Finite Group Databasebeta

← #10858312

Subgroups — #10858312

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic
2#82362C211yesmaximal, characteristic, complemented
3#82334C211yesmaximal, characteristic, complemented
4C2×#4761C211yesmaximal, characteristic, complemented
5#72037C211yesmaximal, characteristic
6#72037C211yesmaximal, characteristic
7#71607C211yesmaximal, characteristic, complemented
8#71606C211yesmaximal, characteristic, complemented
9#70620C211yesmaximal, characteristic, complemented
10#70617C211yesmaximal, characteristic, complemented
11#70182C211yesmaximal, characteristic, complemented
12#70179C211yesmaximal, characteristic, complemented
13#70170C211yesmaximal, characteristic, complemented
14#69784C211yesmaximal, characteristic, complemented
15#69769C211yesmaximal, characteristic, complemented
16#26380—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#924—21no—
23C2×#888C2211yescharacteristic
24C3×C4×D8—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5210—21no—
36#5210—21no—
37C2×C3⋊#95—21no—
38C2×#817C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4984C2211yescharacteristic
42#4977C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4948—21no—
47#4947—21no—
48#4946—21no—
49#4942—21no—
50C4×Dic3⋊C4C2211yescharacteristic
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×D8D311yescharacteristic
64#2452—31no—
65#2349—31no—
66#2092—31no—
67#2085—31no—
68#2065—31no—
69#2057—31no—
70#1941—31no—
71#1916—31no—
72#1879—31no—
73#1879—31no—
74#1864—31no—
75#1864—31no—
76#1862—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C22×Dic6—21no—
84C2×C3×D8—22no—
85C2×C3×D8—22no—
86C2×C3×D8Q814yescomplemented
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—41no—
92C3×C4×D4—41no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104D4×Dic3—22no—
105D4×Dic3—22no—
106D4×Dic3—22no—
107D4×Dic3—41no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113#924—44no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#888—22no—
125#888—41no—
126C3×#107—22no—
127C3×#102—22no—
128C3×#102D414yes—
129C3×C4×C8—22no—
130#832—22no—
131#832—22no—
132#832—22no—
133#832—41no—
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#807—41no—
147C22×D8C2×D311yescharacteristic
148C2×C4⋊Q8—31no—
149C2×C22⋊Q8—61no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154#476—64no—
155C4×D8—24no—
156C2×#107—31no—
157C2×#107C2×D311yescharacteristic
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C4×C8C2×D311yescharacteristic
169#396—61no—
170#393—61no—
171#390—31no—
172#390—31no—
173#389—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#385—61no—
178#385—61no—
179#385—61no—
180#382—31no—
181#380—61no—
182C4×C4⋊C4—31no—
183C2×#95—61no—
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—
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—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×Q812yes—
204C2×C3×D4C2×Q812yes—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4—81no—
213C22×C3×C4—81no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C22×Dic3—21no—
218C22×Dic3—24no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—42no—
223C22×Dic3—42no—
224C22×Dic3—81no—
225C22×Dic3—81no—
226C22×Dic3—81no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C2×Dic6—42no—
232C2×Dic6—42no—
233C2×Dic6—81no—
234C3×D8—28no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C2×D412yes—
240C2×C3×C8C2×Q812yes—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D412yes—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C42—21no—
248C3×C42—21no—
249#217—24no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—42no—
254#217—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3#4112yes—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—81no—
269Dic3⋊C4—81no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C22×Q8—61no—
281C22×D4C22×D311yescharacteristic
282C22×D4C22×D311yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—61no—
286C23×C4—61no—
287C2×D8—22no—
288C2×D8—22no—
289C2×D8Dic614yescomplemented
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—121no—
296C22⋊Q8—124no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—121no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42C22×D311yescharacteristic
329#107—22no—
330#107—121no—
331#106—62no—
332#106—62no—
333#106—62no—
334C4⋊C8—62no—
335#102—22no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—121no—
340#102#6714yes—
341C4×C8—22no—
342#95—31no—
343#95—31no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4C22⋊Q814yescomplemented
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—84no—
385C2×C3×C4—84no—
386C2×C3×C4AGL(1,ℤ/8)12yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×Q811yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
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—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—84no—
417C2×Dic3—84no—
418C2×Dic3#13512yes—
419Dic6—84no—
420C3×C8—22no—
421C3×C8—22no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C24—21no—
425C24—21no—
426C2×Q8—122no—
427C2×Q8—122no—
428C2×Q8—241no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4C2×Dic612yes—
438C2×D4C2×Dic612yes—
439C2×D4#23512yes—
440C2×D4C2×#6712yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—61no—
452C22×C4—64no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—241no—
461C22×C4—241no—
462C22×C4—241no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466D8—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8C2×Dic612yes—
478C2×C8C2×#6712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C4⋊C4—241no—
496C4⋊C4—241no—
497C4⋊C4C2×#6712yes—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—64no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—241no—
506C42—21no—
507C42—21no—
508C42—61no—
509C42—61no—
510C42—61no—
511C42—61no—
512C22×C3—24no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3#52511yescharacteristic
527C22×C3D4211yescharacteristic
528C22×C3D4×Q811yescharacteristic
529C22×C3C2×#13511yescharacteristic
530C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
531C22×C3#57611yescharacteristic
532C22×C3#57711yescharacteristic
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—41no—
536C3×C4—81no—
537C3×C4C2×C22⋊Q811yescharacteristic
538C3×C4#53911yescharacteristic
539C3×C4#54711yescharacteristic
540C3×C4#55411yescharacteristic
541Dic3—22no—
542Dic3—41no—
543Dic3—41no—
544Dic3—42no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548Dic3—81no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D3×D411yescharacteristic
564Q8—244no—
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4#92414yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—32no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—244no—
611C2×C4—244no—
612C2×C4#91112yes—
613C2×C4C22×Dic611yescharacteristic
614C2×C4C2×#23511yescharacteristic
615C2×C4#100411yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4#101011yescharacteristic
618C2×C4C22×#6711yescharacteristic
619C8—22no—
620C8—22no—
621C8—121no—
622C8—121no—
623C2×C3—44no—
624C2×C3—82no—
625C2×C3#244911yescharacteristic
626C2×C3#310611yescharacteristic
627C2×C3#311811yescharacteristic
628C2×C3#326411yescharacteristic
629C2×C3#328211yescharacteristic
630C2×C3#338011yescharacteristic
631C2×C3#338411yescharacteristic
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22D4×Dic611yescharacteristic
647C22#554711yescharacteristic
648C22C2×#91111yescharacteristic
649C22#575711yescharacteristic
650C22D3×#13511yescharacteristic
651C22#577011yescharacteristic
652C22D4×#6711yescharacteristic
653C4—22no—
654C4—42no—
655C4—41no—
656C4—62no—
657C4—81no—
658C4—121no—
659C4—121no—
660C4—121no—
661C4—122no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#553911yescharacteristic
666C4#556411yescharacteristic
667C4C2×#92411yescharacteristic
668C4#582411yescharacteristic
669C3#2638011yescharacteristic, complemented, Sylow
670C2—44no—
671C2—82no—
672C2#7957511yescharacteristic
673C2#7957611yescharacteristic
674C2#7959311yescharacteristic
675C2#7959411yescharacteristic
676C2#8236011yescharacteristic
677C2#8383711yescharacteristic
678C2#8391911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×C4×D8D3
hypocenterC3#26380
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. #82362C2semidirect
3. #82334C2semidirect
4. C2×#4761C2semidirect
7. #71607C2semidirect
8. #71606C2semidirect
9. #70620C2semidirect
10. #70617C2semidirect
11. #70182C2semidirect
12. #70179C2semidirect
13. #70170C2semidirect
14. #69784C2semidirect
15. #69769C2semidirect
86. C2×C3×D8Q8semidirect
289. C2×D8Dic6semidirect
372. C3×D4C22⋊Q8semidirect
572. D4#924semidirect
669. C3#26380semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
#823622
#823342
C2×#47612
#720372
#716072
#716062
#706202
#706172
#701822
#701792
#701702
#697842
#697692
#263803