Finite Group Databasebeta

← #10844743

Subgroups — #10844743

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×C3×#464C211yesmaximal, characteristic
2#82976C211yesmaximal, characteristic
3#82481C211yesmaximal, characteristic, complemented
4C2×#4830C211yesmaximal, characteristic, complemented
5#72068C211yesmaximal, characteristic, complemented
6#72066C211yesmaximal, characteristic, complemented
7#72051C211yesmaximal, characteristic
8#72049C211yesmaximal, characteristic
9#71725C211yesmaximal, characteristic
10C4×#835C211yesmaximal, characteristic, complemented
11#70455C211yesmaximal, characteristic, complemented
12#70254C211yesmaximal, characteristic, complemented
13#70247C211yesmaximal, characteristic, complemented
14#70186C211yesmaximal, characteristic
15#70180C211yesmaximal, characteristic
16#25297—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#464—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26#5229C2211yescharacteristic
27#5229C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43C2×C8×Dic3C2211yescharacteristic
44#4986—21no—
45#4979—21no—
46#4972C2211yescharacteristic
47#4972C2211yescharacteristic
48#4953—21no—
49#4947—21no—
50#4945—21no—
51#4945—21no—
52#4936C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54#4933—21no—
55#4933—21no—
56C2×C4×C3⋊C8C2211yescharacteristic
57#4830—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464D311yescharacteristic
63C2×C8×Q8—31no—
64#2592—31no—
65#2581—31no—
66#2067—31no—
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×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×Dic6D411yescharacteristic
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—41no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#124—22no—
90C3×#124—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×C4⋊C8D414yes—
120C3×#103—22no—
121C3×#102—22no—
122#835—22no—
123#835—22no—
124#835—22no—
125#835—41no—
126#832—22no—
127#832—41no—
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—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#818—41no—
142#817—41no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C8×Dic3—22no—
147C4×C3⋊C8—22no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16C2×D311yescharacteristic
150C2×#124C2×D311yescharacteristic
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#464—24no—
154C8×Q8—64no—
155C2×#107—31no—
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×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
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#390—31no—
181#389—31no—
182#389—31no—
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#340—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8#4112yes—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C3×Q16—44no—
230C3×SD16—44no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D412yes—
236C2×C3×C8C2×D412yes—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C42—42no—
243C3×C42C2×D412yes—
244C3×C42#4112yes—
245#217—42no—
246#217—42no—
247#217—81no—
248#217—81no—
249C4⋊Dic3—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258Dic3⋊C4—22no—
259Dic3⋊C4—22no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×Q8C22×D311yescharacteristic
276C22×D4C22×D311yescharacteristic
277C2×Q16—22no—
278C2×Q16—22no—
279C2×Q16—41no—
280C2×SD16—22no—
281C2×SD16—22no—
282C2×SD16—41no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287#124—22no—
288#124—22no—
289C4×Q8—22no—
290C4×Q8—22no—
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—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
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—21no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#106—121no—
317C4⋊C8—62no—
318C4⋊C8—62no—
319C4⋊C8#6714yes—
320#103—22no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325#102—22no—
326#102—62no—
327#102—121no—
328C4×C8—62no—
329C4×C8—62no—
330C4×C8—62no—
331#95—61no—
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×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
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#12112yes—
378C2×C3×C4#12112yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C42−1+411yescharacteristic
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
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—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3#13512yes—
408C2×Dic3#13512yes—
409Dic6—28no—
410C3×C8—22no—
411C3×C8—22no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C24—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—62no—
427C2×Q8—62no—
428C2×Q8—62no—
429C2×Q8#23712yes—
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—44no—
451SD16—44no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8C2×#6712yes—
463C2×C8C2×#6712yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—241no—
485C22⋊C4—241no—
486C42—42no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—241no—
492C42#23712yes—
493C42C2×#6712yes—
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—22no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52911yescharacteristic
514C3×C4#54211yescharacteristic
515C3×C4#54611yescharacteristic
516C3×C4#54811yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—41no—
520Dic3—41no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—68no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
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—121no—
582C2×C4—121no—
583C2×C4#93812yes—
584C2×C4#93812yes—
585C2×C4C2×#23711yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4#100411yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4#101011yescharacteristic
590C2×C4C22×#6711yescharacteristic
591C8—22no—
592C8—22no—
593C8—62no—
594C8—121no—
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#559811yescharacteristic
611C22#559911yescharacteristic
612C22D3×#13511yescharacteristic
613C22D3×#13511yescharacteristic
614C22#577811yescharacteristic
615C22#577911yescharacteristic
616C22C2×#93811yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—62no—
621C4—62no—
622C4—81no—
623C4—81no—
624C4—121no—
625C4—121no—
626C4—241no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#560011yescharacteristic
631C4#560311yescharacteristic
632C4#580611yescharacteristic
633C4#583111yescharacteristic
634C3#2529711yescharacteristic, complemented, Sylow
635C2—161no—
636C2#8058011yescharacteristic
637C2#8058111yescharacteristic
638C2#8058811yescharacteristic
639C2#8058911yescharacteristic
640C2#8264611yescharacteristic
641C2#8391411yescharacteristic
642C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#464C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#464D3
hypocenterC3#25297
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
3. #82481C2semidirect
4. C2×#4830C2semidirect
5. #72068C2semidirect
6. #72066C2semidirect
10. C4×#835C2semidirect
11. #70455C2semidirect
12. #70254C2semidirect
13. #70247C2semidirect
634. C3#25297semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4642
#829762
#824812
C2×#48302
#720682
#720662
#720512
#720492
#717252
C4×#8352
#704552
#702542
#702472
#701862
#701802
#252973