Finite Group Databasebeta

← #26123250

Subgroups — #26123250

Order 12032.

1070 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689146C211yesmaximal, characteristic, complemented
2C22×#13093951C211yesmaximal, characteristic
3#20245879C211yesmaximal, characteristic
4C2×#15688884C211yesmaximal, characteristic, complemented
5C2×#15688883C211yesmaximal, characteristic, complemented
6C4×C47⋊#95C211yesmaximal, characteristic, complemented
7C22×#104×C47C211yesmaximal, characteristic
8C22×#10625329C2211yescharacteristic
9C2×#13093983C2211yescharacteristic
10C22×C4×Dic47C2211yescharacteristic
11C22×C4×Dic47C2211yescharacteristic
12#15689146—22no—
13#15689146—22no—
14C2×Dic47⋊C8—22no—
15C2×#13093951—22no—
16C2×#13093951—22no—
17C2×#13093951—22no—
18#15689030C411yescharacteristic, complemented
19#15689030C411yescharacteristic, complemented
20#15689026C411yescharacteristic, complemented
21#15689026C411yescharacteristic, complemented
22#15689003—22no—
23C4×C22⋊Dic47—22no—
24C2×C47⋊#95—22no—
25#15688884—22no—
26#15688883—22no—
27C22×#41×C47C2211yescharacteristic
28C22×M16×C47C2211yescharacteristic
29C22×C42×C47C2211yescharacteristic
30C2×#104×C47—22no—
31C2×#104×C47—22no—
32C2×#104×C47—22no—
33C23×Dic47—21no—
34C23×Dic47D412yes—
35C2×#10625329—21no—
36C2×#10625329—21no—
37C2×#10625329—21no—
38C2×#10625329—21no—
39C2×#10625329—21no—
40C2×#10625329D412yes—
41C22×C47⋊C8—21no—
42#13093983—21no—
43#13093983—21no—
44#13093983—21no—
45#13093983—21no—
46C2×C4⋊Dic47C2×C411yescharacteristic
47C2×C4⋊Dic47C2×C411yescharacteristic
48C2×C22⋊Dic47—21no—
49C2×C22⋊Dic47—22no—
50C2×C22⋊Dic47—22no—
51C2×C4×Dic47—21no—
52C2×C4×Dic47—21no—
53C2×C4×Dic47—21no—
54C2×C4×Dic47—21no—
55C2×C4×Dic47—21no—
56C2×C4×Dic47—21no—
57C2×C4×Dic47—21no—
58C2×C4×Dic47—21no—
59C2×C4×Dic47—21no—
60C2×C4×Dic47—21no—
61C2×C4×Dic47C2×C411yescharacteristic
62C2×C4×Dic47C2×C411yescharacteristic
63C2×C4×Dic47D412yes—
64C2×C4×Dic47D412yes—
65Dic47⋊C8—28no—
66#13093951—28no—
67C47⋊#95—28no—
68C47⋊#95—22no—
69C47⋊#95—22no—
70C2×#41×C47—21no—
71C2×#41×C47—21no—
72C2×#41×C47—21no—
73C2×#41×C47—21no—
74C2×#41×C47—21no—
75C2×#41×C47—21no—
76C2×#41×C47—21no—
77C2×#41×C47—21no—
78C2×#41×C47—21no—
79C2×#41×C47—21no—
80C2×#41×C47D412yes—
81C2×#41×C47Q812yes—
82C22×Q8×C47C2×C411yescharacteristic
83C22×D4×C47—21no—
84C22×D4×C47C2×C411yescharacteristic
85C23×C4×C47—21no—
86C23×C4×C47—21no—
87C23×C4×C47C2311yescharacteristic
88C2×M16×C47—21no—
89C2×M16×C47—21no—
90C2×M16×C47—21no—
91C2×M16×C47—21no—
92C2×M16×C47—21no—
93C2×M16×C47D412yes—
94C22×C8×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97C2×C42×C47—21no—
98C2×C42×C47—21no—
99C2×C42×C47—21no—
100C2×C42×C47D412yes—
101#104×C47—28no—
102C24×C47—21no—
103C24×C47C2×D411yescharacteristic
104#41×C47—22no—
105#41×C47—22no—
106#41×C47—22no—
107#41×C47—22no—
108#41×C47—22no—
109#41×C47—22no—
110#41×C47—22no—
111#41×C47—22no—
112#41×C47—41no—
113#41×C47—41no—
114#41×C47—41no—
115#41×C47—41no—
116#41×C47—41no—
117#41×C47—41no—
118#41×C47—41no—
119#41×C47—41no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47C4⋊C412yes—
126C2×D4×C47—22no—
127C2×D4×C47—22no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—41no—
134C2×D4×C47—41no—
135C2×D4×C47—41no—
136C2×D4×C47—41no—
137C2×D4×C47C4⋊C412yes—
138C22×C4×C47—24no—
139C22×C4×C47—21no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—42no—
144C22×C4×C47—41no—
145C22×C4×C47—41no—
146C22×C4×C47—41no—
147C22×C4×C47—41no—
148C22×C4×C47—41no—
149C22×C4×C47—41no—
150C22×C4×C47C22×C411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×Q811yescharacteristic
158C22×C4×C47#4111yescharacteristic
159C22×C4×C47#4111yescharacteristic
160C22×C4×C47#4111yescharacteristic
161C22×C4×C47#4111yescharacteristic
162C22×C4×C47#4111yescharacteristic
163C22×C4×C47#4111yescharacteristic
164C22×Dic47—24no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—24no—
169C22×Dic47—21no—
170C22×Dic47—21no—
171C22×Dic47—42no—
172C22×Dic47—42no—
173C22×Dic47—41no—
174C22×Dic47—41no—
175C22×Dic47C22⋊C412yes—
176C22×Dic47C22⋊C412yes—
177M16×C47—22no—
178M16×C47—22no—
179M16×C47—41no—
180M16×C47—41no—
181C2×C8×C47—22no—
182C2×C8×C47—41no—
183C2×C8×C47—41no—
184C42×C47—22no—
185C42×C47—22no—
186C42×C47—41no—
187C42×C47—41no—
188C22⋊Dic47—44no—
189C22⋊Dic47—41no—
190C22⋊Dic47—41no—
191C22⋊Dic47—44no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C4×Dic47—21no—
199C4×Dic47—21no—
200C4×Dic47—22no—
201C4×Dic47—22no—
202C4×Dic47—22no—
203C4×Dic47—22no—
204C4×Dic47—41no—
205C4×Dic47—41no—
206C4×Dic47—41no—
207C4×Dic47—41no—
208#10625329—22no—
209#10625329—22no—
210#10625329—41no—
211#10625329—41no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—41no—
214C2×C47⋊C8—41no—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47—22no—
218C23×C47—41no—
219C23×C47—41no—
220C23×C47—42no—
221C23×C47—42no—
222C23×C47C2×C22⋊C411yescharacteristic
223C23×C47#12111yescharacteristic
224C23×C47#12111yescharacteristic
225C23×C47#12111yescharacteristic
226C23×C47#12111yescharacteristic
227C23×C47#12111yescharacteristic
228C23×C47C22⋊Q811yescharacteristic
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—41no—
232Q8×C47—41no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47—42no—
236D4×C47—42no—
237D4×C47—42no—
238D4×C47—42no—
239D4×C47—41no—
240D4×C47—41no—
241C2×C4×C47—24no—
242C2×C4×C47—22no—
243C2×C4×C47—44no—
244C2×C4×C47—44no—
245C2×C4×C47—42no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47C2×C4⋊C411yescharacteristic
254C2×C4×C47#11711yescharacteristic
255C2×C4×C47C4×D411yescharacteristic
256C2×C4×C47C4×D411yescharacteristic
257C2×C4×C47C4×D411yescharacteristic
258C2×C4×C47C4×D411yescharacteristic
259C2×C4×C47#12011yescharacteristic
260C2×C4×C47#12011yescharacteristic
261C2×C4×C47#12011yescharacteristic
262C2×C4×C47#12111yescharacteristic
263C2×C4×C47#12111yescharacteristic
264C2×C4×C47#12111yescharacteristic
265C2×C4×C47C22⋊Q811yescharacteristic
266C2×C4×C47C22⋊Q811yescharacteristic
267C2×C4×C47C22⋊Q811yescharacteristic
268C2×C4×C47C22⋊Q811yescharacteristic
269C2×C4×C47C22⋊Q811yescharacteristic
270C2×C4×C47#12311yescharacteristic
271C2×C4×C47#12311yescharacteristic
272C2×C4×C47#12311yescharacteristic
273C2×C4×C47#12311yescharacteristic
274C2×C4×C47#12311yescharacteristic
275C2×C4×C47#12411yescharacteristic
276C2×C4×C47#12411yescharacteristic
277C2×C4×C47#12411yescharacteristic
278C2×C4×C47#12611yescharacteristic
279C2×C4×C47#12611yescharacteristic
280C2×C4×C47#12711yescharacteristic
281C2×Dic47—22no—
282C2×Dic47—22no—
283C2×Dic47—24no—
284C2×Dic47—22no—
285C2×Dic47—22no—
286C2×Dic47—22no—
287C2×Dic47—44no—
288C2×Dic47—44no—
289C2×Dic47—44no—
290C2×Dic47—44no—
291C2×Dic47—42no—
292C2×Dic47—42no—
293C2×Dic47—41no—
294C2×Dic47—41no—
295C2×Dic47—42no—
296C2×Dic47—42no—
297C2×Dic47#10414yescomplemented
298C8×C47—42no—
299C47⋊C8—42no—
300#11053—471nomaximal, Sylow
301C22×C47—22no—
302C22×C47—22no—
303C22×C47—22no—
304C22×C47—22no—
305C22×C47—21no—
306C22×C47—21no—
307C22×C47—21no—
308C22×C47—21no—
309C22×C47—44no—
310C22×C47—42no—
311C22×C47—82no—
312C22×C47—81no—
313C22×C47—81no—
314C22×C47#38011yescharacteristic
315C22×C47#38611yescharacteristic
316C22×C47#39011yescharacteristic
317C22×C47#39411yescharacteristic
318C22×C47#39411yescharacteristic
319C22×C47#39611yescharacteristic
320C22×C47#39611yescharacteristic
321C22×C47C2×#10412yes—
322C22×C47#42112yes—
323C4×C47—44no—
324C4×C47—42no—
325C4×C47#38511yescharacteristic
326C4×C47#38511yescharacteristic
327C4×C47#38811yescharacteristic
328C4×C47#38811yescharacteristic
329C4×C47#39211yescharacteristic
330C4×C47#39311yescharacteristic
331C4×C47#39711yescharacteristic
332C4×C47#39911yescharacteristic
333Dic47—24no—
334Dic47—44no—
335Dic47—42no—
336Dic47—81no—
337Dic47—81no—
338C2×#423—471no—
339C22×#104—471no—
340C22×#104D4711yescharacteristic
341#2297—471no—
342C2×#339—471no—
343C2×#339—471no—
344C4×#95—471no—
345C2×C47—21no—
346C2×C47—21no—
347C2×C47—21no—
348C2×C47—21no—
349C2×C47—82no—
350C2×C47#143911yescharacteristic
351C2×C47#180612yes—
352C2×C47#187112yes—
353C2×C47#196211yescharacteristic
354C2×C47#196311yescharacteristic
355C22×#41C2×D4711yescharacteristic
356C22×M16—471no—
357C22×M16C2×D4711yescharacteristic
358C2×#117—471no—
359C22×C42—471no—
360C22×C42—471no—
361C22×C42C2×D4711yescharacteristic
362#423—942no—
363#423—942no—
364C2×C4⋊C8—942no—
365C2×#104—22no—
366C2×#104—22no—
367C2×#104—22no—
368C2×#104—942no—
369C2×#104—942no—
370C2×#104—942no—
371#391—471no—
372#391—471no—
373#390—471no—
374#390—471no—
375#386—942no—
376C4×C22⋊C4—942no—
377C2×#95—942no—
378#339—942no—
379#339—942no—
380C47#1105311yescharacteristic, complemented, Sylow
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41—21no—
387C2×#41—21no—
388C2×#41—21no—
389C2×#41—21no—
390C2×#41—21no—
391C2×#41Dic9412yes—
392C2×#41C22⋊D4712yes—
393C22×Q8C4×D4711yescharacteristic
394C22×D4—21no—
395C22×D4C4×D4711yescharacteristic
396C23×C4—21no—
397C23×C4—21no—
398C23×C4—472no—
399C23×C4—941no—
400C23×C4C22×D4711yescharacteristic
401C2×M16—21no—
402C2×M16—21no—
403C2×M16—21no—
404C2×M16—21no—
405C2×M16—21no—
406C2×M16—472no—
407C2×M16—941no—
408C2×M16—941no—
409C2×M16—941no—
410C2×M16—941no—
411C2×M16—941no—
412C2×M16C22⋊D4712yes—
413C22×C8—21no—
414C22×C8—941no—
415#117—941no—
416#117—941no—
417#117—941no—
418#117—941no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—471no—
421C2×C22⋊C4—942no—
422C2×C22⋊C4—942no—
423C2×C22⋊C4—941no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—471no—
430C2×C42—471no—
431C2×C42—472no—
432C2×C42—472no—
433C2×C42—941no—
434C2×C42—941no—
435C2×C42—941no—
436C2×C42—941no—
437C2×C42—941no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42—941no—
443C2×C42C22⋊D4712yes—
444C4⋊C8—948no—
445#104—28no—
446#104—948no—
447#95—942no—
448#95—942no—
449#95—948no—
450C24—21no—
451C24D4×D4711yescharacteristic
452#41—22no—
453#41—22no—
454#41—22no—
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—22no—
460#41—41no—
461#41—41no—
462#41—41no—
463#41—41no—
464#41—41no—
465#41—41no—
466#41—41no—
467#41—41no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8Dic47⋊C412yes—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—41no—
482C2×D4—41no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4Dic47⋊C412yes—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—42no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—472no—
499C22×C4—472no—
500C22×C4—944no—
501C22×C4—944no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—941no—
506C22×C4—941no—
507C22×C4—1882no—
508C22×C4—1882no—
509C22×C4—1881no—
510C22×C4—1881no—
511C22×C4C2×Dic9411yescharacteristic
512C22×C4C2×C4×D4711yescharacteristic
513C22×C4#1062535011yescharacteristic
514C22×C4#1062535011yescharacteristic
515C22×C4D4×D4711yescharacteristic
516C22×C4D4×D4711yescharacteristic
517C22×C4D4×D4711yescharacteristic
518C22×C4#1062535211yescharacteristic
519C22×C4#1062535211yescharacteristic
520C22×C4#1062535211yescharacteristic
521C22×C4#1062535211yescharacteristic
522C22×C4C2×C22⋊D4711yescharacteristic
523C22×C4C2×C22⋊D4711yescharacteristic
524C22×C4C2×C22⋊D4711yescharacteristic
525M16—22no—
526M16—22no—
527M16—41no—
528M16—41no—
529M16—942no—
530M16—942no—
531M16—1881no—
532M16—1881no—
533C2×C8—22no—
534C2×C8—41no—
535C2×C8—41no—
536C2×C8—942no—
537C2×C8—1881no—
538C2×C8—1881no—
539C4⋊C4—941no—
540C4⋊C4—941no—
541C4⋊C4—941no—
542C4⋊C4—941no—
543C22⋊C4—1884no—
544C22⋊C4—1884no—
545C22⋊C4—1881no—
546C22⋊C4—1881no—
547C42—22no—
548C42—22no—
549C42—41no—
550C42—41no—
551C42—941no—
552C42—941no—
553C42—941no—
554C42—941no—
555C42—942no—
556C42—942no—
557C42—942no—
558C42—942no—
559C42—1881no—
560C42—1881no—
561C42—1881no—
562C42—1881no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23C22⋊C4×D4711yescharacteristic
571C23#1309399511yescharacteristic
572C23#1309399611yescharacteristic
573C23#1309399611yescharacteristic
574C23#1309399711yescharacteristic
575C23#1309399711yescharacteristic
576C23#1309400111yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581D4—22no—
582D4—22no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—41no—
588D4—41no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—474no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—944no—
605C2×C4—942no—
606C2×C4—942no—
607C2×C4—942no—
608C2×C4—1884no—
609C2×C4—1884no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1882no—
613C2×C4—1882no—
614C2×C4—1884no—
615C2×C4—1884no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4C2×Dic47⋊C411yescharacteristic
619C2×C4#1309398011yescharacteristic
620C2×C4(C22⋊D47)⋊C411yescharacteristic
621C2×C4(C22⋊D47)⋊C411yescharacteristic
622C2×C4C4×C22⋊D4711yescharacteristic
623C2×C4C4×C22⋊D4711yescharacteristic
624C2×C4#1309399211yescharacteristic
625C2×C4#1309399211yescharacteristic
626C2×C4#1309399311yescharacteristic
627C2×C4#1309399511yescharacteristic
628C2×C4#1309399611yescharacteristic
629C2×C4#1309399611yescharacteristic
630C2×C4#1309400111yescharacteristic
631C2×C4#1309400111yescharacteristic
632C2×C4#1309400111yescharacteristic
633C2×C4#1309400311yescharacteristic
634C2×C4#1309400311yescharacteristic
635C2×C4#1309400511yescharacteristic
636C2×C4#1309400511yescharacteristic
637C2×C4#1309400611yescharacteristic
638C2×C4#1309400611yescharacteristic
639C2×C4#1309400711yescharacteristic
640C2×C4#1309401011yescharacteristic
641C2×C4#1309401011yescharacteristic
642C2×C4#1309401111yescharacteristic
643C2×C4#1309401711yescharacteristic
644C2×C4#1309401711yescharacteristic
645C2×C4#1309402011yescharacteristic
646C8—42no—
647C8—1882no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—44no—
657C22—42no—
658C22—82no—
659C22—81no—
660C22—81no—
661C22#1568898111yescharacteristic
662C22#1568899911yescharacteristic
663C22#1568902311yescharacteristic
664C22#1568903711yescharacteristic
665C22#1568903811yescharacteristic
666C22#1568904611yescharacteristic
667C22#1568904611yescharacteristic
668C22#104×D4712yes—
669C22#1568913812yes—
670C4—44no—
671C4—42no—
672C4—944no—
673C4—3761no—
674C4—3761no—
675C4—1884no—
676C4—1882no—
677C4#1568899511yescharacteristic
678C4#1568899611yescharacteristic
679C4#1568901111yescharacteristic
680C4#1568901711yescharacteristic
681C4#1568903111yescharacteristic
682C4#1568903511yescharacteristic
683C4#1568905111yescharacteristic
684C4#1568905911yescharacteristic
685C2—21no—
686C2—21no—
687C2—21no—
688C2—21no—
689C2—82no—
690C2#2024163111yescharacteristic
691C2#2024340612yes—
692C2#2024368712yes—
693C2#2024421411yescharacteristic
694C2#2024421811yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC22×#104×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC22×#41×C47C22
cosocleC23×C4×C47C23
hypercenterC22×#104D47
hypocenterC47#11053
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
1. C2×#15689146C2semidirect
4. C2×#15688884C2semidirect
5. C2×#15688883C2semidirect
6. C4×C47⋊#95C2semidirect
18. #15689030C4semidirect
19. #15689030C4semidirect
20. #15689026C4semidirect
21. #15689026C4semidirect
297. C2×Dic47#104semidirect
380. C47#11053semidirect

Maximal subgroups

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

subgroupindex
C2×#156891462
C22×#130939512
#202458792
C2×#156888842
C2×#156888832
C4×C47⋊#952
C22×#104×C472
#1105347