Finite Group Databasebeta

← #10844647

Subgroups — #10844647

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
1C2×C3×#478C211yesmaximal, characteristic
2#83025C211yesmaximal, characteristic, complemented
3C2×#5069C211yesmaximal, characteristic, complemented
4#82382C211yesmaximal, characteristic
5#71609C211yesmaximal, characteristic
6#70643C211yesmaximal, characteristic, complemented
7#70624C211yesmaximal, characteristic, complemented
8#70589C211yesmaximal, characteristic, complemented
9#70570C211yesmaximal, characteristic, complemented
10#70469C211yesmaximal, characteristic, complemented
11#70456C211yesmaximal, characteristic, complemented
12#70178C211yesmaximal, characteristic
13#70178C211yesmaximal, characteristic
14#70162C211yesmaximal, characteristic
15#70162C211yesmaximal, characteristic
16#25289—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×#478—24no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5212—21no—
31#5212—21no—
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#5069—24no—
39#4984—21no—
40#4984C2211yescharacteristic
41#4984C2211yescharacteristic
42#4980—21no—
43#4980—21no—
44Dic3×C4⋊C4—21no—
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47#4973C2211yescharacteristic
48#4973C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#810C2211yescharacteristic
53#4953—21no—
54#4953—21no—
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57C2×#803C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#491—31no—
63C2×#478D311yescharacteristic
64#2547—31no—
65#2370—31no—
66#2068—31no—
67#2066—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940—31no—
74#1928—31no—
75#1927—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#125—22no—
84C3×#125D414yes—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98#936—22no—
99#936—22no—
100#936—41no—
101#936—41no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#869—22no—
118#869—22no—
119C3×#107—22no—
120C3×#106—22no—
121C3×C4⋊C8—22no—
122C3×#102—22no—
123C3×#102—22no—
124#832—22no—
125#832—22no—
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⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#816—22no—
138#816—22no—
139#816—41no—
140#816—41no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#810—22no—
144#810—22no—
145#810—22no—
146#810—22no—
147#803—22no—
148C2×C4⋊Q8—31no—
149C2×#125C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×D4C2×D311yescharacteristic
153#491—64no—
154#478—24no—
155C2×#107C2×D311yescharacteristic
156C2×#106C2×D311yescharacteristic
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×#103—31no—
167C2×#103—31no—
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—31no—
188C4×C4⋊C4—61no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
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—42no—
205C22×C3×C4—42no—
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—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—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—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#217—42no—
247#217—42no—
248#217—42no—
249#217—42no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—41no—
254C4⋊Dic3—41no—
255C4⋊Dic3—42no—
256C4⋊Dic3—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3—81no—
259C4⋊Dic3#4112yes—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265C4×Dic3—81no—
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—21no—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C22×C8—31no—
278C22×C8—31no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C4⋊Q8—62no—
282C4⋊Q8—62no—
283#125—22no—
284#125#6714yes—
285#124—62no—
286#124—62no—
287#124—121no—
288#124—121no—
289C4×D4—22no—
290C4×D4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
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—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—22no—
316#106—22no—
317C4⋊C8—22no—
318C4⋊C8—121no—
319C4⋊C8—121no—
320#103—62no—
321#103—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#102—62no—
331#102—62no—
332#97—62no—
333#95—31no—
334#95—31no—
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—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×D4—22no—
353C3×D4—22no—
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—22no—
361C2×C3×C4—22no—
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—84no—
370C2×C3×C4#12112yes—
371C2×C3×C4#12112yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C42−1+411yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—22no—
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—
402Dic6—41no—
403Dic6—41no—
404Dic6—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—41no—
411C3×C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—21no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
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×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4#23912yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
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—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—32no—
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⋊C4C2×#6712yes—
479C4⋊C4C2×#6712yes—
480C22⋊C4—42no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C42—61no—
486C42—61no—
487C42—61no—
488C42—61no—
489C42—122no—
490C42—241no—
491C42—241no—
492C42#23712yes—
493C42C2×#6712yes—
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—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4#52911yescharacteristic
514C3×C4#54211yescharacteristic
515C3×C4#54811yescharacteristic
516C3×C4#55011yescharacteristic
517Dic3—42no—
518Dic3—41no—
519Dic3—41no—
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—121no—
534Q8—121no—
535Q8—121no—
536Q8—121no—
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—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
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—84no—
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#93912yes—
584C2×C4#93912yes—
585C2×C4C2×#23711yescharacteristic
586C2×C4C2×#23911yescharacteristic
587C2×C4#100711yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4#100911yescharacteristic
590C2×C4C22×#6711yescharacteristic
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#563011yescharacteristic
611C22#563311yescharacteristic
612C22#577011yescharacteristic
613C22#577111yescharacteristic
614C22#577311yescharacteristic
615C22#577311yescharacteristic
616C22C2×#93911yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—122no—
623C4—121no—
624C4—121no—
625C4—121no—
626C4—121no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#562411yescharacteristic
631C4#564311yescharacteristic
632C4#581711yescharacteristic
633C4#583411yescharacteristic
634C3#2528911yescharacteristic, complemented, Sylow
635C2—82no—
636C2#8115411yescharacteristic
637C2#8115511yescharacteristic
638C2#8118811yescharacteristic
639C2#8119011yescharacteristic
640C2#8275511yescharacteristic
641C2#8390211yescharacteristic
642C2#8393311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#478C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#478D3
hypocenterC3#25289
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. #83025C2semidirect
3. C2×#5069C2semidirect
6. #70643C2semidirect
7. #70624C2semidirect
8. #70589C2semidirect
9. #70570C2semidirect
10. #70469C2semidirect
11. #70456C2semidirect
634. C3#25289semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4782
#830252
C2×#50692
#823822
#716092
#706432
#706242
#705892
#705702
#704692
#704562
#701782
#701622
#252893