Finite Group Databasebeta

← #26123694

Subgroups — #26123694

Order 12032.

930 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
1C22×#13093960C211yesmaximal, characteristic, complemented
2C22×#13093951C211yesmaximal, characteristic
3C2×#15689090C211yesmaximal, characteristic
4C2×#15688882C211yesmaximal, characteristic, complemented
5#20243085C211yesmaximal, characteristic, complemented
6C2×#15688877C211yesmaximal, characteristic, complemented
7C2×C47×#416C211yesmaximal, characteristic
8C22×#10625329C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C22×C4×Dic47C2211yescharacteristic
12C2×#13093960—22no—
13C2×#13093960—22no—
14C2×#13093960—22no—
15C2×#13093951—22no—
16C2×#13093951—22no—
17C2×#13093951—22no—
18#15689090—22no—
19#15689090—22no—
20C2×C22⋊(C47⋊C8)—22no—
21#15688882—22no—
22#15688878—22no—
23#15688877—22no—
24C22×#41×C47C2211yescharacteristic
25C23×C8×C47C2211yescharacteristic
26C2×#117×C47C2211yescharacteristic
27C47×#416—22no—
28C47×#416—22no—
29C2×#103×C47C411yescharacteristic, complemented
30C2×#103×C47C411yescharacteristic, complemented
31C2×#102×C47C411yescharacteristic, complemented
32C2×#102×C47C411yescharacteristic, complemented
33C23×Dic47—21no—
34C2×#10625329—21no—
35C2×#10625329—21no—
36C2×#10625329—21no—
37C2×#10625329—21no—
38C2×#10625329—21no—
39C2×#10625329—21no—
40C2×#10625329—21no—
41C2×#10625329—21no—
42C2×#10625329—21no—
43C2×#10625329—21no—
44C2×#10625329D412yes—
45C2×#10625329D412yes—
46C22×C47⋊C8—21no—
47C22×C47⋊C8—21no—
48C22×C47⋊C8—21no—
49C22×C47⋊C8—21no—
50C22×C47⋊C8—21no—
51C22×C47⋊C8—21no—
52C22×C47⋊C8—21no—
53C22×C47⋊C8—21no—
54C22×C47⋊C8D412yes—
55C2×C4×Dic47—21no—
56C2×C4×Dic47—21no—
57C2×C4×Dic47—21no—
58C2×C4×Dic47—21no—
59C2×C4×Dic47—21no—
60C2×C4×Dic47D412yes—
61#13093960—28no—
62#13093951—28no—
63C22⋊(C47⋊C8)—28no—
64C2×#41×C47—21no—
65C2×#41×C47—21no—
66C2×#41×C47—21no—
67C2×#41×C47—21no—
68C2×#41×C47—21no—
69C2×#41×C47—21no—
70C2×#41×C47—21no—
71C2×#41×C47—21no—
72C2×#41×C47—21no—
73C2×#41×C47—21no—
74C2×#41×C47D412yes—
75C2×#41×C47Q812yes—
76C22×Q8×C47C2×C411yescharacteristic
77C22×D4×C47—21no—
78C22×D4×C47C2×C411yescharacteristic
79C23×C4×C47—21no—
80C23×C4×C47C2311yescharacteristic
81C22×C8×C47—21no—
82C22×C8×C47—21no—
83C22×C8×C47—21no—
84C22×C8×C47—21no—
85C22×C8×C47C2×C411yescharacteristic
86C22×C8×C47C2×C411yescharacteristic
87C22×C8×C47D412yes—
88C22×C8×C47D412yes—
89#117×C47—21no—
90#117×C47—21no—
91#117×C47—21no—
92#117×C47—21no—
93C2×C4⋊C4×C47C2×C411yescharacteristic
94C2×C4⋊C4×C47C2×C411yescharacteristic
95C2×C22⋊C4×C47—21no—
96C2×C42×C47—21no—
97#103×C47—22no—
98#103×C47—22no—
99#102×C47—22no—
100#102×C47—22no—
101C24×C47—21no—
102C24×C47#4111yescharacteristic
103#41×C47—22no—
104#41×C47—22no—
105#41×C47—22no—
106#41×C47—22no—
107#41×C47—22no—
108#41×C47—22no—
109#41×C47—22no—
110#41×C47—22no—
111#41×C47—41no—
112#41×C47—41no—
113#41×C47—41no—
114#41×C47—41no—
115#41×C47—41no—
116#41×C47—41no—
117#41×C47—41no—
118#41×C47—41no—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47C4⋊C412yes—
125C2×D4×C47—22no—
126C2×D4×C47—22no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—41no—
133C2×D4×C47—41no—
134C2×D4×C47—41no—
135C2×D4×C47—41no—
136C2×D4×C47C4⋊C412yes—
137C22×C4×C47—21no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—41no—
143C22×C4×C47—41no—
144C22×C4×C47—41no—
145C22×C4×C47—41no—
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×Q811yescharacteristic
155C22×C4×C47#4111yescharacteristic
156C22×C4×C47#4111yescharacteristic
157C22×C4×C47#4111yescharacteristic
158C22×C4×C47#4111yescharacteristic
159C22×C4×C47#4111yescharacteristic
160C22×Dic47—24no—
161C22×Dic47—22no—
162C22×Dic47—42no—
163C22×Dic47—41no—
164C22×Dic47—41no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C2×C8×C47—22no—
172C2×C8×C47—22no—
173C2×C8×C47C22⋊C412yes—
174C2×C8×C47C22⋊C412yes—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C22⋊C4×C47—41no—
180C22⋊C4×C47—41no—
181C42×C47—41no—
182C42×C47—41no—
183C4×Dic47—22no—
184C4×Dic47—22no—
185C4×Dic47—41no—
186C4×Dic47—41no—
187#10625329—22no—
188#10625329—22no—
189#10625329—22no—
190#10625329—22no—
191#10625329—41no—
192#10625329—41no—
193#10625329—41no—
194#10625329—41no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—41no—
201C2×C47⋊C8—41no—
202C2×C47⋊C8—41no—
203C2×C47⋊C8—41no—
204C2×C47⋊C8—41no—
205C2×C47⋊C8—41no—
206C2×C47⋊C8—41no—
207C2×C47⋊C8—41no—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—42no—
214C23×C47—42no—
215C23×C47C4×D411yescharacteristic
216C23×C47#12111yescharacteristic
217C23×C47C22⋊Q811yescharacteristic
218C23×C47#12311yescharacteristic
219C23×C47#12411yescharacteristic
220C23×C47#12411yescharacteristic
221C23×C47#12611yescharacteristic
222Q8×C47—22no—
223Q8×C47—22no—
224Q8×C47—41no—
225Q8×C47—41no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47—42no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—42no—
232D4×C47—41no—
233D4×C47—41no—
234C2×C4×C47—22no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47C2×C22⋊C411yescharacteristic
246C2×C4×C47C2×C4⋊C411yescharacteristic
247C2×C4×C47#11711yescharacteristic
248C2×C4×C47C4×D411yescharacteristic
249C2×C4×C47C4×D411yescharacteristic
250C2×C4×C47C4×D411yescharacteristic
251C2×C4×C47#12011yescharacteristic
252C2×C4×C47#12011yescharacteristic
253C2×C4×C47#12011yescharacteristic
254C2×C4×C47#12111yescharacteristic
255C2×C4×C47#12111yescharacteristic
256C2×C4×C47#12111yescharacteristic
257C2×C4×C47#12111yescharacteristic
258C2×C4×C47#12111yescharacteristic
259C2×C4×C47#12111yescharacteristic
260C2×C4×C47#12111yescharacteristic
261C2×C4×C47C22⋊Q811yescharacteristic
262C2×C4×C47C22⋊Q811yescharacteristic
263C2×C4×C47C22⋊Q811yescharacteristic
264C2×C4×C47C22⋊Q811yescharacteristic
265C2×C4×C47C22⋊Q811yescharacteristic
266C2×C4×C47#12311yescharacteristic
267C2×C4×C47#12311yescharacteristic
268C2×C4×C47#12311yescharacteristic
269C2×C4×C47#12311yescharacteristic
270C2×C4×C47#12411yescharacteristic
271C2×C4×C47#12611yescharacteristic
272C2×C4×C47#12711yescharacteristic
273C2×Dic47—24no—
274C2×Dic47—44no—
275C2×Dic47—44no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C8×C47—22no—
279C8×C47—22no—
280C47⋊C8—42no—
281C47⋊C8—42no—
282C47⋊C8—42no—
283#11125—471nomaximal, Sylow
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—21no—
289C22×C47—21no—
290C22×C47—21no—
291C22×C47—21no—
292C22×C47—44no—
293C22×C47—42no—
294C22×C47—82no—
295C22×C47—81no—
296C22×C47—81no—
297C22×C47#38511yescharacteristic
298C22×C47#38811yescharacteristic
299C22×C47#39011yescharacteristic
300C22×C47#39311yescharacteristic
301C22×C47#39411yescharacteristic
302C22×C47#39611yescharacteristic
303C22×C47#39711yescharacteristic
304C22×C47#44312yes—
305C22×C47#44412yes—
306C4×C47—42no—
307C4×C47—81no—
308C4×C47—81no—
309C4×C47#38011yescharacteristic
310C4×C47#38511yescharacteristic
311C4×C47#38611yescharacteristic
312C4×C47#38811yescharacteristic
313C4×C47#39211yescharacteristic
314C4×C47#39411yescharacteristic
315C4×C47#39611yescharacteristic
316C4×C47#39911yescharacteristic
317Dic47—44no—
318C22×#108—471no—
319C22×#104—471no—
320C2×#416D4711yescharacteristic
321C2×#408—471no—
322C2×#339—471no—
323#1729—471no—
324C2×#336—471no—
325C2×C47—21no—
326C2×C47—21no—
327C2×C47—21no—
328C2×C47—21no—
329C2×C47—82no—
330C2×C47#143911yescharacteristic
331C2×C47#186612yes—
332C2×C47#192912yes—
333C2×C47#197411yescharacteristic
334C2×C47#197511yescharacteristic
335C22×#41C2×D4711yescharacteristic
336C22×M16—471no—
337C22×M16—471no—
338C23×C8—471no—
339C23×C8C2×D4711yescharacteristic
340C2×#117C2×D4711yescharacteristic
341C22×C42—471no—
342C2×#108—942no—
343C2×#108—942no—
344C2×#108—942no—
345C2×#104—942no—
346C2×#104—942no—
347C2×#104—942no—
348#416—22no—
349#416—22no—
350C2×#103Dic4711yescharacteristic, complemented
351C2×#103Dic4711yescharacteristic, complemented
352C2×#102Dic4711yescharacteristic, complemented
353C2×#102Dic4711yescharacteristic, complemented
354#408—942no—
355#408—942no—
356C2×C22⋊C8—942no—
357#339—942no—
358#336—942no—
359#336—942no—
360C47#1112511yescharacteristic, complemented, Sylow
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41Dic9412yes—
372C2×#41D18812yes—
373C22×Q8C2×Dic4711yescharacteristic
374C22×D4—21no—
375C22×D4C2×Dic4711yescharacteristic
376C23×C4—21no—
377C23×C4—941no—
378C23×C4C22×D4711yescharacteristic
379C2×M16—472no—
380C2×M16—472no—
381C2×M16—941no—
382C2×M16—941no—
383C2×M16—941no—
384C2×M16—941no—
385C2×M16—941no—
386C2×M16—941no—
387C2×M16—941no—
388C2×M16—941no—
389C2×M16—941no—
390C2×M16—941no—
391C22×C8—21no—
392C22×C8—21no—
393C22×C8—21no—
394C22×C8—21no—
395C22×C8—472no—
396C22×C8—941no—
397C22×C8—941no—
398C22×C8—941no—
399C22×C8—941no—
400C22×C8—941no—
401C22×C8—941no—
402C22×C8—941no—
403C22×C8—941no—
404C22×C8C2×Dic4711yescharacteristic
405C22×C8C2×Dic4711yescharacteristic
406C22×C8C22⋊D4712yes—
407C22×C8C22⋊D4712yes—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412C2×C4⋊C4C2×Dic4711yescharacteristic
413C2×C4⋊C4C2×Dic4711yescharacteristic
414C2×C22⋊C4—21no—
415C2×C42—21no—
416C2×C42—472no—
417C2×C42—941no—
418C2×C42—941no—
419C2×C42—941no—
420C2×C42—941no—
421C2×C42—941no—
422#108—948no—
423#104—948no—
424#103—22no—
425#103—22no—
426#102—22no—
427#102—22no—
428C22⋊C8—948no—
429C24—21no—
430C24#1062535211yescharacteristic
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—22no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446#41—41no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C4⋊Dic4712yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C4⋊Dic4712yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—944no—
475C22×C4—942no—
476C22×C4—1882no—
477C22×C4—1881no—
478C22×C4—1881no—
479C22×C4C2×Dic9411yescharacteristic
480C22×C4C2×D18811yescharacteristic
481C22×C4#1062535011yescharacteristic
482C22×C4#1062535011yescharacteristic
483C22×C4D4×D4711yescharacteristic
484C22×C4D4×D4711yescharacteristic
485C22×C4D4×D4711yescharacteristic
486C22×C4D4×D4711yescharacteristic
487C22×C4#1062535211yescharacteristic
488C22×C4#1062535211yescharacteristic
489C22×C4#1062535211yescharacteristic
490C22×C4C22×Dic4711yescharacteristic
491C22×C4C2×C22⋊D4711yescharacteristic
492C22×C4C2×C22⋊D4711yescharacteristic
493M16—942no—
494M16—942no—
495M16—942no—
496M16—942no—
497M16—1881no—
498M16—1881no—
499M16—1881no—
500M16—1881no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—942no—
510C2×C8—942no—
511C2×C8—942no—
512C2×C8—942no—
513C2×C8—942no—
514C2×C8—1881no—
515C2×C8—1881no—
516C2×C8—1881no—
517C2×C8—1881no—
518C2×C8—1881no—
519C2×C8—1881no—
520C2×C8—1881no—
521C2×C8—1881no—
522C2×C8C22⋊Dic4712yes—
523C2×C8C22⋊Dic4712yes—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C42—41no—
531C42—41no—
532C42—942no—
533C42—942no—
534C42—1881no—
535C42—1881no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23D4×Dic4711yescharacteristic
544C23#1309399511yescharacteristic
545C23#1309400111yescharacteristic
546C23#1309400811yescharacteristic
547C23#1309401011yescharacteristic
548C23#1309401111yescharacteristic
549C23#1309401711yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—41no—
553Q8—41no—
554D4—22no—
555D4—22no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—41no—
561D4—41no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—944no—
574C2×C4—1882no—
575C2×C4—1884no—
576C2×C4—1884no—
577C2×C4—1882no—
578C2×C4C2×C22⋊Dic4711yescharacteristic
579C2×C4C2×C4⋊Dic4711yescharacteristic
580C2×C4#1309398311yescharacteristic
581C2×C4D4×Dic4711yescharacteristic
582C2×C4D4×Dic4711yescharacteristic
583C2×C4D4×Dic4711yescharacteristic
584C2×C4#1309399211yescharacteristic
585C2×C4#1309399411yescharacteristic
586C2×C4#1309399411yescharacteristic
587C2×C4#1309399511yescharacteristic
588C2×C4#1309399611yescharacteristic
589C2×C4#1309399611yescharacteristic
590C2×C4#1309399711yescharacteristic
591C2×C4#1309399711yescharacteristic
592C2×C4#1309399811yescharacteristic
593C2×C4#1309399811yescharacteristic
594C2×C4#1309400111yescharacteristic
595C2×C4#1309400111yescharacteristic
596C2×C4#1309400111yescharacteristic
597C2×C4#1309400311yescharacteristic
598C2×C4#1309400311yescharacteristic
599C2×C4#1309400511yescharacteristic
600C2×C4#1309400611yescharacteristic
601C2×C4#1309400611yescharacteristic
602C2×C4#1309400811yescharacteristic
603C2×C4#1309401011yescharacteristic
604C2×C4#1309401711yescharacteristic
605C2×C4#1309402011yescharacteristic
606C8—22no—
607C8—22no—
608C8—1882no—
609C8—1882no—
610C8—1882no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—44no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—82no—
624C22#1568899811yescharacteristic
625C22#1568901611yescharacteristic
626C22#1568902611yescharacteristic
627C22#1568903511yescharacteristic
628C22#1568903911yescharacteristic
629C22#1568904611yescharacteristic
630C22#1568904811yescharacteristic
631C22#1568924012yes—
632C22#1568924512yes—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—1884no—
637C4#1568898311yescharacteristic
638C4#1568899811yescharacteristic
639C4#1568900311yescharacteristic
640C4#1568901611yescharacteristic
641C4#1568903211yescharacteristic
642C4#1568903911yescharacteristic
643C4#1568904611yescharacteristic
644C4#1568905911yescharacteristic
645C2—21no—
646C2—21no—
647C2—21no—
648C2—21no—
649C2—82no—
650C2#2024163311yescharacteristic
651C2#2024366512yes—
652C2#2024403812yes—
653C2#2024427711yescharacteristic
654C2#2024428211yescharacteristic
subgroupG/H
centerC22×C4D4×D47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#416C2
socleC23×C47C4×D4
prime radicalC22×#41×C47C22
cosocleC23×C4×C47C23
hypercenterC2×#416D47
hypocenterC47#11125
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C22×#13093960C2semidirect
4. C2×#15688882C2semidirect
5. #20243085C2semidirect
6. C2×#15688877C2semidirect
29. C2×#103×C47C4semidirect
30. C2×#103×C47C4semidirect
31. C2×#102×C47C4semidirect
32. C2×#102×C47C4semidirect
350. C2×#103Dic47semidirect
351. C2×#103Dic47semidirect
352. C2×#102Dic47semidirect
353. C2×#102Dic47semidirect
360. C47#11125semidirect

Maximal subgroups

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

subgroupindex
C22×#130939602
C22×#130939512
C2×#156890902
C2×#156888822
#202430852
C2×#156888772
C2×C47×#4162
#1112547