Finite Group Databasebeta

← #10858908

Subgroups — #10858908

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82464C211yesmaximal, characteristic
2#76000C211yesmaximal, characteristic
3C3×#1863C211yesmaximal, characteristic, complemented
4#71626C211yesmaximal, characteristic, complemented
5#70633C211yesmaximal, characteristic
6#70610C211yesmaximal, characteristic, complemented
7#70270C211yesmaximal, characteristic
8#70269C211yesmaximal, characteristic
9#70261C211yesmaximal, characteristic, complemented
10#70232C211yesmaximal, characteristic, complemented
11#69610C211yesmaximal, characteristic, complemented
12#68866C211yesmaximal, characteristic
13#68801C211yesmaximal, characteristic, complemented
14#68666C211yesmaximal, characteristic
15#68642C211yesmaximal, characteristic, complemented
16#26419—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#389C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4994—21no—
33#4990—21no—
34#4987C2211yescharacteristic
35#4984—21no—
36#4984C2211yescharacteristic
37#4978—21no—
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4940—21no—
41C4×(C2×D3)⋊C4C2211yescharacteristic
42#4935C2211yescharacteristic
43C4×Dic3⋊C4—21no—
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4677—21no—
47#4653—21no—
48C3×#340C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2441—31no—
61#2362—31no—
62#2099—31no—
63#2088—31no—
64#2066—31no—
65#2026—31no—
66#1989—31no—
67#1921—31no—
68#1892—31no—
69#1873—31no—
70#1863D311yescharacteristic, complemented
71#1854—31no—
72#1850—31no—
73#1803—31no—
74#1735—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88#946—22no—
89#946Q814yes—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×D3×C8—22no—
114C2×D3×C8—41no—
115C2×D3×C8—41no—
116#892—22no—
117#892—22no—
118#892—41no—
119#892—41no—
120#890—22no—
121#890—41no—
122C3×#107—22no—
123C3×#103—22no—
124C3×#103—41no—
125C3×#95—21no—
126#835—41no—
127#835—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136(C2×D3)⋊C8—22no—
137Dic3⋊C8—41no—
138#808—41no—
139C4⋊(C3⋊C8)—41no—
140C23×C8—31no—
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8C2×D311yescharacteristic
144C2×#117—31no—
145C2×#117—31no—
146C2×#107C2×D311yescharacteristic
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×C22⋊C8—31no—
155#399—61no—
156#396—61no—
157#391—31no—
158#389—31no—
159#389—31no—
160#389—61no—
161#389C2×D311yescharacteristic
162#388—61no—
163#388—61no—
164#387—61no—
165#387—61no—
166#384—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#336—31no—
182#336C2×D311yescharacteristic
183C23×D3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3C2×Q812yes—
209C2×C4×D3#4112yes—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4#4112yes—
223C3×C4⋊C4#4112yes—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—42no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—41no—
230(C2×D3)⋊C4—42no—
231(C2×D3)⋊C4—42no—
232(C2×D3)⋊C4—81no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—42no—
241C4⋊Dic3—42no—
242C4⋊Dic3—81no—
243C4⋊Dic3—81no—
244C4⋊Dic3C2×Q812yes—
245Dic3⋊C4—24no—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—42no—
252C4×Dic3—42no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259D3×C8—44no—
260C22×Q8C22×D311yescharacteristic
261C23×C4—31no—
262C22×C8—31no—
263C22×C8—31no—
264C22×C8—62no—
265C22×C8—121no—
266C22×C8—121no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269#125—62no—
270#125—121no—
271C22⋊Q8—34no—
272C22⋊Q8—62no—
273C4×Q8—22no—
274C4×Q8—41no—
275#117—62no—
276#117—62no—
277#117—121no—
278#117—121no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—21no—
302C2×C42—31no—
303C2×C42—61no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42C22×D311yescharacteristic
307#107—22no—
308#106—121no—
309C4⋊C8—121no—
310C4⋊C8—121no—
311#103—22no—
312#103—41no—
313#103—121no—
314#103—121no—
315C22⋊C8—62no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3C22×D411yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3C22⋊Q812yes—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
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#13512yes—
355C2×C3×C4C22×Q811yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3C22⋊Q812yes—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C24—31no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8D3×Q812yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4C23×D311yescharacteristic
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—124no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—32no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—64no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—122no—
448C4⋊C4—122no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C4⋊C4#23512yes—
452C4⋊C4#23712yes—
453C22⋊C4—64no—
454C22⋊C4—62no—
455C22⋊C4—121no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C42—21no—
460C42—21no—
461C42—42no—
462C42—61no—
463C42—61no—
464C42—122no—
465C42—122no—
466C42—122no—
467C22×C3C2×C22⋊Q811yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54611yescharacteristic
470C22×C3C2×#13511yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57811yescharacteristic
474C2×D3—22no—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—81no—
487C3×C4#52911yescharacteristic
488C3×C4#53811yescharacteristic
489C3×C4#55411yescharacteristic
490C3×C4#55411yescharacteristic
491Dic3—22no—
492Dic3—42no—
493Dic3—41no—
494Dic3—41no—
495Dic3—81no—
496Dic3—81no—
497Dic3—81no—
498C23—32no—
499C23—61no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23C2×D3×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—41no—
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—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—62no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4#91912yes—
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4#100411yescharacteristic
556C2×C4C2×D3×Q811yescharacteristic
557C2×C4D3×#4111yescharacteristic
558C2×C4#100911yescharacteristic
559C8—41no—
560C8—41no—
561C8—121no—
562C8—121no—
563C2×C3#245011yescharacteristic
564C2×C3#307611yescharacteristic
565C2×C3#308211yescharacteristic
566C2×C3#330011yescharacteristic
567C2×C3#330411yescharacteristic
568C2×C3#330711yescharacteristic
569C2×C3#331511yescharacteristic
570D3—42no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22#555011yescharacteristic
579C22#559111yescharacteristic
580C22D3×C22⋊Q811yescharacteristic
581C22C2×#91911yescharacteristic
582C22#577011yescharacteristic
583C22#577311yescharacteristic
584C22#577911yescharacteristic
585C4—22no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—62no—
591C4—81no—
592C4—121no—
593C4—121no—
594C4—122no—
595C4—241no—
596C4—241no—
597C4—241no—
598C4#551111yescharacteristic
599C4#555711yescharacteristic
600C4#563511yescharacteristic
601C4#565711yescharacteristic
602C3#2641911yescharacteristic, complemented, Sylow
603C2—122no—
604C2#7461011yescharacteristic
605C2#7973711yescharacteristic
606C2#7980311yescharacteristic
607C2#8010811yescharacteristic
608C2#8019411yescharacteristic
609C2#8110711yescharacteristic
610C2#8123311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1863D3
hypocenterC3#26419
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. C3×#1863C2semidirect
4. #71626C2semidirect
6. #70610C2semidirect
9. #70261C2semidirect
10. #70232C2semidirect
11. #69610C2semidirect
13. #68801C2semidirect
15. #68642C2semidirect
70. #1863D3semidirect
602. C3#26419semidirect

Maximal subgroups

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

subgroupindex
#824642
#760002
C3×#18632
#716262
#706332
#706102
#702702
#702692
#702612
#702322
#696102
#688662
#688012
#686662
#686422
#264193