Finite Group Databasebeta

← #10844619

Subgroups — #10844619

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
1C2×C3×#489C211yesmaximal, characteristic
2#83230C211yesmaximal, characteristic, complemented
3C2×#5094C211yesmaximal, characteristic, complemented
4#82971C211yesmaximal, characteristic
5#71721C211yesmaximal, characteristic
6#70261C211yesmaximal, characteristic, complemented
7#70246C211yesmaximal, characteristic, complemented
8#70241C211yesmaximal, characteristic, complemented
9#70229C211yesmaximal, characteristic, complemented
10#70193C211yesmaximal, characteristic
11#70178C211yesmaximal, characteristic
12#70174C211yesmaximal, characteristic
13#70162C211yesmaximal, characteristic
14#69836C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#25287—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#489—24no—
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5229—21no—
29#5229—21no—
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#5094—24no—
41#4984—21no—
42#4984C2211yescharacteristic
43#4984C2211yescharacteristic
44Dic3×C4⋊C4—21no—
45Dic3×C4⋊C4C2211yescharacteristic
46Dic3×C4⋊C4C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#808C2211yescharacteristic
51#4948—21no—
52#4948—21no—
53#4944—21no—
54#4944—21no—
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#489D311yescharacteristic
63C2×#475—31no—
64#2545—31no—
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×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8D414yes—
85C3×#124—22no—
86C3×#124—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97#944—22no—
98#944—41no—
99#936—22no—
100#936—41no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3D411yescharacteristic
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C4×Dic6—22no—
120C4×Dic6—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#97—22no—
126#835—22no—
127#835—41no—
128#832—22no—
129#832—41no—
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#816—22no—
140#816—22no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#810—22no—
144#810—22no—
145#808—22no—
146#808—22no—
147C4⋊(C3⋊C8)—22no—
148C2×C4⋊Q8—31no—
149C2×C4⋊Q8C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153#489—24no—
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×#103C2×D311yescharacteristic
167C2×#103C2×D311yescharacteristic
168C2×#102—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97C2×D311yescharacteristic
173#396—61no—
174#396—61no—
175#391—31no—
176#391—61no—
177#391—61no—
178#390—31no—
179#389—31no—
180#389—31no—
181#389—61no—
182#388—61no—
183#388—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×Q8—24no—
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—81no—
202C2×C3×Q8—81no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
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#4112yes—
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—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4C2×D412yes—
239C3×C4⋊C4C2×D412yes—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C42C2×D412yes—
243C3×C42#4112yes—
244#217—42no—
245#217—42no—
246#217—42no—
247#217—42no—
248C4⋊Dic3—21no—
249C4⋊Dic3—22no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—41no—
255C4⋊Dic3—81no—
256C4⋊Dic3—81no—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C4×Dic3—42no—
267C4×Dic3—81no—
268C4×Dic3—81no—
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—21no—
276C22×Q8—31no—
277C22×Q8C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C4⋊Q8—22no—
284C4⋊Q8—62no—
285C4⋊Q8—121no—
286C4⋊Q8#6714yes—
287#124—22no—
288#124—22no—
289#124—62no—
290#124—121no—
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—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—62no—
320#106—62no—
321C4⋊C8—62no—
322C4⋊C8—121no—
323C4⋊C8—121no—
324#103—22no—
325#103—22no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—121no—
332#102—22no—
333#102—22no—
334#102—62no—
335#102—121no—
336#97—22no—
337#95—31no—
338#95—31no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
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—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—84no—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
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#12112yes—
377C2×C3×C4#12112yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C42−1+411yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
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#13712yes—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412C3×C8—41no—
413C3×C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—21no—
417C2×Q8—24no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—32no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—81no—
431C2×Q8—81no—
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—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—32no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—121no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C4⋊C4C2×#6712yes—
481C4⋊C4C2×#6712yes—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—61no—
492C42—122no—
493C42—122no—
494C42—241no—
495C42—241no—
496C42#23712yes—
497C42C2×#6712yes—
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—42no—
515C3×C4—42no—
516C3×C4—81no—
517C3×C4#52911yescharacteristic
518C3×C4#54211yescharacteristic
519C3×C4#54811yescharacteristic
520C3×C4#55011yescharacteristic
521Dic3—22no—
522Dic3—42no—
523Dic3—42no—
524Dic3—41no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C2×D3×D411yescharacteristic
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—62no—
542Q8—62no—
543Q8—84no—
544Q8—121no—
545Q8—121no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—32no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
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—121no—
587C2×C4#93912yes—
588C2×C4#93912yes—
589C2×C4C2×#23711yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4#100911yescharacteristic
593C2×C4#101011yescharacteristic
594C2×C4C22×#6711yescharacteristic
595C8—41no—
596C8—41no—
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—
614C22#567611yescharacteristic
615C22#567711yescharacteristic
616C22#577511yescharacteristic
617C22#577511yescharacteristic
618C22D3×#13711yescharacteristic
619C22#577711yescharacteristic
620C22C2×#93911yescharacteristic
621C4—22no—
622C4—22no—
623C4—42no—
624C4—42no—
625C4—62no—
626C4—81no—
627C4—122no—
628C4—121no—
629C4—121no—
630C4—121no—
631C4—122no—
632C4—241no—
633C4—241no—
634C4#568011yescharacteristic
635C4#568211yescharacteristic
636C4#583111yescharacteristic
637C4#583411yescharacteristic
638C3#2528711yescharacteristic, complemented, Sylow
639C2—161no—
640C2#8167111yescharacteristic
641C2#8167611yescharacteristic
642C2#8170111yescharacteristic
643C2#8170311yescharacteristic
644C2#8299511yescharacteristic
645C2#8393511yescharacteristic
646C2#8396611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#489D3
hypocenterC3#25287
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. #83230C2semidirect
3. C2×#5094C2semidirect
6. #70261C2semidirect
7. #70246C2semidirect
8. #70241C2semidirect
9. #70229C2semidirect
14. #69836C2semidirect
15. #69813C2semidirect
638. C3#25287semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
#832302
C2×#50942
#829712
#717212
#702612
#702462
#702412
#702292
#701932
#701782
#701742
#701622
#698362
#698132
#252873