Finite Group Databasebeta

← #10858801

Subgroups — #10858801

Order 768.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76205C211yesmaximal, characteristic
2#74326C211yesmaximal, characteristic
3C3×#2099C211yesmaximal, characteristic, complemented
4#69610C211yesmaximal, characteristic, complemented
5#69588C211yesmaximal, characteristic, complemented
6#68912C211yesmaximal, characteristic, complemented
7#68910C211yesmaximal, characteristic, complemented
8#68882C211yesmaximal, characteristic
9#68878C211yesmaximal, characteristic
10#68875C211yesmaximal, characteristic
11D3×#340C211yesmaximal, characteristic
12#68816C211yesmaximal, characteristic, complemented
13#68800C211yesmaximal, characteristic, complemented
14#68684C211yesmaximal, characteristic, complemented
15#68666C211yesmaximal, characteristic
16#26412—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#896—21no—
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#384C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×#817C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4989—21no—
32#4989—21no—
33#4987C2211yescharacteristic
34#4983—21no—
35#4979—21no—
36Dic3×C4⋊C4C2211yescharacteristic
37#4973C2211yescharacteristic
38#4972—21no—
39#4972C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C4×(C2×D3)⋊C4C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4666—21no—
47#4654—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2440—31no—
61#2360—31no—
62#2099D311yescharacteristic, complemented
63#2088—31no—
64#2066—31no—
65#2024—31no—
66#1988—31no—
67#1919—31no—
68#1892—31no—
69#1872—31no—
70#1865—31no—
71#1853—31no—
72#1850—31no—
73#1803—31no—
74C2×#340—31no—
75C22×C4×D3—21no—
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×D3×C8—22no—
110C2×D3×C8—41no—
111C2×D3×C8—41no—
112#897—22no—
113#897Q814yescomplemented
114#896—44no—
115D3×C4⋊C4—22no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—41no—
118D3×C4⋊C4—41no—
119#888—22no—
120#888—41no—
121C4×Dic6—22no—
122C4×Dic6—41no—
123C3×#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132(C2×D3)⋊C8—22no—
133#817—22no—
134#816—22no—
135#816—41no—
136Dic3⋊C8—41no—
137#810—41no—
138#810—41no—
139#808—41no—
140C4⋊(C3⋊C8)—41no—
141C23×C8—31no—
142C2×C4⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×C4×Q8—31no—
146C22×C4⋊C4—31no—
147C22×C4⋊C4—31no—
148C2×#106—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8—31no—
157#398—61no—
158#395—61no—
159#391—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#389C2×D311yescharacteristic
164#387—61no—
165#385—61no—
166#385—61no—
167#384C2×D311yescharacteristic
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—31no—
171C4×C22⋊C4—61no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C23×D3C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—81no—
208C2×C4×D3—81no—
209C2×C4×D3C2×Q812yes—
210C2×C4×D3#4112yes—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C2×Dic6—81no—
219C2×Dic6C2×Q812yes—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4C2×Q812yes—
237C3×C42—21no—
238C3×C42—21no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4SD1614yes—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—24no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3#4112yes—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—42no—
259C4×Dic3—81no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266D3×C8—44no—
267C22×Q8—31no—
268C22×Q8—61no—
269C23×C4—31no—
270C23×C4—61no—
271C23×C4—61no—
272C22×C8—31no—
273C22×C8—31no—
274C22×C8—62no—
275C22×C8—121no—
276C22×C8—121no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C4⋊Q8—62no—
280C4⋊Q8—121no—
281C22⋊Q8—34no—
282C22⋊Q8—62no—
283C22⋊Q8—124no—
284C4×Q8—62no—
285C4×Q8—121no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—121no—
302C2×C4⋊C4—121no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#106—62no—
316#106—121no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#103—62no—
320#103—121no—
321#103—121no—
322#103—121no—
323C22⋊C8—62no—
324#95—21no—
325#95—31no—
326#95—31no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333C23×C3C22×D411yescharacteristic
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3C22⋊Q812yes—
339C22×D3C2×SD1612yes—
340C22×D3#13712yes—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4C2×SD1612yes—
359C2×C3×C4C22×Q811yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3C22⋊Q812yes—
385C2×Dic3C2×SD1612yes—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389C4×D3—22no—
390C4×D3—84no—
391C4×D3—84no—
392Dic6—22no—
393Dic6—22no—
394Dic6—41no—
395Dic6—41no—
396Dic6—84no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C24—31no—
402C2×Q8—32no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—122no—
409C2×Q8—122no—
410C2×Q8—241no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—31no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—241no—
433C22×C4—241no—
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4C23×D311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—124no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—64no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4D3×Q812yes—
474C22⋊C4—34no—
475C22⋊C4—62no—
476C22⋊C4—64no—
477C22⋊C4—122no—
478C42—21no—
479C42—21no—
480C42—61no—
481C42—61no—
482C42—122no—
483C42—241no—
484C22×C3C2×C22⋊Q811yescharacteristic
485C22×C3#54611yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3C22×SD1611yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3#57611yescharacteristic
490C22×C3#57811yescharacteristic
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—22no—
497C2×D3#47514yes—
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4#52911yescharacteristic
505C3×C4#53811yescharacteristic
506C3×C4#55411yescharacteristic
507C3×C4#55411yescharacteristic
508Dic3—22no—
509Dic3—22no—
510Dic3—42no—
511Dic3—42no—
512Dic3—41no—
513Dic3—42no—
514Dic3—81no—
515C23—32no—
516C23—32no—
517C23—32no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C2×D3×D411yescharacteristic
523Q8—62no—
524Q8—62no—
525Q8—121no—
526Q8—121no—
527Q8—244no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—244no—
571C2×C4—244no—
572C2×C4D3×SD1612yes—
573C2×C4C2×D3×Q811yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C2×C4#100911yescharacteristic
578C2×C4#100911yescharacteristic
579C8—41no—
580C8—41no—
581C8—121no—
582C8—121no—
583C2×C3#245011yescharacteristic
584C2×C3C2×#47511yescharacteristic
585C2×C3#308311yescharacteristic
586C2×C3#329511yescharacteristic
587C2×C3#329811yescharacteristic
588C2×C3#331211yescharacteristic
589C2×C3#331411yescharacteristic
590D3—24no—
591C22—34no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22—62no—
596C22—62no—
597C22—62no—
598C22D3×C22⋊Q811yescharacteristic
599C22#565411yescharacteristic
600C22#567611yescharacteristic
601C22#575711yescharacteristic
602C22C2×D3×SD1611yescharacteristic
603C22D3×#13711yescharacteristic
604C22#577911yescharacteristic
605C4—22no—
606C4—42no—
607C4—41no—
608C4—62no—
609C4—62no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—122no—
614C4—121no—
615C4—122no—
616C4—122no—
617C4—241no—
618C4#569011yescharacteristic
619C4#569011yescharacteristic
620C4#570411yescharacteristic
621C4#570611yescharacteristic
622C3#2641211yescharacteristic, complemented, Sylow
623C2—64no—
624C2#7834411yescharacteristic
625C2#8096811yescharacteristic
626C2D3×#47511yescharacteristic
627C2#8128911yescharacteristic
628C2#8145011yescharacteristic
629C2#8157711yescharacteristic
630C2#8174911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2099C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2099D3
hypocenterC3#26412
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
3. C3×#2099C2semidirect
4. #69610C2semidirect
5. #69588C2semidirect
6. #68912C2semidirect
7. #68910C2semidirect
12. #68816C2semidirect
13. #68800C2semidirect
14. #68684C2semidirect
62. #2099D3semidirect
113. #897Q8semidirect
622. C3#26412semidirect

Maximal subgroups

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

subgroupindex
#762052
#743262
C3×#20992
#696102
#695882
#689122
#689102
#688822
#688782
#688752
D3×#3402
#688162
#688002
#686842
#686662
#264123