Finite Group Databasebeta

← #10844609

Subgroups — #10844609

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2545C211yesmaximal, characteristic
2C2×#5055C211yesmaximal, characteristic
3C2×#5020C211yesmaximal, characteristic, complemented
4#73656C211yesmaximal, characteristic, complemented
5#71781C211yesmaximal, characteristic, complemented
6#71778C211yesmaximal, characteristic, complemented
7#71722C211yesmaximal, characteristic
8#71721C211yesmaximal, characteristic
9#71721C211yesmaximal, characteristic
10#71631C211yesmaximal, characteristic, complemented
11#69682C211yesmaximal, characteristic, complemented
12#69666C211yesmaximal, characteristic, complemented
13#69652C211yesmaximal, characteristic
14#69637C211yesmaximal, characteristic
15#69485C211yesmaximal, characteristic, complemented
16#25287—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×#396—21no—
23C3×#396—21no—
24C3×#391C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#388—21no—
27C3×#388—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41#5055—24no—
42#5020—24no—
43Dic3×C4⋊C4—21no—
44#4972—21no—
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4646—21no—
57#4646—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#475—31no—
64#2545D311yescharacteristic
65#2369—31no—
66#2066—31no—
67#2065—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1927—31no—
75#1926—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×Dic6—21no—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—41no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124—41no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#869—22no—
120#869D414yes—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
125#835—22no—
126#835—22no—
127#835—22no—
128#835—22no—
129#835—22no—
130#835—22no—
131#835—22no—
132#835—41no—
133#832—22no—
134#832—22no—
135#832—22no—
136#832—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#808—22no—
143#808—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8C2×D311yescharacteristic
150C2×#124C2×D311yescharacteristic
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153#489—64no—
154#475—64no—
155C2×#106—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×#103—31no—
167C2×#103—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#396—21no—
174#396—21no—
175#391—61no—
176#391—61no—
177#391C2×D311yescharacteristic
178#390C2×D311yescharacteristic
179#389—31no—
180#389—31no—
181#389—61no—
182#388—21no—
183#388—21no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—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×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C2×Dic6—24no—
227C2×Dic6—81no—
228C2×Dic6—81no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C22⋊C4—42no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—42no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249C4⋊Dic3—24no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D412yes—
256C4⋊Dic3C2×D412yes—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260C4×Dic3—42no—
261C4×Dic3—81no—
262C4×Dic3—81no—
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⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C22×Q8—61no—
276C22×Q8C22×D311yescharacteristic
277C22×Q8C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C22×C8—31no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C4⋊Q8—22no—
284C4⋊Q8—34no—
285C4⋊Q8—41no—
286C4⋊Q8—62no—
287#124—22no—
288#124—22no—
289#124—22no—
290#124—41no—
291C4×Q8—22no—
292C4×Q8—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—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42—21no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—62no—
321C4⋊C8—62no—
322C4⋊C8—121no—
323C4⋊C8—121no—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—121no—
332#102—62no—
333#102—62no—
334#102—62no—
335#102—121no—
336#97—62no—
337#95—21no—
338#95—21no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95#6711yescharacteristic
345#95#6711yescharacteristic
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
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—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4#12112yes—
391C2×C3×C4#12112yes—
392C2×C3×C4#13712yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411Dic6—84no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—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—21no—
427C2×Q8—21no—
428C2×Q8—64no—
429C2×Q8—241no—
430C2×Q8—241no—
431C2×Q8#23712yes—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—32no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—64no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—42no—
493C42—122no—
494C42—241no—
495C42—241no—
496C42D3×D412yes—
497C42#23712yes—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3C2×#12111yescharacteristic
506C22×C3#54611yescharacteristic
507C22×C3#54811yescharacteristic
508C22×C3C2×#13711yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57711yescharacteristic
511C22×C3#57711yescharacteristic
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4#52911yescharacteristic
522C3×C4#54211yescharacteristic
523C3×C4#54811yescharacteristic
524C3×C4#55011yescharacteristic
525Dic3—42no—
526Dic3—42no—
527Dic3—42no—
528Dic3—81no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C22×#6711yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—244no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
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#93812yes—
587C2×C4#93812yes—
588C2×C4#95112yes—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4C2×#23711yescharacteristic
592C2×C4#100411yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#101011yescharacteristic
595C8—121no—
596C8—121no—
597C8—121no—
598C8—121no—
599C2×C3—161no—
600C2×C3#242611yescharacteristic
601C2×C3#303711yescharacteristic
602C2×C3#304211yescharacteristic
603C2×C3#331611yescharacteristic
604C2×C3#332111yescharacteristic
605C2×C3#336911yescharacteristic
606C2×C3#337611yescharacteristic
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×#93811yescharacteristic
615C22#580611yescharacteristic
616C22#582211yescharacteristic
617C22C2×#95111yescharacteristic
618C22#583111yescharacteristic
619C22#583711yescharacteristic
620C22#583711yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—122no—
631C4—122no—
632C4—122no—
633C4—241no—
634C4#559811yescharacteristic
635C4#559911yescharacteristic
636C4#560011yescharacteristic
637C4#560311yescharacteristic
638C3#2528711yescharacteristic, complemented, Sylow
639C2—161no—
640C2#8264611yescharacteristic
641C2#8331411yescharacteristic
642C2#8340511yescharacteristic
643C2#8353811yescharacteristic
644C2#8353911yescharacteristic
645C2#8366911yescharacteristic
646C2#8367711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2545C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2545D3
hypocenterC3#25287
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2545

Complements

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

NH ≅ G/Ntype
3. C2×#5020C2semidirect
4. #73656C2semidirect
5. #71781C2semidirect
6. #71778C2semidirect
10. #71631C2semidirect
11. #69682C2semidirect
12. #69666C2semidirect
15. #69485C2semidirect
638. C3#25287semidirect

Maximal subgroups

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

subgroupindex
C3×#25452
C2×#50552
C2×#50202
#736562
#717812
#717782
#717222
#717212
#716312
#696822
#696662
#696522
#696372
#694852
#252873