Finite Group Databasebeta

← #26223537

Subgroups — #26223537

Order 12032.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689442C211yesmaximal, characteristic
2#20246081C211yesmaximal, characteristic
3#20245744C211yesmaximal, characteristic, complemented
4#20244016C211yesmaximal, characteristic, complemented
5#20244009C211yesmaximal, characteristic, complemented
6#20243974C211yesmaximal, characteristic, complemented
7#20243967C211yesmaximal, characteristic, complemented
8#20243708C211yesmaximal, characteristic
9#20243589C211yesmaximal, characteristic, complemented
10#20243475C211yesmaximal, characteristic, complemented
11#20243285C211yesmaximal, characteristic, complemented
12C4×#13093956C211yesmaximal, characteristic, complemented
13#20243214C211yesmaximal, characteristic, complemented
14C4×#13093946C211yesmaximal, characteristic, complemented
15C47×#1750C211yesmaximal, characteristic, complemented
16C2×#13093998C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C22×C4⋊Dic47C2211yescharacteristic
20#15689442C418yescomplemented
21C2×#13093958C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093956C2211yescharacteristic
25C2×#13093946C2211yescharacteristic
26C2×#13093946C2211yescharacteristic
27C2×#13093946C2211yescharacteristic
28C2×#13093946C2211yescharacteristic
29#15689025C2211yescharacteristic
30#15689023—21no—
31#15689021C2211yescharacteristic
32#15689013—21no—
33#15688999—21no—
34#15688994—21no—
35#15688983C2211yescharacteristic
36C4⋊C4×Dic47—21no—
37C4⋊C4×Dic47—21no—
38C4×C4⋊Dic47C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C4⋊Dic47C2211yescharacteristic
42C4×C4⋊Dic47C2211yescharacteristic
43C22⋊C4×Dic47—21no—
44C22⋊C4×Dic47—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53C2×#117×C47C2211yescharacteristic
54C2×C22⋊C8×C47C2211yescharacteristic
55C2×C22⋊C8×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C2×C4×C8×C47C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C22⋊D47—21no—
62C23×Dic47—21no—
63C22×C4×D47—21no—
64#13093998—22no—
65#13093998C2×C414yes—
66C4×D188—22no—
67C4×D188—22no—
68#13093983—22no—
69#13093983—22no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47C2×C414yescomplemented
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×Dic47⋊C4—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93#13093958—22no—
94#13093956—22no—
95#13093956C2×C414yescomplemented
96#13093956C2×C414yescomplemented
97#13093946—22no—
98#13093946—22no—
99#13093946C2×C414yes—
100#13093946C2×C414yes—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113#117×C47—22no—
114#117×C47—22no—
115C2×C4⋊C4×C47D411yescharacteristic
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C22⋊C4×C47D411yescharacteristic
118C2×C22⋊C4×C47D411yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121C22⋊C8×C47—22no—
122C22⋊C8×C47C2×C414yescomplemented
123#97×C47—22no—
124C4×C8×C47—22no—
125C24×C47C2×D411yescharacteristic
126C23×D47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47#4112yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—24no—
140C2×C22⋊D47—81no—
141C2×C22⋊D47—81no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—24no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—81no—
151C22×Dic47—81no—
152C2×D188—21no—
153C2×D188—21no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188—21no—
157C2×D188C22×C412yes—
158C2×C4×D47—42no—
159C2×C4×D47—81no—
160C2×C4×D47—81no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47C22×C412yes—
172C2×C8×C47C22×C412yes—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—22no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47#4112yes—
180C42×C47#4112yes—
181C22⋊Dic47—24no—
182C22⋊Dic47—81no—
183(C2×D47)⋊C4—24no—
184(C2×D47)⋊C4—42no—
185(C2×D47)⋊C4—81no—
186(C2×D47)⋊C4—81no—
187C4⋊Dic47—24no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47C22×C412yes—
198C4⋊Dic47C22×C412yes—
199C4⋊Dic47C22×C412yes—
200C4⋊Dic47#4112yes—
201C4⋊Dic47#4112yes—
202Dic47⋊C4—81no—
203C4×Dic47—24no—
204C4×Dic47—81no—
205C4×Dic47—81no—
206C4×Dic47—81no—
207C4×Dic47—81no—
208C4×Dic47—81no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47C4×D412yes—
216C23×C47C22×D411yescharacteristic
217C22×D47—22no—
218C22×D47—41no—
219C22×D47—41no—
220C22×D47—41no—
221C22×D47—41no—
222C22×D47—41no—
223C22×D47—41no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47C4×D412yes—
241C2×C4×C47#12312yes—
242C2×C4×C47#12312yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C22⋊D47—84no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
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—84no—
272D188—22no—
273D188—22no—
274D188—41no—
275D188—41no—
276C4×D47—84no—
277C8×C47—22no—
278C8×C47—22no—
279C8×C47—41no—
280C8×C47—41no—
281#21584—471nomaximal, Sylow
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47C2×C4×D411yescharacteristic
290C22×C47C2×#12311yescharacteristic
291C22×C47#52511yescharacteristic
292C22×C47#57611yescharacteristic
293C22×C47#57711yescharacteristic
294C22×C47#57711yescharacteristic
295C22×C47#57711yescharacteristic
296C2×D47—42no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—81no—
300C2×D47—81no—
301C2×D47—81no—
302C2×D47—81no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—22no—
306C4×C47—41no—
307C4×C47—41no—
308C4×C47C4×#4111yescharacteristic
309C4×C47C4×#4111yescharacteristic
310C4×C47#52911yescharacteristic
311C4×C47#52911yescharacteristic
312Dic47—42no—
313Dic47—42no—
314Dic47—42no—
315Dic47—42no—
316Dic47—42no—
317Dic47—81no—
318Dic47—81no—
319C2×#481—471no—
320#2324—471no—
321#2274—471no—
322#1926—471no—
323#1925—471no—
324#1920—471no—
325#1919—471no—
326#1875—471no—
327#1854—471no—
328#1823—471no—
329#1776—471no—
330C4×#106—471no—
331#1762—471no—
332C4×#102—471no—
333#1750D4711yescharacteristic, complemented
334C2×C47—42no—
335C2×C47C4×#12311yescharacteristic
336C2×C47#294611yescharacteristic
337C2×C47#294911yescharacteristic
338C2×C47#309611yescharacteristic
339C2×C47#309911yescharacteristic
340C2×C47#311911yescharacteristic
341C2×C47#312511yescharacteristic
342D47—82no—
343C2×#121—471no—
344C2×C4×D4—471no—
345C2×#117—471no—
346C2×#117C2×D4711yescharacteristic
347C22×C4⋊C4—471no—
348#481—478no—
349C2×#107—471no—
350C2×#106—471no—
351C2×#106—471no—
352C2×#106—471no—
353C2×#102—471no—
354C2×#102—471no—
355C2×#102—471no—
356C2×#102—471no—
357C2×C22⋊C8C2×D4711yescharacteristic
358C2×C22⋊C8C2×D4711yescharacteristic
359C2×#97C2×D4711yescharacteristic
360C2×C4×C8C2×D4711yescharacteristic
361#390—471no—
362#390—941no—
363#389—471no—
364#388—941no—
365#386—941no—
366#385—941no—
367#380—471no—
368C4×C4⋊C4—471no—
369C4×C4⋊C4—471no—
370C4×C4⋊C4—471no—
371C4×C4⋊C4—471no—
372C4×C4⋊C4—471no—
373C4×C4⋊C4—941no—
374C4×C4⋊C4—941no—
375C4×C22⋊C4—941no—
376C4×C22⋊C4—941no—
377#340—471no—
378#340—471no—
379#340—471no—
380#340—471no—
381#340—471no—
382#340—471no—
383#340—471no—
384#340—471no—
385#336C2×D4711yescharacteristic
386#336C2×D4711yescharacteristic
387C47#2158411yescharacteristic, complemented, Sylow
388C22×D4—471no—
389C22×D4—941no—
390C23×C4—941no—
391C23×C4—941no—
392C23×C4C22×D4711yescharacteristic
393C22×C8C22×D4711yescharacteristic
394C22×C8C22×D4711yescharacteristic
395C22×C8C22×D4711yescharacteristic
396C22×C8C22×D4711yescharacteristic
397#121—474no—
398#121—942no—
399C4×D4—942no—
400C4×D4—942no—
401#117—22no—
402#117—22no—
403#117—942no—
404#117—942no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—471no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4D18811yescharacteristic
417C2×C4⋊C4D18811yescharacteristic
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C22⋊C4D18811yescharacteristic
425C2×C22⋊C4D18811yescharacteristic
426C2×C42—941no—
427C2×C42—941no—
428C2×C42—941no—
429C2×C42—941no—
430C2×C42—941no—
431C2×C42—941no—
432C2×C42C22×D4711yescharacteristic
433C2×C42C22×D4711yescharacteristic
434#107—942no—
435#106—474no—
436#106—474no—
437#106—942no—
438#102—474no—
439#102—474no—
440#102—942no—
441#102—942no—
442C22⋊C8—22no—
443C22⋊C8C4×D4714yescomplemented
444#97—22no—
445C4×C8—22no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95—941no—
451#95—941no—
452#95—941no—
453C24—941no—
454C24C2×D18811yescharacteristic
455C2×D4—472no—
456C2×D4—944no—
457C2×D4—941no—
458C2×D4—941no—
459C2×D4—941no—
460C2×D4—941no—
461C2×D4—941no—
462C2×D4—3761no—
463C2×D4—3761no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—944no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—3761no—
477C22×C4—3761no—
478C22×C4—3761no—
479C22×C4—3761no—
480C22×C4—1882no—
481C22×C4C2×C4×D4712yes—
482C22×C4C2×D18811yescharacteristic
483C22×C4C2×D18811yescharacteristic
484C22×C4C2×D18811yescharacteristic
485C22×C4C2×D18811yescharacteristic
486C22×C4C2×D18811yescharacteristic
487C22×C4#1062535012yes—
488C22×C4C23×D4711yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8C2×C4×D4712yes—
500C2×C8C2×C4×D4712yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—472no—
504C4⋊C4—472no—
505C4⋊C4—472no—
506C4⋊C4—472no—
507C4⋊C4—472no—
508C4⋊C4—944no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—941no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—3761no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—944no—
522C22⋊C4—944no—
523C22⋊C4—3761no—
524C22⋊C4—3761no—
525C22⋊C4—3761no—
526C22⋊C4—1882no—
527C42—21no—
528C42—21no—
529C42—944no—
530C42—3761no—
531C42—3761no—
532C42—3761no—
533C42—3761no—
534C42—3761no—
535C42#1062535212yes—
536C42#1062535212yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—942no—
544C23—1881no—
545C23—1881no—
546C23—1881no—
547C23—1881no—
548C23—1881no—
549C23—1881no—
550C23C4×D18812yes—
551C23C22×D18811yescharacteristic
552D4—942no—
553D4—942no—
554D4—3764no—
555D4—1881no—
556D4—1881no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—942no—
574C2×C4—942no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—942no—
578C2×C4—3764no—
579C2×C4—3764no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
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×C4—1881no—
595C2×C4—1881no—
596C2×C4C4×D18812yes—
597C2×C4#1309400812yes—
598C2×C4#1309400812yes—
599C2×C4C22×C4×D4711yescharacteristic
600C2×C4#41×D4711yescharacteristic
601C2×C4#41×D4711yescharacteristic
602C2×C4C2×#1062535211yescharacteristic
603C2×C4C2×#1062535211yescharacteristic
604C2×C4C2×#1062535011yescharacteristic
605C8—22no—
606C8—22no—
607C8—41no—
608C8—41no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—3761no—
617C22—3761no—
618C22—3761no—
619C22—3761no—
620C22—1882no—
621C22—1882no—
622C22—1882no—
623C22C2×C4×D18811yescharacteristic
624C22C2×#1309400811yescharacteristic
625C22#1568963511yescharacteristic
626C22#1568992811yescharacteristic
627C22#1568993511yescharacteristic
628C22#1568993511yescharacteristic
629C22#1568993511yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—3761no—
636C4—3761no—
637C4—1882no—
638C4—1882no—
639C4—1882no—
640C4—1882no—
641C4—1882no—
642C4C4×#1062535211yescharacteristic
643C4C4×#1062535211yescharacteristic
644C4#1568965311yescharacteristic
645C4#1568965311yescharacteristic
646C2—42no—
647C2—3762no—
648C2C4×#1309400811yescharacteristic
649C2#2025222211yescharacteristic
650C2#2025224911yescharacteristic
651C2#2025334711yescharacteristic
652C2#2025337911yescharacteristic
653C2#2025358911yescharacteristic
654C2#2025366711yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1750C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#1750D47
hypocenterC47#21584
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
3. #20245744C2semidirect
4. #20244016C2semidirect
5. #20244009C2semidirect
6. #20243974C2semidirect
7. #20243967C2semidirect
9. #20243589C2semidirect
10. #20243475C2semidirect
11. #20243285C2semidirect
12. C4×#13093956C2semidirect
13. #20243214C2semidirect
14. C4×#13093946C2semidirect
15. C47×#1750C2semidirect
20. #15689442C4semidirect
72. C2×C4⋊Dic47C2×C4semidirect
95. #13093956C2×C4semidirect
96. #13093956C2×C4semidirect
122. C22⋊C8×C47C2×C4semidirect
333. #1750D47semidirect
387. C47#21584semidirect
443. C22⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894422
#202460812
#202457442
#202440162
#202440092
#202439742
#202439672
#202437082
#202435892
#202434752
#202432852
C4×#130939562
#202432142
C4×#130939462
C47×#17502
#2158447