Finite Group Databasebeta

← #10844742

Subgroups — #10844742

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4830C211yesmaximal, characteristic, complemented
2C2×#4767C211yesmaximal, characteristic
3#76365C211yesmaximal, characteristic
4#76076C211yesmaximal, characteristic, complemented
5C3×#2067C211yesmaximal, characteristic
6#70627C211yesmaximal, characteristic, complemented
7#70616C211yesmaximal, characteristic, complemented
8#70487C211yesmaximal, characteristic, complemented
9#70217C211yesmaximal, characteristic
10#70186C211yesmaximal, characteristic
11#70158C211yesmaximal, characteristic
12#69646C211yesmaximal, characteristic
13#69614C211yesmaximal, characteristic, complemented
14#68912C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#25297—31nomaximal, Sylow
17C2×#936C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C22×Dic12C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#389C2211yescharacteristic
28#5221C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4980—21no—
37#4979—21no—
38#4977C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40#4973C2211yescharacteristic
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4952—21no—
48#4951—21no—
49#4947—21no—
50#4944—21no—
51#4929C2211yescharacteristic
52#4929C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54#4830—24no—
55#4767—24no—
56#4658—21no—
57#4652—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×C8×Q8—31no—
64#2592—31no—
65#2581—31no—
66#2067D311yescharacteristic
67#2066—31no—
68#2056—31no—
69#2055—31no—
70#1969—31no—
71#1968—31no—
72#1958—31no—
73#1949—31no—
74#1934—31no—
75#1933—31no—
76C4×#103—31no—
77C22×C3×D4C2311yescharacteristic
78C22×Dic6D411yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94#936—22no—
95#936—22no—
96C2×#214—22no—
97C2×#214—22no—
98C2×#214—41no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×Dic12—22no—
113C2×Dic12—22no—
114C2×Dic12—41no—
115#887—22no—
116#887—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#107—41no—
120C3×C4⋊C8—22no—
121C3×#102—22no—
122C3×#102—41no—
123C3×#95—21no—
124#832—22no—
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—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#816—22no—
138#816—41no—
139Dic3⋊C8—22no—
140Dic3⋊C8D414yes—
141C8×Dic3—22no—
142C8×Dic3—22no—
143#810—22no—
144#810—22no—
145#810—22no—
146#808—41no—
147C4×C3⋊C8—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153#464—64no—
154C8×Q8—64no—
155C2×#107C2×D311yescharacteristic
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171#400—61no—
172#397—61no—
173#395—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#390—31no—
180#390C2×D311yescharacteristic
181#389—31no—
182#389C2×D311yescharacteristic
183#388—61no—
184#388—61no—
185#387—61no—
186#387—61no—
187#382—31no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
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—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—21no—
218C2×Dic6—21no—
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#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D412yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—81no—
244C3×C42—21no—
245C3×C42—21no—
246#217—42no—
247#217—42no—
248#217—42no—
249#217—81no—
250#214—44no—
251C4⋊Dic3—22no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—42no—
263C4×Dic3—81no—
264C4×Dic3C2×D412yes—
265C4×Dic3#4112yes—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8C2×D412yes—
271C2×C3⋊C8#4112yes—
272Dic12—44no—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×Q8—31no—
276C22×D4C22×D311yescharacteristic
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16—121no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—121no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287#124—62no—
288#124—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—41no—
316#106—121no—
317C4⋊C8—22no—
318C4⋊C8—34no—
319C4⋊C8—62no—
320#103—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325#102—22no—
326#102—41no—
327#102—62no—
328C4×C8—62no—
329C4×C8—62no—
330C4×C8—62no—
331#95—21no—
332#95—61no—
333#95—61no—
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—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
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#13512yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C42−1+411yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3#12112yes—
399C2×Dic3#12112yes—
400C2×Dic3#13512yes—
401Dic6—28no—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C24—21no—
415C2×Q8—32no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×Q8—62no—
419C2×Q8—61no—
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×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—124no—
451SD16—124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—32no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8D3×D412yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—241no—
486C42—21no—
487C42—21no—
488C42—32no—
489C42—32no—
490C42—61no—
491C42—61no—
492C42—122no—
493C42—241no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12111yescharacteristic
502C22×C3#54811yescharacteristic
503C22×C3#55011yescharacteristic
504C22×C3C2×#13511yescharacteristic
505C22×C3C2×#13511yescharacteristic
506C22×C3#57711yescharacteristic
507C22×C3#57811yescharacteristic
508C3×C4—22no—
509C3×C4—41no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52911yescharacteristic
514C3×C4#54211yescharacteristic
515C3×C4#54611yescharacteristic
516C3×C4#54811yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—22no—
520Dic3—42no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533Q8—68no—
534Q8—62no—
535Q8—62no—
536Q8—121no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540Q8—121no—
541Q8—121no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—32no—
554C2×C4—41no—
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—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
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—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4#91212yes—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4#100411yescharacteristic
587C2×C4#100711yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—62no—
595C2×C3—161no—
596C2×C3#242611yescharacteristic
597C2×C3#305011yescharacteristic
598C2×C3#306311yescharacteristic
599C2×C3#336411yescharacteristic
600C2×C3#336511yescharacteristic
601C2×C3#339311yescharacteristic
602C2×C3#339411yescharacteristic
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#559911yescharacteristic
611C22D3×#12111yescharacteristic
612C22#563311yescharacteristic
613C22C2×#91211yescharacteristic
614C22D3×#13511yescharacteristic
615C22#577511yescharacteristic
616C22#577911yescharacteristic
617C4—22no—
618C4—41no—
619C4—62no—
620C4—62no—
621C4—62no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—121no—
626C4—122no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#561011yescharacteristic
631C4#563311yescharacteristic
632C4#564011yescharacteristic
633C4#570611yescharacteristic
634C3#2529711yescharacteristic, complemented, Sylow
635C2—161no—
636C2#7728611yescharacteristic
637C2#8032511yescharacteristic
638C2#8058811yescharacteristic
639C2#8071411yescharacteristic
640C2#8081011yescharacteristic
641C2#8099411yescharacteristic
642C2#8115111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2067C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2067D3
hypocenterC3#25297
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. C2×#4830C2semidirect
4. #76076C2semidirect
6. #70627C2semidirect
7. #70616C2semidirect
8. #70487C2semidirect
13. #69614C2semidirect
14. #68912C2semidirect
15. #68781C2semidirect
634. C3#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#48302
C2×#47672
#763652
#760762
C3×#20672
#706272
#706162
#704872
#702172
#701862
#701582
#696462
#696142
#689122
#687812
#252973