Finite Group Databasebeta

← #10877342

Subgroups — #10877342

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic
2#82233C211yesmaximal, characteristic
3#71693C211yesmaximal, characteristic
4#71692C211yesmaximal, characteristic
5#71627C211yesmaximal, characteristic, complemented
6#71626C211yesmaximal, characteristic, complemented
7#71583C211yesmaximal, characteristic
8#69686C211yesmaximal, characteristic, complemented
9#69684C211yesmaximal, characteristic, complemented
10#69613C211yesmaximal, characteristic, complemented
11#69611C211yesmaximal, characteristic, complemented
12#69506C211yesmaximal, characteristic
13#69504C211yesmaximal, characteristic
14#69468C211yesmaximal, characteristic, complemented
15#69466C211yesmaximal, characteristic, complemented
16#27690—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C3×#391C2211yescharacteristic
23C3×#389—21no—
24C3×#389C2211yescharacteristic
25C3×#388—21no—
26C3×#388—21no—
27C3×#387—21no—
28C3×#380C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#4984—21no—
37#4984—21no—
38C2×#808C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4950—21no—
43#4950—21no—
44#4936C2211yescharacteristic
45#4936C2211yescharacteristic
46#4935C2211yescharacteristic
47#4935C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2323D311yescharacteristic
61#2317—31no—
62#2101—31no—
63#2099—31no—
64#2097—31no—
65#2095—31no—
66#1949—31no—
67#1947—31no—
68#1873—31no—
69#1872—31no—
70#1854—31no—
71#1852—31no—
72#1805—31no—
73#1803—31no—
74#1798—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C23×Dic3—21no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8Q814yes—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101#926—41no—
102#926—41no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—41no—
109C2×C4⋊Dic3—41no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124#835—41no—
125#835—41no—
126#835—41no—
127#835—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#830—22no—
134#808—41no—
135#807—41no—
136#807—41no—
137#807—41no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×C22⋊Q8C2×D311yescharacteristic
141C2×C4×Q8C2×D311yescharacteristic
142C2×#117—31no—
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#107—31no—
148C2×#106—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157#391C2×D311yescharacteristic
158#389—21no—
159#389—61no—
160#389—61no—
161#389C2×D311yescharacteristic
162#388—21no—
163#388—21no—
164#387—21no—
165#385—61no—
166#385—61no—
167#384—31no—
168#384—31no—
169#382—31no—
170#382—31no—
171#380C2×D311yescharacteristic
172C4×C4⋊C4C2×D311yescharacteristic
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×Q812yes—
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—42no—
211C22×Dic3—81no—
212C22×Dic3—81no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4C2×Q812yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—81no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—42no—
231C3×C42—81no—
232C3×C42#4112yes—
233C3×C42#4112yes—
234#217—42no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—81no—
245C4⋊Dic3—81no—
246C4⋊Dic3—81no—
247C4⋊Dic3—81no—
248C4×Dic3—42no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×Q8C22×D311yescharacteristic
262C23×C4—61no—
263C23×C4C22×D311yescharacteristic
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C22⋊Q8—22no—
269C22⋊Q8Dic614yes—
270C4×Q8—22no—
271C4×Q8—22no—
272#117—22no—
273#117—22no—
274#117—121no—
275#117—121no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—121no—
289C2×C4⋊C4—121no—
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4#6711yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4#6711yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#107—121no—
307#107—121no—
308#106—121no—
309C4⋊C8—62no—
310C4⋊C8—62no—
311#103—121no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3C22⋊Q812yes—
333C23×C3C22×D411yescharacteristic
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—41no—
337C3×Q8—41no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C22⋊Q812yes—
367C2×C3×C4#12312yes—
368C2×C3×C4#12312yes—
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
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—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—84no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24C2×#6711yescharacteristic
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C2×Dic612yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—122no—
412C22×C4—241no—
413C22×C4—241no—
414C22×C4C2×Dic612yes—
415C22×C4#23512yes—
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C23×D311yescharacteristic
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—81no—
452C4⋊C4—122no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4—241no—
457C4⋊C4C2×Dic612yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—81no—
462C22⋊C4—122no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—81no—
467C42—122no—
468C42#23712yes—
469C42#23712yes—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×C22⋊Q811yescharacteristic
478C22×C3C2×#12311yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3#57611yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C22×C3#57811yescharacteristic
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4#55311yescharacteristic
494C3×C4#55311yescharacteristic
495C3×C4#55411yescharacteristic
496C3×C4#55411yescharacteristic
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23#92412yes—
508C23C22×#6711yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—41no—
512Q8—41no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
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—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—244no—
554C2×C4#92412yes—
555C2×C4#93512yes—
556C2×C4#93512yes—
557C2×C4C22×Dic611yescharacteristic
558C2×C4C2×#23511yescharacteristic
559C2×C4C2×#23711yescharacteristic
560C2×C4C2×#23711yescharacteristic
561C2×C4#100911yescharacteristic
562C2×C4#100911yescharacteristic
563C8—121no—
564C8—121no—
565C8—121no—
566C8—121no—
567C2×C3—42no—
568C2×C3#248711yescharacteristic
569C2×C3#308211yescharacteristic
570C2×C3#308311yescharacteristic
571C2×C3#309711yescharacteristic
572C2×C3#309811yescharacteristic
573C2×C3#322311yescharacteristic
574C2×C3#322411yescharacteristic
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×#92411yescharacteristic
583C22C2×#93511yescharacteristic
584C22#583011yescharacteristic
585C22#583511yescharacteristic
586C22#583711yescharacteristic
587C22#583811yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—241no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#554311yescharacteristic
603C4#554311yescharacteristic
604C4#556511yescharacteristic
605C4#556511yescharacteristic
606C3#2769011yescharacteristic, complemented, Sylow
607C2—42no—
608C2#8237411yescharacteristic
609C2#8255011yescharacteristic
610C2#8255911yescharacteristic
611C2#8339711yescharacteristic
612C2#8341811yescharacteristic
613C2#8345211yescharacteristic
614C2#8346311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2323D3
hypocenterC3#27690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
5. #71627C2semidirect
6. #71626C2semidirect
8. #69686C2semidirect
9. #69684C2semidirect
10. #69613C2semidirect
11. #69611C2semidirect
14. #69468C2semidirect
15. #69466C2semidirect
606. C3#27690semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
#822332
#716932
#716922
#716272
#716262
#715832
#696862
#696842
#696132
#696112
#695062
#695042
#694682
#694662
#276903