Finite Group Databasebeta

← #10858252

Subgroups — #10858252

Order 768.

900 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×C4×SD16C211yesmaximal, characteristic
2#82470C211yesmaximal, characteristic, complemented
3D4×Dic3⋊C4C211yesmaximal, characteristic
4C2×#4761C211yesmaximal, characteristic
5#72047C211yesmaximal, characteristic
6#72047C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic, complemented
8#71607C211yesmaximal, characteristic
9#70567C211yesmaximal, characteristic, complemented
10#70565C211yesmaximal, characteristic, complemented
11#70250C211yesmaximal, characteristic, complemented
12#70247C211yesmaximal, characteristic, complemented
13#70170C211yesmaximal, characteristic
14#69784C211yesmaximal, characteristic, complemented
15#69769C211yesmaximal, characteristic, complemented
16#26376—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×Dic3⋊C4—21no—
23C2×#888C2211yescharacteristic
24C3×C4×SD16—24no—
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5229C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5210—21no—
36C2×#817C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4984C2211yescharacteristic
40#4982—21no—
41#4979—21no—
42#4975C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4946—21no—
47#4942—21no—
48#4933—21no—
49C4×Dic3⋊C4C2211yescharacteristic
50#4929—21no—
51#4928C2211yescharacteristic
52#4928C2211yescharacteristic
53#4761—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#476—31no—
63C2×C4×SD16D311yescharacteristic
64#2451—31no—
65D4×C4⋊C4—31no—
66#2096—31no—
67#2084—31no—
68#2066—31no—
69#2058—31no—
70#1941—31no—
71#1916—31no—
72#1877—31no—
73#1877—31no—
74#1865—31no—
75#1864—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×Dic6—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16Q814yescomplemented
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—41no—
90C3×C4×D4—22no—
91C3×C4×D4—41no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100D4×Dic3—22no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103D4×Dic3—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—44no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#888—22no—
123#888—41no—
124C3×#106—22no—
125C3×#103—22no—
126C3×#102D414yes—
127C3×C4×C8—22no—
128#835—22no—
129#835—41no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#817—22no—
143#817—22no—
144Dic3⋊C8—22no—
145#808—22no—
146#808—41no—
147C22×SD16C2×D311yescharacteristic
148C2×C4⋊Q8—31no—
149C2×C4×Q8C2×D311yescharacteristic
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C22×C4⋊C4—61no—
154#476—64no—
155C4×SD16—24no—
156C2×#106—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106C2×D311yescharacteristic
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×C4×C8C2×D311yescharacteristic
169#398—61no—
170#395—61no—
171#391—31no—
172#391—31no—
173#391—61no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—31no—
178#387—61no—
179#385—61no—
180#385—61no—
181#384—31no—
182#380—61no—
183C4×C4⋊C4—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192C24×C3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8C2×Q812yes—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×Q812yes—
204C2×C3×D4#4112yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—81no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—24no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—42no—
222C22×Dic3—42no—
223C22×Dic3—81no—
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C2×Dic6—42no—
229C2×Dic6—42no—
230C2×Dic6—81no—
231C3×SD16—28no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C2×D412yes—
237C2×C3×C8C2×Q812yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—42no—
247#217—24no—
248#217—42no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—81no—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—41no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265Dic3⋊C4SD1614yescomplemented
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—61no—
281C23×C4—61no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×SD16Dic614yescomplemented
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289C4⋊Q8—62no—
290C4⋊Q8—121no—
291C4×Q8—22no—
292C4×Q8—41no—
293C4×D4—22no—
294C4×D4—41no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—62no—
298C4×D4—121no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—124no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—21no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42C22×D311yescharacteristic
323#106—22no—
324#106—62no—
325#106—62no—
326#106—62no—
327#106—121no—
328C4⋊C8—62no—
329#103—22no—
330#103—62no—
331#103—121no—
332#102—62no—
333#102—62no—
334#102#6714yes—
335C4×C8—22no—
336#95—31no—
337#95—31no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4C22⋊Q814yes—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
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—84no—
377C2×C3×C4C2×SD1612yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×Q811yescharacteristic
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—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—48no—
396C2×Dic3—41no—
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—84no—
408C2×Dic3C2×SD1612yes—
409C2×Dic3C2×SD1612yes—
410Dic6—84no—
411C3×C8—22no—
412C3×C8—22no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C24—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—122no—
422C2×Q8—122no—
423C2×Q8—241no—
424C2×Q8C2×Dic612yes—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×Dic612yes—
429C2×D4#23512yes—
430C2×D4C2×#6712yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—64no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—81no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—241no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454SD16—28no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8C2×Dic612yes—
466C2×C8C2×#6712yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—34no—
471C4⋊C4—32no—
472C4⋊C4—42no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—121no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C4⋊C4—241no—
488C4⋊C4C2×#6712yes—
489C22⋊C4—42no—
490C22⋊C4—64no—
491C22⋊C4—122no—
492C42—21no—
493C42—21no—
494C42—42no—
495C42—61no—
496C42—61no—
497C42—61no—
498C42—61no—
499C22×C3—24no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3#52511yescharacteristic
507C22×C3D4211yescharacteristic
508C22×C3D4×Q811yescharacteristic
509C22×C3C22×SD1611yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57711yescharacteristic
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—81no—
518C3×C4C2×C22⋊Q811yescharacteristic
519C3×C4#53911yescharacteristic
520C3×C4#54711yescharacteristic
521C3×C4#55411yescharacteristic
522Dic3—22no—
523Dic3—22no—
524Dic3—42no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×D3×D411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—41no—
541Q8—41no—
542Q8—244no—
543D4—22no—
544D4—22no—
545D4—22no—
546D4#92414yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
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—61no—
573C2×C4—62no—
574C2×C4—84no—
575C2×C4—128no—
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—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—244no—
588C2×C4D3×SD1612yes—
589C2×C4C22×Dic611yescharacteristic
590C2×C4C2×#23511yescharacteristic
591C2×C4#100411yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4#101011yescharacteristic
594C2×C4C22×#6711yescharacteristic
595C8—22no—
596C8—22no—
597C8—121no—
598C8—121no—
599C2×C3—44no—
600C2×C3#244911yescharacteristic
601C2×C3#310111yescharacteristic
602C2×C3#312511yescharacteristic
603C2×C3#326711yescharacteristic
604C2×C3D4×SD1611yescharacteristic
605C2×C3Q8×SD1611yescharacteristic
606C2×C3#338811yescharacteristic
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22D4×Dic611yescharacteristic
615C22#554711yescharacteristic
616C22C2×D3×SD1611yescharacteristic
617C22#576211yescharacteristic
618C22#576811yescharacteristic
619C22#577011yescharacteristic
620C22D4×#6711yescharacteristic
621C4—22no—
622C4—42no—
623C4—41no—
624C4—42no—
625C4—62no—
626C4—62no—
627C4—81no—
628C4—122no—
629C4—121no—
630C4—121no—
631C4—241no—
632C4—241no—
633C4—241no—
634C4#553911yescharacteristic
635C4#556411yescharacteristic
636C4C2×#92411yescharacteristic
637C4#582411yescharacteristic
638C3#2637611yescharacteristic, complemented, Sylow
639C2—44no—
640C2SD16×Dic611yescharacteristic
641C2#7963611yescharacteristic
642C2#7965111yescharacteristic
643C2#7966011yescharacteristic
644C2#8236011yescharacteristic
645C2SD16×#6711yescharacteristic
646C2#8392111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×C4×SD16D3
hypocenterC3#26376
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. #82470C2semidirect
7. #71629C2semidirect
9. #70567C2semidirect
10. #70565C2semidirect
11. #70250C2semidirect
12. #70247C2semidirect
14. #69784C2semidirect
15. #69769C2semidirect
85. C2×C3×SD16Q8semidirect
265. Dic3⋊C4SD16semidirect
284. C2×SD16Dic6semidirect
638. C3#26376semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SD162
#824702
D4×Dic3⋊C42
C2×#47612
#720472
#716292
#716072
#705672
#705652
#702502
#702472
#701702
#697842
#697692
#263763