Finite Group Databasebeta

← #26138828

Subgroups — #26138828

Order 12032.

2374 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic47C211yesmaximal, characteristic
2C2×#15689026C211yesmaximal, characteristic
3C2×#15688988C211yesmaximal, characteristic, complemented
4C2×C4×Dic47⋊C4C211yesmaximal, characteristic, complemented
5#20241631C214yesmaximal, complemented
6#20241618C211yesmaximal, characteristic, complemented
7#20241617C211yesmaximal, characteristic, complemented
8#20241610C214yesmaximal, complemented
9C2×C47×#390C211yesmaximal, characteristic
10C24×Dic47C2212yes—
11C2×D4×Dic47—24no—
12C2×D4×Dic47—24no—
13C2×D4×Dic47C418yescomplemented
14C22×C4⋊Dic47C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×C22⋊Dic47C2214yes—
18C22×C22⋊Dic47C2212yes—
19C22×C4×Dic47C2211yescharacteristic
20C22×C4×Dic47C2211yescharacteristic
21#15689026—24no—
22#15688988—24no—
23C4×Dic47⋊C4—24no—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2214yes—
28C2×C47⋊#95C2214yes—
29C23×D4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C22⋊C4×C47C2214yes—
32C22×C42×C47C2211yescharacteristic
33C47×#390—24no—
34C23×Dic47—24no—
35C23×Dic47—24no—
36C23×Dic47—44no—
37C23×Dic47—44no—
38C23×Dic47C2×C418yes—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44D4×Dic47—232no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47C2×C414yes—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×C22⋊Dic47—24no—
55C2×C22⋊Dic47—24no—
56C2×C22⋊Dic47—28no—
57C2×C22⋊Dic47—28no—
58C2×C22⋊Dic47—44no—
59C2×C22⋊Dic47C2×C418yes—
60C2×C4×Dic47—22no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47C2×C414yes—
64C2×C4×Dic47D414yes—
65C2×C4×Dic47D414yes—
66C47⋊#95—24no—
67C47⋊#95—24no—
68C47⋊#95—24no—
69C47⋊#95—44no—
70C47⋊#95—44no—
71C25×C47C2312yes—
72C22×D4×C47—22no—
73C22×D4×C47—22no—
74C22×D4×C47—22no—
75C22×D4×C47—22no—
76C22×D4×C47C2×C414yes—
77C22×D4×C47D414yes—
78C22×D4×C47Q814yes—
79C23×C4×C47C2312yes—
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47—22no—
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—22no—
93C24×C47—24no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—24no—
97C24×C47—24no—
98C24×C47—24no—
99C24×C47C22×C414yes—
100C24×C47C2×D414yes—
101C24×C47C2×Q814yes—
102C24×C47C2411yescharacteristic
103C2×D4×C47—28no—
104C2×D4×C47—28no—
105C2×D4×C47—28no—
106C2×D4×C47—28no—
107C2×D4×C47—28no—
108C2×D4×C47—28no—
109C2×D4×C47C4⋊C4116yescomplemented
110C22×C4×C47—22no—
111C22×C4×C47—22no—
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×C47—24no—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×Q812yes—
126C22×C4×C47#4114yes—
127C22×C4×C47#4112yes—
128C22×C4×C47#4112yes—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—24no—
141C22×Dic47—24no—
142C22×Dic47—22no—
143C22×Dic47—216no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—24no—
147C22×Dic47—22no—
148C22×Dic47—24no—
149C22×Dic47—416no—
150C22×Dic47—48no—
151C22×Dic47—416no—
152C22×Dic47—48no—
153C22×Dic47C22×C414yes—
154C22×Dic47C22×C412yes—
155C22×Dic47C22×C412yes—
156C22×Dic47C2×D412yes—
157C22×Dic47C2×D412yes—
158C22×Dic47C2×D412yes—
159C22×Dic47C2×D412yes—
160C22×Dic47#4114yes—
161C4⋊C4×C47—44no—
162C22⋊C4×C47—416no—
163C42×C47—44no—
164C22⋊Dic47—216no—
165C22⋊Dic47—416no—
166C4⋊Dic47—28no—
167C4⋊Dic47—44no—
168Dic47⋊C4—216no—
169C4×Dic47—44no—
170C4×Dic47C22⋊C4116yescomplemented
171C23×C47—24no—
172C23×C47—24no—
173C23×C47—28no—
174C23×C47—28no—
175C23×C47—24no—
176C23×C47—24no—
177C23×C47—24no—
178C23×C47—24no—
179C23×C47—28no—
180C23×C47—28no—
181C23×C47—28no—
182C23×C47—28no—
183C23×C47—44no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—44no—
188C23×C47—42no—
189C23×C47—44no—
190C23×C47—44no—
191C23×C47—44no—
192C23×C47—44no—
193C23×C47C2×C4⋊C418yes—
194C23×C47C23×C411yescharacteristic
195C23×C47C22×D411yescharacteristic
196C23×C47C22×D411yescharacteristic
197C23×C47C22×D411yescharacteristic
198C23×C47C22×Q811yescharacteristic
199C23×C47C2×#4112yes—
200C23×C47C2×#4111yescharacteristic
201C23×C47C2×#4111yescharacteristic
202C23×C47C2×#4112yes—
203C23×C47H(5,𝔽2)12yes—
204C23×C47H(5,𝔽2)12yes—
205D4×C47—232no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—44no—
217C2×C4×C47—44no—
218C2×C4×C47C2×C22⋊C414yes—
219C2×C4×C47C2×C4⋊C414yes—
220C2×C4×C47#11714yes—
221C2×C4×C47#12114yes—
222C2×C4×C47#12114yes—
223C2×C4×C47C22⋊Q814yes—
224C2×C4×C47C22⋊Q814yes—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—28no—
235C2×Dic47—24no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—432no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—44no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—44no—
250C2×Dic47—44no—
251C2×Dic47C2×C22⋊C414yes—
252C2×Dic47C2×C22⋊C414yes—
253#13941—471nomaximal, Sylow
254C22×C47—216no—
255C22×C47—28no—
256C22×C47—44no—
257C22×C47—44no—
258C22×C47—44no—
259C22×C47—48no—
260C22×C47—48no—
261C22×C47—48no—
262C22×C47—44no—
263C22×C47—44no—
264C22×C47—44no—
265C22×C47C22×C22⋊C411yescharacteristic
266C22×C47C22×C4⋊C411yescharacteristic
267C22×C47C2×#11711yescharacteristic
268C22×C47C4×#4112yes—
269C22×C47#51812yes—
270C22×C47C2×#12111yescharacteristic
271C22×C47C2×#12111yescharacteristic
272C22×C47C2×C22⋊Q811yescharacteristic
273C22×C47C2×C22⋊Q811yescharacteristic
274C22×C47#52512yes—
275C22×C47#53211yescharacteristic
276C22×C47#53211yescharacteristic
277C22×C47#53511yescharacteristic
278C22×C47#53512yes—
279C22×C47#53511yescharacteristic
280C22×C47#54612yes—
281C22×C47#54612yes—
282C22×C47#55312yes—
283C22×C47#55312yes—
284C22×C47#55412yes—
285C22×C47#55412yes—
286C22×C47#55912yes—
287C22×C47#55912yes—
288C4×C47—44no—
289C4×C47—82no—
290C4×C47#38018yes—
291Dic47—24no—
292Dic47—24no—
293Dic47—44no—
294Dic47—44no—
295Dic47—82no—
296C22×C4×D4—471no—
297C2×#390—471no—
298C2×#390D4711yescharacteristic
299C2×#382—471no—
300C2×C4×C4⋊C4—471no—
301#1439—474no—
302C4⋊#95—471no—
303C4⋊#95—471no—
304C22⋊#95—474no—
305C2×C47—48no—
306C2×C47C2×#38011yescharacteristic
307C2×C47#229711yescharacteristic
308C2×C47#230911yescharacteristic
309C2×C47#234912yes—
310C2×C47#235712yes—
311C2×C47#253512yes—
312C2×C47#254411yescharacteristic
313C2×C47#254411yescharacteristic
314C2×C47#256611yescharacteristic
315C2×C47#256611yescharacteristic
316C2×C47#258012yes—
317C23×D4C2×D4711yescharacteristic
318C24×C4—472no—
319C2×C4×D4—478no—
320C2×C4×D4—944no—
321C2×C4×D4—944no—
322C22×C4⋊C4—471no—
323C22×C4⋊C4—472no—
324C22×C4⋊C4—471no—
325C22×C4⋊C4C2×D4711yescharacteristic
326C22×C22⋊C4—472no—
327C22×C22⋊C4—474no—
328C22×C22⋊C4C2×D4714yes—
329C22×C42—471no—
330C22×C42—471no—
331C22×C42C2×D4711yescharacteristic
332#390—24no—
333#390—944no—
334#382—944no—
335C4×C4⋊C4—944no—
336C2×#95—472no—
337C2×#95—472no—
338C2×#95—472no—
339C2×#95—474no—
340C2×#95—474no—
341C47#1394111yescharacteristic, complemented, Sylow
342C25C22×D4712yes—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4Dic9414yes—
348C22×D4C4×D4714yes—
349C22×D4C22⋊D4714yes—
350C23×C4—471no—
351C23×C4—471no—
352C23×C4—472no—
353C23×C4—472no—
354C23×C4—478no—
355C23×C4—472no—
356C23×C4—944no—
357C23×C4—944no—
358C23×C4—1884no—
359C23×C4—1884no—
360C23×C4C22×D4712yes—
361C23×C4C22×D4712yes—
362C23×C4C22×D4711yescharacteristic
363C4×D4—9432no—
364C2×C4⋊C4—22no—
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—942no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—942no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—28no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4—478no—
380C2×C22⋊C4—944no—
381C2×C22⋊C4—944no—
382C2×C22⋊C4—948no—
383C2×C22⋊C4—948no—
384C2×C22⋊C4—1884no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—474no—
389C2×C42—474no—
390C2×C42—474no—
391C2×C42—942no—
392C2×C42—942no—
393C2×C42—942no—
394#95—944no—
395#95—944no—
396#95—944no—
397#95—1884no—
398#95—1884no—
399C24—22no—
400C24—22no—
401C24—24no—
402C24—24no—
403C24—22no—
404C24—22no—
405C24—24no—
406C24—24no—
407C24C2×Dic9414yes—
408C24C2×C4×D4714yes—
409C24C2×C22⋊D4714yes—
410C24C23×D4711yescharacteristic
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4Dic47⋊C4116yescomplemented
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—472no—
430C22×C4—472no—
431C22×C4—472no—
432C22×C4—472no—
433C22×C4—472no—
434C22×C4—472no—
435C22×C4—474no—
436C22×C4—474no—
437C22×C4—9416no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—944no—
455C22×C4—942no—
456C22×C4—944no—
457C22×C4—18816no—
458C22×C4—1888no—
459C22×C4—18816no—
460C22×C4—1888no—
461C22×C4C2×Dic9412yes—
462C22×C4C2×C4×D4712yes—
463C22×C4D4×D4712yes—
464C22×C4D4×D4712yes—
465C22×C4#1062535214yes—
466C22×C4#1062535212yes—
467C22×C4#1062535212yes—
468C22×C4C2×C22⋊D4712yes—
469C4⋊C4—44no—
470C4⋊C4—9416no—
471C4⋊C4—948no—
472C4⋊C4—1884no—
473C22⋊C4—416no—
474C22⋊C4—9416no—
475C22⋊C4—18816no—
476C42—44no—
477C42—4716no—
478C42—1884no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—28no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23C2×Dic47⋊C418yes—
502C23C22×C4×D4711yescharacteristic
503C23C2×D4×D4711yescharacteristic
504C23C2×D4×D4711yescharacteristic
505C23C22×C22⋊D4711yescharacteristic
506C23C22×Dic9411yescharacteristic
507C23#41×D4712yes—
508C23C2×#1062535212yes—
509C23C2×#1062535211yescharacteristic
510C23C2×#1062535211yescharacteristic
511C23#1309406912yes—
512C23#1309407012yes—
513D4—232no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—474no—
527C2×C4—474no—
528C2×C4—944no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—942no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—944no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—948no—
540C2×C4—944no—
541C2×C4—18832no—
542C2×C4—1884no—
543C2×C4—1884no—
544C2×C4—1882no—
545C2×C4—1882no—
546C2×C4—1882no—
547C2×C4—1882no—
548C2×C4—1882no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4—1884no—
553C2×C4—1882no—
554C2×C4C22⋊C4×D4714yes—
555C2×C4C2×Dic47⋊C414yes—
556C2×C4#1309398014yes—
557C2×C4#1309399614yes—
558C2×C4#1309399614yes—
559C2×C4#1309400114yes—
560C2×C4#1309400114yes—
561C22—216no—
562C22—28no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—48no—
568C22—44no—
569C22—48no—
570C22—44no—
571C22—48no—
572C22C2×C22⋊C4×D4711yescharacteristic
573C22C22×Dic47⋊C411yescharacteristic
574C22C2×#1309398011yescharacteristic
575C22C4×#1062535212yes—
576C22#1568959512yes—
577C22C2×#1309399611yescharacteristic
578C22C2×#1309399611yescharacteristic
579C22C2×#1309400111yescharacteristic
580C22C2×#1309400111yescharacteristic
581C22#1568963312yes—
582C22#1568967111yescharacteristic
583C22#1568967211yescharacteristic
584C22#1568968611yescharacteristic
585C22#1568968912yes—
586C22#1568969111yescharacteristic
587C22#1568976112yes—
588C22#1568976712yes—
589C22#1568982012yes—
590C22#1568982512yes—
591C22#1568983012yes—
592C22#1568983212yes—
593C22#1568985512yes—
594C22#1568985912yes—
595C4—44no—
596C4—82no—
597C4—944no—
598C4—944no—
599C4—3762no—
600C4—1884no—
601C4—1884no—
602C4#1568898118yes—
603C2—48no—
604C2C2×#1568898111yescharacteristic
605C2#2024587511yescharacteristic
606C2#2024596711yescharacteristic
607C2#2024629312yes—
608C2#2024636012yes—
609C2#2024828712yes—
610C2#2024835611yescharacteristic
611C2#2024835811yescharacteristic
612C2#2024859011yescharacteristic
613C2#2024859111yescharacteristic
614C2#2024874412yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#390C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC2×#390D47
hypocenterC47#13941
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
3. C2×#15688988C2semidirect
4. C2×C4×Dic47⋊C4C2semidirect
5. #20241631C2semidirect
6. #20241618C2semidirect
7. #20241617C2semidirect
8. #20241610C2semidirect
13. C2×D4×Dic47C4semidirect
109. C2×D4×C47C4⋊C4semidirect
170. C4×Dic47C22⋊C4semidirect
341. C47#13941semidirect
417. C2×D4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic472
C2×#156890262
C2×#156889882
C2×C4×Dic47⋊C42
#202416312
#202416182
#202416172
#202416102
C2×C47×#3902
#1394147