Finite Group Databasebeta

← #26232953

Subgroups — #26232953

Order 12032.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689832C211yesmaximal, characteristic
2C2×#15689250C211yesmaximal, characteristic, complemented
3#20246340C211yesmaximal, characteristic, complemented
4#20244359C211yesmaximal, characteristic, complemented
5#20244355C211yesmaximal, characteristic, complemented
6#20244352C211yesmaximal, characteristic, complemented
7#20244344C211yesmaximal, characteristic, complemented
8#20244343C211yesmaximal, characteristic, complemented
9#20244338C211yesmaximal, characteristic, complemented
10#20243396C211yesmaximal, characteristic
11#20243386C211yesmaximal, characteristic
12#20243215C211yesmaximal, characteristic
13#20243215C211yesmaximal, characteristic
14C4×#13093945C211yesmaximal, characteristic
15C47×#1804C211yesmaximal, characteristic
16#15689832—24no—
17#15689832C418yescomplemented
18C2×#13094023C2211yescharacteristic
19C2×#13094016C2211yescharacteristic
20C2×#13094015C2211yescharacteristic
21C2×#13094003—21no—
22C2×#13094001—21no—
23C2×#13094001—21no—
24C2×C4×Dic94C2211yescharacteristic
25C2×D4×Dic47C2211yescharacteristic
26C2×D4×Dic47C2211yescharacteristic
27C22×Dic47⋊C4—21no—
28#15689250—24no—
29C2×Dic47⋊C8C2211yescharacteristic
30C2×Dic47⋊C8C2211yescharacteristic
31C2×#13093945C2211yescharacteristic
32C2×#13093945C2211yescharacteristic
33C2×#13093934C2211yescharacteristic
34C2×#13093934C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36#15689027—21no—
37#15689008—21no—
38#15689005—21no—
39#15688992C2211yescharacteristic
40#15688988C2211yescharacteristic
41C4⋊C4×Dic47C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4×C4⋊Dic47C2211yescharacteristic
44#15688974—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53#15688885C2211yescharacteristic
54#15688885C2211yescharacteristic
55C2×C4×D4×C47C2211yescharacteristic
56C2×C4⋊C8×C47C2211yescharacteristic
57C2×#102×C47C2211yescharacteristic
58C2×#102×C47C2211yescharacteristic
59C47×#382C2211yescharacteristic
60C47×#340C2211yescharacteristic
61C47×#340C2211yescharacteristic
62C22×Dic94—21no—
63C22×Dic94D411yescharacteristic
64C23×Dic47—21no—
65C23×Dic47—21no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094023—22no—
69#13094023C2×C414yes—
70#13094016—22no—
71#13094016C2×C414yes—
72#13094015—22no—
73#13094015C2×C414yes—
74#13094003—28no—
75#13094001—28no—
76#13094001—28no—
77C4×Dic94C2×C414yes—
78C4×Dic94D414yes—
79D4×Dic47—22no—
80D4×Dic47—22no—
81D4×Dic47C2×C414yes—
82D4×Dic47C2×C414yes—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47D411yescharacteristic
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2311yescharacteristic
87C2×C4⋊Dic47C2311yescharacteristic
88C2×C4⋊Dic47C2311yescharacteristic
89C2×Dic47⋊C4—21no—
90C2×Dic47⋊C4—21no—
91C2×Dic47⋊C4—21no—
92C2×Dic47⋊C4—21no—
93C2×Dic47⋊C4—28no—
94C2×Dic47⋊C4—21no—
95C2×Dic47⋊C4—21no—
96C2×Dic47⋊C4D411yescharacteristic
97C2×Dic47⋊C4D411yescharacteristic
98C2×C22⋊Dic47—21no—
99C2×C22⋊Dic47—21no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—21no—
102C2×C4×Dic47—21no—
103C2×C4×Dic47C2311yescharacteristic
104C2×C4×Dic47C2311yescharacteristic
105Dic47⋊C8—22no—
106Dic47⋊C8—22no—
107#13093945—22no—
108#13093945Q814yescomplemented
109#13093934—22no—
110#13093934—22no—
111C4×C47⋊C8—22no—
112C47⋊#95—21no—
113C47⋊#95—21no—
114C47⋊#95—21no—
115C47⋊#95—21no—
116C47⋊#95—21no—
117C22×D4×C47C2311yescharacteristic
118C23×C4×C47—21no—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C4×D4×C47—22no—
122C4×D4×C47C2×C414yes—
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C4⋊C4×C47C2311yescharacteristic
125C2×C4⋊C4×C47C2311yescharacteristic
126C2×C22⋊C4×C47—21no—
127C2×C42×C47C2311yescharacteristic
128C4⋊C8×C47—22no—
129#102×C47—22no—
130#102×C47—22no—
131#95×C47—21no—
132#95×C47—21no—
133C24×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47C22×C412yes—
138C2×D4×C47C2×Q812yes—
139C2×D4×C47#4112yes—
140C22×C4×C47—21no—
141C22×C4×C47—24no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—81no—
145C22×C4×C47—81no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—24no—
153C22×Dic47—24no—
154C22×Dic47—21no—
155C22×Dic47—42no—
156C22×Dic47—42no—
157C22×Dic47—81no—
158C22×Dic47—81no—
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162C22×Dic47C2×D411yescharacteristic
163C2×Dic94—24no—
164C2×Dic94—41no—
165C2×Dic94—41no—
166C2×Dic94—81no—
167C2×Dic94—81no—
168C2×Dic94C22⋊C414yes—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47C22×C412yes—
181C22⋊C4×C47—24no—
182C42×C47C22×C412yes—
183C42×C47C2×D412yes—
184C22⋊Dic47—24no—
185C22⋊Dic47—24no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—24no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47C22⋊C414yes—
193C4⋊Dic47C22×C412yes—
194C4⋊Dic47C22×C412yes—
195C4⋊Dic47C2×Q812yes—
196Dic47⋊C4—24no—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199Dic47⋊C4—24no—
200Dic47⋊C4—42no—
201Dic47⋊C4—81no—
202Dic47⋊C4C22⋊C414yes—
203Dic47⋊C4C22⋊C414yes—
204C4×Dic47—42no—
205C4×Dic47—81no—
206C4×Dic47—81no—
207C4×Dic47C22×C412yes—
208C4×Dic47C22×C412yes—
209C4×Dic47C2×D412yes—
210C4×Dic47C2×D412yes—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8C2×Q812yes—
217C23×C47—22no—
218C23×C47—22no—
219C23×C47—22no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47C22×D411yescharacteristic
225D4×C47—22no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47C4×Q814yes—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—22no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—84no—
243C2×C4×C47C2×C22⋊C412yes—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×Q811yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C472−1+411yescharacteristic
250C2×C4×C472−1+411yescharacteristic
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—48no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—84no—
273C2×Dic47C2×C22⋊C412yes—
274C2×Dic47C2×C22⋊C412yes—
275C2×Dic47C2×C22⋊C412yes—
276C2×Dic47C2×C22⋊C412yes—
277Dic94—44no—
278Dic94—84no—
279C8×C47—41no—
280C8×C47—41no—
281C47⋊C8—22no—
282C47⋊C8—41no—
283#22526—471nomaximal, Sylow
284C22×C47—24no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—42no—
291C22×C47C22×C22⋊C411yescharacteristic
292C22×C47#54811yescharacteristic
293C22×C47D4×Q811yescharacteristic
294C22×C47#57511yescharacteristic
295C22×C47#57511yescharacteristic
296C22×C47#57611yescharacteristic
297C22×C47#57711yescharacteristic
298C4×C47—22no—
299C4×C47—22no—
300C4×C47—42no—
301C4×C47—81no—
302C4×C47—81no—
303C4×C47C2×C4×Q811yescharacteristic
304C4×C47#51911yescharacteristic
305C4×C47#54811yescharacteristic
306C4×C47D4×Q811yescharacteristic
307Dic47—22no—
308Dic47—22no—
309Dic47—22no—
310Dic47—22no—
311Dic47—42no—
312Dic47—42no—
313Dic47—42no—
314Dic47—81no—
315C2×#554—471no—
316C2×#446—471no—
317#2355—471no—
318#1993—471no—
319#1992—471no—
320#1991—471no—
321#1989—471no—
322#1989—471no—
323#1988—471no—
324#1804—471no—
325#1804D4711yescharacteristic
326#1802—471no—
327#1762—471no—
328#1762—471no—
329C4×#102—471no—
330C2×C47—44no—
331C2×C47Q8×C22⋊C411yescharacteristic
332C2×C47#289611yescharacteristic
333C2×C47#289711yescharacteristic
334C2×C47#337011yescharacteristic
335C2×C47#337611yescharacteristic
336C2×C47#338811yescharacteristic
337C2×C47#338911yescharacteristic
338#554—478no—
339#554—944no—
340C2×C4⋊Q8—471no—
341C2×#125—471no—
342C2×#125—471no—
343C2×C22⋊Q8—941no—
344C2×C22⋊Q8—941no—
345C2×C22⋊Q8—941no—
346C2×C4×Q8—471no—
347C2×C4×D4—471no—
348C2×C4×D4—471no—
349C2×C4×D4C2×D4711yescharacteristic
350C22×C4⋊C4—941no—
351#446—944no—
352C2×C4⋊C8—471no—
353C2×C4⋊C8—471no—
354C2×C4⋊C8C2×D4711yescharacteristic
355C2×#102—471no—
356C2×#102—471no—
357C2×#102C2×D4711yescharacteristic
358C2×#102C2×D4711yescharacteristic
359C2×#97—471no—
360C2×#97—471no—
361C2×C4×C8—471no—
362#391—941no—
363#387—941no—
364#387—941no—
365#384—471no—
366#382—471no—
367#382C2×D4711yescharacteristic
368C4×C4⋊C4—471no—
369C4×C4⋊C4—471no—
370C4×C4⋊C4—471no—
371C4×C4⋊C4—941no—
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340C2×D4711yescharacteristic
383#340C2×D4711yescharacteristic
384C47#2252611yescharacteristic, complemented, Sylow
385C22×Q8—471no—
386C22×Q8—941no—
387C22×D4C22×D4711yescharacteristic
388C23×C4—21no—
389C23×C4—941no—
390C23×C4—941no—
391C22×C8—471no—
392C22×C8—471no—
393C22×C8C22×D4711yescharacteristic
394C22×C8C22×D4711yescharacteristic
395C4⋊Q8—474no—
396C4⋊Q8—942no—
397#125—474no—
398#125—474no—
399#125—942no—
400#125—942no—
401C22⋊Q8—948no—
402C22⋊Q8—948no—
403C22⋊Q8—948no—
404C4×Q8—474no—
405C4×Q8—474no—
406C4×D4—22no—
407C4×D4—474no—
408C4×D4—474no—
409C4×D4—942no—
410C4×D4—942no—
411C4×D4C4×D4714yes—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—471no—
414C2×C4⋊C4—471no—
415C2×C4⋊C4—471no—
416C2×C4⋊C4—471no—
417C2×C4⋊C4—471no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—948no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4—941no—
424C2×C4⋊C4—941no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4C22×D4711yescharacteristic
428C2×C4⋊C4C22×D4711yescharacteristic
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—941no—
432C2×C22⋊C4—941no—
433C2×C42—471no—
434C2×C42—471no—
435C2×C42—941no—
436C2×C42—941no—
437C2×C42—941no—
438C2×C42C22×D4711yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—942no—
441C4⋊C8—942no—
442#102—22no—
443#102—22no—
444#102—474no—
445#102—942no—
446#97—942no—
447#97—942no—
448C4×C8—942no—
449#95—21no—
450#95—21no—
451#95—941no—
452#95—941no—
453#95—941no—
454#95—941no—
455#95—941no—
456C24—21no—
457C2×Q8—474no—
458C2×Q8—944no—
459C2×Q8—3761no—
460C2×Q8—3761no—
461C2×Q8—1881no—
462C2×Q8—1881no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×C4×D4712yes—
467C2×D4#1062535212yes—
468C2×D4Q8×D4712yes—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—944no—
480C22×C4—944no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—3761no—
486C22×C4—3761no—
487C22×C4—1882no—
488C22×C4—1882no—
489C22×C4D4×D4711yescharacteristic
490C22×C4D4×D4711yescharacteristic
491C22×C4C23×D4711yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—472no—
499C2×C8—941no—
500C2×C8—941no—
501C2×C8—941no—
502C2×C8—941no—
503C2×C8—941no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—474no—
510C4⋊C4—474no—
511C4⋊C4—474no—
512C4⋊C4—472no—
513C4⋊C4—472no—
514C4⋊C4—472no—
515C4⋊C4—944no—
516C4⋊C4—944no—
517C4⋊C4—944no—
518C4⋊C4—944no—
519C4⋊C4—944no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—941no—
523C4⋊C4—941no—
524C4⋊C4—941no—
525C4⋊C4—3761no—
526C4⋊C4—1882no—
527C4⋊C4C2×C4×D4712yes—
528C22⋊C4—24no—
529C22⋊C4—944no—
530C22⋊C4—944no—
531C42—472no—
532C42—472no—
533C42—472no—
534C42—472no—
535C42—3761no—
536C42—3761no—
537C42—1882no—
538C42C2×C4×D4712yes—
539C42D4×D4712yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×D4×D4711yescharacteristic
548Q8—3764no—
549Q8—1884no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4#1309398914yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—472no—
569C2×C4—472no—
570C2×C4—472no—
571C2×C4—472no—
572C2×C4—941no—
573C2×C4—941no—
574C2×C4—941no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—942no—
583C2×C4—3764no—
584C2×C4—1888no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4C22⋊C4×D4712yes—
595C2×C4C22⋊C4×D4712yes—
596C2×C4C22×C4×D4711yescharacteristic
597C2×C4C2×D4×D4711yescharacteristic
598C2×C4C2×Q8×D4711yescharacteristic
599C2×C4C2×#1062535211yescharacteristic
600C2×C4#1309407111yescharacteristic
601C2×C4#1309407111yescharacteristic
602C8—41no—
603C8—41no—
604C8—942no—
605C8—1881no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×C22⋊C4×D4711yescharacteristic
614C22#1568977911yescharacteristic
615C22#1568978611yescharacteristic
616C22#1568992011yescharacteristic
617C22#1568992011yescharacteristic
618C22#1568992711yescharacteristic
619C22#1568993311yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—942no—
626C4—942no—
627C4—942no—
628C4—942no—
629C4—3761no—
630C4—1882no—
631C4—1882no—
632C4—1882no—
633C4C2×#1309398911yescharacteristic
634C4#1568960011yescharacteristic
635C4#1568977911yescharacteristic
636C4#1568978611yescharacteristic
637C2—44no—
638C2#2024622011yescharacteristic
639C2#2025190311yescharacteristic
640C2#2025191311yescharacteristic
641C2#2025614511yescharacteristic
642C2#2025621911yescharacteristic
643C2#2025633711yescharacteristic
644C2#2025634611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1804C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1804D47
hypocenterC47#22526
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#15689250C2semidirect
3. #20246340C2semidirect
4. #20244359C2semidirect
5. #20244355C2semidirect
6. #20244352C2semidirect
7. #20244344C2semidirect
8. #20244343C2semidirect
9. #20244338C2semidirect
17. #15689832C4semidirect
108. #13093945Q8semidirect
384. C47#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#156898322
C2×#156892502
#202463402
#202443592
#202443552
#202443522
#202443442
#202443432
#202443382
#202433962
#202433862
#202432152
C4×#130939452
C47×#18042
#2252647