Finite Group Databasebeta

← #26222678

Subgroups — #26222678

Order 12032.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689224C211yesmaximal, characteristic
2C2×SD16×Dic47C211yesmaximal, characteristic
3#20246108C211yesmaximal, characteristic, complemented
4#20245749C211yesmaximal, characteristic
5#20244312C211yesmaximal, characteristic, complemented
6#20244158C211yesmaximal, characteristic, complemented
7#20244031C211yesmaximal, characteristic, complemented
8#20244024C211yesmaximal, characteristic, complemented
9#20243724C211yesmaximal, characteristic
10#20243724C211yesmaximal, characteristic
11#20243398C211yesmaximal, characteristic
12#20243247C211yesmaximal, characteristic, complemented
13#20243220C211yesmaximal, characteristic, complemented
14#103×Dic47C211yesmaximal, characteristic, complemented
15C2×C47×#461C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C2×D4×Dic47C2211yescharacteristic
20#15689224—24no—
21SD16×Dic47—24no—
22C2×#13093958C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093945C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31C2×#13093934C2211yescharacteristic
32C2×C8×Dic47C2211yescharacteristic
33#15689030C2211yescharacteristic
34#15689030C2211yescharacteristic
35#15689009—21no—
36#15689009—21no—
37#15689009—21no—
38#15689009—21no—
39#15688989C2211yescharacteristic
40C4⋊C4×Dic47C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C22⋊C4×Dic47—21no—
44C22⋊C4×Dic47—21no—
45C4×C22⋊Dic47—21no—
46#15688969C2211yescharacteristic
47C2×C47⋊#95—21no—
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688877C2211yescharacteristic
53#15688877C2211yescharacteristic
54C22×SD16×C47C2211yescharacteristic
55C22×SD16×C47C2211yescharacteristic
56C2×C4⋊Q8×C47C2211yescharacteristic
57C2×C4×D4×C47C2211yescharacteristic
58C47×#461C418yescomplemented
59C2×C4⋊C8×C47C2211yescharacteristic
60C2×#103×C47C2211yescharacteristic
61C2×#103×C47C2211yescharacteristic
62C23×Dic47—21no—
63C23×Dic47—21no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66Q8×Dic47—22no—
67Q8×Dic47—22no—
68Q8×Dic47—22no—
69Q8×Dic47—22no—
70D4×Dic47—22no—
71D4×Dic47—22no—
72D4×Dic47—22no—
73D4×Dic47D414yes—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47C2311yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89#13093958—22no—
90#13093956—22no—
91C4⋊(C47⋊C8)—22no—
92#13093948—22no—
93#13093948—22no—
94#13093948—22no—
95#13093948—22no—
96#13093945—22no—
97#13093945—22no—
98#13093934—22no—
99C8×Dic47—22no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95D411yescharacteristic
110C47⋊#95D411yescharacteristic
111C47⋊#95D411yescharacteristic
112C47⋊#95D411yescharacteristic
113C22×Q8×C47C2311yescharacteristic
114C22×Q8×C47C2311yescharacteristic
115C22×D4×C47C2311yescharacteristic
116C23×C4×C47—21no—
117C2×SD16×C47—22no—
118C2×SD16×C47—22no—
119C2×SD16×C47—22no—
120C2×SD16×C47—22no—
121C2×SD16×C47C2×C414yes—
122C2×SD16×C47C2×C414yes—
123C22×C8×C47C2311yescharacteristic
124C22×C8×C47C2311yescharacteristic
125C4⋊Q8×C47—22no—
126C4⋊Q8×C47C2×C414yescomplemented
127C4×D4×C47—22no—
128C4×D4×C47C2×C414yes—
129C2×C4⋊C4×C47—21no—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47C2311yescharacteristic
132C2×C22⋊C4×C47—21no—
133C2×C42×C47C2311yescharacteristic
134C4⋊C8×C47C2×C414yescomplemented
135#103×C47C2×C414yescomplemented
136#103×C47C2×C414yescomplemented
137C24×C47—21no—
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47—21no—
147C2×Q8×C47—21no—
148C2×Q8×C47C22×C412yes—
149C2×Q8×C47C22×C412yes—
150C2×D4×C47—21no—
151C2×D4×C47—21no—
152C2×D4×C47—21no—
153C2×D4×C47C22×C412yes—
154C2×D4×C47C2×D412yes—
155C2×D4×C47#4112yes—
156C22×C4×C47—21no—
157C22×C4×C47—21no—
158C22×C4×C47—21no—
159C22×C4×C47—24no—
160C22×C4×C47—81no—
161C22×C4×C47—81no—
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47C2411yescharacteristic
165C22×Dic47—21no—
166C22×Dic47—21no—
167C22×Dic47—24no—
168C22×Dic47—24no—
169C22×Dic47—21no—
170C22×Dic47—21no—
171C22×Dic47—81no—
172C22×Dic47—81no—
173C22×Dic47—81no—
174C22×Dic47—81no—
175C22×Dic47C2×D411yescharacteristic
176C22×Dic47C2×D411yescharacteristic
177C22×Dic47C2×D411yescharacteristic
178C22×Dic47C2×D411yescharacteristic
179SD16×C47—28no—
180SD16×C47—28no—
181C2×C8×C47—21no—
182C2×C8×C47—21no—
183C2×C8×C47—21no—
184C2×C8×C47—21no—
185C2×C8×C47C22×C412yes—
186C2×C8×C47C22×C412yes—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—24no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47C22×C412yes—
191C22⋊C4×C47—24no—
192C42×C47C22×C412yes—
193C42×C47#4112yes—
194C22⋊Dic47—24no—
195C22⋊Dic47—24no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—42no—
203C4⋊Dic47—42no—
204C4⋊Dic47C2×D412yes—
205C4⋊Dic47#4112yes—
206C4×Dic47—21no—
207C4×Dic47—21no—
208C4×Dic47—42no—
209C4×Dic47—42no—
210C4×Dic47—81no—
211C4×Dic47—81no—
212C4×Dic47—81no—
213C4×Dic47C2×D412yes—
214C4×Dic47#4112yes—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—21no—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47C22×D411yescharacteristic
229Q8×C47—22no—
230Q8×C47—22no—
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—41no—
234Q8×C47—41no—
235Q8×C47—41no—
236Q8×C47—41no—
237D4×C47—22no—
238D4×C47—22no—
239D4×C47—22no—
240D4×C47C4×D414yes—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—84no—
255C2×C4×C47C4×D412yes—
256C2×C4×C47C4×D412yes—
257C2×C4×C47C23×C411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×C4×C47C2×#4111yescharacteristic
261C2×C4×C47C2×#4111yescharacteristic
262C2×C4×C47C2×#4111yescharacteristic
263C2×Dic47—21no—
264C2×Dic47—21no—
265C2×Dic47—21no—
266C2×Dic47—21no—
267C2×Dic47—22no—
268C2×Dic47—21no—
269C2×Dic47—21no—
270C2×Dic47—22no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47—21no—
274C2×Dic47—21no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47—84no—
286C2×Dic47—84no—
287C2×Dic47#12112yes—
288C2×Dic47#12112yes—
289C8×C47—22no—
290C8×C47—22no—
291C47⋊C8—41no—
292C47⋊C8—41no—
293#21514—471nomaximal, Sylow
294C22×C47—24no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—42no—
300C22×C47—42no—
301C22×C47C2×C4×D411yescharacteristic
302C22×C47C2×#12111yescharacteristic
303C22×C47#52511yescharacteristic
304C22×C47#57711yescharacteristic
305C22×C47#57711yescharacteristic
306C22×C47#57711yescharacteristic
307C22×C47#57811yescharacteristic
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—42no—
311C4×C47—42no—
312C4×C47—42no—
313C4×C47C2×C4×D411yescharacteristic
314C4×C47C4×#4111yescharacteristic
315C4×C47#52911yescharacteristic
316C4×C47#53211yescharacteristic
317Dic47—22no—
318Dic47—22no—
319Dic47—41no—
320Dic47—41no—
321Dic47—42no—
322Dic47—42no—
323Dic47—81no—
324Dic47—81no—
325C2×#461D4711yescharacteristic
326C2×#440—471no—
327C2×C4×SD16—471no—
328#2327—471no—
329#2275—471no—
330#1981—471no—
331#1949—471no—
332#1928—471no—
333#1927—471no—
334#1877—471no—
335#1877—471no—
336#1804—471no—
337#1768—471no—
338#1763—471no—
339C4×#103—471no—
340C2×C47—44no—
341C2×C47C4×#12111yescharacteristic
342C2×C47#294711yescharacteristic
343C2×C47#294911yescharacteristic
344C2×C47#304211yescharacteristic
345C2×C47#304611yescharacteristic
346C2×C47#312011yescharacteristic
347C2×C47#312311yescharacteristic
348C22×SD16C2×D4711yescharacteristic
349C22×SD16C2×D4711yescharacteristic
350C2×C4⋊Q8C2×D4711yescharacteristic
351C2×C4×Q8—471no—
352C2×C4×Q8—471no—
353C2×C4×D4—471no—
354C2×C4×D4—471no—
355C2×C4×D4C2×D4711yescharacteristic
356#461Dic4718yescomplemented
357#440—944no—
358C4×SD16—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8C2×D4711yescharacteristic
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103C2×D4711yescharacteristic
368C2×#103C2×D4711yescharacteristic
369C2×#102—471no—
370C2×#102—471no—
371C2×#97—471no—
372C2×C4×C8—471no—
373#391—471no—
374#391—471no—
375#387—941no—
376#387—941no—
377#387—941no—
378#387—941no—
379#382—471no—
380C4×C4⋊C4—471no—
381C4×C4⋊C4—471no—
382C4×C4⋊C4—471no—
383C4×C22⋊C4—941no—
384C4×C22⋊C4—941no—
385C4×C22⋊C4—941no—
386#376—471no—
387C2×#95—941no—
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#336—471no—
393#336—471no—
394C47#2151411yescharacteristic, complemented, Sylow
395C22×Q8C22×D4711yescharacteristic
396C22×Q8C22×D4711yescharacteristic
397C22×D4C22×D4711yescharacteristic
398C23×C4—21no—
399C23×C4—941no—
400C23×C4—941no—
401C2×SD16—22no—
402C2×SD16—22no—
403C2×SD16—22no—
404C2×SD16—22no—
405C2×SD16C2×Dic4714yes—
406C2×SD16C2×Dic4714yes—
407C22×C8—471no—
408C22×C8—471no—
409C22×C8C22×D4711yescharacteristic
410C22×C8C22×D4711yescharacteristic
411C4⋊Q8—22no—
412C4⋊Q8C2×Dic4714yescomplemented
413C4×Q8—942no—
414C4×Q8—942no—
415C4×Q8—942no—
416C4×Q8—942no—
417C4×D4—22no—
418C4×D4—474no—
419C4×D4—942no—
420C4×D4—942no—
421C4×D4—942no—
422C4×D4C2×Dic4714yes—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—941no—
434C2×C22⋊C4—941no—
435C2×C42—471no—
436C2×C42—471no—
437C2×C42—941no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42—941no—
441C2×C42—941no—
442C2×C42C22×D4711yescharacteristic
443#107—942no—
444#106—942no—
445C4⋊C8—942no—
446C4⋊C8C2×Dic4714yescomplemented
447#103—942no—
448#103—942no—
449#103—942no—
450#103—942no—
451#103C2×Dic4714yescomplemented
452#103C2×Dic4714yescomplemented
453#102—942no—
454#102—942no—
455#97—942no—
456C4×C8—942no—
457#95—471no—
458#95—471no—
459#95—471no—
460#95—471no—
461#95—941no—
462#95—941no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95—941no—
467#95—941no—
468#95—941no—
469#95—941no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8C22×Dic4712yes—
482C2×Q8C22×Dic4712yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4D4×D4712yes—
487C2×D4#1062535212yes—
488C2×D4C22×Dic4712yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—471no—
496C22×C4—471no—
497C22×C4—471no—
498C22×C4—471no—
499C22×C4—944no—
500C22×C4—944no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4—3761no—
506C22×C4—3761no—
507C22×C4—3761no—
508C22×C4—3761no—
509C22×C4D4×D4711yescharacteristic
510C22×C4D4×D4711yescharacteristic
511C22×C4C23×D4711yescharacteristic
512SD16—28no—
513SD16—28no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8—941no—
522C2×C8—941no—
523C2×C8—941no—
524C2×C8C22×Dic4712yes—
525C2×C8C22×Dic4712yes—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—472no—
530C4⋊C4—472no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—941no—
535C4⋊C4—941no—
536C4⋊C4—941no—
537C4⋊C4—1882no—
538C4⋊C4—1882no—
539C4⋊C4C22×Dic4712yes—
540C22⋊C4—24no—
541C22⋊C4—944no—
542C22⋊C4—944no—
543C42—472no—
544C42—472no—
545C42—941no—
546C42—941no—
547C42—3761no—
548C42—3761no—
549C42—3761no—
550C42—1882no—
551C42—1882no—
552C42#1062535212yes—
553C42C22×Dic4712yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C2×D4×D4711yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4D4×Dic4714yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—472no—
589C2×C4—472no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—941no—
593C2×C4—941no—
594C2×C4—942no—
595C2×C4—941no—
596C2×C4—941no—
597C2×C4—942no—
598C2×C4—941no—
599C2×C4—941no—
600C2×C4—941no—
601C2×C4—941no—
602C2×C4—3764no—
603C2×C4—3764no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4D4×Dic4712yes—
615C2×C4D4×Dic4712yes—
616C2×C4C23×Dic4711yescharacteristic
617C2×C4C2×D4×D4711yescharacteristic
618C2×C4#41×D4711yescharacteristic
619C2×C4#41×D4711yescharacteristic
620C2×C4C2×#1062535211yescharacteristic
621C2×C4C2×#1062535211yescharacteristic
622C8—22no—
623C8—22no—
624C8—1881no—
625C8—1881no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22C2×D4×Dic4711yescharacteristic
634C22#121×D4711yescharacteristic
635C22#1568963311yescharacteristic
636C22#1568992911yescharacteristic
637C22#1568992911yescharacteristic
638C22#1568993211yescharacteristic
639C22#1568993911yescharacteristic
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—942no—
646C4—942no—
647C4—3761no—
648C4—3761no—
649C4—1881no—
650C4—1881no—
651C4—1882no—
652C4—1882no—
653C4C2×D4×Dic4711yescharacteristic
654C4#41×Dic4711yescharacteristic
655C4#1568965411yescharacteristic
656C4#1568966811yescharacteristic
657C2—44no—
658C2#121×Dic4711yescharacteristic
659C2#2025223211yescharacteristic
660C2#2025224811yescharacteristic
661C2#2025289511yescharacteristic
662C2#2025293311yescharacteristic
663C2#2025359911yescharacteristic
664C2#2025363811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#461C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenterC2×#461D47
hypocenterC47#21514
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. #20246108C2semidirect
5. #20244312C2semidirect
6. #20244158C2semidirect
7. #20244031C2semidirect
8. #20244024C2semidirect
12. #20243247C2semidirect
13. #20243220C2semidirect
14. #103×Dic47C2semidirect
58. C47×#461C4semidirect
126. C4⋊Q8×C47C2×C4semidirect
134. C4⋊C8×C47C2×C4semidirect
135. #103×C47C2×C4semidirect
136. #103×C47C2×C4semidirect
356. #461Dic47semidirect
394. C47#21514semidirect
412. C4⋊Q8C2×Dic47semidirect
446. C4⋊C8C2×Dic47semidirect
451. #103C2×Dic47semidirect
452. #103C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892242
C2×SD16×Dic472
#202461082
#202457492
#202443122
#202441582
#202440312
#202440242
#202437242
#202433982
#202432472
#202432202
#103×Dic472
C2×C47×#4612
#2151447