Finite Group Databasebeta

← #26214054

Subgroups — #26214054

Order 12032.

2018 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D47C211yesmaximal, characteristic
2C2×#15689126C212yesmaximal
3C2×#15689075C211yesmaximal, characteristic, complemented
4C4×#13093982C211yesmaximal, characteristic, complemented
5#20243216C211yesmaximal, characteristic, complemented
6#20243214C211yesmaximal, characteristic, complemented
7#20243213C211yesmaximal, characteristic, complemented
8#102×Dic47C212yesmaximal, complemented
9#20243202C212yesmaximal, complemented
10#20243106C212yesmaximal, complemented
11C47×#1762C211yesmaximal, characteristic, complemented
12C22×D4×D47C2211yescharacteristic
13C23×C4×D47—21no—
14C22×#10625324C2212yes—
15C2×C4×D188C2211yescharacteristic
16C2×D4×Dic47C2211yescharacteristic
17C2×#13093986C2211yescharacteristic
18C2×C4×C22⋊D47—21no—
19C2×(C22⋊D47)⋊C4—21no—
20C4×D4×D47—24no—
21C4×D4×D47C418yescomplemented
22C2×#13093982C2212yes—
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C22⋊C4×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689126C4116yescomplemented
27C2×#13093946C2212yes—
28C2×#13093945C2212yes—
29C2×#13093944C2212yes—
30#15689075—24no—
31C2×#13093935C2211yescharacteristic, complemented
32C2×C8×Dic47C2212yescomplemented
33C4⋊C4×Dic47C2212yescomplemented
34C4×C4⋊Dic47C2211yescharacteristic, complemented
35C4×(C2×D47)⋊C4—21no—
36#15688970—21no—
37#15688968—21no—
38C42×Dic47—21no—
39#15688886C2212yescomplemented
40#15688885C2212yescomplemented
41#15688885C2212yescomplemented
42C2×C4×D4×C47C2211yescharacteristic
43C2×#102×C47C2212yes—
44C2×#97×C47C2211yescharacteristic, complemented
45C4×C4⋊C4×C47C2211yescharacteristic, complemented
46C47×#340C2212yescomplemented
47C24×D47—21no—
48C22×D188C2311yescharacteristic
49C22×C22⋊D47—21no—
50C2×D4×D47—22no—
51C2×D4×D47—24no—
52C2×D4×D47C2×C414yes—
53C2×D4×D47C2×C418yes—
54C23×Dic47—21no—
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47—21no—
58C22×C4×D47—28no—
59C22×C4×D47—48no—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C2×#10625324—24no—
64C2×#10625324—24no—
65C2×#10625324C2×C418yescomplemented
66C22×C47⋊C8C2312yes—
67C4×D188—22no—
68C4×D188C2×C414yes—
69D4×Dic47—22no—
70D4×Dic47C2×C414yes—
71#13093986—22no—
72#13093986C2×C414yes—
73C4×C22⋊D47—28no—
74(C22⋊D47)⋊C4—28no—
75#13093982—24no—
76#13093982C2×C418yescomplemented
77C2×C4⋊Dic47C2311yescharacteristic
78C2×C4⋊Dic47C2312yes—
79C2×Dic47⋊C4—21no—
80C4⋊C4×D47—22no—
81C4⋊C4×D47C2×C414yes—
82C2×C22⋊Dic47—21no—
83C2×(C2×D47)⋊C4—22no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C22⋊C4×D47—28no—
87C2×C4×Dic47—22no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—22no—
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93C42×D47C2×C414yes—
94C42×D47D414yes—
95#13093946C2×C418yescomplemented
96#13093945C2×C418yescomplemented
97#13093944C2×C418yescomplemented
98#13093935—22no—
99C8×Dic47—24no—
100C47⋊#95—21no—
101C47⋊#95—22no—
102C22×D4×C47C2311yescharacteristic
103C23×C4×C47—21no—
104C22×C8×C47C2312yes—
105C4×D4×C47—22no—
106C4×D4×C47C2×C414yes—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2312yes—
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112#102×C47C2×C418yescomplemented
113#97×C47—22no—
114#95×C47—21no—
115C24×C47—21no—
116C23×D47—21no—
117C23×D47—24no—
118C23×D47—28no—
119C23×D47—44no—
120C23×D47—42no—
121C23×D47—42no—
122C23×D47C2×D411yescharacteristic
123C2×D4×C47—21no—
124C2×D4×C47—22no—
125C2×D4×C47C22×C412yes—
126C2×D4×C47C22×C414yes—
127C22×C4×C47—22no—
128C22×C4×C47—21no—
129C22×C4×C47—24no—
130C22×C4×C47—44no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—24no—
135C2×C22⋊D47—28no—
136C22×Dic47—22no—
137C22×Dic47—21no—
138C22×Dic47—24no—
139C22×Dic47—44no—
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143D4×D47—28no—
144D4×D47—216no—
145D4×D47C42116yescomplemented
146C2×D188—21no—
147C2×D188—22no—
148C2×D188C22×C414yes—
149C2×D188C22×C412yes—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—24no—
153C2×C4×D47—24no—
154C2×C4×D47—432no—
155C2×C4×D47—41no—
156C2×C4×D47—41no—
157C2×C4×D47—44no—
158C2×C4×D47—44no—
159C2×C4×D47—44no—
160C2×C4×D47C22⋊C414yes—
161C2×C4×D47C22⋊C414yes—
162C2×C4×D47C22×C412yes—
163C2×C4×D47C22×C414yes—
164C2×C4×D47C2×D412yes—
165C2×C4×D47#4114yes—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47C22×C414yescomplemented
169C4⋊C4×C47—21no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47C22×C412yes—
172C4⋊C4×C47C22×C414yescomplemented
173C22⋊C4×C47—24no—
174C42×C47—42no—
175C42×C47C22×C412yes—
176C42×C47C2×D412yes—
177C22⋊Dic47—24no—
178(C2×D47)⋊C4—28no—
179(C2×D47)⋊C4—24no—
180(C2×D47)⋊C4—24no—
181C4⋊Dic47—21no—
182C4⋊Dic47—22no—
183C4⋊Dic47C22×C412yes—
184C4⋊Dic47C22×C414yescomplemented
185Dic47⋊C4—24no—
186C4×Dic47—28no—
187C4×Dic47—42no—
188C4×Dic47—44no—
189C4×Dic47C22⋊C414yes—
190C4×Dic47C22⋊C414yes—
191C4×Dic47C22×C412yes—
192C4×Dic47C2×D412yes—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8C22×C414yescomplemented
196#10625324—216no—
197C23×C47—22no—
198C23×C47—24no—
199C23×C47—42no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47C22×D411yescharacteristic
203C22×D47—22no—
204C22×D47—21no—
205C22×D47—21no—
206C22×D47—22no—
207C22×D47—24no—
208C22×D47—216no—
209C22×D47—48no—
210C22×D47—48no—
211C22×D47—416no—
212C22×D47—416no—
213C22×D47—42no—
214C22×D47—41no—
215C22×D47—41no—
216C22×D47C2×C22⋊C412yes—
217C22×D47C4×D414yes—
218D4×C47—22no—
219D4×C47—24no—
220D4×C47C2×C4214yes—
221C2×C4×C47—22no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—48no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47C2×C22⋊C412yes—
231C2×C4×C47C2×C22⋊C412yes—
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C23×C412yes—
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C2×#4112yes—
236C22⋊D47—216no—
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—48no—
245C2×Dic47C2×C22⋊C412yes—
246C2×Dic47C2×C22⋊C412yes—
247C2×Dic47C2×C22⋊C412yes—
248C2×Dic47C4×D414yes—
249C2×Dic47AGL(1,ℤ/8)14yes—
250D188—22no—
251D188—24no—
252D188C2×C4214yes—
253C4×D47—28no—
254C4×D47—44no—
255C4×D47—48no—
256C4×D47—48no—
257C4×D47—48no—
258C4×D47C2×C4214yes—
259C4×D47#11714yes—
260C4×D47C4×D418yes—
261C8×C47—24no—
262C47⋊C8—24no—
263C4×#418—471nomaximal, Sylow
264C22×C47—24no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—44no—
268C22×C47—44no—
269C22×C47C22×C22⋊C411yescharacteristic
270C22×C47C2×C4×D412yes—
271C22×C47C2×AGL(1,ℤ/8)12yes—
272C22×C47#57511yescharacteristic
273C22×C47#57511yescharacteristic
274C2×D47—22no—
275C2×D47—22no—
276C2×D47—24no—
277C2×D47—24no—
278C2×D47—24no—
279C2×D47—432no—
280C2×D47—42no—
281C2×D47—42no—
282C2×D47—44no—
283C2×D47—44no—
284C2×D47C4×C22⋊C414yes—
285C4×C47—22no—
286C4×C47—22no—
287C4×C47—42no—
288C4×C47—44no—
289C4×C47C22×C4211yescharacteristic
290C4×C47C2×#11711yescharacteristic
291C4×C47C2×C4×D412yes—
292Dic47—22no—
293Dic47—42no—
294Dic47—44no—
295Dic47C4×C22⋊C414yes—
296Dic47#41814yescomplemented
297C22×C4×D4—471no—
298C2×#418—472no—
299C2×C4×M16—471no—
300C4×#117—471no—
301#1762—471no—
302#1762—471no—
303#1762—471no—
304#1762D4711yescharacteristic, complemented
305C4×#102—472no—
306C4×#102—472no—
307#1736—472no—
308C2×C47—44no—
309C2×C47C2×C4×C22⋊C411yescharacteristic
310C2×C47C2×#41811yescharacteristic
311C2×C47#289611yescharacteristic
312C2×C47C4×AGL(1,ℤ/8)12yes—
313C2×C47C4×AGL(1,ℤ/8)12yes—
314D47—24no—
315D47—44no—
316C23×D4—471no—
317C24×C4—941no—
318C22×M16—472no—
319C2×C4×D4—471no—
320C2×C4×D4—478no—
321C2×C4×D4—471no—
322C2×C4×D4—471no—
323C2×C4×D4—941no—
324C2×C4×D4—944no—
325C2×C4×D4—941no—
326C2×C4×D4C2×D4711yescharacteristic
327C2×#117—472no—
328C22×C4⋊C4—471no—
329C22×C22⋊C4—941no—
330C22×C42—471no—
331#418—4716no—
332C2×#102—472no—
333C2×#102—472no—
334C2×#102—472no—
335C2×#102C2×D4712yes—
336C4×M16—944no—
337C2×#97—471no—
338C2×#97C2×D4711yescharacteristic, complemented
339C2×C4×C8—472no—
340C4×C4⋊C4—472no—
341C4×C4⋊C4—471no—
342C4×C4⋊C4C2×D4711yescharacteristic, complemented
343C4×C22⋊C4—941no—
344C4×C22⋊C4—941no—
345#376—941no—
346C43—941no—
347#340—472no—
348#340—472no—
349#340—472no—
350#340C2×D4712yescomplemented
351C47C4×#41811yescharacteristic, complemented, Sylow
352C25—941no—
353C22×D4—474no—
354C22×D4—478no—
355C22×D4—471no—
356C22×D4—942no—
357C22×D4—944no—
358C22×D4—941no—
359C22×D4C22×D4711yescharacteristic
360C23×C4—21no—
361C23×C4—471no—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—941no—
365C23×C4—948no—
366C23×C4—941no—
367C23×C4—941no—
368C23×C4—941no—
369C23×C4—1888no—
370C2×M16—478no—
371C2×M16—944no—
372C2×M16—944no—
373C22×C8—472no—
374C22×C8C22×D4712yes—
375C4×D4—22no—
376C4×D4—474no—
377C4×D4—474no—
378C4×D4—474no—
379C4×D4—948no—
380C4×D4—942no—
381C4×D4—948no—
382C4×D4—942no—
383C4×D4—942no—
384C4×D4C4×D4714yes—
385#117—478no—
386#117—944no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—472no—
389C2×C4⋊C4—474no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4C22×D4711yescharacteristic
393C2×C4⋊C4C22×D4712yes—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—948no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—942no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C42—21no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—942no—
407C2×C42—942no—
408C2×C42—941no—
409C2×C42C22×D4711yescharacteristic
410#102—478no—
411#102—478no—
412#102—478no—
413#102C4×D4718yescomplemented
414#97—22no—
415#97—942no—
416C4×C8—944no—
417#95—21no—
418#95—942no—
419#95—941no—
420C24—21no—
421C24—471no—
422C24—944no—
423C24—948no—
424C24—941no—
425C24—1884no—
426C24—1882no—
427C24—1882no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—4716no—
431C2×D4—474no—
432C2×D4—472no—
433C2×D4—948no—
434C2×D4—9416no—
435C2×D4—944no—
436C2×D4—948no—
437C2×D4—941no—
438C2×D4—942no—
439C2×D4C2×C4×D4714yes—
440C2×D4C2×C4×D4712yes—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—472no—
448C22×C4—474no—
449C22×C4—472no—
450C22×C4—474no—
451C22×C4—471no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—944no—
455C22×C4—944no—
456C22×C4—944no—
457C22×C4—944no—
458C22×C4—942no—
459C22×C4—941no—
460C22×C4—944no—
461C22×C4—1881no—
462C22×C4—1881no—
463C22×C4—1884no—
464C22×C4—1884no—
465C22×C4—18832no—
466C22×C4—1884no—
467C22×C4—1884no—
468C22×C4D4×D4711yescharacteristic
469C22×C4D4×D4711yescharacteristic
470C22×C4C23×D4711yescharacteristic
471M16—9416no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—474no—
475C2×C8—942no—
476C2×C8—942no—
477C2×C8C2×C4×D4714yescomplemented
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—472no—
481C4⋊C4—474no—
482C4⋊C4—944no—
483C4⋊C4—941no—
484C4⋊C4—942no—
485C4⋊C4C2×C4×D4712yes—
486C4⋊C4C2×C4×D4714yescomplemented
487C22⋊C4—24no—
488C22⋊C4—944no—
489C22⋊C4—948no—
490C22⋊C4—944no—
491C22⋊C4—944no—
492C42—42no—
493C42—474no—
494C42—474no—
495C42—472no—
496C42—472no—
497C42—948no—
498C42—1882no—
499C42—1884no—
500C42C2×C4×D4712yes—
501C42D4×D4712yes—
502C23—22no—
503C23—24no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—472no—
508C23—474no—
509C23—9416no—
510C23—942no—
511C23—941no—
512C23—941no—
513C23—942no—
514C23—944no—
515C23—1888no—
516C23—1888no—
517C23—18816no—
518C23—18816no—
519C23—1882no—
520C23—1881no—
521C23—1881no—
522C23C2×D4×D4711yescharacteristic
523D4—22no—
524D4—24no—
525D4—474no—
526D4—9416no—
527D4—942no—
528D4—944no—
529D4C42×D4714yes—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—472no—
540C2×C4—472no—
541C2×C4—474no—
542C2×C4—474no—
543C2×C4—474no—
544C2×C4—474no—
545C2×C4—478no—
546C2×C4—472no—
547C2×C4—948no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—942no—
551C2×C4—944no—
552C2×C4—1884no—
553C2×C4—1888no—
554C2×C4—1888no—
555C2×C4—1888no—
556C2×C4—1888no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4—1882no—
560C2×C4C22⋊C4×D4712yes—
561C2×C4C22⋊C4×D4712yes—
562C2×C4C22×C4×D4711yescharacteristic
563C2×C4C22×C4×D4712yes—
564C2×C4C2×D4×D4711yescharacteristic
565C2×C4#41×D4712yes—
566C8—24no—
567C8—944no—
568C22—24no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—474no—
574C22—942no—
575C22—944no—
576C22—944no—
577C22—942no—
578C22—944no—
579C22—18832no—
580C22—1882no—
581C22—1884no—
582C22—1884no—
583C22—1882no—
584C22C2×C22⋊C4×D4711yescharacteristic
585C22C4×D4×D4712yes—
586C22AGL(1,ℤ/8)×D4712yes—
587C22#1568991911yescharacteristic
588C22#1568992211yescharacteristic
589C4—22no—
590C4—22no—
591C4—44no—
592C4—42no—
593C4—474no—
594C4—474no—
595C4—942no—
596C4—1884no—
597C4—1882no—
598C4C2×C42×D4711yescharacteristic
599C4#117×D4711yescharacteristic
600C4C4×D4×D4712yes—
601C2—44no—
602C2—944no—
603C2—1884no—
604C2C4×C22⋊C4×D4711yescharacteristic
605C2#418×D4711yescharacteristic
606C2#2025190411yescharacteristic
607C2#2025219312yes—
608C2#2025219912yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC22×D4×D47C22
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47C4×#418
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C2×#15689075C2semidirect
4. C4×#13093982C2semidirect
5. #20243216C2semidirect
6. #20243214C2semidirect
7. #20243213C2semidirect
8. #102×Dic47C2semidirect
9. #20243202C2semidirect
10. #20243106C2semidirect
11. C47×#1762C2semidirect
21. C4×D4×D47C4semidirect
26. #15689126C4semidirect
31. C2×#13093935C22semidirect
32. C2×C8×Dic47C22semidirect
33. C4⋊C4×Dic47C22semidirect
34. C4×C4⋊Dic47C22semidirect
39. #15688886C22semidirect
40. #15688885C22semidirect
41. #15688885C22semidirect
44. C2×#97×C47C22semidirect
45. C4×C4⋊C4×C47C22semidirect
46. C47×#340C22semidirect
65. C2×#10625324C2×C4semidirect
76. #13093982C2×C4semidirect
95. #13093946C2×C4semidirect
96. #13093945C2×C4semidirect
97. #13093944C2×C4semidirect
112. #102×C47C2×C4semidirect
145. D4×D47C42semidirect
168. C2×C8×C47C22×C4semidirect
172. C4⋊C4×C47C22×C4semidirect
184. C4⋊Dic47C22×C4semidirect
195. C2×C47⋊C8C22×C4semidirect
296. Dic47#418semidirect
304. #1762D47semidirect
338. C2×#97C2×D47semidirect
342. C4×C4⋊C4C2×D47semidirect
350. #340C2×D47semidirect
351. C47C4×#418semidirect
413. #102C4×D47semidirect
477. C2×C8C2×C4×D47semidirect
486. C4⋊C4C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×D472
C2×#156891262
C2×#156890752
C4×#130939822
#202432162
#202432142
#202432132
#102×Dic472
#202432022
#202431062
C47×#17622
C4×#41847