Finite Group Databasebeta

← #10858204

Subgroups — #10858204

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×SD16C211yesmaximal, characteristic
2#82333C211yesmaximal, characteristic
3C2×#4759C211yesmaximal, characteristic
4#76509C211yesmaximal, characteristic, complemented
5C3×#1877C211yesmaximal, characteristic
6#72047C211yesmaximal, characteristic
7#71606C211yesmaximal, characteristic
8#70262C211yesmaximal, characteristic, complemented
9#70236C211yesmaximal, characteristic, complemented
10#70169C211yesmaximal, characteristic
11#69686C211yesmaximal, characteristic, complemented
12#68859C211yesmaximal, characteristic, complemented
13#68827C211yesmaximal, characteristic, complemented
14#68641C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#26372—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#391C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5212—21no—
32C2×C3⋊#95—21no—
33Dic3×SD16—24no—
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4950—21no—
45#4950—21no—
46#4936C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48#4928C2211yescharacteristic
49#4759—24no—
50#4657—21no—
51#4656—21no—
52#4654—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#474—31no—
63C2×C4×SD16—31no—
64#2451—31no—
65#2349—31no—
66#2097—31no—
67#2085—31no—
68#2068—31no—
69#2058—31no—
70#1942—31no—
71#1916—31no—
72#1877—31no—
73#1877D311yescharacteristic
74#1864—31no—
75#1863—31no—
76#1862—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×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96Q8×Dic3—22no—
97Q8×Dic3—41no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—41no—
101D4×Dic3—41no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3D411yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#888—22no—
120#888—41no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#95—21no—
124C3×#95—21no—
125#835—22no—
126#835—41no—
127#832—22no—
128#832D414yes—
129C3⋊#95—21no—
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#818—22no—
140#817—22no—
141Dic3⋊C8—22no—
142C8×Dic3—22no—
143#808—41no—
144#807—22no—
145#807—22no—
146C22×SD16C2×D311yescharacteristic
147C2×C4⋊Q8—31no—
148C2×C4×Q8—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4C2×D311yescharacteristic
152#474—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×C4×C8—31no—
167#398—61no—
168#395—61no—
169#391—31no—
170#391—61no—
171#391C2×D311yescharacteristic
172#389—31no—
173#389—31no—
174#389—31no—
175#389—31no—
176#387—61no—
177#385—61no—
178#385—61no—
179#382—31no—
180#380—61no—
181C4×C4⋊C4—31no—
182C2×#95—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340C2×D311yescharacteristic
187#336—31no—
188#336—31no—
189#336—31no—
190#336C2×D311yescharacteristic
191C24×C3—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×Q812yes—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C2×D412yes—
202C2×C3×D4C2×Q812yes—
203C2×C3×D4#4112yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—81no—
209C22×C3×C4—81no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—81no—
222C22×Dic3—81no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C2×Dic6—42no—
228C2×Dic6—42no—
229C2×Dic6—81no—
230C3×SD16—28no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C2×Q812yes—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4#4112yes—
241C3×C22⋊C4—24no—
242C3×C42—21no—
243C3×C42—21no—
244#217—42no—
245#217—42no—
246C4⋊Dic3—21no—
247C4⋊Dic3—22no—
248C4⋊Dic3—22no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—42no—
256C4⋊Dic3—81no—
257C4⋊Dic3—81no—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—42no—
260Dic3⋊C4—81no—
261Dic3⋊C4—81no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273C22×Q8—61no—
274C22×Q8C22×D311yescharacteristic
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C23×C4—61no—
278C23×C4—61no—
279C2×SD16—22no—
280C2×SD16—22no—
281C2×SD16Dic614yescomplemented
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C4⋊Q8—62no—
287C4⋊Q8—121no—
288C4×Q8—62no—
289C4×Q8—121no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—121no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#107—62no—
319#107—62no—
320#106—62no—
321#106—121no—
322C4⋊C8—62no—
323#103—22no—
324#103—62no—
325#103—121no—
326#102—22no—
327#102—34no—
328#102—62no—
329C4×C8—62no—
330#95—21no—
331#95—21no—
332#95—31no—
333#95—31no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3—22no—
343C23×C3—22no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4C22⋊Q814yes—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—84no—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C22×Q811yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C42−1+411yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
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—84no—
403C2×Dic3—84no—
404C2×Dic3#13512yes—
405C2×Dic3#13712yes—
406Dic6—84no—
407C3×C8—22no—
408C3×C8—41no—
409C3⋊C8—22no—
410C3⋊C8—41no—
411C24—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—122no—
418C2×Q8—122no—
419C2×Q8—241no—
420C2×Q8C2×Dic612yes—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×Dic612yes—
425C2×D4D3×D412yes—
426C2×D4#23712yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—241no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—28no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8C2×Dic612yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C4⋊C4—241no—
481C4⋊C4—241no—
482C4⋊C4#23712yes—
483C22⋊C4—24no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C42—21no—
487C42—21no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—61no—
492C42—122no—
493C22×C3—24no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3#52511yescharacteristic
501C22×C3D4211yescharacteristic
502C22×C3D4×Q811yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×#13711yescharacteristic
505C22×C3#57611yescharacteristic
506C22×C3#57711yescharacteristic
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4C2×C22⋊Q811yescharacteristic
513C3×C4#53911yescharacteristic
514C3×C4#54711yescharacteristic
515C3×C4#55411yescharacteristic
516Dic3—22no—
517Dic3—22no—
518Dic3—41no—
519Dic3—42no—
520Dic3—41no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—244no—
537D4—22no—
538D4—22no—
539D4—22no—
540D4#87814yes—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—84no—
570C2×C4—121no—
571C2×C4—121no—
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—244no—
582C2×C4—244no—
583C2×C4C22×Dic611yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4C2×#23711yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C2×C4#100911yescharacteristic
588C2×C4#101011yescharacteristic
589C8—22no—
590C8—41no—
591C8—62no—
592C8—121no—
593C2×C3—44no—
594C2×C3#244911yescharacteristic
595C2×C3#310711yescharacteristic
596C2×C3#311811yescharacteristic
597C2×C3#326611yescharacteristic
598C2×C3#328011yescharacteristic
599C2×C3#338111yescharacteristic
600C2×C3#338511yescharacteristic
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22D4×Dic611yescharacteristic
609C22#558611yescharacteristic
610C22#560911yescharacteristic
611C22#576211yescharacteristic
612C22D3×#13511yescharacteristic
613C22#577411yescharacteristic
614C22D3×#13711yescharacteristic
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—62no—
620C4—62no—
621C4—81no—
622C4—121no—
623C4—121no—
624C4—122no—
625C4—241no—
626C4—241no—
627C4—241no—
628C4C2×#87811yescharacteristic
629C4#554311yescharacteristic
630C4#561111yescharacteristic
631C4#566511yescharacteristic
632C3#2637211yescharacteristic, complemented, Sylow
633C2—44no—
634C2#7471011yescharacteristic
635C2#7963311yescharacteristic
636C2#7975911yescharacteristic
637C2#8013611yescharacteristic
638C2#8021311yescharacteristic
639C2#8044411yescharacteristic
640C2#8055711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1877D3
hypocenterC3#26372
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
4. #76509C2semidirect
8. #70262C2semidirect
9. #70236C2semidirect
11. #69686C2semidirect
12. #68859C2semidirect
13. #68827C2semidirect
14. #68641C2semidirect
15. #68610C2semidirect
85. C2×C3×SD16Q8semidirect
281. C2×SD16Dic6semidirect
632. C3#26372semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SD162
#823332
C2×#47592
#765092
C3×#18772
#720472
#716062
#702622
#702362
#701692
#696862
#688592
#688272
#686412
#686102
#263723