Finite Group Databasebeta

← #26152995

Subgroups — #26152995

Order 12032.

2214 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094023C211yesmaximal, characteristic, complemented
2C2×#15689035C214yesmaximal, complemented
3C2×#15689026C211yesmaximal, characteristic
4C2×#15689026C211yesmaximal, characteristic
5#20241631C214yesmaximal, complemented
6#20241618C211yesmaximal, characteristic, complemented
7#20241617C211yesmaximal, characteristic, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C2×C47×#390C211yesmaximal, characteristic
10C23×Dic94C2212yes—
11C2×#13094023—24no—
12C2×#13094023—24no—
13C2×#13094023—24no—
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×C22⋊Dic47C2212yes—
18C22×C22⋊Dic47C2212yes—
19C22×C22⋊Dic47C2214yes—
20C22×C4×Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22#15689035—216no—
23#15689026—24no—
24#15689026—24no—
25C2×C47⋊#95C2214yes—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C23×D4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2214yes—
32C22×C42×C47C2211yescharacteristic
33C47×#390—24no—
34C22×Dic94—24no—
35C22×Dic94—24no—
36C22×Dic94—24no—
37C22×Dic94—28no—
38C22×Dic94—44no—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44#13094023—232no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47D414yes—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×C22⋊Dic47—28no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—24no—
57C2×C22⋊Dic47—28no—
58C2×C22⋊Dic47—28no—
59C2×C22⋊Dic47—28no—
60C2×C22⋊Dic47—44no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47—22no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47D414yes—
67C47⋊#95—28no—
68C47⋊#95—28no—
69C47⋊#95—24no—
70C47⋊#95D418yescomplemented
71C25×C47C2312yes—
72C22×D4×C47—22no—
73C22×D4×C47—22no—
74C22×D4×C47—22no—
75C22×D4×C47—22no—
76C22×D4×C47—22no—
77C22×D4×C47—22no—
78C22×D4×C47Q814yescomplemented
79C23×C4×C47C2312yes—
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47D414yes—
85C2×C22⋊C4×C47—28no—
86C2×C22⋊C4×C47—28no—
87C2×C22⋊C4×C47—44no—
88C2×C42×C47—22no—
89C2×C42×C47—22no—
90C2×C42×C47—22no—
91C24×C47—22no—
92C24×C47—24no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—24no—
99C24×C47—24no—
100C24×C47—24no—
101C24×C47C2×Q814yes—
102C24×C47C2411yescharacteristic
103C2×D4×C47—28no—
104C2×D4×C47—28no—
105C2×D4×C47—28no—
106C2×D4×C47—44no—
107C2×D4×C47—44no—
108C2×D4×C47—44no—
109C2×D4×C47—44no—
110C22×C4×C47—22no—
111C22×C4×C47—24no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47C2×D414yes—
121C22×C4×C47C2×D414yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×Q812yes—
126C22×C4×C47#4112yes—
127C22×C4×C47#4112yes—
128C22×C4×C47#4112yes—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—21no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—24no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—24no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—22no—
147C22×Dic47—24no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47C2×D414yes—
151C22×Dic47C2×D414yes—
152C22×Dic47C2×D412yes—
153C22×Dic47C2×D414yes—
154C22×Dic47C2×D412yes—
155C22×Dic47#4114yes—
156C2×Dic94—216no—
157C2×Dic94—416no—
158C2×Dic94—416no—
159C2×Dic94—48no—
160C2×Dic94—48no—
161C4⋊C4×C47—28no—
162C22⋊C4×C47—416no—
163C42×C47—44no—
164C22⋊Dic47—416no—
165C22⋊Dic47—48no—
166C22⋊Dic47—48no—
167C4⋊Dic47—28no—
168C4⋊Dic47—44no—
169Dic47⋊C4—216no—
170C4×Dic47—28no—
171C4×Dic47—44no—
172C23×C47—24no—
173C23×C47—24no—
174C23×C47—24no—
175C23×C47—28no—
176C23×C47—28no—
177C23×C47—42no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—44no—
184C23×C47—44no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—44no—
188C23×C47—44no—
189C23×C47—44no—
190C23×C47—44no—
191C23×C47—44no—
192C23×C47—44no—
193C23×C47—44no—
194C23×C47—44no—
195C23×C47C22×D411yescharacteristic
196C23×C47C22×D411yescharacteristic
197C23×C47C22×D412yes—
198C23×C47C22×D411yescharacteristic
199C23×C47C22×Q811yescharacteristic
200C23×C47C2×#4112yes—
201C23×C47C2×#4111yescharacteristic
202C23×C47C2×#4111yescharacteristic
203C23×C47C2×#4111yescharacteristic
204C23×C47H(5,𝔽2)12yes—
205C23×C472−1+412yes—
206D4×C47—416no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—28no—
211C2×C4×C47—24no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—44no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47#12014yes—
220C2×C4×C47C22⋊Q814yes—
221C2×C4×C47C22⋊Q814yes—
222C2×C4×C47C22⋊Q814yes—
223C2×C4×C47#12414yes—
224C2×C4×C47#12414yes—
225C2×C4×C47#12414yes—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—28no—
238C2×Dic47—24no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—44no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—44no—
250C2×Dic47—44no—
251C2×Dic47#12414yes—
252C2×Dic47#12714yes—
253Dic94—432no—
254#15144—471nomaximal, Sylow
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—48no—
258C22×C47—48no—
259C22×C47—44no—
260C22×C47—82no—
261C22×C47—82no—
262C22×C47—82no—
263C22×C47—82no—
264C22×C47—84no—
265C22×C47—84no—
266C22×C47C2×#12011yescharacteristic
267C22×C47C2×C22⋊Q811yescharacteristic
268C22×C47C2×C22⋊Q811yescharacteristic
269C22×C47C2×C22⋊Q811yescharacteristic
270C22×C47C2×#12411yescharacteristic
271C22×C47C2×#12411yescharacteristic
272C22×C47C2×#12411yescharacteristic
273C22×C47C2×#12411yescharacteristic
274C22×C47C2×#12711yescharacteristic
275C22×C47#53212yes—
276C22×C47#53212yes—
277C22×C47#53511yescharacteristic
278C22×C47#53611yescharacteristic
279C22×C47#54212yes—
280C22×C47#54212yes—
281C22×C47D4212yes—
282C22×C47D4212yes—
283C22×C47#54712yes—
284C22×C47#54812yes—
285C22×C47#54812yes—
286C22×C47D4×Q812yes—
287C22×C47#55412yes—
288C22×C47#55512yes—
289C4×C47—44no—
290C4×C47—44no—
291C4×C47#39318yescomplemented
292Dic47—24no—
293Dic47—24no—
294Dic47—44no—
295Dic47—44no—
296Dic47—82no—
297C22×C4⋊Q8—471no—
298C2×#393—474no—
299C2×#390—471no—
300C2×#390—471no—
301C2×#390D4711yescharacteristic
302#1439—474no—
303C4⋊#95—471no—
304C4⋊#95—471no—
305C4×#95—471no—
306C2×C47—84no—
307C2×C47C2×#39311yescharacteristic
308C2×C47#243311yescharacteristic
309C2×C47#243611yescharacteristic
310C2×C47#245212yes—
311C2×C47#245212yes—
312C2×C47#249112yes—
313C2×C47#249112yes—
314C2×C47#254211yescharacteristic
315C2×C47#256311yescharacteristic
316C2×C47#264011yescharacteristic
317C2×C47#285811yescharacteristic
318C23×Q8—472no—
319C23×D4C2×D4711yescharacteristic
320C2×C4⋊Q8—944no—
321C2×C4⋊Q8—944no—
322C2×C4⋊Q8—944no—
323C22×C4⋊C4—471no—
324C22×C4⋊C4—472no—
325C22×C4⋊C4—471no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—472no—
328C22×C22⋊C4—472no—
329C22×C22⋊C4—474no—
330C22×C22⋊C4C2×D4714yes—
331C22×C42—471no—
332C22×C42—471no—
333C22×C42C2×D4711yescharacteristic
334#393—9416no—
335#390—24no—
336#390—944no—
337#390—944no—
338C2×#95—472no—
339C2×#95—474no—
340C2×#95—474no—
341C2×#95—472no—
342C47#1514411yescharacteristic, complemented, Sylow
343C25C22×D4712yes—
344C22×Q8—944no—
345C22×Q8—944no—
346C22×Q8—948no—
347C22×Q8—944no—
348C22×Q8—1884no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4Dic9414yescomplemented
356C23×C4—471no—
357C23×C4—471no—
358C23×C4—472no—
359C23×C4—472no—
360C23×C4—472no—
361C23×C4C22×D4712yes—
362C23×C4C22×D4712yes—
363C23×C4C22×D4711yescharacteristic
364C4⋊Q8—9432no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—944no—
369C2×C4⋊C4—944no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4C22⋊D4714yes—
377C2×C22⋊C4—28no—
378C2×C22⋊C4—28no—
379C2×C22⋊C4—44no—
380C2×C22⋊C4—948no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—944no—
383C2×C22⋊C4—948no—
384C2×C22⋊C4—948no—
385C2×C22⋊C4—948no—
386C2×C22⋊C4—1884no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—474no—
391C2×C42—942no—
392C2×C42—942no—
393C2×C42—942no—
394C2×C42—942no—
395C2×C42—942no—
396#95—478no—
397#95—944no—
398#95—948no—
399#95—948no—
400C24—24no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—24no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—24no—
409C24—24no—
410C24C2×Dic9414yes—
411C24C23×D4711yescharacteristic
412C2×Q8—9416no—
413C2×Q8—18816no—
414C2×Q8—1888no—
415C2×Q8—18816no—
416C2×Q8—1888no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—474no—
435C22×C4—472no—
436C22×C4—472no—
437C22×C4—474no—
438C22×C4—474no—
439C22×C4—474no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—944no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—944no—
455C22×C4—942no—
456C22×C4—944no—
457C22×C4—942no—
458C22×C4—944no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4C2×Dic9412yes—
462C22×C4#1062535012yes—
463C22×C4D4×D4714yes—
464C22×C4D4×D4712yes—
465C22×C4D4×D4712yes—
466C22×C4#1062535212yes—
467C22×C4#1062535212yes—
468C22×C4C2×C22⋊D4714yes—
469C22×C4C2×C22⋊D4712yes—
470C4⋊C4—28no—
471C4⋊C4—9416no—
472C4⋊C4—948no—
473C4⋊C4—1884no—
474C22⋊C4—416no—
475C22⋊C4—1888no—
476C22⋊C4—1888no—
477C22⋊C4—18816no—
478C42—44no—
479C42—948no—
480C42—1884no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23C2×D4×D4712yes—
505C23C2×D4×D4711yescharacteristic
506C23C2×D4×D4711yescharacteristic
507C23C22×C22⋊D4711yescharacteristic
508C23C22×Dic9411yescharacteristic
509C23#41×D4712yes—
510C23C2×#1062535211yescharacteristic
511C23C2×#1062535211yescharacteristic
512C23C2×#1062535011yescharacteristic
513C23#1309406912yes—
514C23#1309407112yes—
515Q8—18832no—
516D4—416no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—474no—
530C2×C4—474no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—944no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—944no—
540C2×C4—948no—
541C2×C4—944no—
542C2×C4—944no—
543C2×C4—944no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C2×C4—1884no—
548C2×C4—1884no—
549C2×C4—1882no—
550C2×C4—1884no—
551C2×C4—1882no—
552C2×C4—1882no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1882no—
556C2×C4#1309399214yes—
557C2×C4#1309400114yes—
558C2×C4#1309400114yes—
559C2×C4#1309400314yes—
560C2×C4#1309401014yes—
561C2×C4#1309401014yes—
562C2×C4#1309401114yes—
563C22—44no—
564C22—44no—
565C22—48no—
566C22—48no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—84no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22C2×#1309399211yescharacteristic
575C22C2×#1309400111yescharacteristic
576C22C2×#1309400111yescharacteristic
577C22C2×#1309400311yescharacteristic
578C22#124×D4711yescharacteristic
579C22C2×#1309401011yescharacteristic
580C22C2×#1309401011yescharacteristic
581C22C2×#1309401111yescharacteristic
582C22#127×D4711yescharacteristic
583C22#1568966812yes—
584C22#1568967012yes—
585C22#1568968611yescharacteristic
586C22#1568969311yescharacteristic
587C22#1568973812yes—
588C22#1568974012yes—
589C22D4×C22⋊D4712yes—
590C22#1568975812yes—
591C22#1568977312yes—
592C22#1568978012yes—
593C22#1568978112yes—
594C22D4×Dic9412yes—
595C22#1568983212yes—
596C22#1568983712yes—
597C4—44no—
598C4—44no—
599C4—944no—
600C4—944no—
601C4—3762no—
602C4—1884no—
603C4—1884no—
604C4#1568903518yescomplemented
605C2—84no—
606C2C2×#1568903511yescharacteristic
607C2#2024702211yescharacteristic
608C2#2024704311yescharacteristic
609C2#2024723212yes—
610C2#2024724112yes—
611C2#2024775212yes—
612C2#2024775312yes—
613C2#2024833911yescharacteristic
614C2#2024855911yescharacteristic
615C2#2024932511yescharacteristic
616C2#2025174211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutator, socle, cosocleC24×C47C24
FittingC2×C47×#390C2
prime radicalC23×D4×C47C22
hypercenterC2×#390D47
hypocenterC47#15144
LCSG ▷ C24×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×#13094023C2semidirect
2. C2×#15689035C2semidirect
5. #20241631C2semidirect
6. #20241618C2semidirect
7. #20241617C2semidirect
8. C4×C47⋊#95C2semidirect
70. C47⋊#95D4semidirect
78. C22×D4×C47Q8semidirect
291. C4×C47#393semidirect
342. C47#15144semidirect
355. C22×D4Dic94semidirect
604. C4#15689035semidirect

Maximal subgroups

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

subgroupindex
C22×#130940232
C2×#156890352
C2×#156890262
#202416312
#202416182
#202416172
C4×C47⋊#952
C2×C47×#3902
#1514447