Finite Group Databasebeta

← #10824704

Subgroups — #10824704

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic
2C2×C4×#216C211yesmaximal, characteristic, complemented
3#82235C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic
5#71700C211yesmaximal, characteristic
6#71695C211yesmaximal, characteristic
7#71631C211yesmaximal, characteristic, complemented
8#69683C211yesmaximal, characteristic, complemented
9#69681C211yesmaximal, characteristic, complemented
10#69635C211yesmaximal, characteristic, complemented
11#69622C211yesmaximal, characteristic, complemented
12#69527C211yesmaximal, characteristic
13#69518C211yesmaximal, characteristic
14C4×#807C211yesmaximal, characteristic, complemented
15#69475C211yesmaximal, characteristic, complemented
16#23705—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#388—21no—
27C3×#388—21no—
28C3×#387—21no—
29C3×#380C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C4×#216—24no—
38#5000—24no—
39#4978—21no—
40#4976—21no—
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4945—21no—
48#4944—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50#4929C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×Q16—31no—
63C2×#433—31no—
64#2323—31no—
65#2323D311yescharacteristic
66#2086—31no—
67#2075—31no—
68#2068—31no—
69#2068—31no—
70#2056—31no—
71#2036—31no—
72#2036—31no—
73#2001—31no—
74C4×#107—31no—
75#1763—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#217—21no—
103C2×#217—21no—
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—22no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110#924—22no—
111#924—22no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#810—22no—
139#810—22no—
140#810—22no—
141#807—22no—
142#807—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146#803—22no—
147C4×C3⋊C8D414yes—
148C22×Q16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×#117C2×D311yescharacteristic
154C4×Q16—64no—
155#433—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#391—31no—
172#391C2×D311yescharacteristic
173#389—21no—
174#389—31no—
175#389—61no—
176#389C2×D311yescharacteristic
177#388—21no—
178#388—21no—
179#388—61no—
180#388—61no—
181#387—21no—
182#387—61no—
183#380—31no—
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186C4×C4⋊C4C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6#4112yes—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—81no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—42no—
244C3×C42—81no—
245C3×C42C2×D412yes—
246C3×C42#4112yes—
247#217—24no—
248#217—81no—
249#216—28no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—24no—
257Dic3⋊C4—42no—
258Dic3⋊C4—81no—
259C4×Dic3—42no—
260C4×Dic3—81no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272C2×C3⋊C8C2×D412yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C2×Q16—62no—
277C2×Q16—62no—
278C2×Q16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—62no—
286C22⋊Q8—62no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—62no—
290C4×Q8—62no—
291#117—22no—
292#117—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4#6711yescharacteristic
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4#6711yescharacteristic
312C2×C22⋊C4#6711yescharacteristic
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#107—62no—
320#107—62no—
321#107—62no—
322C4⋊C8—62no—
323C4⋊C8—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330C22⋊C8—62no—
331C22⋊C8—62no—
332#97—62no—
333C4×C8—34no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3C22×D411yescharacteristic
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409C3⋊C8—22no—
410C3⋊C8—22no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C24C2×#6711yescharacteristic
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—32no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8#23512yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C23×D311yescharacteristic
446Q16—68no—
447C2×C8—32no—
448C2×C8—32no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—32no—
468C4⋊C4—42no—
469C4⋊C4—64no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—81no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4#23512yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—64no—
483C22⋊C4—81no—
484C22⋊C4—241no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—81no—
489C42—122no—
490C42—241no—
491C42D3×D412yes—
492C42#23712yes—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3C2×#12111yescharacteristic
501C22×C3#52511yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#57611yescharacteristic
504C22×C3#57811yescharacteristic
505C22×C3#57811yescharacteristic
506C22×C3#57811yescharacteristic
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4#52511yescharacteristic
517C3×C4#52911yescharacteristic
518C3×C4#52911yescharacteristic
519C3×C4#53211yescharacteristic
520Dic3—42no—
521Dic3—42no—
522Dic3—81no—
523Dic3—81no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23C22×#6711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—62no—
537Q8—62no—
538Q8—121no—
539Q8—121no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4#93912yes—
582C2×C4#93912yes—
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×#23511yescharacteristic
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—62no—
590C8—62no—
591C8—121no—
592C8—121no—
593C2×C3—81no—
594C2×C3#239211yescharacteristic
595C2×C3#305911yescharacteristic
596C2×C3#306111yescharacteristic
597C2×C3#312411yescharacteristic
598C2×C3#312411yescharacteristic
599C2×C3#312711yescharacteristic
600C2×C3#312711yescharacteristic
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22#579111yescharacteristic
609C22C2×#93911yescharacteristic
610C22#582811yescharacteristic
611C22#583511yescharacteristic
612C22#583811yescharacteristic
613C22#583811yescharacteristic
614C22#583811yescharacteristic
615C4—22no—
616C4—22no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—122no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4#553811yescharacteristic
629C4#554711yescharacteristic
630C4#554811yescharacteristic
631C4#555711yescharacteristic
632C3#2370511yescharacteristic, complemented, Sylow
633C2—81no—
634C2#8234511yescharacteristic
635C2#8255511yescharacteristic
636C2#8256911yescharacteristic
637C2#8339911yescharacteristic
638C2#8342111yescharacteristic
639C2#8345111yescharacteristic
640C2#8346111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2323D3
hypocenterC3#23705
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
2. C2×C4×#216C2semidirect
7. #71631C2semidirect
8. #69683C2semidirect
9. #69681C2semidirect
10. #69635C2semidirect
11. #69622C2semidirect
14. C4×#807C2semidirect
15. #69475C2semidirect
632. C3#23705semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×C4×#2162
#822352
C2×#50002
#717002
#716952
#716312
#696832
#696812
#696352
#696222
#695272
#695182
C4×#8072
#694752
#237053