Finite Group Databasebeta

← #10798252

Subgroups — #10798252

Order 768.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#440C211yesmaximal, characteristic
2C4×Q8×Dic3C211yesmaximal, characteristic, complemented
3#82319C211yesmaximal, characteristic
4#72048C211yesmaximal, characteristic
5#72048C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic, complemented
7#71609C211yesmaximal, characteristic
8#70610C211yesmaximal, characteristic, complemented
9#70609C211yesmaximal, characteristic, complemented
10#70238C211yesmaximal, characteristic, complemented
11#70236C211yesmaximal, characteristic, complemented
12#70156C211yesmaximal, characteristic
13#70155C211yesmaximal, characteristic
14#69814C211yesmaximal, characteristic, complemented
15#69805C211yesmaximal, characteristic, complemented
16#21124—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#926—21no—
23C3×#440C418yescomplemented
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C4×#217—21no—
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37#4984C2211yescharacteristic
38#4984C2211yescharacteristic
39Dic3×C4⋊C4—21no—
40Dic3×C4⋊C4—21no—
41Dic3×C4⋊C4C2211yescharacteristic
42Dic3×C4⋊C4C2211yescharacteristic
43#4951—21no—
44#4951—21no—
45C4×C4⋊Dic3—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C42×Dic3—21no—
49C42×Dic3C2211yescharacteristic
50C2×#803C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#440D311yescharacteristic
62#2293—31no—
63C42×Q8—31no—
64#1926—31no—
65#1925—31no—
66#1921—31no—
67#1918—31no—
68#1880—31no—
69#1880—31no—
70#1865—31no—
71#1863—31no—
72#1776—31no—
73#1763—31no—
74#1762—31no—
75#1745—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16C2×C414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×Q8—22no—
85C3×C4×Q8C2×C414yes—
86C3×C4×D4—22no—
87C3×C4×D4C2×C414yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102#926—44no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C3×#107C2×C414yes—
126C3×#103C2×C414yes—
127C3×#102C2×C414yes—
128C3×#97C2×C414yes—
129#835—22no—
130#835—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#818—22no—
140#817—22no—
141#803—22no—
142C22×SD16C2×D311yescharacteristic
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8C2×D311yescharacteristic
146C2×C4×D4C2×D311yescharacteristic
147C2×#117—61no—
148#440Dic318yescomplemented
149C2×#107—31no—
150C2×#107C2×D311yescharacteristic
151C2×#106—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102C2×D311yescharacteristic
158C2×#97—31no—
159C2×#97C2×D311yescharacteristic
160#390—31no—
161#390—31no—
162#389—31no—
163#389—31no—
164#388—61no—
165#388—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4—61no—
173C4×C22⋊C4—61no—
174C43—31no—
175C43—61no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8C22×C412yes—
191C2×C3×Q8#4112yes—
192C2×C3×Q8#4112yes—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4C22×C412yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—24no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C3×SD16—28no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8C22×C412yes—
222C2×C3×C8C22×C412yes—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4C22×C412yes—
227C3×C4⋊C4C22×C412yes—
228C3×C22⋊C4—24no—
229C3×C42—24no—
230C3×C42C22×C412yes—
231C3×C42#4112yes—
232#217—42no—
233#217—42no—
234#217—42no—
235#217—42no—
236C4⋊Dic3—24no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—42no—
246C4⋊Dic3#4112yes—
247C4⋊Dic3#4112yes—
248C4×Dic3—24no—
249C4×Dic3—24no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3—41no—
257C4×Dic3—41no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C22×Q8C22×D311yescharacteristic
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C2×SD16—22no—
272C2×SD16—22no—
273C2×SD16C2×Dic314yes—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4×Q8—22no—
279C4×Q8—62no—
280C4×Q8—62no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×Q8C2×Dic314yes—
284C4×D4—22no—
285C4×D4C2×Dic314yes—
286#117—124no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C42—21no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107C2×Dic314yes—
319#106—62no—
320#103—62no—
321#103—62no—
322#103C2×Dic314yes—
323#102—62no—
324#102—62no—
325#102C2×Dic314yes—
326#97—62no—
327#97C2×Dic314yes—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334C23×C3—22no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3C22×D411yescharacteristic
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8#11714yescomplemented
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—48no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4C4×D412yes—
365C2×C3×C4C4×D412yes—
366C2×C3×C4C23×C411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3AGL(1,ℤ/8)12yes—
395C2×Dic3#13712yes—
396C3×C8—22no—
397C3×C8—22no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C24—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8#23512yes—
405C2×Q8#23512yes—
406C2×Q8C22×Dic312yes—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4C22×Dic312yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429C22×C4C23×D311yescharacteristic
430SD16—28no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8C22×Dic312yes—
442C2×C8C22×Dic312yes—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—32no—
447C4⋊C4—32no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4C22×Dic312yes—
459C4⋊C4C22×Dic312yes—
460C22⋊C4—24no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C42—24no—
466C42—62no—
467C42—62no—
468C42—64no—
469C42—64no—
470C42—61no—
471C42—61no—
472C42—61no—
473C42—61no—
474C42—121no—
475C42—121no—
476C42—122no—
477C42—122no—
478C42—122no—
479C42—122no—
480C42#23712yes—
481C42C22×Dic312yes—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3C2×C4×D411yescharacteristic
490C22×C3#52511yescharacteristic
491C22×C3#52511yescharacteristic
492C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
493C22×C3C2×#13711yescharacteristic
494C22×C3#57511yescharacteristic
495C22×C3#57511yescharacteristic
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—24no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4C2×#11711yescharacteristic
502C3×C4C4×#4111yescharacteristic
503C3×C4#52911yescharacteristic
504C3×C4#52911yescharacteristic
505Dic3—22no—
506Dic3—22no—
507Dic3—41no—
508Dic3—42no—
509Dic3—41no—
510Dic3—42no—
511Dic3—81no—
512Dic3—81no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D3×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8#92614yescomplemented
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4D4×Dic312yes—
568C2×C4D4×Dic312yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23511yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4C23×Dic311yescharacteristic
575C8—22no—
576C8—22no—
577C8—121no—
578C8—121no—
579C2×C3—82no—
580C2×C3#229011yescharacteristic
581C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
582C2×C3C4×#13711yescharacteristic
583C2×C3#311111yescharacteristic
584C2×C3#311211yescharacteristic
585C2×C3#311511yescharacteristic
586C2×C3#311711yescharacteristic
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22#554711yescharacteristic
595C22#554711yescharacteristic
596C22D3×AGL(1,ℤ/8)11yescharacteristic
597C22#577311yescharacteristic
598C22D3×#13711yescharacteristic
599C22#577711yescharacteristic
600C22C2×D4×Dic311yescharacteristic
601C4—24no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—62no—
607C4—62no—
608C4—121no—
609C4—122no—
610C4—121no—
611C4—122no—
612C4—241no—
613C4—241no—
614C4#553811yescharacteristic
615C4#553811yescharacteristic
616C4C2×#92611yescharacteristic
617C4Dic3×#4111yescharacteristic
618C3#2112411yescharacteristic, complemented, Sylow
619C2—82no—
620C2#7978111yescharacteristic
621C2#7978411yescharacteristic
622C2#7981011yescharacteristic
623C2#7981111yescharacteristic
624C2#8231811yescharacteristic
625C2Dic3×AGL(1,ℤ/8)11yescharacteristic
626C2Dic3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenterC2×#440D3
hypocenterC3#21124
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×Dic3C2semidirect
6. #71631C2semidirect
8. #70610C2semidirect
9. #70609C2semidirect
10. #70238C2semidirect
11. #70236C2semidirect
14. #69814C2semidirect
15. #69805C2semidirect
23. C3×#440C4semidirect
148. #440Dic3semidirect
345. C3×Q8#117semidirect
524. Q8#926semidirect
618. C3#21124semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4402
C4×Q8×Dic32
#823192
#720482
#716312
#716092
#706102
#706092
#702382
#702362
#701562
#701552
#698142
#698052
#211243