Finite Group Databasebeta

← #10798327

Subgroups — #10798327

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
1#82317C211yesmaximal, characteristic
2C2×#4758C211yesmaximal, characteristic
3#75970C211yesmaximal, characteristic, complemented
4C3×#1862C211yesmaximal, characteristic
5#71612C211yesmaximal, characteristic
6#70624C211yesmaximal, characteristic, complemented
7Dic3×#107C211yesmaximal, characteristic, complemented
8#70171C211yesmaximal, characteristic
9#70168C211yesmaximal, characteristic
10#70152C211yesmaximal, characteristic
11#69602C211yesmaximal, characteristic, complemented
12#68819C211yesmaximal, characteristic, complemented
13#68783C211yesmaximal, characteristic, complemented
14#68628C211yesmaximal, characteristic, complemented
15#68598C211yesmaximal, characteristic, complemented
16#21130—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C4×#217—21no—
28C2×C3⋊#95—21no—
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4984C2211yescharacteristic
35#4978—21no—
36#4977—21no—
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43Dic3×C22⋊C4—21no—
44Dic3×C22⋊C4—21no—
45C4×C4⋊Dic3C2211yescharacteristic
46#4932C2211yescharacteristic
47#4929C2211yescharacteristic
48#4758C418yescomplemented
49#4655—21no—
50#4651—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#1928—31no—
66#1920—31no—
67#1920—31no—
68#1877—31no—
69#1877—31no—
70#1862—31no—
71#1862D311yescharacteristic
72C4×#107—31no—
73#1763—31no—
74#1762—31no—
75#1746—31no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×Dic3—21no—
79C23×Dic3—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95D4×Dic3—22no—
96D4×Dic3C2×C414yes—
97C2×#214—22no—
98C2×#214—22no—
99C2×#214C2×C414yes—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—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#887C2×C414yescomplemented
116C3×#107—22no—
117C3×#102—22no—
118C3×#102C2×C414yes—
119C3×#95—21no—
120#832—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
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#818—22no—
135#816—22no—
136#816C2×C414yescomplemented
137#815—22no—
138C8×Dic3C2×C414yescomplemented
139#810—22no—
140#808C2×C414yescomplemented
141C22×SD16—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4C2×D311yescharacteristic
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#391—31no—
159#391—31no—
160#389—31no—
161#389C2×D311yescharacteristic
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⋊C4—31no—
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
185C24×C3—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4C22×C412yes—
190C2×C3×D4#4112yes—
191C2×C3×D4#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—21no—
196C22×C3×C4—81no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—21no—
207C22×Dic3—81no—
208C22×Dic3—81no—
209C22×Dic3—81no—
210C22×Dic3—81no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C22×C412yes—
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⋊C4—81no—
232C3×C4⋊C4C22×C412yes—
233C3×C4⋊C4#4112yes—
234C3×C22⋊C4—24no—
235C3×C42—21no—
236C3×C42—21no—
237#217—24no—
238#217—24no—
239#214—28no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—81no—
245C4⋊Dic3C22×C412yes—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—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×Q8—31no—
264C22×D4C22×D311yescharacteristic
265C23×C4—21no—
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—22no—
278C4×D4—22no—
279C4×D4—34no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
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—61no—
294C2×C22⋊C4—61no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—61no—
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—34no—
307#103—62no—
308#103—62no—
309#102—22no—
310#102—62no—
311#102C4×D314yes—
312#97—62no—
313C4×C8—34no—
314#95—21no—
315#95—31no—
316#95—31no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C22×D411yescharacteristic
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4#11714yes—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
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—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×C4—84no—
356C2×C3×C4C23×C411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—22no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385C2×Dic3—84no—
386C2×Dic3C4×D412yes—
387C2×Dic3C4×D412yes—
388Dic6—22no—
389Dic6—22no—
390Dic6—41no—
391Dic6—41no—
392C3×C8—22no—
393C3×C8—41no—
394C3⋊C8—22no—
395C3⋊C8—41no—
396C24—21no—
397C2×Q8—32no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C2×C4×D312yes—
407C2×D4#23712yes—
408C2×D4#23912yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—64no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—64no—
422C22×C4—61no—
423C22×C4—81no—
424C22×C4—81no—
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—81no—
457C4⋊C4—241no—
458C4⋊C4C2×C4×D312yes—
459C4⋊C4#23712yes—
460C22⋊C4—24no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C42—21no—
464C42—21no—
465C42—32no—
466C42—32no—
467C42—64no—
468C42—61no—
469C42—61no—
470C42—241no—
471C42—241no—
472C42—241no—
473C42—241no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3C2×C4×D411yescharacteristic
482C22×C3#52511yescharacteristic
483C22×C3#52511yescharacteristic
484C22×C3#57611yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57711yescharacteristic
487C22×C3#57811yescharacteristic
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4C2×#11711yescharacteristic
494C3×C4C4×#4111yescharacteristic
495C3×C4#52911yescharacteristic
496C3×C4#52911yescharacteristic
497Dic3—22no—
498Dic3—22no—
499Dic3—41no—
500Dic3—42no—
501Dic3—41no—
502Dic3—42no—
503Dic3—42no—
504Dic3—81no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C2×D3×D411yescharacteristic
513Q8—62no—
514Q8—62no—
515Q8—121no—
516Q8—121no—
517D4—22no—
518D4—22no—
519D4—22no—
520D4#89214yes—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
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—41no—
537C2×C4—41no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—62no—
551C2×C4—84no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—244no—
562C2×C4—244no—
563C2×C4C22×C4×D311yescharacteristic
564C2×C4C2×#23711yescharacteristic
565C2×C4C2×#23711yescharacteristic
566C2×C4C2×#23911yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C8—22no—
570C8—41no—
571C8—62no—
572C8—121no—
573C2×C3—44no—
574C2×C3#229011yescharacteristic
575C2×C3#294911yescharacteristic
576C2×C3#295111yescharacteristic
577C2×C3#311811yescharacteristic
578C2×C3#311811yescharacteristic
579C2×C3#312011yescharacteristic
580C2×C3#312011yescharacteristic
581C22—24no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22C4×D3×D411yescharacteristic
589C22#558611yescharacteristic
590C22#562711yescharacteristic
591C22#575711yescharacteristic
592C22#577111yescharacteristic
593C22#577511yescharacteristic
594C22#577811yescharacteristic
595C4—41no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—62no—
600C4—62no—
601C4—81no—
602C4—121no—
603C4—122no—
604C4—122no—
605C4—122no—
606C4—121no—
607C4—241no—
608C4C2×#89211yescharacteristic
609C4C4×#23711yescharacteristic
610C4#562411yescharacteristic
611C4#564911yescharacteristic
612C3#2113011yescharacteristic, complemented, Sylow
613C2—44no—
614C2#7437711yescharacteristic
615C2#7958911yescharacteristic
616C2#7977711yescharacteristic
617C2#7999211yescharacteristic
618C2#8021711yescharacteristic
619C2#8095211yescharacteristic
620C2#8114511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1862C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#1862D3
hypocenterC3#21130
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
3. #75970C2semidirect
6. #70624C2semidirect
7. Dic3×#107C2semidirect
11. #69602C2semidirect
12. #68819C2semidirect
13. #68783C2semidirect
14. #68628C2semidirect
15. #68598C2semidirect
48. #4758C4semidirect
115. #887C2×C4semidirect
136. #816C2×C4semidirect
138. C8×Dic3C2×C4semidirect
140. #808C2×C4semidirect
612. C3#21130semidirect

Maximal subgroups

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

subgroupindex
#823172
C2×#47582
#759702
C3×#18622
#716122
#706242
Dic3×#1072
#701712
#701682
#701522
#696022
#688192
#687832
#686282
#685982
#211303