Finite Group Databasebeta

← #10801786

Subgroups — #10801786

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic
2C2×#5051C211yesmaximal, characteristic
3C2×#5030C211yesmaximal, characteristic, complemented
4#82235C211yesmaximal, characteristic
5C2×#4997C211yesmaximal, characteristic
6#71697C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic, complemented
8#69678C211yesmaximal, characteristic, complemented
9#69674C211yesmaximal, characteristic, complemented
10#69620C211yesmaximal, characteristic, complemented
11#69614C211yesmaximal, characteristic, complemented
12#69516C211yesmaximal, characteristic
13#69511C211yesmaximal, characteristic
14#69481C211yesmaximal, characteristic, complemented
15C4×#810C211yesmaximal, characteristic, complemented
16#21530—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×#924C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#388—21no—
30C3×#387—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5212C2211yescharacteristic
37#5051C418yescomplemented
38#5030—24no—
39#4997—24no—
40#4978—21no—
41#4976—21no—
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4945—21no—
49#4944—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928C2211yescharacteristic
53C2×#803C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#467—31no—
64C2×#441—31no—
65C2×#405—31no—
66#2323—31no—
67#2323D311yescharacteristic
68#1971—31no—
69#1943—31no—
70#1933—31no—
71#1933—31no—
72#1894—31no—
73#1894—31no—
74#1807—31no—
75#1776—31no—
76C4×#103—31no—
77C4×#103—31no—
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8C2×C414yes—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×#217—21no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216C2×C414yes—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111#924—22no—
112#924C2×C414yes—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×#208—22no—
119C2×#208—22no—
120C2×#208—22no—
121C22×C3⋊C8—22no—
122C22×C3⋊C8—22no—
123C22×C3⋊C8C2×C414yes—
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C4×Dic6—22no—
129C4×Dic6—22no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135#835—22no—
136#835—22no—
137#835C2×C414yes—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#810—22no—
144#810—22no—
145#810C2×C414yes—
146#808—22no—
147#808—22no—
148#807C2×C414yes—
149#803—22no—
150#803D414yes—
151C4×C3⋊C8—22no—
152C4×C3⋊C8—22no—
153C22×Q16—31no—
154C22×M16—31no—
155C23×C8—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×C4×Q8—31no—
159C2×C4×Q8C2×D311yescharacteristic
160C2×#117C2×D311yescharacteristic
161#467—38no—
162#441—64no—
163C2×#107—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177#391—31no—
178#391C2×D311yescharacteristic
179#389—21no—
180#389—31no—
181#389—61no—
182#389C2×D311yescharacteristic
183#388—21no—
184#388—21no—
185#388—61no—
186#388—61no—
187#387—21no—
188#387—61no—
189#380—31no—
190#380C2×D311yescharacteristic
191C4×C4⋊C4—31no—
192C4×C4⋊C4C2×D311yescharacteristic
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C22×C412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C22×C412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4#4112yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C22×C412yes—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4C22×C412yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—81no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250C3×C42—81no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#217—24no—
254#217—81no—
255#216—28no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3C22×C412yes—
262Dic3⋊C4—24no—
263Dic3⋊C4—42no—
264Dic3⋊C4—81no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267#208—28no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—28no—
275C2×C3⋊C8C22×C412yes—
276C2×C3⋊C8C22×C412yes—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8C2×D412yes—
279C2×C3⋊C8#4112yes—
280C2×C3⋊C8#4112yes—
281C22×Q8—31no—
282C22×Q8C22×D311yescharacteristic
283C23×C4C22×D311yescharacteristic
284C2×Q16—34no—
285C2×Q16—62no—
286C2×Q16—62no—
287C2×M16—62no—
288C2×M16—62no—
289C2×M16—62no—
290C22×C8—31no—
291C22×C8—34no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—62no—
296C22×C8—62no—
297C22⋊Q8—22no—
298C22⋊Q8—34no—
299C22⋊Q8—62no—
300C22⋊Q8C4×D314yes—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—62no—
304C4×Q8—62no—
305#117—22no—
306#117—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4#6711yescharacteristic
317C2×C4⋊C4#6711yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4#6711yescharacteristic
326C2×C22⋊C4#6711yescharacteristic
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—61no—
330C2×C42—61no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—34no—
334#106—62no—
335#106—62no—
336#103—34no—
337#103—34no—
338#103—62no—
339#103—62no—
340#103—62no—
341#103—62no—
342#97—34no—
343#97—62no—
344C4×C8—62no—
345C4×C8—62no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3C4×D412yes—
363C23×C3C22×D411yescharacteristic
364C3×Q8—22no—
365C3×Q8—22no—
366C3×Q8—41no—
367C3×Q8—41no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4C4×D412yes—
397C2×C3×C4#12112yes—
398C2×C3×C4#12112yes—
399C2×C3×C4C23×C411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3⋊C8—22no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8C4×D414yes—
425C24C2×#6711yescharacteristic
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8C2×C4×D312yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4C2×C4×D312yes—
451C22×C4#23512yes—
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459M16—68no—
460C2×C8—32no—
461C2×C8—32no—
462C2×C8—32no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—68no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—32no—
482C4⋊C4—42no—
483C4⋊C4—64no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—81no—
490C4⋊C4—122no—
491C4⋊C4—241no—
492C4⋊C4C2×C4×D312yes—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—64no—
497C22⋊C4—81no—
498C22⋊C4—241no—
499C42—21no—
500C42—21no—
501C42—42no—
502C42—81no—
503C42—122no—
504C42—241no—
505C42D3×D412yes—
506C42#23712yes—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3C2×C4×D411yescharacteristic
515C22×C3C2×#12111yescharacteristic
516C22×C3#53211yescharacteristic
517C22×C3#57711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57811yescharacteristic
520C22×C3#57811yescharacteristic
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—42no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—42no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4C2×C4×D411yescharacteristic
531C3×C4C4×#4111yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#52911yescharacteristic
534Dic3—42no—
535Dic3—42no—
536Dic3—81no—
537Dic3—81no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23C4×#6712yes—
545C23C22×#6711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—62no—
551Q8—62no—
552Q8—121no—
553Q8—121no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4C4×#6712yes—
595C2×C4#93912yes—
596C2×C4#93912yes—
597C2×C4C22×C4×D311yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C8—34no—
604C8—62no—
605C8—62no—
606C8—62no—
607C2×C3—42no—
608C2×C3C4×#12111yescharacteristic
609C2×C3#295011yescharacteristic
610C2×C3#295211yescharacteristic
611C2×C3#306311yescharacteristic
612C2×C3#306311yescharacteristic
613C2×C3#317811yescharacteristic
614C2×C3#317811yescharacteristic
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22C2×C4×#6711yescharacteristic
623C22C2×#93911yescharacteristic
624C22#583211yescharacteristic
625C22#583611yescharacteristic
626C22#583711yescharacteristic
627C22#583811yescharacteristic
628C22#583811yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—122no—
639C4—122no—
640C4—241no—
641C4—241no—
642C4C4×#23711yescharacteristic
643C4#553811yescharacteristic
644C4C4×D3×D411yescharacteristic
645C4#554711yescharacteristic
646C3#2153011yescharacteristic, complemented, Sylow
647C2—42no—
648C2C4×#93911yescharacteristic
649C2#8254911yescharacteristic
650C2#8255811yescharacteristic
651C2#8340011yescharacteristic
652C2#8342311yescharacteristic
653C2#8344511yescharacteristic
654C2#8345811yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenter#2323D3
hypocenterC3#21530
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. C2×#5030C2semidirect
7. C4×#835C2semidirect
8. #69678C2semidirect
9. #69674C2semidirect
10. #69620C2semidirect
11. #69614C2semidirect
14. #69481C2semidirect
15. C4×#810C2semidirect
37. #5051C4semidirect
646. C3#21530semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×#50512
C2×#50302
#822352
C2×#49972
#716972
C4×#8352
#696782
#696742
#696202
#696142
#695162
#695112
#694812
C4×#8102
#215303