Finite Group Databasebeta

← C2×#20243759 #26217770

Subgroups — C2×#20243759 #26217770

Order 12032.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C22×#13093951C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4C2×#15689030C211yesmaximal, characteristic, complemented
5#20243759C218yesmaximal, direct product factor
6C2×#15688891C211yesmaximal, characteristic, complemented
7C2×#15688884C211yesmaximal, characteristic, complemented
8C22×#137×C47C211yesmaximal, characteristic
9C22×#10625329C2211yescharacteristic
10#15689606—22no—
11#15689606—22no—
12#15689606—22no—
13#15689606C2214yes—
14C2×Q8×Dic47—21no—
15C2×D4×Dic47—22no—
16C2×D4×Dic47—21no—
17C2×#13093983—21no—
18C2×#13093983C2211yescharacteristic
19C22×C4⋊Dic47—21no—
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22C2×#13093951—24no—
23C2×#13093951C2214yes—
24#15689132—22no—
25#15689132C2214yescomplemented
26C2×#13093948—21no—
27C2×#13093948—21no—
28#15689030—22no—
29#15689030C2214yescomplemented
30C2×C47⋊#95—22no—
31#15688891C2214yescomplemented
32#15688884C2214yescomplemented
33C22×#41×C47C2211yescharacteristic
34C23×Q8×C47C2211yescharacteristic
35C2×#137×C47—24no—
36C2×#137×C47—22no—
37C2×#137×C47C418yescomplemented
38C2×#137×C47C2214yes—
39C22×Q16×C47—21no—
40C22×SD16×C47—21no—
41C22×M16×C47C2211yescharacteristic
42C23×Dic47—21no—
43C23×Dic47—22no—
44C23×Dic47D412yes—
45C2×#10625329—21no—
46C2×#10625329—22no—
47C2×#10625329—22no—
48C2×#10625329C2312yes—
49C22×C47⋊C8—21no—
50Q8×Dic47—24no—
51Q8×Dic47—24no—
52D4×Dic47—28no—
53D4×Dic47—24no—
54D4×Dic47—24no—
55D4×Dic47—28no—
56#13093983—24no—
57#13093983—21no—
58#13093983—21no—
59#13093983—24no—
60#13093983D412yes—
61#13093983C2312yes—
62C2×C4⋊Dic47—22no—
63C2×C4⋊Dic47—24no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47D412yes—
70C2×C4⋊Dic47D411yescharacteristic
71C2×C4⋊Dic47D411yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×C22⋊Dic47—22no—
74C2×C22⋊Dic47—21no—
75C2×C4×Dic47—22no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—22no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47C2312yes—
81#13093951—28no—
82#13093948—24no—
83#13093948—24no—
84C47⋊#95—28no—
85C2×#41×C47—21no—
86C2×#41×C47—21no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47C2×C414yes—
90C2×#41×C47D414yes—
91C2×#41×C47D412yes—
92C2×#41×C47C2312yes—
93C22×Q8×C47—21no—
94C22×Q8×C47—22no—
95C22×Q8×C47—21no—
96C22×Q8×C47—21no—
97C22×Q8×C47—22no—
98C22×Q8×C47—21no—
99C22×Q8×C47—21no—
100C22×Q8×C47—22no—
101C22×Q8×C47C2×C414yescomplemented
102C22×Q8×C47D411yescharacteristic
103C22×Q8×C47C2312yes—
104C22×D4×C47—21no—
105C22×D4×C47D411yescharacteristic
106C23×C4×C47—21no—
107C23×C4×C47—21no—
108C23×C4×C47C2311yescharacteristic
109#137×C47—216no—
110#137×C47—28no—
111#137×C47C2×C4116yescomplemented
112C2×Q16×C47—28no—
113C2×Q16×C47—24no—
114C2×SD16×C47—28no—
115C2×SD16×C47—24no—
116C2×M16×C47—22no—
117C2×M16×C47—21no—
118C2×M16×C47C2×C414yescomplemented
119C2×M16×C47C2312yes—
120C22×C8×C47—21no—
121C24×C47—21no—
122C24×C47C2×D411yescharacteristic
123#41×C47—24no—
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—22no—
129#41×C47—22no—
130#41×C47C22⋊C418yes—
131#41×C47C22×C414yes—
132#41×C47C2×D414yes—
133C2×Q8×C47—24no—
134C2×Q8×C47—21no—
135C2×Q8×C47—22no—
136C2×Q8×C47—22no—
137C2×Q8×C47—24no—
138C2×Q8×C47—24no—
139C2×Q8×C47—24no—
140C2×Q8×C47—24no—
141C2×Q8×C47—22no—
142C2×Q8×C47—22no—
143C2×Q8×C47—22no—
144C2×Q8×C47—22no—
145C2×Q8×C47—41no—
146C2×Q8×C47—41no—
147C2×Q8×C47—42no—
148C2×Q8×C47—42no—
149C2×Q8×C47—42no—
150C2×Q8×C47—42no—
151C2×Q8×C47—42no—
152C2×Q8×C47C22⋊C414yes—
153C2×Q8×C47C22×C414yescomplemented
154C2×Q8×C47C2×D412yes—
155C2×D4×C47—24no—
156C2×D4×C47—22no—
157C2×D4×C47—24no—
158C2×D4×C47—22no—
159C2×D4×C47—21no—
160C2×D4×C47—22no—
161C2×D4×C47C22⋊C414yes—
162C2×D4×C47C2×D412yes—
163C22×C4×C47—21no—
164C22×C4×C47—24no—
165C22×C4×C47—22no—
166C22×C4×C47—24no—
167C22×C4×C47—22no—
168C22×C4×C47—24no—
169C22×C4×C47—22no—
170C22×C4×C47—41no—
171C22×C4×C47—41no—
172C22×C4×C47—42no—
173C22×C4×C47C22×C412yes—
174C22×C4×C47C2×D411yescharacteristic
175C22×C4×C47C2×D411yescharacteristic
176C22×C4×C47C2×D412yes—
177C22×C4×C47C2×D411yescharacteristic
178C22×C4×C47C2×D411yescharacteristic
179C22×C4×C47C2×D411yescharacteristic
180C22×C4×C47#4112yes—
181C22×C4×C47#4112yes—
182C22×C4×C47C2411yescharacteristic
183C22×Dic47—24no—
184C22×Dic47—21no—
185C22×Dic47—22no—
186C22×Dic47—21no—
187C22×Dic47—22no—
188C22×Dic47—28no—
189C22×Dic47—22no—
190C22×Dic47—42no—
191C22×Dic47—42no—
192C22×Dic47—44no—
193C22×Dic47—84no—
194C22×Dic47C2×D414yes—
195Q16×C47—216no—
196SD16×C47—216no—
197M16×C47—24no—
198M16×C47—22no—
199M16×C47C22×C414yescomplemented
200C2×C8×C47—24no—
201C2×C8×C47—22no—
202C22⋊Dic47—28no—
203C22⋊Dic47—22no—
204C22⋊Dic47—22no—
205C4⋊Dic47—28no—
206C4⋊Dic47—22no—
207C4⋊Dic47—21no—
208C4⋊Dic47—22no—
209C4⋊Dic47—21no—
210C4⋊Dic47—22no—
211C4⋊Dic47—22no—
212C4⋊Dic47C2×D412yes—
213C4⋊Dic47C2×D412yes—
214C4×Dic47—28no—
215C4×Dic47—22no—
216C4×Dic47—22no—
217C4×Dic47—22no—
218C4×Dic47—22no—
219C4×Dic47—42no—
220#10625329—22no—
221#10625329—22no—
222#10625329—42no—
223C2×C47⋊C8—22no—
224C2×C47⋊C8—42no—
225C23×C47—22no—
226C23×C47—22no—
227C23×C47—22no—
228C23×C47—24no—
229C23×C47—42no—
230C23×C47—42no—
231C23×C47C4×D412yes—
232C23×C47#12011yescharacteristic
233C23×C47#12011yescharacteristic
234C23×C47#12112yes—
235C23×C47C22×D411yescharacteristic
236Q8×C47—22no—
237Q8×C47—24no—
238Q8×C47—24no—
239Q8×C47—24no—
240Q8×C47—42no—
241Q8×C47—42no—
242Q8×C47—44no—
243Q8×C47—44no—
244Q8×C47C2×C22⋊C414yes—
245D4×C47—24no—
246D4×C47—24no—
247D4×C47—28no—
248D4×C47—22no—
249D4×C47—24no—
250D4×C47C2×C22⋊C414yes—
251C2×C4×C47—22no—
252C2×C4×C47—24no—
253C2×C4×C47—24no—
254C2×C4×C47—24no—
255C2×C4×C47—28no—
256C2×C4×C47—24no—
257C2×C4×C47—42no—
258C2×C4×C47—42no—
259C2×C4×C47—44no—
260C2×C4×C47—44no—
261C2×C4×C47C2×C22⋊C412yes—
262C2×C4×C47C2×C22⋊C412yes—
263C2×C4×C47C4×D412yes—
264C2×C4×C47C4×D412yes—
265C2×C4×C47C4×D412yes—
266C2×C4×C47#12011yescharacteristic
267C2×C4×C47#12011yescharacteristic
268C2×C4×C47#12112yes—
269C2×C4×C47#12112yes—
270C2×C4×C47#12112yes—
271C2×C4×C47#12312yes—
272C2×C4×C47#12312yes—
273C2×C4×C47C23×C411yescharacteristic
274C2×C4×C47C22×D411yescharacteristic
275C2×C4×C47C22×D411yescharacteristic
276C2×C4×C47C22×D411yescharacteristic
277C2×C4×C47C2×#4111yescharacteristic
278C2×C4×C47C2×#4111yescharacteristic
279C2×Dic47—24no—
280C2×Dic47—22no—
281C2×Dic47—24no—
282C2×Dic47—22no—
283C2×Dic47—48no—
284C2×Dic47—42no—
285C2×Dic47—42no—
286C2×Dic47—44no—
287C2×Dic47—42no—
288C2×Dic47—42no—
289C2×Dic47—44no—
290C2×Dic47—82no—
291C2×Dic47—88no—
292C8×C47—24no—
293C47⋊C8—42no—
294C2×#1882—471nomaximal, Sylow
295C22×C47—24no—
296C22×C47—24no—
297C22×C47—21no—
298C22×C47—21no—
299C22×C47—21no—
300C22×C47—21no—
301C22×C47—24no—
302C22×C47—48no—
303C22×C47—42no—
304C22×C47—42no—
305C22×C47#38612yes—
306C22×C47#38612yes—
307C22×C47#45512yes—
308C22×C47#45612yes—
309C22×C47C2×C4×D411yescharacteristic
310C22×C47C2×#12011yescharacteristic
311C22×C47C2×#12111yescharacteristic
312C4×C47—24no—
313C4×C47—44no—
314C4×C47#38612yes—
315C4×C47#38612yes—
316C4×C47C22×C22⋊C411yescharacteristic
317C4×C47C2×C4×D411yescharacteristic
318C4×C47C2×#12111yescharacteristic
319C4×C47C2×#12311yescharacteristic
320Dic47—44no—
321Dic47—44no—
322Dic47—42no—
323Dic47—42no—
324C22×#137D4711yescharacteristic
325C2×#520—471no—
326C22×#104—471no—
327C2×#419—471no—
328C2×#391—471no—
329#1882—478no—
330C2×#343—471no—
331C2×#339—471no—
332C2×C47—21no—
333C2×C47—21no—
334C2×C47—22no—
335C2×C47—44no—
336C2×C47#188214yescomplemented
337C2×C47C2×#38611yescharacteristic
338C2×C47C2×#45511yescharacteristic
339C2×C47C2×#45611yescharacteristic
340C22×#41C2×D4711yescharacteristic
341C23×Q8C2×D4711yescharacteristic
342C2×#137—24no—
343C2×#137—22no—
344C2×#137Dic4718yescomplemented
345C2×#137C2×D4714yes—
346C22×Q16—21no—
347C22×SD16—21no—
348C22×M16—471no—
349C22×M16C2×D4711yescharacteristic
350#520—474no—
351#520—942no—
352#520—942no—
353#520—942no—
354C2×C4×Q8—941no—
355C2×C4×D4—942no—
356C2×C4×D4—941no—
357C2×#117—471no—
358C2×#117—941no—
359C22×C4⋊C4—471no—
360C22×C4⋊C4—941no—
361C22×C42—471no—
362C2×#104—474no—
363C2×#104—944no—
364#419—474no—
365#419—942no—
366C2×#103—941no—
367C2×#103—941no—
368#391—474no—
369#391—942no—
370C2×#95—942no—
371#343—474no—
372#339—474no—
373C47C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C2×#41—21no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41C2×Dic4714yes—
379C2×#41C22⋊D4714yes—
380C2×#41C22⋊D4712yes—
381C2×#41C22×D4712yes—
382C22×Q8—21no—
383C22×Q8—22no—
384C22×Q8—21no—
385C22×Q8—21no—
386C22×Q8—22no—
387C22×Q8—21no—
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8C2×Dic4714yescomplemented
391C22×Q8C22⋊D4711yescharacteristic
392C22×Q8C22×D4712yes—
393C22×D4—21no—
394C22×D4C22⋊D4711yescharacteristic
395C23×C4—21no—
396C23×C4—21no—
397C23×C4—472no—
398C23×C4—942no—
399C23×C4—941no—
400C23×C4C22×D4711yescharacteristic
401#137—216no—
402#137—28no—
403#137C2×Dic47116yescomplemented
404C2×Q16—28no—
405C2×Q16—24no—
406C2×SD16—28no—
407C2×SD16—24no—
408C2×M16—22no—
409C2×M16—21no—
410C2×M16—472no—
411C2×M16—941no—
412C2×M16—942no—
413C2×M16—942no—
414C2×M16C2×Dic4714yescomplemented
415C2×M16C22×D4712yes—
416C22×C8—21no—
417C22×C8—941no—
418C4×Q8—944no—
419C4×Q8—944no—
420C4×D4—948no—
421C4×D4—948no—
422C4×D4—944no—
423C4×D4—944no—
424#117—472no—
425#117—472no—
426#117—944no—
427#117—944no—
428#117—941no—
429#117—941no—
430C2×C4⋊C4—471no—
431C2×C4⋊C4—471no—
432C2×C4⋊C4—472no—
433C2×C4⋊C4—472no—
434C2×C4⋊C4—944no—
435C2×C4⋊C4—942no—
436C2×C4⋊C4—944no—
437C2×C4⋊C4—941no—
438C2×C4⋊C4—941no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—941no—
441C2×C22⋊C4—942no—
442C2×C22⋊C4—941no—
443C2×C42—472no—
444C2×C42—942no—
445C2×C42—941no—
446C2×C42—942no—
447C2×C42—942no—
448C2×C42—941no—
449#104—948no—
450#103—944no—
451#103—944no—
452#95—948no—
453C24—21no—
454C24D4×D4711yescharacteristic
455#41—24no—
456#41—24no—
457#41—24no—
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41C22⋊Dic4718yes—
463#41C22×Dic4714yes—
464#41C2×C22⋊D4714yes—
465C2×Q8—24no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—24no—
470C2×Q8—24no—
471C2×Q8—24no—
472C2×Q8—24no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—41no—
478C2×Q8—41no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8C22⋊Dic4714yes—
485C2×Q8C22×Dic4714yescomplemented
486C2×Q8C2×C22⋊D4712yes—
487C2×D4—24no—
488C2×D4—22no—
489C2×D4—24no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4C22⋊Dic4714yes—
494C2×D4C2×C22⋊D4712yes—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—42no—
505C22×C4—474no—
506C22×C4—948no—
507C22×C4—942no—
508C22×C4—944no—
509C22×C4—941no—
510C22×C4—941no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—3764no—
514C22×C4—1884no—
515C22×C4—1882no—
516C22×C4—1882no—
517C22×C4D4×D4711yescharacteristic
518C22×C4D4×D4711yescharacteristic
519C22×C4D4×D4711yescharacteristic
520C22×C4#1062535212yes—
521C22×C4#1062535212yes—
522C22×C4C22×Dic4712yes—
523C22×C4C2×C22⋊D4711yescharacteristic
524C22×C4C2×C22⋊D4711yescharacteristic
525C22×C4C2×C22⋊D4712yes—
526C22×C4C23×D4711yescharacteristic
527Q16—216no—
528SD16—216no—
529M16—24no—
530M16—22no—
531M16—942no—
532M16—942no—
533M16—1882no—
534M16C22×Dic4714yescomplemented
535C2×C8—24no—
536C2×C8—22no—
537C2×C8—942no—
538C2×C8—1882no—
539C4⋊C4—472no—
540C4⋊C4—472no—
541C4⋊C4—948no—
542C4⋊C4—941no—
543C4⋊C4—942no—
544C4⋊C4—942no—
545C4⋊C4—941no—
546C4⋊C4—942no—
547C4⋊C4—942no—
548C22⋊C4—948no—
549C22⋊C4—942no—
550C22⋊C4—942no—
551C42—948no—
552C42—942no—
553C42—942no—
554C42—942no—
555C42—942no—
556C42—1882no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23D4×Dic4712yes—
564C23#1309399211yescharacteristic
565C23#1309399211yescharacteristic
566C23#1309399612yes—
567C23C2×D4×D4711yescharacteristic
568Q8—24no—
569Q8—22no—
570Q8—24no—
571Q8—24no—
572Q8—44no—
573Q8—44no—
574Q8—42no—
575Q8—42no—
576Q8C2×C22⋊Dic4714yes—
577D4—28no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—22no—
582D4C2×C22⋊Dic4714yes—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—944no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—944no—
597C2×C4—3762no—
598C2×C4—3768no—
599C2×C4—1882no—
600C2×C4—1888no—
601C2×C4—1882no—
602C2×C4—1884no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1884no—
606C2×C4C2×C22⋊Dic4712yes—
607C2×C4C2×C22⋊Dic4712yes—
608C2×C4D4×Dic4712yes—
609C2×C4D4×Dic4712yes—
610C2×C4D4×Dic4712yes—
611C2×C4#1309399211yescharacteristic
612C2×C4#1309399211yescharacteristic
613C2×C4#1309399612yes—
614C2×C4#1309399612yes—
615C2×C4#1309399612yes—
616C2×C4#1309400512yes—
617C2×C4#1309400512yes—
618C2×C4C23×Dic4711yescharacteristic
619C2×C4C2×D4×D4711yescharacteristic
620C2×C4C22×C22⋊D4711yescharacteristic
621C2×C4C22×C22⋊D4711yescharacteristic
622C2×C4C2×#1062535211yescharacteristic
623C2×C4C2×#1062535211yescharacteristic
624C8—24no—
625C8—1882no—
626C22—21no—
627C22—21no—
628C22—24no—
629C22—24no—
630C22—21no—
631C22—21no—
632C22—24no—
633C22—42no—
634C22—48no—
635C22—42no—
636C22#1568900312yes—
637C22#1568900312yes—
638C22#1568929912yes—
639C22#1568930412yes—
640C22C2×D4×Dic4711yescharacteristic
641C22C2×#1309399211yescharacteristic
642C22C2×#1309399611yescharacteristic
643C4—24no—
644C4—44no—
645C4—1884no—
646C4—1882no—
647C4—1884no—
648C4—1882no—
649C4#1568900312yes—
650C4#1568900312yes—
651C4C22×C22⋊Dic4711yescharacteristic
652C4C2×D4×Dic4711yescharacteristic
653C4C2×#1309399611yescharacteristic
654C4C2×#1309400511yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—44no—
659C2#2024375914yesdirect product factor
660C2C2×#1568900311yescharacteristic
661C2C2×#1568929911yescharacteristic
662C2C2×#1568930411yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC22×#137D47
hypocenterC47C2×#1882
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C2×#15689132C2semidirect
4. C2×#15689030C2semidirect
5. #20243759C2direct (N × H)
6. C2×#15688891C2semidirect
7. C2×#15688884C2semidirect
25. #15689132C22semidirect
29. #15689030C22semidirect
31. #15688891C22semidirect
32. #15688884C22semidirect
37. C2×#137×C47C4semidirect
101. C22×Q8×C47C2×C4semidirect
111. #137×C47C2×C4semidirect
118. C2×M16×C47C2×C4semidirect
153. C2×Q8×C47C22×C4semidirect
199. M16×C47C22×C4semidirect
336. C2×C47#1882semidirect
344. C2×#137Dic47semidirect
373. C47C2×#1882semidirect
390. C22×Q8C2×Dic47semidirect
403. #137C2×Dic47semidirect
414. C2×M16C2×Dic47semidirect
485. C2×Q8C22×Dic47semidirect
534. M16C22×Dic47semidirect
659. C2#20243759direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896062
C22×#130939512
C2×#156891322
C2×#156890302
#202437592
C2×#156888912
C2×#156888842
C22×#137×C472
C2×#188247