Finite Group Databasebeta

← #26123256

Subgroups — #26123256

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689129C211yesmaximal, characteristic
2C2×#15689127C211yesmaximal, characteristic
3#20246004C211yesmaximal, characteristic, complemented
4C2×#15689025C211yesmaximal, characteristic
5C2×#15688891C211yesmaximal, characteristic, complemented
6C2×#15688883C211yesmaximal, characteristic, complemented
7C2×C47×#339C211yesmaximal, characteristic, complemented
8C23×D188C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×(C2×D47)⋊C4—22no—
13#15689129—22no—
14#15689129—22no—
15#15689127—22no—
16#15689127—22no—
17C2×#13093946—22no—
18C2×#13093944—22no—
19#15689025—22no—
20#15689025C414yescomplemented
21#15689021—22no—
22#15689019—22no—
23#15688981—22no—
24#15688891—22no—
25#15688883—22no—
26C22×M16×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C22×C42×C47C2211yescharacteristic
29C47×#339—22no—
30C24×D47—21no—
31C22×D188—21no—
32C22×D188—21no—
33C22×D188—21no—
34C22×D188—21no—
35C22×D188—21no—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188—21no—
39C22×D188—21no—
40C22×D188—21no—
41C22×D188—21no—
42C22×D188C2×C412yes—
43C22×D188D412yes—
44C22×D188Q812yes—
45C23×Dic47D412yes—
46C2×#10625329—21no—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329D412yes—
52C22×C47⋊C8—21no—
53#13093983—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47C2×C412yes—
62C2×C4⋊Dic47D412yes—
63C2×C4⋊Dic47D412yes—
64C2×Dic47⋊C4—24no—
65C2×C22⋊Dic47—21no—
66C2×(C2×D47)⋊C4—28no—
67C2×(C2×D47)⋊C4—48no—
68C2×C4×Dic47—21no—
69#13093946—28no—
70#13093944—28no—
71C47⋊#95—24no—
72C23×C4×C47—21no—
73C23×C4×C47C2311yescharacteristic
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47D412yes—
80C22×C8×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47—21no—
85C2×C4⋊C4×C47D412yes—
86C2×C22⋊C4×C47—21no—
87C2×C42×C47—21no—
88C2×C42×C47—21no—
89C2×C42×C47—21no—
90C2×C42×C47—21no—
91C2×C42×C47—21no—
92C2×C42×C47—21no—
93C2×C42×C47C2×C412yes—
94C24×C47C2×D411yescharacteristic
95C23×D47—22no—
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—44no—
99C23×D47—44no—
100C23×D47—41no—
101C23×D47—41no—
102C23×D47—41no—
103C23×D47—41no—
104C22×C4×C47—24no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—42no—
108C22×C4×C47—41no—
109C22×C4×C47—41no—
110C22×C4×C47C22×C411yescharacteristic
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×Q811yescharacteristic
118C22×C4×C47#4111yescharacteristic
119C22×C4×C47#4111yescharacteristic
120C22×C4×C47#4111yescharacteristic
121C22×C4×C47#4111yescharacteristic
122C22×C4×C47#4111yescharacteristic
123C22×C4×C47#4111yescharacteristic
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—44no—
128C22×Dic47C22⋊C414yes—
129C2×D188—22no—
130C2×D188—22no—
131C2×D188—22no—
132C2×D188—22no—
133C2×D188—22no—
134C2×D188—22no—
135C2×D188—22no—
136C2×D188—22no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188—22no—
144C2×D188—41no—
145C2×D188—41no—
146C2×D188—41no—
147C2×D188—41no—
148C2×D188—41no—
149C2×D188—41no—
150C2×D188—41no—
151C2×D188—41no—
152C2×D188—41no—
153C2×D188—41no—
154C2×D188—41no—
155C2×D188—41no—
156C2×D188C4⋊C414yescomplemented
157M16×C47—22no—
158M16×C47—22no—
159M16×C47—41no—
160M16×C47—41no—
161C2×C8×C47—22no—
162C2×C8×C47—41no—
163C2×C8×C47—41no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C22⋊C4×C47—41no—
167C22⋊C4×C47—41no—
168C42×C47—22no—
169C42×C47—22no—
170C42×C47—41no—
171C42×C47—41no—
172C42×C47—41no—
173C42×C47—41no—
174C22⋊Dic47—41no—
175C22⋊Dic47—41no—
176(C2×D47)⋊C4—416no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183Dic47⋊C4—84no—
184C4×Dic47—41no—
185C4×Dic47—41no—
186#10625329—22no—
187#10625329—22no—
188#10625329—41no—
189#10625329—41no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—41no—
192C2×C47⋊C8—41no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47C2×C22⋊C411yescharacteristic
196C23×C47#12011yescharacteristic
197C23×C47#12111yescharacteristic
198C23×C47C22⋊Q811yescharacteristic
199C23×C47#12311yescharacteristic
200C23×C47#12411yescharacteristic
201C23×C47#12711yescharacteristic
202C22×D47—24no—
203C22×D47—44no—
204C22×D47—44no—
205C22×D47—48no—
206C22×D47—48no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—82no—
214C22×D47—82no—
215C22×D47—84no—
216C22×D47—84no—
217C22×D47—82no—
218C22×D47—82no—
219C2×C4×C47—24no—
220C2×C4×C47—42no—
221C2×C4×C47—44no—
222C2×C4×C47—44no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47C2×C4⋊C411yescharacteristic
228C2×C4×C47#11711yescharacteristic
229C2×C4×C47C4×D411yescharacteristic
230C2×C4×C47C4×D411yescharacteristic
231C2×C4×C47C4×D411yescharacteristic
232C2×C4×C47C4×D411yescharacteristic
233C2×C4×C47#12011yescharacteristic
234C2×C4×C47#12011yescharacteristic
235C2×C4×C47#12111yescharacteristic
236C2×C4×C47#12111yescharacteristic
237C2×C4×C47#12111yescharacteristic
238C2×C4×C47#12111yescharacteristic
239C2×C4×C47#12111yescharacteristic
240C2×C4×C47#12111yescharacteristic
241C2×C4×C47#12111yescharacteristic
242C2×C4×C47C22⋊Q811yescharacteristic
243C2×C4×C47C22⋊Q811yescharacteristic
244C2×C4×C47C22⋊Q811yescharacteristic
245C2×C4×C47C22⋊Q811yescharacteristic
246C2×C4×C47C22⋊Q811yescharacteristic
247C2×C4×C47#12311yescharacteristic
248C2×C4×C47#12311yescharacteristic
249C2×C4×C47#12311yescharacteristic
250C2×C4×C47#12311yescharacteristic
251C2×C4×C47#12411yescharacteristic
252C2×C4×C47#12411yescharacteristic
253C2×C4×C47#12611yescharacteristic
254C2×C4×C47#12611yescharacteristic
255C2×Dic47—24no—
256C2×Dic47—44no—
257C2×Dic47—44no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—82no—
262D188—24no—
263D188—24no—
264D188—42no—
265D188—42no—
266D188—42no—
267D188—42no—
268D188—42no—
269D188—42no—
270C8×C47—42no—
271C47⋊C8—42no—
272#11054—471nomaximal, Sylow
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—21no—
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—22no—
281C22×C47#38011yescharacteristic
282C22×C47#38611yescharacteristic
283C22×C47#39011yescharacteristic
284C22×C47#39211yescharacteristic
285C22×C47#39311yescharacteristic
286C22×C47#39711yescharacteristic
287C22×C47#39911yescharacteristic
288C22×C47#45312yes—
289C22×C47#45412yes—
290C2×D47—48no—
291C2×D47—44no—
292C2×D47—84no—
293C2×D47—84no—
294C2×D47—84no—
295C2×D47—82no—
296C2×D47—82no—
297C2×D47—84no—
298C2×D47—84no—
299C4×C47—44no—
300C4×C47—82no—
301C4×C47#38511yescharacteristic
302C4×C47#38511yescharacteristic
303C4×C47#38811yescharacteristic
304C4×C47#38811yescharacteristic
305C4×C47#39411yescharacteristic
306C4×C47#39411yescharacteristic
307C4×C47#39611yescharacteristic
308C4×C47#39611yescharacteristic
309Dic47—44no—
310Dic47—82no—
311C2×#418—471no—
312C2×#418—471no—
313#2316—471no—
314C2×#390—471no—
315C2×#343—471no—
316C2×#339—471no—
317C2×#339D4711yescharacteristic, complemented
318C2×C47—21no—
319C2×C47—21no—
320C2×C47—21no—
321C2×C47—21no—
322C2×C47#143911yescharacteristic
323C2×C47#188711yescharacteristic
324C2×C47#188811yescharacteristic
325C2×C47#200612yes—
326C2×C47#202712yes—
327D47—84no—
328C23×D4—471no—
329C22×M16—471no—
330C22×M16C2×D4711yescharacteristic
331C2×#117—471no—
332C2×#117C2×D4711yescharacteristic
333C22×C4⋊C4—471no—
334C22×C22⋊C4—942no—
335C22×C42C2×D4711yescharacteristic
336#418—942no—
337#418—942no—
338#418—942no—
339#418—942no—
340C2×#102—942no—
341C2×#102—942no—
342#390—474no—
343#390—942no—
344#389—942no—
345#389—942no—
346#380—942no—
347#343—942no—
348#339—22no—
349#339—942no—
350C47#1105411yescharacteristic, complemented, Sylow
351C25—941no—
352C22×D4—472no—
353C22×D4—472no—
354C22×D4—472no—
355C22×D4—941no—
356C22×D4—941no—
357C22×D4—941no—
358C22×D4—941no—
359C22×D4—941no—
360C22×D4—941no—
361C22×D4—941no—
362C22×D4—941no—
363C22×D4—941no—
364C22×D4—941no—
365C22×D4—941no—
366C23×C4—21no—
367C23×C4—472no—
368C23×C4C22×D4711yescharacteristic
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—472no—
375C2×M16—941no—
376C2×M16—941no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—941no—
380C2×M16C22⋊D4712yes—
381C22×C8—21no—
382C22×C8—941no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—941no—
388#117—941no—
389#117—941no—
390#117—941no—
391C2×C4⋊C4—472no—
392C2×C4⋊C4—472no—
393C2×C4⋊C4—472no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—944no—
399C2×C4⋊C4D18812yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—948no—
402C2×C22⋊C4—941no—
403C2×C22⋊C4—1888no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—941no—
411C2×C42C4×D4712yes—
412#102—948no—
413#102—948no—
414#95—944no—
415C24—942no—
416C24—942no—
417C24—942no—
418C24—1884no—
419C24—1884no—
420C24—1881no—
421C24—1881no—
422C24—1881no—
423C24—1881no—
424C24D4×D4711yescharacteristic
425C2×D4—474no—
426C2×D4—942no—
427C2×D4—942no—
428C2×D4—942no—
429C2×D4—942no—
430C2×D4—942no—
431C2×D4—942no—
432C2×D4—942no—
433C2×D4—942no—
434C2×D4—942no—
435C2×D4—942no—
436C2×D4—942no—
437C2×D4—942no—
438C2×D4—942no—
439C2×D4—942no—
440C2×D4—942no—
441C2×D4—1881no—
442C2×D4—1881no—
443C2×D4—1881no—
444C2×D4—1881no—
445C2×D4—1881no—
446C2×D4—1881no—
447C2×D4—1881no—
448C2×D4—1881no—
449C2×D4—1881no—
450C2×D4—1881no—
451C2×D4—1881no—
452C2×D4—1881no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—474no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—1884no—
464C22×C4C2×C4×D4711yescharacteristic
465C22×C4C2×D18811yescharacteristic
466C22×C4#1062535011yescharacteristic
467C22×C4#1062535011yescharacteristic
468C22×C4#1062535011yescharacteristic
469C22×C4#1062535011yescharacteristic
470C22×C4D4×D4711yescharacteristic
471C22×C4D4×D4711yescharacteristic
472C22×C4D4×D4711yescharacteristic
473C22×C4D4×D4711yescharacteristic
474C22×C4#1062535211yescharacteristic
475C22×C4Q8×D4711yescharacteristic
476C22×C4#1062535411yescharacteristic
477C22×C4C2×C22⋊D4711yescharacteristic
478M16—22no—
479M16—22no—
480M16—41no—
481M16—41no—
482M16—942no—
483M16—942no—
484M16—1881no—
485M16—1881no—
486C2×C8—22no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—942no—
490C2×C8—1881no—
491C2×C8—1881no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—942no—
495C4⋊C4—942no—
496C4⋊C4—942no—
497C4⋊C4—942no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—3764no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—18816no—
504C22⋊C4—1881no—
505C22⋊C4—1881no—
506C42—22no—
507C42—22no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—1881no—
513C42—1881no—
514C23—22no—
515C23—22no—
516C23—944no—
517C23—3764no—
518C23—3762no—
519C23—3762no—
520C23—3764no—
521C23—3762no—
522C23—3762no—
523C23—1884no—
524C23—1888no—
525C23—1884no—
526C23—1888no—
527C23—1882no—
528C23—1882no—
529C23—1882no—
530C23—1882no—
531C23—1882no—
532C23—1882no—
533C23C22⋊C4×D4711yescharacteristic
534C23#1309399411yescharacteristic
535C23#1309399711yescharacteristic
536C23#1309400211yescharacteristic
537C23#1309400911yescharacteristic
538C23#1309401011yescharacteristic
539C23#1309402011yescharacteristic
540D4—944no—
541D4—944no—
542D4—1882no—
543D4—1882no—
544D4—1882no—
545D4—1882no—
546D4—1882no—
547D4—1882no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—944no—
557C2×C4—3762no—
558C2×C4—1882no—
559C2×C4—1882no—
560C2×C4—1882no—
561C2×C4—1884no—
562C2×C4—1884no—
563C2×C4C4⋊C4×D4711yescharacteristic
564C2×C4#1309398111yescharacteristic
565C2×C4(C22⋊D47)⋊C411yescharacteristic
566C2×C4C4×C22⋊D4711yescharacteristic
567C2×C4#1309398611yescharacteristic
568C2×C4C4×D18811yescharacteristic
569C2×C4#1309399211yescharacteristic
570C2×C4#1309399411yescharacteristic
571C2×C4#1309399611yescharacteristic
572C2×C4#1309399711yescharacteristic
573C2×C4#1309399711yescharacteristic
574C2×C4#1309399711yescharacteristic
575C2×C4#1309399711yescharacteristic
576C2×C4#1309399811yescharacteristic
577C2×C4#1309399911yescharacteristic
578C2×C4#1309400011yescharacteristic
579C2×C4#1309400211yescharacteristic
580C2×C4#1309400211yescharacteristic
581C2×C4#1309400211yescharacteristic
582C2×C4#1309400411yescharacteristic
583C2×C4#1309400611yescharacteristic
584C2×C4#1309400711yescharacteristic
585C2×C4#1309400911yescharacteristic
586C2×C4#1309400911yescharacteristic
587C2×C4#1309401011yescharacteristic
588C2×C4#1309401211yescharacteristic
589C2×C4#1309401811yescharacteristic
590C2×C4#1309401911yescharacteristic
591C8—42no—
592C8—1882no—
593C22—22no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—3764no—
602C22—3764no—
603C22—3764no—
604C22—3764no—
605C22—3762no—
606C22—3762no—
607C22—3764no—
608C22—1888no—
609C22—1884no—
610C22#1568898211yescharacteristic
611C22#1568900111yescharacteristic
612C22#1568902311yescharacteristic
613C22#1568903211yescharacteristic
614C22#1568903611yescharacteristic
615C22#1568905211yescharacteristic
616C22#1568905811yescharacteristic
617C22#1568929212yes—
618C22#1568929712yes—
619C4—44no—
620C4—82no—
621C4—3762no—
622C4—1884no—
623C4#1568899411yescharacteristic
624C4#1568899711yescharacteristic
625C4#1568901211yescharacteristic
626C4#1568901411yescharacteristic
627C4#1568903811yescharacteristic
628C4#1568904111yescharacteristic
629C4#1568904511yescharacteristic
630C4#1568904711yescharacteristic
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—3764no—
636C2#2024163211yescharacteristic
637C2#2024378511yescharacteristic
638C2#2024379011yescharacteristic
639C2#2024443112yes—
640C2#2024454312yes—
subgroupG/H
centerC23#13093994
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#339C2
socleC23×C47#120
prime radicalC23×D188C22
cosocleC23×C4×C47C23
hypercenterC2×#339D47
hypocenterC47#11054
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
3. #20246004C2semidirect
5. C2×#15688891C2semidirect
6. C2×#15688883C2semidirect
7. C2×C47×#339C2semidirect
20. #15689025C4semidirect
156. C2×D188C4⋊C4semidirect
317. C2×#339D47semidirect
350. C47#11054semidirect

Maximal subgroups

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

subgroupindex
C2×#156891292
C2×#156891272
#202460042
C2×#156890252
C2×#156888912
C2×#156888832
C2×C47×#3392
#1105447