Finite Group Databasebeta

← #10845197

Subgroups — #10845197

Order 768.

824 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2#83229C211yesmaximal, characteristic, complemented
3C2×#5082C211yesmaximal, characteristic, complemented
4#82386C211yesmaximal, characteristic
5#72051C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic
7#72048C211yesmaximal, characteristic
8#72048C211yesmaximal, characteristic
9#71609C211yesmaximal, characteristic
10#70469C211yesmaximal, characteristic, complemented
11#70461C211yesmaximal, characteristic, complemented
12#70234C211yesmaximal, characteristic, complemented
13#70233C211yesmaximal, characteristic, complemented
14#70222C211yesmaximal, characteristic, complemented
15#70221C211yesmaximal, characteristic, complemented
16#25331—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C3×#461—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5210—21no—
39#5210—21no—
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#5082—24no—
43#4984—21no—
44#4980—21no—
45#4980—21no—
46#4977C2211yescharacteristic
47#4975C2211yescharacteristic
48Dic3×C4⋊C4—21no—
49#4948—21no—
50#4948—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#491—31no—
63C2×#461D311yescharacteristic
64#2566—31no—
65#2365—31no—
66#2074—31no—
67#2072—31no—
68#2055—31no—
69#2055—31no—
70#1989—31no—
71#1940—31no—
72#1880—31no—
73#1880—31no—
74#1805—31no—
75#1803—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C2×C3×SD16—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4⋊Q8—22no—
90C3×C4⋊Q8D414yes—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102Q8×Dic3—22no—
103Q8×Dic3—22no—
104Q8×Dic3—41no—
105Q8×Dic3—41no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#870—22no—
126#870—22no—
127C3×C4⋊C8—22no—
128C3×#103—22no—
129C3×#103—22no—
130#835—22no—
131#835—22no—
132#835—22no—
133#835—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95D411yescharacteristic
144C3⋊#95D411yescharacteristic
145Dic3⋊C8—41no—
146Dic3⋊C8—41no—
147#803—22no—
148C22×SD16C2×D311yescharacteristic
149C22×SD16C2×D311yescharacteristic
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#125—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
155#491—64no—
156#461—24no—
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×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169#396—61no—
170#396—61no—
171#391—31no—
172#391—31no—
173#390—31no—
174#390—31no—
175#389—61no—
176#387—61no—
177#387—61no—
178#385—61no—
179#385—61no—
180#384—31no—
181#382—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
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×Q8C2×D412yes—
205C2×C3×Q8C2×D412yes—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4#4112yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—81no—
217C22×C3×C4—81no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C3×SD16—44no—
230C3×SD16—44no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C22⋊C4—42no—
242C3×C42C2×D412yes—
243C3×C42#4112yes—
244#217—42no—
245#217—42no—
246#217—81no—
247#217—81no—
248C4⋊Dic3—21no—
249C4⋊Dic3—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—41no—
257C4⋊Dic3—42no—
258C4⋊Dic3—42no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—81no—
263Dic3⋊C4—81no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C22×Q8C22×D311yescharacteristic
278C22×Q8C22×D311yescharacteristic
279C22×D4C22×D311yescharacteristic
280C23×C4—21no—
281C2×SD16—22no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×SD16—22no—
285C2×SD16—41no—
286C2×SD16—41no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291C4⋊Q8—22no—
292C4⋊Q8#6714yes—
293#125—62no—
294#125—62no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—121no—
298C4×Q8—121no—
299C4×D4—22no—
300C4×D4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C4⋊C8—22no—
328C4⋊C8—121no—
329C4⋊C8—121no—
330#103—22no—
331#103—22no—
332#103—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#97—62no—
339#95—31no—
340#95—31no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—22no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3C22×D411yescharacteristic
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—41no—
367C3×D4—41no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—84no—
382C2×C3×C4#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
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—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3#13712yes—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—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—21no—
429C2×Q8C2×#6712yes—
430C2×Q8C2×#6712yes—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4#23712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—42no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454SD16—44no—
455SD16—44no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—32no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—121no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C4⋊C4—122no—
486C4⋊C4—241no—
487C4⋊C4—241no—
488C22⋊C4—42no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—241no—
492C22⋊C4—241no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—61no—
497C42—122no—
498C42—122no—
499C42—122no—
500C42#23712yes—
501C42C2×#6712yes—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3C2×#12111yescharacteristic
510C22×C3#54611yescharacteristic
511C22×C3#54811yescharacteristic
512C22×C3C2×#13711yescharacteristic
513C22×C3#57511yescharacteristic
514C22×C3#57711yescharacteristic
515C22×C3#57711yescharacteristic
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—81no—
521C3×C4#52911yescharacteristic
522C3×C4#54211yescharacteristic
523C3×C4#54611yescharacteristic
524C3×C4#54711yescharacteristic
525Dic3—22no—
526Dic3—41no—
527Dic3—41no—
528Dic3—42no—
529Dic3—41no—
530Dic3—81no—
531Dic3—81no—
532Dic3—81no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×D3×D411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—32no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—84no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4#93912yes—
592C2×C4#93912yes—
593C2×C4C2×#23711yescharacteristic
594C2×C4C2×#23711yescharacteristic
595C2×C4#100411yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C2×C4#101011yescharacteristic
598C2×C4C22×#6711yescharacteristic
599C8—41no—
600C8—41no—
601C8—121no—
602C8—121no—
603C2×C3—82no—
604C2×C3#242711yescharacteristic
605C2×C3#303711yescharacteristic
606C2×C3#304311yescharacteristic
607C2×C3#330711yescharacteristic
608C2×C3#332011yescharacteristic
609C2×C3#337211yescharacteristic
610C2×C3#337611yescharacteristic
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22#559811yescharacteristic
619C22#561211yescharacteristic
620C22#576211yescharacteristic
621C22#576211yescharacteristic
622C22D3×#13711yescharacteristic
623C22#577711yescharacteristic
624C22C2×#93911yescharacteristic
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—62no—
630C4—81no—
631C4—121no—
632C4—121no—
633C4—122no—
634C4—121no—
635C4—241no—
636C4—241no—
637C4—241no—
638C4#560011yescharacteristic
639C4#560311yescharacteristic
640C4#580611yescharacteristic
641C4#582411yescharacteristic
642C3#2533111yescharacteristic, complemented, Sylow
643C2—82no—
644C2#8041811yescharacteristic
645C2#8041911yescharacteristic
646C2#8045211yescharacteristic
647C2#8045311yescharacteristic
648C2#8264711yescharacteristic
649C2#8384911yescharacteristic
650C2#8396611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#461D3
hypocenterC3#25331
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
2. #83229C2semidirect
3. C2×#5082C2semidirect
10. #70469C2semidirect
11. #70461C2semidirect
12. #70234C2semidirect
13. #70233C2semidirect
14. #70222C2semidirect
15. #70221C2semidirect
642. C3#25331semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
#832292
C2×#50822
#823862
#720512
#720482
#716092
#704692
#704612
#702342
#702332
#702222
#702212
#253313