Finite Group Databasebeta

← #10858826

Subgroups — #10858826

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#73493C211yesmaximal, characteristic
2#73465C211yesmaximal, characteristic
3C3×#1850C211yesmaximal, characteristic
4#71972C211yesmaximal, characteristic
5C2×#4551C211yesmaximal, characteristic
6#70489C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70463C211yesmaximal, characteristic, complemented
9#70459C211yesmaximal, characteristic, complemented
10#69930C211yesmaximal, characteristic
11#69927C211yesmaximal, characteristic
12#68689C211yesmaximal, characteristic, complemented
13#68683C211yesmaximal, characteristic, complemented
14#68643C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#26414—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#878—21no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×C4×C22⋊C4C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#4950—21no—
36#4946—21no—
37Dic3×C22⋊C4—21no—
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4661—21no—
45#4656—21no—
46#4655—21no—
47#4647—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2440—31no—
61#2360—31no—
62#2100—31no—
63#2087—31no—
64#2068—31no—
65#2024—31no—
66#1987—31no—
67#1921—31no—
68#1892—31no—
69#1872—31no—
70#1863—31no—
71#1852—31no—
72#1850D311yescharacteristic
73#1803—31no—
74C2×#340—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C23×Dic3—21no—
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8—22no—
81C22×C3×C8—41no—
82C22×C3×C8—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—41no—
101C2×C4⋊Dic3—41no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109#924—22no—
110#924Q814yescomplemented
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118#878—44no—
119#869—22no—
120#869—41no—
121C4×Dic6—22no—
122C4×Dic6—41no—
123C3×C4⋊C8—41no—
124C3×C4⋊C8—41no—
125C3×C22⋊C8—22no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#818—22no—
136#818—41no—
137#816—22no—
138#816—41no—
139#816—41no—
140#816—41no—
141C23×C8C2×D311yescharacteristic
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×#107—31no—
149C2×#107—31no—
150C2×C4⋊C8C2×D311yescharacteristic
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×C22⋊C8C2×D311yescharacteristic
157#398—61no—
158#395—61no—
159#391—31no—
160#389—31no—
161#389—31no—
162#389—31no—
163#389—61no—
164#387—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#380—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4—61no—
171C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
181#340—31no—
182#336C2×D311yescharacteristic
183#336C2×D311yescharacteristic
184C24×C3C2×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×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×Q812yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3—42no—
209C22×Dic3—81no—
210C22×Dic3—81no—
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—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—44no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4Q1614yes—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239#217—24no—
240#217—42no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—42no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3—42no—
256C4⋊Dic3—81no—
257C4⋊Dic3C2×Q812yes—
258C4⋊Dic3#4112yes—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—81no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C22×Q8—31no—
268C22×Q8—61no—
269C23×C4—61no—
270C23×C4—61no—
271C23×C4C22×D311yescharacteristic
272C22×C8—22no—
273C22×C8—41no—
274C22×C8—41no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
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—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—62no—
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—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—121no—
306C2×C4⋊C4—121no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4D1211yescharacteristic
310C2×C22⋊C4D1211yescharacteristic
311C2×C42—61no—
312C2×C42—61no—
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#107—121no—
317C4⋊C8—41no—
318C4⋊C8—41no—
319#103—62no—
320#103—121no—
321#103—121no—
322#103—121no—
323C22⋊C8—22no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95D1211yescharacteristic
332#95D1211yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C2×Q1612yes—
339C23×C3AGL(1,ℤ/8)12yes—
340C23×C3C22×D411yescharacteristic
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4C22⋊Q812yes—
358C2×C3×C4C2×Q1612yes—
359C2×C3×C4C2×Q1612yes—
360C2×C3×C4C22×Q811yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
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×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—84no—
391C2×Dic3—84no—
392Dic6—22no—
393Dic6—22no—
394Dic6—41no—
395Dic6—41no—
396Dic6—84no—
397C3×C8—41no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C24C2×D1211yescharacteristic
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—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—122no—
427C22×C4—241no—
428C22×C4—241no—
429C22×C4C2×D1211yescharacteristic
430C22×C4C2×D1211yescharacteristic
431C22×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4#23712yes—
435C22×C4D3×Q812yes—
436C22×C4C23×D311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—44no—
450C4⋊C4—32no—
451C4⋊C4—32no—
452C4⋊C4—32no—
453C4⋊C4—64no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—241no—
473C4⋊C4—241no—
474C22⋊C4—22no—
475C22⋊C4—64no—
476C22⋊C4—122no—
477C22⋊C4Dic1214yes—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—122no—
483C42—241no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3#47714yes—
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3#54611yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3C22×Q1611yescharacteristic
495C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
496C22×C3#57711yescharacteristic
497C22×C3#57711yescharacteristic
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4#52911yescharacteristic
504C3×C4#53811yescharacteristic
505C3×C4#55411yescharacteristic
506C3×C4#55411yescharacteristic
507Dic3—42no—
508Dic3—42no—
509Dic3—42no—
510Dic3—42no—
511Dic3—81no—
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#89712yes—
520C23C2×Dic1212yes—
521C23#90812yes—
522C23C22×D1211yescharacteristic
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—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—244no—
569C2×C4—244no—
570C2×C4#89712yes—
571C2×C4C2×Dic1212yes—
572C2×C4C2×Dic1212yes—
573C2×C4C2×#23711yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4#100411yescharacteristic
576C2×C4C2×D3×Q811yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C2×C3—24no—
584C2×C3#245011yescharacteristic
585C2×C3C2×#47711yescharacteristic
586C2×C3#308411yescharacteristic
587C2×C3#329911yescharacteristic
588C2×C3#329911yescharacteristic
589C2×C3#330611yescharacteristic
590C2×C3#331011yescharacteristic
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22#485014yes—
598C22C2×#89711yescharacteristic
599C22#555411yescharacteristic
600C22#555411yescharacteristic
601C22C22×Dic1211yescharacteristic
602C22C2×#90811yescharacteristic
603C22#575111yescharacteristic
604C22#575111yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—122no—
611C4—122no—
612C4—122no—
613C4—122no—
614C4—241no—
615C4—241no—
616C4—241no—
617C4—241no—
618C4#559311yescharacteristic
619C4#563611yescharacteristic
620C4#563611yescharacteristic
621C4#564911yescharacteristic
622C3#2641411yescharacteristic, complemented, Sylow
623C2—24no—
624C2#7442411yescharacteristic
625C2C2×#485011yescharacteristic
626C2#7909211yescharacteristic
627C2#7944811yescharacteristic
628C2#7944811yescharacteristic
629C2#7949911yescharacteristic
630C2#7950111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1850C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1850D3
hypocenterC3#26414
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
6. #70489C2semidirect
7. #70485C2semidirect
8. #70463C2semidirect
9. #70459C2semidirect
12. #68689C2semidirect
13. #68683C2semidirect
14. #68643C2semidirect
15. #68624C2semidirect
110. #924Q8semidirect
622. C3#26414semidirect

Maximal subgroups

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

subgroupindex
#734932
#734652
C3×#18502
#719722
C2×#45512
#704892
#704852
#704632
#704592
#699302
#699272
#686892
#686832
#686432
#686242
#264143