Finite Group Databasebeta

← #10853307

Subgroups — #10853307

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83242C211yesmaximal, characteristic, complemented
2#76302C211yesmaximal, characteristic
3C3×#2023C211yesmaximal, characteristic
4#71779C211yesmaximal, characteristic, complemented
5#70179C211yesmaximal, characteristic
6#70163C211yesmaximal, characteristic
7#70153C211yesmaximal, characteristic
8Dic3×#102C211yesmaximal, characteristic
9#69652C211yesmaximal, characteristic
10#68847C211yesmaximal, characteristic, complemented
11#68799C211yesmaximal, characteristic, complemented
12#68796C211yesmaximal, characteristic, complemented
13#68789C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68617C211yesmaximal, characteristic, complemented
16#26024—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#944—21no—
19C2×D4×Dic3C2211yescharacteristic
20C2×#888C2211yescharacteristic
21C2×#877—21no—
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25#5230—21no—
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4979—21no—
34#4976—21no—
35#4975C2211yescharacteristic
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41#4945—21no—
42#4944—21no—
43#4942—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2551—31no—
61#2354—31no—
62#2065—31no—
63#2065—31no—
64#2023—31no—
65#2023D311yescharacteristic
66#1987—31no—
67#1987—31no—
68#1948—31no—
69#1939—31no—
70#1920—31no—
71#1920—31no—
72#1834—31no—
73#1802—31no—
74C4×#102—31no—
75C22×C3×Q8—21no—
76C22×C3×D4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#944—44no—
94D4×Dic3—22no—
95D4×Dic3Q814yescomplemented
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#888—22no—
117#888—41no—
118#877—44no—
119C3×#102—22no—
120C3×#102—41no—
121#832—22no—
122#832—41no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130#817—22no—
131Dic3⋊C8—22no—
132Dic3⋊C8—41no—
133C8×Dic3—22no—
134#808—22no—
135C4⋊(C3⋊C8)—41no—
136C2×C4⋊Q8—31no—
137C2×C4⋊Q8—61no—
138C2×C4⋊Q8C2×D311yescharacteristic
139C2×C4×D4—31no—
140C2×#117—61no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×C4⋊C8—31no—
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×C4×C8—31no—
151#395—61no—
152#395—61no—
153#390—31no—
154#390C2×D311yescharacteristic
155#389—31no—
156#389—31no—
157#389—61no—
158#388—61no—
159#388—61no—
160#384—31no—
161#384—31no—
162#380—61no—
163C4×C4⋊C4—31no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C24×C3—21no—
180C2×C3×Q8—42no—
181C2×C3×Q8—42no—
182C2×C3×Q8—81no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C2×Q812yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—24no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×Dic6—42no—
207C2×Dic6—42no—
208C2×Dic6—81no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4#4112yes—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—81no—
224C3×C42—21no—
225C3×C42—21no—
226#217—24no—
227#217—42no—
228#217—81no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3C2×Q812yes—
236C4⋊Dic3#4112yes—
237Dic3⋊C4—22no—
238Dic3⋊C4—42no—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—81no—
244Dic3⋊C4—81no—
245Dic3⋊C4SD1614yes—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C4×Dic3—42no—
250C4×Dic3—42no—
251C4×Dic3C2×Q812yes—
252C4×Dic3#4112yes—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C22×Q8—21no—
260C22×Q8—61no—
261C22×D4C22×D311yescharacteristic
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C4⋊Q8—22no—
268C4⋊Q8—41no—
269C4⋊Q8—62no—
270C4⋊Q8—121no—
271C4⋊Q8—124no—
272C4×D4—34no—
273C4×D4—62no—
274#117—124no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#106—62no—
304#106—62no—
305C4⋊C8—62no—
306C4⋊C8—121no—
307C4⋊C8—121no—
308#102—22no—
309#102—41no—
310#102—62no—
311#102—121no—
312C4×C8—62no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3C22×D411yescharacteristic
328C3×Q8—84no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—41no—
332C3×D4—41no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4C2×SD1612yes—
348C2×C3×C4C22×Q811yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C2×C3×C42−1+411yescharacteristic
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—22no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—48no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3C22⋊Q812yes—
375C2×Dic3C22⋊Q812yes—
376C2×Dic3C2×SD1612yes—
377C2×Dic3C2×SD1612yes—
378C2×Dic3AGL(1,ℤ/8)12yes—
379Dic6—84no—
380C3×C8—41no—
381C3×C8—41no—
382C3⋊C8—41no—
383C3⋊C8—41no—
384C24—21no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—81no—
388C2×Q8—122no—
389C2×Q8—122no—
390C2×Q8—241no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4D3×Q812yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—64no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4C23×D311yescharacteristic
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—34no—
432C4⋊C4—32no—
433C4⋊C4—32no—
434C4⋊C4—42no—
435C4⋊C4—62no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C4⋊C4#23712yes—
450C22⋊C4—42no—
451C22⋊C4—64no—
452C22⋊C4—81no—
453C22⋊C4—122no—
454C22⋊C4—241no—
455C42—21no—
456C42—21no—
457C42—32no—
458C42—32no—
459C42—61no—
460C42—61no—
461C42—122no—
462C42—122no—
463C42—122no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×C22⋊Q811yescharacteristic
472C22×C3#54611yescharacteristic
473C22×C3#54811yescharacteristic
474C22×C3C22×SD1611yescharacteristic
475C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
476C22×C3#57611yescharacteristic
477C22×C3#57811yescharacteristic
478C3×C4—22no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4#53311yescharacteristic
484C3×C4#54411yescharacteristic
485C3×C4#55511yescharacteristic
486C3×C4#55611yescharacteristic
487Dic3—22no—
488Dic3—22no—
489Dic3—42no—
490Dic3—41no—
491Dic3—42no—
492Dic3—81no—
493Dic3—81no—
494Dic3#47614yes—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C2×D3×D411yescharacteristic
503Q8—84no—
504Q8—244no—
505D4—22no—
506D4—22no—
507D4—41no—
508D4—41no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—32no—
516C2×C4—32no—
517C2×C4—32no—
518C2×C4—32no—
519C2×C4—32no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—62no—
537C2×C4—128no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4D3×SD1612yes—
549C2×C4C2×#23711yescharacteristic
550C2×C4#100411yescharacteristic
551C2×C4C2×D3×Q811yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C2×C4D3×#4111yescharacteristic
554C2×C4#101011yescharacteristic
555C8—41no—
556C8—41no—
557C8—121no—
558C8—121no—
559C2×C3—82no—
560C2×C3#243911yescharacteristic
561C2×C3C2×#47611yescharacteristic
562C2×C3#308311yescharacteristic
563C2×C3#329811yescharacteristic
564C2×C3#332511yescharacteristic
565C2×C3#336211yescharacteristic
566C2×C3#336711yescharacteristic
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22#559111yescharacteristic
575C22D3×C22⋊Q811yescharacteristic
576C22#567711yescharacteristic
577C22#575711yescharacteristic
578C22C2×D3×SD1611yescharacteristic
579C22D3×AGL(1,ℤ/8)11yescharacteristic
580C22#577911yescharacteristic
581C4—22no—
582C4—34no—
583C4—41no—
584C4—42no—
585C4—62no—
586C4—62no—
587C4—81no—
588C4—81no—
589C4—122no—
590C4—121no—
591C4—122no—
592C4—241no—
593C4—241no—
594C4#562211yescharacteristic
595C4#562311yescharacteristic
596C4#567011yescharacteristic
597C4#567111yescharacteristic
598C3#2602411yescharacteristic, complemented, Sylow
599C2—82no—
600C2#7678211yescharacteristic
601C2#8002111yescharacteristic
602C2#8004911yescharacteristic
603C2#8096811yescharacteristic
604C2D3×#47611yescharacteristic
605C2#8158811yescharacteristic
606C2#8167011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2023D3
hypocenterC3#26024
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #83242C2semidirect
4. #71779C2semidirect
10. #68847C2semidirect
11. #68799C2semidirect
12. #68796C2semidirect
13. #68789C2semidirect
14. #68623C2semidirect
15. #68617C2semidirect
95. D4×Dic3Q8semidirect
598. C3#26024semidirect

Maximal subgroups

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

subgroupindex
#832422
#763022
C3×#20232
#717792
#701792
#701632
#701532
Dic3×#1022
#696522
#688472
#687992
#687962
#687892
#686232
#686172
#260243