Finite Group Databasebeta

← #10844648

Subgroups — #10844648

Order 768.

792 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
1#82973C211yesmaximal, characteristic
2C2×#4797C211yesmaximal, characteristic, complemented
3C2×#4766C211yesmaximal, characteristic
4#73660C211yesmaximal, characteristic, complemented
5C3×#1983C211yesmaximal, characteristic
6#71722C211yesmaximal, characteristic
7#70489C211yesmaximal, characteristic, complemented
8#70452C211yesmaximal, characteristic, complemented
9#70261C211yesmaximal, characteristic, complemented
10#70229C211yesmaximal, characteristic, complemented
11#70224C211yesmaximal, characteristic, complemented
12#70178C211yesmaximal, characteristic
13#70162C211yesmaximal, characteristic
14#70152C211yesmaximal, characteristic
15#69464C211yesmaximal, characteristic, complemented
16#25289—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4984C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4952—21no—
46#4952—21no—
47#4942—21no—
48#4942—21no—
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929C2211yescharacteristic
52#4928—21no—
53#4928C2211yescharacteristic
54#4797—24no—
55#4766—24no—
56#4655—21no—
57#4654—21no—
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#491—31no—
63C2×#478—31no—
64#2547—31no—
65#2370—31no—
66#2068—31no—
67#2066—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983D311yescharacteristic
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#124—22no—
84C3×#124—41no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94#944—22no—
95#944—22no—
96#936—22no—
97#936—41no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#890—22no—
118#890D414yes—
119C3×C4⋊C8—41no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#95—21no—
125#835—22no—
126#835—41no—
127#832—22no—
128#832—22no—
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⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#818—22no—
140#816—22no—
141#816—41no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—41no—
145#810—22no—
146#810—22no—
147#808—22no—
148C2×C4⋊Q8—31no—
149C2×#125—31no—
150C2×#124—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×D4—31no—
153#491—64no—
154#478—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×#103—31no—
165C2×#103—31no—
166C2×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97—31no—
173#397—61no—
174#397—61no—
175#391—31no—
176#391—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#387—61no—
181#387—61no—
182#380—61no—
183#380—61no—
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×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4#4112yes—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—42no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—81no—
244#217—42no—
245#217—42no—
246#217—42no—
247C4⋊Dic3—21no—
248C4⋊Dic3—24no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—81no—
254C4⋊Dic3C2×D412yes—
255Dic3⋊C4—24no—
256Dic3⋊C4—41no—
257Dic3⋊C4—41no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—42no—
264C4×Dic3—81no—
265C4×Dic3C2×D412yes—
266C4×Dic3#4112yes—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×D4C22×D311yescharacteristic
276C23×C4—61no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C4⋊Q8—62no—
282C4⋊Q8—62no—
283#125—34no—
284#125—62no—
285#124—22no—
286#124—41no—
287#124—62no—
288#124—121no—
289C4×D4—62no—
290C4×D4—62no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
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×C42—21no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#106—62no—
317C4⋊C8—41no—
318C4⋊C8—62no—
319C4⋊C8—121no—
320#103—22no—
321#103—22no—
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#102—62no—
331#102—62no—
332#97—62no—
333#95—21no—
334#95—31no—
335#95—31no—
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—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—41no—
359C3×D4—41no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
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—84no—
404C2×Dic3#12112yes—
405C2×Dic3#12112yes—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—41no—
411C3×C8—41no—
412C3⋊C8—41no—
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—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4#23512yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—32no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—121no—
473C4⋊C4—121no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4D3×D412yes—
479C4⋊C4#23712yes—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—21no—
486C42—21no—
487C42—32no—
488C42—32no—
489C42—61no—
490C42—61no—
491C42—81no—
492C42—122no—
493C42—241no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12111yescharacteristic
502C22×C3#54611yescharacteristic
503C22×C3#54811yescharacteristic
504C22×C3#57511yescharacteristic
505C22×C3#57511yescharacteristic
506C22×C3#57711yescharacteristic
507C22×C3#57811yescharacteristic
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—81no—
513C3×C4#52911yescharacteristic
514C3×C4#54211yescharacteristic
515C3×C4#54811yescharacteristic
516C3×C4#55011yescharacteristic
517Dic3—22no—
518Dic3—22no—
519Dic3—42no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540Q8—121no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
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—61no—
572C2×C4—62no—
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—244no—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4C2×#23711yescharacteristic
588C2×C4#100411yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4#101011yescharacteristic
591C8—41no—
592C8—41no—
593C8—121no—
594C8—121no—
595C2×C3—82no—
596C2×C3#242611yescharacteristic
597C2×C3#304011yescharacteristic
598C2×C3#304611yescharacteristic
599C2×C3#330711yescharacteristic
600C2×C3#331511yescharacteristic
601C2×C3#337611yescharacteristic
602C2×C3#337711yescharacteristic
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#549011yescharacteristic
611C22#559811yescharacteristic
612C22D3×#12111yescharacteristic
613C22#576811yescharacteristic
614C22#577111yescharacteristic
615C22#577511yescharacteristic
616C22#577711yescharacteristic
617C4—41no—
618C4—41no—
619C4—42no—
620C4—42no—
621C4—62no—
622C4—62no—
623C4—81no—
624C4—121no—
625C4—121no—
626C4—122no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#553811yescharacteristic
631C4#555211yescharacteristic
632C4#566911yescharacteristic
633C4#571811yescharacteristic
634C3#2528911yescharacteristic, complemented, Sylow
635C2—82no—
636C2#7631211yescharacteristic
637C2#7885711yescharacteristic
638C2#7889511yescharacteristic
639C2#8052211yescharacteristic
640C2#8053911yescharacteristic
641C2#8058711yescharacteristic
642C2#8059911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1983C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1983D3
hypocenterC3#25289
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#4797C2semidirect
4. #73660C2semidirect
7. #70489C2semidirect
8. #70452C2semidirect
9. #70261C2semidirect
10. #70229C2semidirect
11. #70224C2semidirect
15. #69464C2semidirect
634. C3#25289semidirect

Maximal subgroups

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

subgroupindex
#829732
C2×#47972
C2×#47662
#736602
C3×#19832
#717222
#704892
#704522
#702612
#702292
#702242
#701782
#701622
#701522
#694642
#252893