Finite Group Databasebeta

← #26244972

Subgroups — #26244972

Order 12032.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689165C211yesmaximal, characteristic, complemented
2C2×#15689125C211yesmaximal, characteristic
3C2×#15689125C211yesmaximal, characteristic
4#20245878C211yesmaximal, characteristic
5C2×#15689021C211yesmaximal, characteristic, complemented
6#20244782C212yesmaximal, complemented
7#20244775C212yesmaximal, complemented
8#20244768C212yesmaximal, complemented
9#20244761C212yesmaximal, complemented
10#20243108C211yesmaximal, characteristic, complemented
11C2×C47×#423C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4—21no—
17#15689165—24no—
18C2×#13093958C2212yes—
19C2×#13093956C2212yes—
20#15689125—24no—
21#15689125—24no—
22C2×#13093949C2212yes—
23C2×#13093949C2212yes—
24C2×#13093946C2212yes—
25C2×#13093946C2212yes—
26#15689029C2211yescharacteristic
27#15689029C2211yescharacteristic
28#15689025C2211yescharacteristic
29#15689025C2211yescharacteristic
30#15689021—21no—
31#15689021—21no—
32#15689021C2212yes—
33#15689021C2212yes—
34#15689002—21no—
35#15689002—21no—
36C4×(C2×D47)⋊C4—21no—
37C4×(C2×D47)⋊C4—21no—
38C2×C47⋊#95—21no—
39#15688886C2212yes—
40#15688886C2212yes—
41C22×M16×C47C2211yescharacteristic
42C22×M16×C47C2211yescharacteristic
43C22×C42×C47C2211yescharacteristic
44C47×#423—24no—
45C2×C4⋊C8×C47C2212yes—
46C2×C4⋊C8×C47C2212yes—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350—22no—
52C2×#10625350Q814yes—
53C22×Dic94C2311yescharacteristic
54C22×D188C2311yescharacteristic
55C22×C22⋊D47—21no—
56C23×Dic47—21no—
57C23×Dic47—21no—
58C22×C4×D47—21no—
59#13093983—22no—
60#13093983—22no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47—22no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47D414yes—
67C2×C4⋊Dic47C2312yes—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—22no—
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—44no—
74C2×C22⋊Dic47—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×C4×Dic47—21no—
80#13093958—24no—
81#13093956—24no—
82#13093949—24no—
83#13093949—24no—
84#13093946—24no—
85#13093946—24no—
86C47⋊#95—22no—
87C47⋊#95—22no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C23×C4×C47D411yescharacteristic
93C23×C4×C47D411yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C2×M16×C47—22no—
96C2×M16×C47—22no—
97C2×M16×C47—22no—
98C2×M16×C47—22no—
99C2×M16×C47—22no—
100C2×M16×C47D414yes—
101C22×C8×C47C2312yes—
102C22×C8×C47C2312yes—
103C2×C42×C47—22no—
104C2×C42×C47—22no—
105C2×C42×C47D411yescharacteristic
106C2×C42×C47D411yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109C4⋊C8×C47—24no—
110C4⋊C8×C47—24no—
111C24×C47C2×D411yescharacteristic
112C23×D47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—41no—
118C22×C4×C47—41no—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×Q812yes—
127C22×C4×C47#4112yes—
128C22×C4×C47#4112yes—
129C22×C4×C47#4112yes—
130C22×C4×C47C2411yescharacteristic
131C2×C22⋊D47—24no—
132C2×C22⋊D47—42no—
133C2×C22⋊D47—42no—
134C22×Dic47—24no—
135C22×Dic47—21no—
136C22×Dic47—22no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—42no—
141C22×Dic47—42no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47—42no—
145#10625350—28no—
146#10625350—28no—
147#10625350—44no—
148#10625350—44no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188C2×Q812yes—
155C2×C4×D47—24no—
156C2×C4×D47—42no—
157C2×C4×D47—42no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94C2×Q812yes—
164M16×C47—28no—
165M16×C47—44no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C2×C8×C47C2×D414yes—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C22⋊Dic47—42no—
179(C2×D47)⋊C4—42no—
180(C2×D47)⋊C4—42no—
181(C2×D47)⋊C4—42no—
182(C2×D47)⋊C4—42no—
183C4⋊Dic47—24no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—22no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—42no—
192C4⋊Dic47C2×D414yes—
193Dic47⋊C4—44no—
194Dic47⋊C4—44no—
195C4×Dic47—42no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47#12112yes—
201C23×C47#12112yes—
202C23×C47C22⋊Q812yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—22no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—41no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—41no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—44no—
222C2×C4×C47—44no—
223C2×C4×C47#12114yes—
224C2×C4×C47#12112yes—
225C2×C4×C47#12112yes—
226C2×C4×C47C22⋊Q814yes—
227C2×C4×C47C22⋊Q814yes—
228C2×C4×C47C22⋊Q812yes—
229C2×C4×C47#12614yes—
230C2×C4×C47AGL(1,ℤ/8)12yes—
231C2×C4×C47#13712yes—
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×Q811yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C22⋊D47—48no—
239C2×Dic47—24no—
240C2×Dic47—22no—
241C2×Dic47—48no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—42no—
255C2×Dic47—84no—
256D188—22no—
257D188—22no—
258D188—41no—
259D188—41no—
260C4×D47—48no—
261Dic94—22no—
262Dic94—22no—
263Dic94—41no—
264Dic94—41no—
265C8×C47—24no—
266C8×C47—42no—
267#23617—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—22no—
274C22×C47#39614yes—
275C22×C47C2×#12111yescharacteristic
276C22×C47C2×#12111yescharacteristic
277C22×C47C2×C22⋊Q811yescharacteristic
278C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C47C2×#13711yescharacteristic
280C22×C47#57511yescharacteristic
281C22×C47#57511yescharacteristic
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—81no—
286C2×D47—81no—
287C2×D47—81no—
288C2×D47—81no—
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—41no—
292C4×C47—41no—
293C4×C47#39614yes—
294C4×C47C2×#12111yescharacteristic
295C4×C47C2×C22⋊Q811yescharacteristic
296C4×C47C2×C22⋊Q811yescharacteristic
297C4×C47C2×#12611yescharacteristic
298Dic47—42no—
299Dic47—44no—
300Dic47—81no—
301Dic47—81no—
302Dic47—82no—
303C2×#428—471no—
304C2×#423D4711yescharacteristic, complemented
305C2×#417—471no—
306C2×#417—471no—
307#2297—471no—
308C2×#389—471no—
309#2068—472no—
310#2067—472no—
311#2066—472no—
312#2065—472no—
313#1736—471no—
314C2×C47—24no—
315C2×C47C2×#39611yescharacteristic
316C2×C47#303611yescharacteristic
317C2×C47#303711yescharacteristic
318C2×C47#305511yescharacteristic
319C2×C47#305611yescharacteristic
320C2×C47#307911yescharacteristic
321C2×C47#308011yescharacteristic
322D47—82no—
323C22×#41—471no—
324C22×M16C2×D4711yescharacteristic
325C22×M16C2×D4711yescharacteristic
326C2×#117—471no—
327C22×C4⋊C4—471no—
328C22×C4⋊C4—471no—
329C22×C4⋊C4—941no—
330C22×C42C2×D4711yescharacteristic
331#428—944no—
332C2×#107—472no—
333C2×#106—472no—
334#423—24no—
335C2×C4⋊C8C2×D4712yes—
336C2×C4⋊C8C2×D4712yes—
337#417—944no—
338#417—944no—
339C2×#103—472no—
340C2×#103—472no—
341C2×#102—472no—
342C2×#102—472no—
343#391—471no—
344#391—471no—
345#390—471no—
346#390—471no—
347#389—472no—
348#389—472no—
349#389—941no—
350#389—941no—
351#386—941no—
352#386—941no—
353C4×C22⋊C4—941no—
354C4×C22⋊C4—941no—
355C2×#95—941no—
356#340—472no—
357#340—472no—
358C47#2361711yescharacteristic, complemented, Sylow
359C2×#41—474no—
360C2×#41—942no—
361C2×#41—942no—
362C2×#41—942no—
363C2×#41—942no—
364C2×#41—942no—
365C22×Q8—471no—
366C22×D4—471no—
367C22×D4—941no—
368C23×C4—941no—
369C23×C4—941no—
370C23×C4—941no—
371C23×C4D18811yescharacteristic
372C23×C4D18811yescharacteristic
373C23×C4C22×D4711yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16C22⋊D4714yes—
380C22×C8C22×D4712yes—
381C22×C8C22×D4712yes—
382#117—942no—
383#117—942no—
384C2×C4⋊C4—474no—
385C2×C4⋊C4—472no—
386C2×C4⋊C4—472no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—942no—
395C2×C4⋊C4—942no—
396C2×C4⋊C4—1884no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—941no—
405C2×C42D18811yescharacteristic
406C2×C42D18811yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4711yescharacteristic
409#107—944no—
410#106—944no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—944no—
414#103—944no—
415#102—944no—
416#102—944no—
417#95—942no—
418#95—942no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423C24—941no—
424C24C2×D18811yescharacteristic
425#41—948no—
426#41—948no—
427#41—1884no—
428#41—1884no—
429C2×Q8—472no—
430C2×Q8—941no—
431C2×Q8—941no—
432C2×Q8—941no—
433C2×Q8—941no—
434C2×Q8—941no—
435C2×D4—472no—
436C2×D4—944no—
437C2×D4—941no—
438C2×D4—941no—
439C2×D4—941no—
440C2×D4—941no—
441C2×D4—941no—
442C2×D4—1882no—
443C2×D4—1882no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—944no—
451C22×C4—941no—
452C22×C4—942no—
453C22×C4—944no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—942no—
457C22×C4—1882no—
458C22×C4—1882no—
459C22×C4—1882no—
460C22×C4—1882no—
461C22×C4—1882no—
462C22×C4—1882no—
463C22×C4—1882no—
464C22×C4C2×D18811yescharacteristic
465C22×C4C2×D18811yescharacteristic
466C22×C4C2×D18811yescharacteristic
467C22×C4C2×D18811yescharacteristic
468C22×C4C2×D18811yescharacteristic
469C22×C4#1062535012yes—
470C22×C4D4×D4712yes—
471C22×C4#1062535212yes—
472C22×C4Q8×D4712yes—
473C22×C4#1062535412yes—
474C22×C4C2×C22⋊D4712yes—
475C22×C4C23×D4711yescharacteristic
476M16—28no—
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8C2×C22⋊D4714yes—
484C4⋊C4—474no—
485C4⋊C4—944no—
486C4⋊C4—942no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—942no—
490C4⋊C4—942no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—1884no—
494C4⋊C4—1882no—
495C4⋊C4—1884no—
496C22⋊C4—1882no—
497C22⋊C4—1882no—
498C22⋊C4—1882no—
499C22⋊C4—1882no—
500C22⋊C4—1882no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—1882no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—942no—
513C23—1881no—
514C23—1881no—
515C23—1881no—
516C23—1881no—
517C23—1881no—
518C23—1881no—
519C23#1309399812yes—
520C23#1309399912yes—
521C23#1309400412yes—
522C23C22×D18811yescharacteristic
523Q8—942no—
524Q8—942no—
525Q8—1881no—
526Q8—1881no—
527D4—942no—
528D4—942no—
529D4—1888no—
530D4—1881no—
531D4—1881no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—944no—
545C2×C4—942no—
546C2×C4—3764no—
547C2×C4—1882no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4—1881no—
551C2×C4—1882no—
552C2×C4—1881no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1882no—
556C2×C4—1888no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1882no—
560C2×C4—1882no—
561C2×C4—1888no—
562C2×C4#1309399614yes—
563C2×C4#1309399812yes—
564C2×C4#1309399912yes—
565C2×C4#1309400014yes—
566C2×C4#1309400214yes—
567C2×C4#1309400412yes—
568C2×C4#1309401914yes—
569C2×C4#1309405212yes—
570C2×C4#1309405712yes—
571C2×C4C2×D4×D4711yescharacteristic
572C2×C4C22×C22⋊D4711yescharacteristic
573C2×C4C2×Q8×D4711yescharacteristic
574C2×C4C2×#1062535211yescharacteristic
575C2×C4C2×#1062535011yescharacteristic
576C2×C4C2×#1062535411yescharacteristic
577C8—24no—
578C8—42no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—3761no—
586C22—3761no—
587C22—3761no—
588C22—3761no—
589C22—1882no—
590C22—1882no—
591C22—1882no—
592C22#1568904514yes—
593C22C2×#1309399811yescharacteristic
594C22C2×#1309399911yescharacteristic
595C22C2×#1309400411yescharacteristic
596C22C2×#1309405211yescharacteristic
597C22C2×#1309405711yescharacteristic
598C22#1568992311yescharacteristic
599C22#1568992311yescharacteristic
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—3762no—
605C4—3761no—
606C4—3761no—
607C4—1882no—
608C4—1884no—
609C4#1568904514yes—
610C4C2×#1309399611yescharacteristic
611C4C2×#1309400011yescharacteristic
612C4C2×#1309400211yescharacteristic
613C4C2×#1309401911yescharacteristic
614C2—24no—
615C2—3762no—
616C2C2×#1568904511yescharacteristic
617C2#2025285411yescharacteristic
618C2#2025286211yescharacteristic
619C2#2025300711yescharacteristic
620C2#2025301811yescharacteristic
621C2#2025320411yescharacteristic
622C2#2025321111yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#423C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenterC2×#423D47
hypocenterC47#23617
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
1. C2×#15689165C2semidirect
5. C2×#15689021C2semidirect
6. #20244782C2semidirect
7. #20244775C2semidirect
8. #20244768C2semidirect
9. #20244761C2semidirect
10. #20243108C2semidirect
11. C2×C47×#423C2semidirect
304. C2×#423D47semidirect
358. C47#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#156891652
C2×#156891252
#202458782
C2×#156890212
#202447822
#202447752
#202447682
#202447612
#202431082
C2×C47×#4232
#2361747