⋊C4—241no—
458C4⋊C4C2×C4×D312yes—
459C4⋊C4#23712yes—
460C22⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—64no—
468C42—61no—
469C42—61no—
470C42—81no—
471C42—241no—
472C42—241no—
473C42—241no—
474C22×C3C2×C4×D411yescharacteristic
475C22×C3#52511yescharacteristic
476C22×C3#52511yescharacteristic
477C22×C3#57611yescharacteristic
478C22×C3#57711yescharacteristic
479C22×C3#57711yescharacteristic
480C22×C3#57811yescharacteristic
481C2×D3—24no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—81no—
494C3×C4C2×#11711yescharacteristic
495C3×C4C4×#4111yescharacteristic
496C3×C4#52911yescharacteristic
497C3×C4#52911yescharacteristic
498Dic3—22no—
499Dic3—22no—
500Dic3—41no—
501Dic3—42no—
502Dic3—41no—
503Dic3—42no—
504Dic3—81no—
505C23—62no—
506C23—62no—
507C23—62no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—121no—
512C23C2×D3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517D4—34no—
518D4—62no—
519D4—62no—
520D4—62no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—32no—
531C2×C4—32no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—244no—
561C2×C4—244no—
562C2×C4—244no—
563C2×C4C22×C4×D311yescharacteristic
564C2×C4C2×#23711yescharacteristic
565C2×C4D3×#4111yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C8—22no—
570C8—41no—
571C8—62no—
572C8—121no—
573C2×C3#229011yescharacteristic
574C2×C3#294911yescharacteristic
575C2×C3#295111yescharacteristic
576C2×C3#311811yescharacteristic
577C2×C3#311811yescharacteristic
578C2×C3#312011yescharacteristic
579C2×C3#312011yescharacteristic
580D3—44no—
581C22—64no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22—122no—
588C22C4×D3×D411yescharacteristic
589C22#560611yescharacteristic
590C22#565611yescharacteristic
591C22#576611yescharacteristic
592C22#576911yescharacteristic
593C22#577511yescharacteristic
594C22#577811yescharacteristic
595C4—41no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—62no—
601C4—62no—
602C4—81no—
603C4—121no—
604C4—122no—
605C4—122no—
606C4—121no—
607C4—241no—
608C4D3×#11711yescharacteristic
609C4C4×#23711yescharacteristic
610C4#568011yescharacteristic
611C4#570611yescharacteristic
612C3#2113011yescharacteristic, complemented, Sylow
613C2—124no—
614C2#7544911yescharacteristic
615C2#7972111yescharacteristic
616C2#7977711yescharacteristic
617C2#8046011yescharacteristic
618C2#8050811yescharacteristic
619C2#8145611yescharacteristic
620C2#8149611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1928C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1928D3
hypocenterC3#21130
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
2. #76483C2semidirect
4. C3×#1928C2semidirect
6. Dic3×#107C2semidirect
8. #70224C2semidirect
9. #70220C2semidirect
11. #69460C2semidirect
12. #68789C2semidirect
15. #68598C2semidirect
44. #4788C4semidirect
65. #1928D3semidirect
89. Q8×Dic3C2×C4semidirect
118. C3×#103C2×C4semidirect
138. C8×Dic3C2×C4semidirect
140. #808C2×C4semidirect
308. #103C4×D3semidirect
612. C3#21130semidirect

Maximal subgroups

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

subgroupindex
C2×#47882
#764832
#737512
C3×#19282
#706382
Dic3×#1072
#702762
#702242
#702202
#696872
#694602
#687892
#687822
#686702
#685982
#211303