Finite Group Databasebeta

← #26276204

Subgroups — #26276204

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689464C211yesmaximal, characteristic, complemented
2C2×#15689166C211yesmaximal, characteristic, complemented
3#20246839C211yesmaximal, characteristic
4#20244932C211yesmaximal, characteristic, complemented
5#20244865C211yesmaximal, characteristic, complemented
6#20244825C211yesmaximal, characteristic, complemented
7#20244584C211yesmaximal, characteristic, complemented
8#20244029C211yesmaximal, characteristic, complemented
9#20243980C211yesmaximal, characteristic, complemented
10#20243819C211yesmaximal, characteristic, complemented
11#20243700C211yesmaximal, characteristic, complemented
12#20243686C211yesmaximal, characteristic
13#20243587C211yesmaximal, characteristic, complemented
14#20243580C211yesmaximal, characteristic, complemented
15C47×#2354C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×#13094003C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20#15689464—24no—
21#15689166—24no—
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093955C2211yescharacteristic
27C2×#13093955C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×C22⋊(C47⋊C8)C2211yescharacteristic
33C2×C22⋊(C47⋊C8)C2211yescharacteristic
34C2×C22⋊(C47⋊C8)C2211yescharacteristic
35#15689056—21no—
36#15689046—21no—
37#15689029C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689003—21no—
40#15688983C2211yescharacteristic
41#15688981—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C4×D4×C47C2211yescharacteristic
51C2×#117×C47C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C47×#390—21no—
54C47×#388—21no—
55C47×#388—21no—
56C47×#384C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C47×#380C2211yescharacteristic
59C4×C4⋊C4×C47—21no—
60C4×C4⋊C4×C47C2211yescharacteristic
61C22×Dic94C2311yescharacteristic
62C23×Dic47—21no—
63C2×#10625329—22no—
64C2×#10625329—22no—
65C2×#10625329—22no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70#13094003—22no—
71#13094003—22no—
72#13094003—22no—
73#13094003—41no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47—21no—
76C2×C4⋊Dic47—41no—
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×C22⋊Dic47—21no—
84C2×C22⋊Dic47—21no—
85C2×C22⋊Dic47—21no—
86#13093957—22no—
87#13093957D414yescomplemented
88#13093955—22no—
89#13093955—22no—
90#13093955—22no—
91#13093955—41no—
92#13093947—22no—
93#13093947—22no—
94#13093947—22no—
95#13093947—41no—
96C22⋊(C47⋊C8)—22no—
97C22⋊(C47⋊C8)—22no—
98C22⋊(C47⋊C8)—22no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C22×D4×C47D411yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C23×C4×C47C2311yescharacteristic
106C4×D4×C47—22no—
107C4×D4×C47—22no—
108#117×C47—22no—
109#117×C47—22no—
110#117×C47—41no—
111#117×C47—41no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47D411yescharacteristic
124C2×C22⋊C4×C47D411yescharacteristic
125C2×C42×C47—21no—
126C2×C42×C47—21no—
127C2×C42×C47—21no—
128C2×C42×C47C2311yescharacteristic
129#95×C47—21no—
130#95×C47—21no—
131#95×C47—21no—
132C24×C47C2×D411yescharacteristic
133C24×C47C2×D411yescharacteristic
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47—41no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47#4112yes—
156C22×C4×C47#4112yes—
157C22×C4×C47C2411yescharacteristic
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C22×Dic47—42no—
163C22×Dic47—42no—
164C22×Dic47—81no—
165C2×Dic94—21no—
166C2×Dic94—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—42no—
180C4⋊C4×C47—81no—
181C4⋊C4×C47C2×D412yes—
182C22⋊C4×C47—22no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47—42no—
185C22⋊C4×C47—42no—
186C22⋊C4×C47—81no—
187C22⋊C4×C47—81no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—42no—
191C42×C47—42no—
192C42×C47—42no—
193C22⋊Dic47—42no—
194C22⋊Dic47—42no—
195C22⋊Dic47—81no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—42no—
202C4⋊Dic47C2×D412yes—
203Dic47⋊C4—42no—
204Dic47⋊C4—42no—
205Dic47⋊C4—81no—
206#10625329—28no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8C2×D412yes—
218C2×C47⋊C8#4112yes—
219C23×C47—21no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—21no—
226C23×C47—21no—
227C23×C47—21no—
228C23×C47—21no—
229C23×C47—21no—
230C23×C47—21no—
231C23×C47—21no—
232C23×C47#12312yes—
233C23×C47C22×D411yescharacteristic
234D4×C47—44no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—21no—
246C2×C4×C47—22no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
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—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—41no—
263C2×C4×C47—41no—
264C2×C4×C47—41no—
265C2×C4×C47#12312yes—
266C2×C4×C47#13712yes—
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47H(5,𝔽2)11yescharacteristic
272C2×C4×C472−1+411yescharacteristic
273C2×Dic47—22no—
274C2×Dic47—41no—
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—
286Dic94—41no—
287Dic94—41no—
288Dic94—41no—
289Dic94—41no—
290C47⋊C8—22no—
291C47⋊C8—22no—
292C47⋊C8—41no—
293C47⋊C8—41no—
294#26155—471nomaximal, Sylow
295C22×C47—22no—
296C22×C47—22no—
297C22×C47—22no—
298C22×C47—41no—
299C22×C47—41no—
300C22×C47—41no—
301C22×C47—41no—
302C22×C47—41no—
303C22×C47—41no—
304C22×C47—41no—
305C22×C47—41no—
306C22×C47—41no—
307C22×C47—41no—
308C22×C47—41no—
309C22×C47C2×#12311yescharacteristic
310C22×C47D4211yescharacteristic
311C22×C47#54711yescharacteristic
312C22×C47C2×#13711yescharacteristic
313C22×C47#57511yescharacteristic
314C22×C47#57711yescharacteristic
315C22×C47#57711yescharacteristic
316C4×C47—22no—
317C4×C47—22no—
318C4×C47—41no—
319C4×C47—41no—
320C4×C47—42no—
321C4×C47—81no—
322C4×C47—81no—
323C4×C47—81no—
324C4×C47#52511yescharacteristic
325C4×C47#53911yescharacteristic
326C4×C47#54811yescharacteristic
327C4×C47#55311yescharacteristic
328Dic47—42no—
329Dic47—81no—
330Dic47—81no—
331Dic47—81no—
332C2×#485—471no—
333C2×#428—471no—
334#2417—471no—
335#2354D4711yescharacteristic
336#2096—471no—
337#2084—471no—
338#2077—471no—
339#2035—471no—
340#1928—471no—
341#1921—471no—
342#1894—471no—
343#1873—471no—
344#1870—471no—
345#1854—471no—
346#1853—471no—
347C2×C47—42no—
348C2×C47—81no—
349C2×C47#244311yescharacteristic
350C2×C47#309511yescharacteristic
351C2×C47#309911yescharacteristic
352C2×C47#326711yescharacteristic
353C2×C47#327811yescharacteristic
354C2×C47#334811yescharacteristic
355C2×C47#335611yescharacteristic
356C22×M16—471no—
357C2×C22⋊Q8—471no—
358C2×C22⋊Q8—471no—
359C2×C4×D4C2×D4711yescharacteristic
360C2×#117C2×D4711yescharacteristic
361C2×#117C2×D4711yescharacteristic
362C22×C4⋊C4—471no—
363#485—944no—
364#428—944no—
365C2×#107—471no—
366C2×#107—471no—
367C2×#106—471no—
368C2×#106—471no—
369C2×#106—471no—
370C2×#106—471no—
371C2×#103—471no—
372C2×#103—471no—
373C2×#103—471no—
374C2×#103—471no—
375C2×C22⋊C8—471no—
376C2×C22⋊C8—471no—
377C2×C22⋊C8—471no—
378#398—941no—
379#396—941no—
380#391—471no—
381#390—21no—
382#389—471no—
383#388—21no—
384#388—21no—
385#386—941no—
386#384C2×D4711yescharacteristic
387#380—471no—
388#380—941no—
389#380C2×D4711yescharacteristic
390#380C2×D4711yescharacteristic
391C4×C4⋊C4—21no—
392C4×C4⋊C4C2×D4711yescharacteristic
393#340—471no—
394#340—471no—
395#340—471no—
396#340—471no—
397#340—471no—
398#340—471no—
399#336—471no—
400#336—471no—
401C47#2615511yescharacteristic, complemented, Sylow
402C22×Q8—471no—
403C22×D4C22⋊D4711yescharacteristic
404C23×C4—941no—
405C23×C4C22×D4711yescharacteristic
406C23×C4C22×D4711yescharacteristic
407C2×M16—942no—
408C2×M16—942no—
409C2×M16—942no—
410C22×C8—471no—
411C22×C8—471no—
412C22×C8—471no—
413C22×C8—471no—
414C22⋊Q8—942no—
415C22⋊Q8—942no—
416C22⋊Q8—942no—
417C22⋊Q8—1881no—
418C4×D4—22no—
419C4×D4—22no—
420#117—22no—
421#117—22no—
422#117—41no—
423#117—41no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—471no—
429C2×C4⋊C4—942no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4—941no—
432C2×C4⋊C4—941no—
433C2×C4⋊C4—941no—
434C2×C4⋊C4—1881no—
435C2×C4⋊C4C22⋊D4711yescharacteristic
436C2×C4⋊C4C22×D4711yescharacteristic
437C2×C4⋊C4C22×D4711yescharacteristic
438C2×C4⋊C4C22×D4711yescharacteristic
439C2×C4⋊C4C22×D4711yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—941no—
445C2×C22⋊C4—941no—
446C2×C22⋊C4—941no—
447C2×C22⋊C4C22⋊D4711yescharacteristic
448C2×C22⋊C4C22⋊D4711yescharacteristic
449C2×C42—21no—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42C22×D4711yescharacteristic
453#107—474no—
454#107—942no—
455#106—942no—
456#106—942no—
457#106—942no—
458#106—1881no—
459#103—942no—
460#103—942no—
461#103—942no—
462#103—1881no—
463C22⋊C8—942no—
464C22⋊C8—942no—
465C22⋊C8—942no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—941no—
470#95—941no—
471#95—941no—
472#95—941no—
473C24C2×C22⋊D4711yescharacteristic
474C24C2×C22⋊D4711yescharacteristic
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×Q8—941no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—3761no—
503C22×C4—1882no—
504C22×C4—1882no—
505C22×C4#1062535012yes—
506C22×C4#1062535012yes—
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C2×C22⋊D4711yescharacteristic
510C22×C4C2×C22⋊D4711yescharacteristic
511C22×C4C23×D4711yescharacteristic
512M16—948no—
513C2×C8—472no—
514C2×C8—472no—
515C2×C8—941no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8—941no—
522C2×C8—941no—
523C2×C8—941no—
524C2×C8—941no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—472no—
536C4⋊C4—941no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4—941no—
540C4⋊C4—941no—
541C4⋊C4—3761no—
542C4⋊C4—1882no—
543C4⋊C4—1882no—
544C4⋊C4—1882no—
545C4⋊C4D4×D4712yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—3761no—
553C22⋊C4—1882no—
554C22⋊C4—1882no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—42no—
559C42—42no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1309400712yes—
574C23C22×C22⋊D4711yescharacteristic
575Q8—1881no—
576Q8—1881no—
577Q8—1881no—
578Q8—1881no—
579D4—44no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—942no—
611C2×C4—3764no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1881no—
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#1309400712yes—
624C2×C4#1309405412yes—
625C2×C4C2×D4×D4711yescharacteristic
626C2×C4#41×D4711yescharacteristic
627C2×C4C2×#1062535011yescharacteristic
628C2×C4C2×#1062535011yescharacteristic
629C2×C4#1309406911yescharacteristic
630C2×C4#1309407111yescharacteristic
631C8—942no—
632C8—942no—
633C8—1881no—
634C8—1881no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1309400711yescharacteristic
650C22D4×C22⋊D4711yescharacteristic
651C22#1568977211yescharacteristic
652C22C2×#1309405411yescharacteristic
653C22#1568992111yescharacteristic
654C22#1568993111yescharacteristic
655C22#1568993411yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—3761no—
665C4—3761no—
666C4—3761no—
667C4—1882no—
668C4#1568963211yescharacteristic
669C4#1568971511yescharacteristic
670C4#1568978211yescharacteristic
671C4#1568982211yescharacteristic
672C2—42no—
673C2—81no—
674C2#2024711711yescharacteristic
675C2#2025333611yescharacteristic
676C2#2025337711yescharacteristic
677C2#2025491411yescharacteristic
678C2#2025505411yescharacteristic
679C2#2025590311yescharacteristic
680C2#2025599711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2354C2
socleC23×C47C22×D4
prime radicalC22×D4×C47D4
hypercenter#2354D47
hypocenterC47#26155
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. C2×#15689464C2semidirect
2. C2×#15689166C2semidirect
4. #20244932C2semidirect
5. #20244865C2semidirect
6. #20244825C2semidirect
7. #20244584C2semidirect
8. #20244029C2semidirect
9. #20243980C2semidirect
10. #20243819C2semidirect
11. #20243700C2semidirect
13. #20243587C2semidirect
14. #20243580C2semidirect
87. #13093957D4semidirect
401. C47#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#156894642
C2×#156891662
#202468392
#202449322
#202448652
#202448252
#202445842
#202440292
#202439802
#202438192
#202437002
#202436862
#202435872
#202435802
C47×#23542
#2615547