Finite Group Databasebeta

← #10844896

Subgroups — #10844896

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82388C211yesmaximal, characteristic
2C2×#5019C211yesmaximal, characteristic
3C2×#4701C211yesmaximal, characteristic, complemented
4#73722C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic
6#70217C211yesmaximal, characteristic
7#70217C211yesmaximal, characteristic
8#70192C211yesmaximal, characteristic
9#70189C211yesmaximal, characteristic
10#69822C211yesmaximal, characteristic, complemented
11#69468C211yesmaximal, characteristic, complemented
12#68913C211yesmaximal, characteristic, complemented
13#68912C211yesmaximal, characteristic, complemented
14#68854C211yesmaximal, characteristic, complemented
15#68850C211yesmaximal, characteristic, complemented
16#25308—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5221C2211yescharacteristic
27#5221C2211yescharacteristic
28#5210—21no—
29#5210—21no—
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#5019—24no—
39#4973C2211yescharacteristic
40#4973C2211yescharacteristic
41#4973C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4949—21no—
50#4947—21no—
51#4936—21no—
52#4936C2211yescharacteristic
53#4929—21no—
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4701—24no—
56#4659—21no—
57#4659—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461—31no—
63C2×#446—31no—
64#2580—31no—
65#2568—31no—
66#2068—31no—
67#2066—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1960—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1762D311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×Dic6D411yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#214—22no—
97C2×#214—22no—
98C2×#214—22no—
99C2×#214—22no—
100C2×#214—41no—
101C2×#214—41no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#869—22no—
116#869—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#97—22no—
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⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#816—22no—
136#816—22no—
137#816—41no—
138#816—41no—
139#815—22no—
140Dic3⋊C8—22no—
141Dic3⋊C8—22no—
142C8×Dic3—22no—
143#810—22no—
144#810—22no—
145#808—41no—
146#807—41no—
147C4⋊(C3⋊C8)D414yes—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×D4C2×D311yescharacteristic
153#461—64no—
154#446—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169C2×#97C2×D311yescharacteristic
170C2×C4×C8—31no—
171#400—61no—
172#400—61no—
173#399—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#391—61no—
180#390—31no—
181#390—31no—
182#389—31no—
183#389—31no—
184#385—61no—
185#385—61no—
186#382—31no—
187#382—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×Dic6—22no—
218C2×Dic6—22no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—41no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8#4112yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C22⋊C4—42no—
245C3×C42—42no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248#217—42no—
249#217—42no—
250#217—81no—
251#217—81no—
252#214—44no—
253#214—44no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
260C4⋊Dic3—42no—
261Dic3⋊C4—22no—
262Dic3⋊C4—22no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8C2×D412yes—
275C22×Q8—31no—
276C22×Q8—31no—
277C22×Q8—31no—
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C2×SD16—121no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C4⋊Q8—62no—
291C4⋊Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—21no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—34no—
320C4⋊C8—62no—
321C4⋊C8—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—121no—
327#103—121no—
328#102—22no—
329#102—22no—
330#97—22no—
331#97—62no—
332C4×C8—62no—
333#95—21no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—41no—
357C3×D4—41no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—84no—
372C2×C3×C4#12112yes—
373C2×C3×C4#12112yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C42−1+411yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
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#13712yes—
404Dic6—44no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413C3×C8—22no—
414C3×C8—41no—
415C3⋊C8—22no—
416C3⋊C8—22no—
417C24—21no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—121no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4#23512yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—42no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—124no—
457SD16—124no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8#23512yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C4⋊C4—62no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C22⋊C4—42no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—241no—
491C22⋊C4—241no—
492C42—42no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—61no—
497C42#23512yes—
498C42D3×D412yes—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12111yescharacteristic
507C22×C3#54811yescharacteristic
508C22×C3#55011yescharacteristic
509C22×C3C2×#13711yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57711yescharacteristic
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54611yescharacteristic
521C3×C4#54811yescharacteristic
522Dic3—22no—
523Dic3—41no—
524Dic3—41no—
525Dic3—41no—
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×D3×D411yescharacteristic
538Q8—124no—
539Q8—121no—
540Q8—121no—
541Q8—121no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546Q8—121no—
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—32no—
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—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—84no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4#88412yes—
590C2×C4#88412yes—
591C2×C4C2×#23511yescharacteristic
592C2×C4C2×#23511yescharacteristic
593C2×C4C2×#23511yescharacteristic
594C2×C4C2×D3×D411yescharacteristic
595C2×C4#100911yescharacteristic
596C2×C4#101011yescharacteristic
597C8—22no—
598C8—41no—
599C8—62no—
600C8—62no—
601C2×C3—82no—
602C2×C3#242611yescharacteristic
603C2×C3#305611yescharacteristic
604C2×C3#305711yescharacteristic
605C2×C3#337211yescharacteristic
606C2×C3#337611yescharacteristic
607C2×C3#340011yescharacteristic
608C2×C3#340111yescharacteristic
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#88411yescharacteristic
617C22#555211yescharacteristic
618C22#555311yescharacteristic
619C22#577311yescharacteristic
620C22#577511yescharacteristic
621C22#577511yescharacteristic
622C22D3×#13711yescharacteristic
623C4—22no—
624C4—22no—
625C4—62no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4#547911yescharacteristic
637C4#553411yescharacteristic
638C4#555011yescharacteristic
639C4#555211yescharacteristic
640C3#2530811yescharacteristic, complemented, Sylow
641C2—82no—
642C2#7348511yescharacteristic
643C2#7888911yescharacteristic
644C2#7893111yescharacteristic
645C2#7975511yescharacteristic
646C2#7976411yescharacteristic
647C2#7985411yescharacteristic
648C2#7985811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1762D3
hypocenterC3#25308
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C2×#4701C2semidirect
4. #73722C2semidirect
10. #69822C2semidirect
11. #69468C2semidirect
12. #68913C2semidirect
13. #68912C2semidirect
14. #68854C2semidirect
15. #68850C2semidirect
640. C3#25308semidirect

Maximal subgroups

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

subgroupindex
#823882
C2×#50192
C2×#47012
#737222
C3×#17622
#702172
#701922
#701892
#698222
#694682
#689132
#689122
#688542
#688502
#253083