Finite Group Databasebeta

← #10858219

Subgroups — #10858219

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic
2Q8×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#4791C211yesmaximal, characteristic, complemented
4#76323C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic
6#72048C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic, complemented
8#70237C211yesmaximal, characteristic, complemented
9#70193C211yesmaximal, characteristic
10#70170C211yesmaximal, characteristic
11#69656C211yesmaximal, characteristic
12#68858C211yesmaximal, characteristic, complemented
13#68827C211yesmaximal, characteristic, complemented
14#68640C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#26373—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×#878—21no—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32Dic3×SD16—24no—
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4984—21no—
38#4984—21no—
39#4984C2211yescharacteristic
40#4984C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45#4948—21no—
46#4947—21no—
47#4946—21no—
48#4935—21no—
49#4935C2211yescharacteristic
50C4×C4⋊Dic3—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4791—24no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#475—31no—
63C2×C4×SD16—31no—
64#2452—31no—
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2084—31no—
68#2065—31no—
69#2058—31no—
70#1941—31no—
71#1916—31no—
72#1880—31no—
73#1880D311yescharacteristic
74#1865—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6—21no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16Q814yescomplemented
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100Q8×Dic3—41no—
101Q8×Dic3—41no—
102D4×Dic3—22no—
103D4×Dic3—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#888—22no—
126#888—41no—
127#878—44no—
128C3×#103—22no—
129C3×#102—22no—
130#835—22no—
131#835D414yes—
132#832—22no—
133#832—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#817—22no—
141#817—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#808—22no—
145#808—22no—
146#808—41no—
147C22×SD16C2×D311yescharacteristic
148C2×C4⋊Q8—31no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154#475—64no—
155C4×SD16—64no—
156C2×#106—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169#396—61no—
170#393—61no—
171#390—31no—
172#390C2×D311yescharacteristic
173#389—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#385—61no—
180#384—31no—
181#384—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192C24×C3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8C2×D412yes—
197C2×C3×Q8C2×Q812yes—
198C2×C3×Q8#4112yes—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4C2×Q812yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×Dic6—42no—
223C2×Dic6—42no—
224C2×Dic6—81no—
225C3×SD16—28no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C2×Q812yes—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—81no—
239C3×C42—24no—
240C3×C42—21no—
241C3×C42—21no—
242#217—42no—
243#217—42no—
244#217—42no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—41no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3—42no—
256C4⋊Dic3—81no—
257C4⋊Dic3—81no—
258C4⋊Dic3—81no—
259C4⋊Dic3SD1614yescomplemented
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D412yes—
276C22×Q8—61no—
277C22×Q8C22×D311yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—61no—
280C2×SD16—22no—
281C2×SD16—22no—
282C2×SD16Dic614yescomplemented
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C4⋊Q8—62no—
288C4⋊Q8—121no—
289C22⋊Q8—124no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—121no—
295C4×Q8—121no—
296C4×D4—62no—
297C4×D4—121no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
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—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—21no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—62no—
328#106—62no—
329#106—62no—
330#106—62no—
331#106—121no—
332C4⋊C8—62no—
333#103—22no—
334#103—34no—
335#103—62no—
336#102—22no—
337#102—62no—
338#102—121no—
339C4×C8—62no—
340#95—31no—
341#95—31no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8C22⋊Q814yescomplemented
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—41no—
361C3×D4—41no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4C2×SD1612yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×Q811yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C42−1+411yescharacteristic
383C2×Dic3—21no—
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—41no—
395C2×Dic3—41no—
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—84no—
406C2×Dic3C2×SD1612yes—
407C2×Dic3C2×SD1612yes—
408Dic6—84no—
409C3×C8—22no—
410C3×C8—41no—
411C3⋊C8—22no—
412C3⋊C8—41no—
413C24—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—122no—
418C2×Q8—122no—
419C2×Q8—241no—
420C2×Q8C2×Dic612yes—
421C2×Q8D3×D412yes—
422C2×Q8#23712yes—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×Dic612yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—241no—
443C22×C4D3×D411yescharacteristic
444C22×C4D3×D411yescharacteristic
445C22×C4C23×D311yescharacteristic
446SD16—28no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—32no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8C2×Dic612yes—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—34no—
464C4⋊C4—32no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—121no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4#23712yes—
483C22⋊C4—42no—
484C22⋊C4—81no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C42—24no—
489C42—21no—
490C42—21no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42—61no—
495C42—122no—
496C42—122no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3#52511yescharacteristic
505C22×C3D4211yescharacteristic
506C22×C3D4×Q811yescharacteristic
507C22×C3C22×SD1611yescharacteristic
508C22×C3#57511yescharacteristic
509C22×C3#57711yescharacteristic
510C22×C3#57711yescharacteristic
511C3×C4—22no—
512C3×C4—24no—
513C3×C4—41no—
514C3×C4—42no—
515C3×C4—81no—
516C3×C4C2×C22⋊Q811yescharacteristic
517C3×C4#53911yescharacteristic
518C3×C4#54711yescharacteristic
519C3×C4#55411yescharacteristic
520Dic3—22no—
521Dic3—22no—
522Dic3—41no—
523Dic3—41no—
524Dic3—42no—
525Dic3—81no—
526Dic3—81no—
527Dic3—81no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×D3×D411yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—244no—
540Q8#87814yescomplemented
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
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—61no—
562C2×C4—61no—
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—62no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—244no—
584C2×C4D3×SD1612yes—
585C2×C4C22×Dic611yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4C2×#23711yescharacteristic
588C2×C4C2×#23711yescharacteristic
589C2×C4#100911yescharacteristic
590C2×C4#101011yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—121no—
595C2×C3—82no—
596C2×C3#244911yescharacteristic
597C2×C3#310311yescharacteristic
598C2×C3#312511yescharacteristic
599C2×C3#327111yescharacteristic
600C2×C3D4×SD1611yescharacteristic
601C2×C3Q8×SD1611yescharacteristic
602C2×C3#338811yescharacteristic
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22D4×Dic611yescharacteristic
611C22#558611yescharacteristic
612C22#560911yescharacteristic
613C22C2×D3×SD1611yescharacteristic
614C22#577011yescharacteristic
615C22#577511yescharacteristic
616C22#577711yescharacteristic
617C4—22no—
618C4—24no—
619C4—41no—
620C4—42no—
621C4—62no—
622C4—62no—
623C4—81no—
624C4—122no—
625C4—121no—
626C4—121no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4C2×#87811yescharacteristic
631C4#554311yescharacteristic
632C4#561111yescharacteristic
633C4#566511yescharacteristic
634C3#2637311yescharacteristic, complemented, Sylow
635C2—82no—
636C2#7471011yescharacteristic
637C2SD16×Dic611yescharacteristic
638C2#7976111yescharacteristic
639C2#8002811yescharacteristic
640C2#8007511yescharacteristic
641C2#8040611yescharacteristic
642C2#8061211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#26373
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic3C2semidirect
3. C2×#4791C2semidirect
7. #71629C2semidirect
8. #70237C2semidirect
12. #68858C2semidirect
13. #68827C2semidirect
14. #68640C2semidirect
15. #68610C2semidirect
83. C2×C3×SD16Q8semidirect
259. C4⋊Dic3SD16semidirect
282. C2×SD16Dic6semidirect
357. C3×Q8C22⋊Q8semidirect
540. Q8#878semidirect
634. C3#26373semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
Q8×C4⋊Dic32
C2×#47912
#763232
C3×#18802
#720482
#716292
#702372
#701932
#701702
#696562
#688582
#688272
#686402
#686102
#263733