Finite Group Databasebeta

← #26226068

Subgroups — #26226068

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689733C211yesmaximal, characteristic
2C2×#15689181C211yesmaximal, characteristic
3#20246012C211yesmaximal, characteristic
4#20244355C211yesmaximal, characteristic, complemented
5#20244351C211yesmaximal, characteristic, complemented
6#20244206C211yesmaximal, characteristic, complemented
7#20244188C211yesmaximal, characteristic, complemented
8#20243701C211yesmaximal, characteristic
9#20243698C211yesmaximal, characteristic
10#20243695C211yesmaximal, characteristic
11#20243402C211yesmaximal, characteristic, complemented
12#20243401C211yesmaximal, characteristic, complemented
13#20243392C211yesmaximal, characteristic, complemented
14#20243391C211yesmaximal, characteristic, complemented
15C47×#1872C211yesmaximal, characteristic
16#15689733—24no—
17#15689733C418yescomplemented
18C2×#13094022C2211yescharacteristic
19C2×#13094017—21no—
20C2×#13094017—21no—
21C2×#13094015—21no—
22C2×#13094015C2211yescharacteristic
23C2×#13094014—21no—
24C2×#13094003C2211yescharacteristic
25C2×Q8×Dic47C2211yescharacteristic
26C2×#13093989C2211yescharacteristic
27C2×#13093983C2211yescharacteristic
28C2×#13093983C2211yescharacteristic
29#15689181—24no—
30C2×Dic47⋊C8C2211yescharacteristic
31C2×Dic47⋊C8C2211yescharacteristic
32C2×#13093949C2211yescharacteristic
33C2×#13093949C2211yescharacteristic
34C2×#13093948C2211yescharacteristic
35C2×#13093948C2211yescharacteristic
36C2×#13093947C2211yescharacteristic
37C2×#13093947C2211yescharacteristic
38C2×C22⋊(C47⋊C8)C2211yescharacteristic
39C2×#13093934C2211yescharacteristic
40C2×C8×Dic47C2211yescharacteristic
41#15689019—21no—
42#15689019—21no—
43#15688996—21no—
44#15688996—21no—
45#15688992C2211yescharacteristic
46#15688992C2211yescharacteristic
47#15688988C2211yescharacteristic
48#15688988C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C22⋊Q8×C47C2211yescharacteristic
56C22×C4⋊C4×C47C2211yescharacteristic
57C2×#103×C47C2211yescharacteristic
58C2×#103×C47C2211yescharacteristic
59C2×C22⋊C8×C47C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094022—22no—
66#13094022C2×C414yescomplemented
67#13094017—28no—
68#13094017—28no—
69#13094015—28no—
70#13094015—22no—
71#13094015C2×C414yescomplemented
72#13094014—28no—
73#13094003—22no—
74#13094003C2×C414yes—
75Q8×Dic47—22no—
76Q8×Dic47C2×C414yescomplemented
77#13093989—22no—
78#13093989C2×C414yescomplemented
79#13093983—41no—
80#13093983—41no—
81#13093983C2×C414yes—
82#13093983D414yes—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47D411yescharacteristic
86C2×C4⋊Dic47D411yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2311yescharacteristic
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—21no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4—21no—
97C2×Dic47⋊C4—21no—
98C2×Dic47⋊C4—21no—
99C2×C22⋊Dic47—21no—
100C2×C22⋊Dic47—21no—
101C2×C22⋊Dic47D411yescharacteristic
102C2×C22⋊Dic47D411yescharacteristic
103C2×C4×Dic47—21no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—21no—
106C2×C4×Dic47C2311yescharacteristic
107C2×C4×Dic47C2311yescharacteristic
108Dic47⋊C8—22no—
109Dic47⋊C8—22no—
110#13093949—41no—
111#13093949—41no—
112#13093948—41no—
113#13093948—41no—
114#13093947—41no—
115#13093947—41no—
116C22⋊(C47⋊C8)—22no—
117#13093934—22no—
118C8×Dic47—22no—
119C47⋊#95—21no—
120C47⋊#95—21no—
121C47⋊#95—21no—
122C47⋊#95—21no—
123C22×Q8×C47C2311yescharacteristic
124C23×C4×C47—21no—
125C23×C4×C47C2311yescharacteristic
126C22×C8×C47C2311yescharacteristic
127C22×C8×C47C2311yescharacteristic
128C22⋊Q8×C47—22no—
129C22⋊Q8×C47C2×C414yes—
130C2×C4⋊C4×C47—21no—
131C2×C4⋊C4×C47—21no—
132C2×C4⋊C4×C47—41no—
133C2×C4⋊C4×C47—41no—
134C2×C4⋊C4×C47C2311yescharacteristic
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C4⋊C4×C47C2311yescharacteristic
137C2×C22⋊C4×C47—21no—
138#103×C47—41no—
139#103×C47—41no—
140C22⋊C8×C47—22no—
141C24×C47C2×D411yescharacteristic
142C2×Q8×C47—21no—
143C2×Q8×C47—21no—
144C2×Q8×C47—21no—
145C2×Q8×C47—21no—
146C2×Q8×C47—21no—
147C2×Q8×C47C22×C412yes—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47—21no—
155C22×C4×C47—21no—
156C22×C4×C47—42no—
157C22×C4×C47—81no—
158C22×C4×C47—81no—
159C22×C4×C47C22×C412yes—
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2411yescharacteristic
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C22×Dic47C2×D411yescharacteristic
168C22×Dic47C2×D411yescharacteristic
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94—21no—
176C2×Dic94C22×C412yes—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47—21no—
180C2×C8×C47—21no—
181C2×C8×C47—21no—
182C2×C8×C47—21no—
183C4⋊C4×C47—24no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—21no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—42no—
190C4⋊C4×C47C22×C412yes—
191C22⋊C4×C47—24no—
192C22⋊Dic47—24no—
193C22⋊Dic47—42no—
194C22⋊Dic47C22⋊C414yes—
195C22⋊Dic47C22⋊C414yes—
196C4⋊Dic47—21no—
197C4⋊Dic47—24no—
198C4⋊Dic47—24no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47C22⋊C414yes—
204C4⋊Dic47C22⋊C414yes—
205C4⋊Dic47C22×C412yes—
206Dic47⋊C4—24no—
207Dic47⋊C4—24no—
208Dic47⋊C4—24no—
209Dic47⋊C4—24no—
210Dic47⋊C4—24no—
211Dic47⋊C4—81no—
212Dic47⋊C4—81no—
213Dic47⋊C4—81no—
214Dic47⋊C4—81no—
215C4×Dic47—24no—
216C4×Dic47—24no—
217C4×Dic47—42no—
218C4×Dic47C22×C412yes—
219C4×Dic47C22×C412yes—
220C4×Dic47C2×D412yes—
221C4×Dic47C2×D412yes—
222C2×C47⋊C8—21no—
223C2×C47⋊C8—21no—
224C2×C47⋊C8—21no—
225C2×C47⋊C8—21no—
226C2×C47⋊C8—21no—
227C2×C47⋊C8—21no—
228C23×C47—21no—
229C23×C47—21no—
230C23×C47—21no—
231C23×C47—21no—
232C23×C47—21no—
233C23×C47—21no—
234C23×C47C2×C22⋊C412yes—
235C23×C47C22×D411yescharacteristic
236Q8×C47—22no—
237Q8×C47—22no—
238Q8×C47—41no—
239Q8×C47—41no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
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—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—84no—
259C2×C4×C47C2×C22⋊C412yes—
260C2×C4×C47C23×C411yescharacteristic
261C2×C4×C47C22×D411yescharacteristic
262C2×C4×C47H(5,𝔽2)11yescharacteristic
263C2×C4×C47H(5,𝔽2)11yescharacteristic
264C2×C4×C472−1+411yescharacteristic
265C2×C4×C472−1+411yescharacteristic
266C2×Dic47—21no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—21no—
270C2×Dic47—21no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47—21no—
274C2×Dic47—22no—
275C2×Dic47—22no—
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—41no—
286C2×Dic47C2×C22⋊C412yes—
287C2×Dic47C2×C22⋊C412yes—
288C2×Dic47C2×C22⋊C412yes—
289C2×Dic47C2×C22⋊C412yes—
290Dic94—22no—
291Dic94—22no—
292Dic94—41no—
293Dic94—41no—
294C8×C47—41no—
295C8×C47—41no—
296C47⋊C8—41no—
297C47⋊C8—41no—
298#21873—471nomaximal, Sylow
299C22×C47—22no—
300C22×C47—22no—
301C22×C47—22no—
302C22×C47—41no—
303C22×C47—41no—
304C22×C47—41no—
305C22×C47—41no—
306C22×C47C22×C22⋊C411yescharacteristic
307C22×C47#53511yescharacteristic
308C22×C47#53611yescharacteristic
309C22×C47#57711yescharacteristic
310C22×C47#57711yescharacteristic
311C22×C47#57811yescharacteristic
312C22×C47#57811yescharacteristic
313C4×C47—22no—
314C4×C47—42no—
315C4×C47—42no—
316C4×C47—81no—
317C4×C47—81no—
318C4×C47#52011yescharacteristic
319C4×C47#52011yescharacteristic
320C4×C47#53711yescharacteristic
321C4×C47#53711yescharacteristic
322Dic47—22no—
323Dic47—22no—
324Dic47—22no—
325Dic47—22no—
326Dic47—42no—
327Dic47—42no—
328Dic47—81no—
329Dic47—81no—
330C2×#541—471no—
331C2×#433—471no—
332#2317—471no—
333#1992—471no—
334#1991—471no—
335#1960—471no—
336#1956—471no—
337#1873—471no—
338#1873—471no—
339#1872—471no—
340#1872D4711yescharacteristic
341#1805—471no—
342#1805—471no—
343#1803—471no—
344#1803—471no—
345C2×C47—42no—
346C2×C47#231111yescharacteristic
347C2×C47#289811yescharacteristic
348C2×C47#289811yescharacteristic
349C2×C47#322211yescharacteristic
350C2×C47#322311yescharacteristic
351C2×C47#324211yescharacteristic
352C2×C47#324211yescharacteristic
353#541—478no—
354#541—944no—
355C2×C4⋊Q8—471no—
356C2×#126—941no—
357C2×#126—941no—
358C2×#125—471no—
359C2×#125—941no—
360C2×#125—941no—
361C2×C22⋊Q8—471no—
362C2×C22⋊Q8C2×D4711yescharacteristic
363C2×C4×Q8—471no—
364C2×C4×Q8—471no—
365C2×#117—471no—
366C2×#117—471no—
367C22×C4⋊C4C2×D4711yescharacteristic
368#433—944no—
369C2×C4⋊C8—471no—
370C2×C4⋊C8—471no—
371C2×#103—471no—
372C2×#103—471no—
373C2×#103—471no—
374C2×#103—471no—
375C2×#103—471no—
376C2×#103—471no—
377C2×#103C2×D4711yescharacteristic
378C2×#103C2×D4711yescharacteristic
379C2×C22⋊C8—471no—
380C2×C22⋊C8C2×D4711yescharacteristic
381C2×#97—471no—
382C2×C4×C8—471no—
383#389—941no—
384#389—941no—
385#385—941no—
386#385—941no—
387#384—471no—
388#384—471no—
389#382—471no—
390#382—471no—
391#340—471no—
392#340—471no—
393#340—471no—
394#340—471no—
395#340—471no—
396#340—471no—
397#340C2×D4711yescharacteristic
398#340C2×D4711yescharacteristic
399C47#2187311yescharacteristic, complemented, Sylow
400C22×Q8—471no—
401C22×Q8C22×D4711yescharacteristic
402C23×C4—21no—
403C23×C4C22×D4711yescharacteristic
404C22×C8—471no—
405C22×C8—471no—
406C22×C8C22×D4711yescharacteristic
407C22×C8C22×D4711yescharacteristic
408C4⋊Q8—474no—
409C4⋊Q8—942no—
410#126—948no—
411#126—948no—
412#125—474no—
413#125—942no—
414#125—948no—
415#125—948no—
416C22⋊Q8—22no—
417C22⋊Q8—474no—
418C22⋊Q8—942no—
419C22⋊Q8C4×D4714yes—
420C4×Q8—474no—
421C4×Q8—474no—
422C4×Q8—942no—
423C4×Q8—942no—
424#117—474no—
425#117—474no—
426#117—1881no—
427#117—1881no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—41no—
431C2×C4⋊C4—41no—
432C2×C4⋊C4—471no—
433C2×C4⋊C4—471no—
434C2×C4⋊C4—471no—
435C2×C4⋊C4—471no—
436C2×C4⋊C4—471no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C4⋊C4—941no—
442C2×C4⋊C4—941no—
443C2×C4⋊C4—941no—
444C2×C4⋊C4—941no—
445C2×C4⋊C4—941no—
446C2×C4⋊C4—941no—
447C2×C4⋊C4—941no—
448C2×C4⋊C4C22×D4711yescharacteristic
449C2×C4⋊C4C22×D4711yescharacteristic
450C2×C4⋊C4C22×D4711yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—471no—
453C2×C22⋊C4—471no—
454C2×C22⋊C4—941no—
455C2×C22⋊C4—941no—
456C2×C42—471no—
457C2×C42—471no—
458C2×C42—941no—
459C2×C42—941no—
460C2×C42—941no—
461C4⋊C8—942no—
462C4⋊C8—942no—
463#103—41no—
464#103—41no—
465#103—1881no—
466#103—1881no—
467#103—1881no—
468#103—1881no—
469#103—1881no—
470#103—1881no—
471C22⋊C8—22no—
472C22⋊C8—942no—
473#97—942no—
474C4×C8—942no—
475#95—941no—
476#95—941no—
477#95—941no—
478#95—941no—
479C24D4×D4711yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—472no—
486C2×Q8—941no—
487C2×Q8—941no—
488C2×Q8—941no—
489C2×Q8—941no—
490C2×Q8—941no—
491C2×Q8C2×C4×D4712yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—471no—
504C22×C4—471no—
505C22×C4—471no—
506C22×C4—471no—
507C22×C4—941no—
508C22×C4—941no—
509C22×C4—941no—
510C22×C4—941no—
511C22×C4C2×C4×D4712yes—
512C22×C4D4×D4711yescharacteristic
513C22×C4D4×D4712yes—
514C22×C4C23×D4711yescharacteristic
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—941no—
522C2×C8—941no—
523C2×C8—941no—
524C2×C8—941no—
525C2×C8—941no—
526C2×C8—941no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—42no—
534C4⋊C4—474no—
535C4⋊C4—474no—
536C4⋊C4—472no—
537C4⋊C4—944no—
538C4⋊C4—944no—
539C4⋊C4—944no—
540C4⋊C4—944no—
541C4⋊C4—944no—
542C4⋊C4—944no—
543C4⋊C4—944no—
544C4⋊C4—941no—
545C4⋊C4—941no—
546C4⋊C4—941no—
547C4⋊C4—941no—
548C4⋊C4—941no—
549C4⋊C4—3761no—
550C4⋊C4—3761no—
551C4⋊C4—3761no—
552C4⋊C4—3761no—
553C4⋊C4C2×C4×D4712yes—
554C22⋊C4—24no—
555C22⋊C4—474no—
556C22⋊C4—474no—
557C22⋊C4—944no—
558C22⋊C4—1882no—
559C42—472no—
560C42—472no—
561C42—472no—
562C42—472no—
563C42—944no—
564C42—944no—
565C42—1882no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23C22⋊C4×D4712yes—
573C23C2×D4×D4711yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—942no—
579Q8—942no—
580Q8—1881no—
581Q8—1881no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—472no—
602C2×C4—472no—
603C2×C4—472no—
604C2×C4—472no—
605C2×C4—941no—
606C2×C4—941no—
607C2×C4—941no—
608C2×C4—941no—
609C2×C4—941no—
610C2×C4—941no—
611C2×C4—941no—
612C2×C4—941no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—1881no—
616C2×C4—1881no—
617C2×C4—1881no—
618C2×C4—1881no—
619C2×C4—1881no—
620C2×C4—1881no—
621C2×C4—1881no—
622C2×C4—1881no—
623C2×C4—1881no—
624C2×C4—1881no—
625C2×C4C22⋊C4×D4712yes—
626C2×C4C22×C4×D4711yescharacteristic
627C2×C4C2×D4×D4711yescharacteristic
628C2×C4#1309406911yescharacteristic
629C2×C4#1309406911yescharacteristic
630C2×C4#1309407111yescharacteristic
631C2×C4#1309407111yescharacteristic
632C8—41no—
633C8—41no—
634C8—1881no—
635C8—1881no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22C2×C22⋊C4×D4711yescharacteristic
644C22#1568968811yescharacteristic
645C22#1568969511yescharacteristic
646C22#1568992911yescharacteristic
647C22#1568993311yescharacteristic
648C22#1568993611yescharacteristic
649C22#1568993911yescharacteristic
650C4—22no—
651C4—42no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—942no—
656C4—942no—
657C4—942no—
658C4—942no—
659C4—3761no—
660C4—3761no—
661C4—1882no—
662C4—1882no—
663C4#1568960211yescharacteristic
664C4#1568960211yescharacteristic
665C4#1568970011yescharacteristic
666C4#1568970011yescharacteristic
667C2—42no—
668C2#2024597511yescharacteristic
669C2#2025191611yescharacteristic
670C2#2025191911yescharacteristic
671C2#2025445611yescharacteristic
672C2#2025446711yescharacteristic
673C2#2025465211yescharacteristic
674C2#2025466011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1872C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1872D47
hypocenterC47#21873
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #20244355C2semidirect
5. #20244351C2semidirect
6. #20244206C2semidirect
7. #20244188C2semidirect
11. #20243402C2semidirect
12. #20243401C2semidirect
13. #20243392C2semidirect
14. #20243391C2semidirect
17. #15689733C4semidirect
66. #13094022C2×C4semidirect
71. #13094015C2×C4semidirect
76. Q8×Dic47C2×C4semidirect
78. #13093989C2×C4semidirect
399. C47#21873semidirect

Maximal subgroups

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

subgroupindex
C2×#156897332
C2×#156891812
#202460122
#202443552
#202443512
#202442062
#202441882
#202437012
#202436982
#202436952
#202434022
#202434012
#202433922
#202433912
C47×#18722
#2187347