Finite Group Databasebeta

← C2×#20243623 #26217142

Subgroups — C2×#20243623 #26217142

Order 12032.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3C2×#15689124C211yesmaximal, characteristic
4C2×#15688989C211yesmaximal, characteristic, complemented
5#20243623C218yesmaximal, direct product factor
6C2×#15688891C211yesmaximal, characteristic, complemented
7C2×#15688883C211yesmaximal, characteristic, complemented
8C22×#104×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
22#15689165—22no—
23#15689165C2214yescomplemented
24C2×#13093958—21no—
25C2×#13093956—21no—
26#15689124—22no—
27#15689124C2214yes—
28C2×#13093948—21no—
29C2×#13093945—21no—
30#15688989—22no—
31#15688989C2214yescomplemented
32C2×C47⋊#95—22no—
33#15688891C2214yescomplemented
34#15688883C2214yescomplemented
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C2×#104×C47C418yescomplemented
39C2×#104×C47C2214yes—
40C23×Dic47—22no—
41C23×Dic47—21no—
42C23×Dic47D412yes—
43C2×#10625329—21no—
44C2×#10625329—22no—
45C2×#10625329—22no—
46C2×#10625329C2312yes—
47C22×C47⋊C8—21no—
48Q8×Dic47—24no—
49Q8×Dic47—24no—
50D4×Dic47—28no—
51D4×Dic47—24no—
52D4×Dic47—24no—
53D4×Dic47—28no—
54#13093983—24no—
55#13093983—21no—
56#13093983—21no—
57#13093983—24no—
58#13093983D412yes—
59#13093983C2312yes—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47—24no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47D412yes—
73C2×C4⋊Dic47D411yescharacteristic
74C2×C4⋊Dic47D411yescharacteristic
75C2×C4⋊Dic47C2312yes—
76C2×C4⋊Dic47C2312yes—
77C2×C22⋊Dic47—22no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—22no—
80C2×C4×Dic47—21no—
81#13093958—24no—
82#13093956—24no—
83#13093948—24no—
84#13093945—24no—
85C47⋊#95—28no—
86C2×#41×C47—21no—
87C2×#41×C47—21no—
88C2×#41×C47—22no—
89C2×#41×C47—22no—
90C2×#41×C47C2×C414yes—
91C2×#41×C47D412yes—
92C2×#41×C47Q814yes—
93C2×#41×C47C2312yes—
94C22×Q8×C47D411yescharacteristic
95C22×D4×C47—21no—
96C22×D4×C47D411yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47—21no—
99C23×C4×C47C2311yescharacteristic
100C2×M16×C47—21no—
101C2×M16×C47—22no—
102C2×M16×C47C2×C414yescomplemented
103C2×M16×C47C2312yes—
104C22×C8×C47—21no—
105C2×C42×C47—21no—
106C2×C42×C47—22no—
107C2×C42×C47C2×C414yescomplemented
108C2×C42×C47C2312yes—
109#104×C47C2×C4116yescomplemented
110C24×C47—21no—
111C24×C47C2×D411yescharacteristic
112#41×C47—24no—
113#41×C47—24no—
114#41×C47—24no—
115#41×C47—24no—
116#41×C47—24no—
117#41×C47—22no—
118#41×C47—22no—
119#41×C47C4⋊C418yes—
120#41×C47C22×C414yes—
121#41×C47C2×Q814yes—
122C2×Q8×C47—21no—
123C2×Q8×C47—22no—
124C2×Q8×C47C4⋊C414yes—
125C2×Q8×C47C2×D412yes—
126C2×D4×C47—24no—
127C2×D4×C47—22no—
128C2×D4×C47—24no—
129C2×D4×C47—22no—
130C2×D4×C47—21no—
131C2×D4×C47—22no—
132C2×D4×C47C4⋊C414yes—
133C2×D4×C47C2×D412yes—
134C22×C4×C47—21no—
135C22×C4×C47—24no—
136C22×C4×C47—22no—
137C22×C4×C47—24no—
138C22×C4×C47—22no—
139C22×C4×C47—24no—
140C22×C4×C47—22no—
141C22×C4×C47—42no—
142C22×C4×C47—42no—
143C22×C4×C47C22×C412yes—
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×Q812yes—
150C22×C4×C47#4112yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—21no—
154C22×Dic47—22no—
155C22×Dic47—21no—
156C22×Dic47—28no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—44no—
160C22×Dic47—44no—
161C22×Dic47—41no—
162C22×Dic47—41no—
163C22×Dic47—84no—
164C22×Dic47C2×D414yes—
165M16×C47—24no—
166M16×C47—22no—
167M16×C47C22×C414yescomplemented
168C2×C8×C47—24no—
169C2×C8×C47—22no—
170C42×C47—24no—
171C42×C47—22no—
172C42×C47C22×C414yescomplemented
173C22⋊Dic47—28no—
174C22⋊Dic47—22no—
175C22⋊Dic47—22no—
176C4⋊Dic47—28no—
177C4⋊Dic47—22no—
178C4⋊Dic47—21no—
179C4⋊Dic47—22no—
180C4⋊Dic47—21no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47—22no—
185C4⋊Dic47—41no—
186C4⋊Dic47—41no—
187C4⋊Dic47C2×D412yes—
188C4⋊Dic47C2×D412yes—
189C4×Dic47—28no—
190C4×Dic47—22no—
191C4×Dic47—22no—
192#10625329—22no—
193#10625329—22no—
194#10625329—42no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—42no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—24no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47C4×D412yes—
204C23×C47#12011yescharacteristic
205C23×C47#12011yescharacteristic
206C23×C47C22⋊Q812yes—
207C23×C47C22×D411yescharacteristic
208Q8×C47—22no—
209Q8×C47—24no—
210Q8×C47C2×C4⋊C414yes—
211D4×C47—24no—
212D4×C47—24no—
213D4×C47—28no—
214D4×C47—22no—
215D4×C47—24no—
216D4×C47C2×C4⋊C414yes—
217C2×C4×C47—22no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—28no—
222C2×C4×C47—24no—
223C2×C4×C47—48no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47C2×C4⋊C412yes—
227C2×C4×C47C2×C4⋊C412yes—
228C2×C4×C47C4×D412yes—
229C2×C4×C47C4×D412yes—
230C2×C4×C47C4×D412yes—
231C2×C4×C47#12011yescharacteristic
232C2×C4×C47#12011yescharacteristic
233C2×C4×C47C22⋊Q812yes—
234C2×C4×C47C22⋊Q812yes—
235C2×C4×C47C22⋊Q812yes—
236C2×C4×C47#12312yes—
237C2×C4×C47#12312yes—
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×Q811yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—22no—
247C2×Dic47—42no—
248C2×Dic47—44no—
249C2×Dic47—42no—
250C2×Dic47—44no—
251C2×Dic47—44no—
252C2×Dic47—42no—
253C2×Dic47—82no—
254C2×Dic47—88no—
255C2×Dic47—82no—
256C2×Dic47—82no—
257C8×C47—24no—
258C47⋊C8—42no—
259C2×#1860—471nomaximal, Sylow
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—21no—
263C22×C47—21no—
264C22×C47—21no—
265C22×C47—21no—
266C22×C47—24no—
267C22×C47—48no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47#38512yes—
271C22×C47#38512yes—
272C22×C47#45412yes—
273C22×C47#45512yes—
274C22×C47C2×C4×D411yescharacteristic
275C22×C47C2×#12011yescharacteristic
276C22×C47C2×C22⋊Q811yescharacteristic
277C4×C47—24no—
278C4×C47—44no—
279C4×C47#38512yes—
280C4×C47#38512yes—
281C4×C47C22×C4⋊C411yescharacteristic
282C4×C47C2×C4×D411yescharacteristic
283C4×C47C2×C22⋊Q811yescharacteristic
284C4×C47C2×#12311yescharacteristic
285Dic47—44no—
286Dic47—42no—
287Dic47—42no—
288Dic47—82no—
289C2×#520—471no—
290C2×#428—471no—
291C22×#104D4711yescharacteristic
292C2×#417—471no—
293C2×#382—471no—
294#1860—478no—
295C2×#343—471no—
296C2×#339—471no—
297C2×C47—21no—
298C2×C47—21no—
299C2×C47—22no—
300C2×C47—44no—
301C2×C47#186014yescomplemented
302C2×C47C2×#38511yescharacteristic
303C2×C47C2×#45511yescharacteristic
304C2×C47C2×#45411yescharacteristic
305C22×#41C2×D4711yescharacteristic
306C22×M16—471no—
307C22×M16C2×D4711yescharacteristic
308#520—474no—
309#520—942no—
310#520—942no—
311#520—942no—
312C2×C4×Q8—941no—
313C2×C4×D4—942no—
314C2×C4×D4—941no—
315C2×#117—471no—
316C2×#117—941no—
317C22×C4⋊C4—471no—
318C22×C4⋊C4—471no—
319C22×C4⋊C4—941no—
320C22×C42C2×D4711yescharacteristic
321#428—474no—
322#428—942no—
323C2×#107—941no—
324C2×#106—941no—
325C2×#104Dic4718yescomplemented
326C2×#104C2×D4714yes—
327#417—474no—
328#417—942no—
329C2×#103—941no—
330C2×#102—941no—
331#382—474no—
332#382—942no—
333C2×#95—942no—
334#343—474no—
335#339—474no—
336C47C2×#186011yescharacteristic, complemented, Sylow
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41Dic9414yes—
342C2×#41D18812yes—
343C2×#41C2×Dic4714yes—
344C2×#41C22×D4712yes—
345C22×Q8D18811yescharacteristic
346C22×D4—21no—
347C22×D4D18811yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—472no—
351C23×C4—942no—
352C23×C4—941no—
353C23×C4C22×D4711yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—472no—
357C2×M16—941no—
358C2×M16—942no—
359C2×M16—942no—
360C2×M16C2×Dic4714yescomplemented
361C2×M16C22×D4712yes—
362C22×C8—21no—
363C22×C8—941no—
364C4×Q8—944no—
365C4×Q8—944no—
366C4×D4—948no—
367C4×D4—948no—
368C4×D4—944no—
369C4×D4—944no—
370#117—472no—
371#117—472no—
372#117—944no—
373#117—944no—
374#117—941no—
375#117—941no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—472no—
379C2×C4⋊C4—472no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—944no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C22⋊C4—942no—
394C2×C22⋊C4—941no—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—942no—
398C2×C42—941no—
399C2×C42C2×Dic4714yescomplemented
400C2×C42C22×D4712yes—
401#107—944no—
402#106—944no—
403#104C2×Dic47116yescomplemented
404#103—944no—
405#102—944no—
406#95—948no—
407C24—21no—
408C24D4×D4711yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41C4⋊Dic4718yes—
417#41C2×Dic9414yes—
418#41C22×Dic4714yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8C4⋊Dic4714yes—
422C2×Q8C2×D18812yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4C4⋊Dic4714yes—
430C2×D4C2×D18812yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—474no—
441C22×C4—948no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—3764no—
448C22×C4—1884no—
449C22×C4—1881no—
450C22×C4—1881no—
451C22×C4—1884no—
452C22×C4C2×Dic9412yes—
453C22×C4C2×D18811yescharacteristic
454C22×C4C2×D18811yescharacteristic
455C22×C4D4×D4711yescharacteristic
456C22×C4D4×D4711yescharacteristic
457C22×C4D4×D4711yescharacteristic
458C22×C4#1062535212yes—
459C22×C4#1062535212yes—
460C22×C4C22×Dic4712yes—
461C22×C4C23×D4711yescharacteristic
462M16—24no—
463M16—22no—
464M16—942no—
465M16—942no—
466M16—1882no—
467M16C22×Dic4714yescomplemented
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—942no—
471C2×C8—1882no—
472C4⋊C4—472no—
473C4⋊C4—472no—
474C4⋊C4—948no—
475C4⋊C4—941no—
476C4⋊C4—942no—
477C4⋊C4—942no—
478C4⋊C4—941no—
479C4⋊C4—942no—
480C4⋊C4—942no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—1881no—
484C4⋊C4—1881no—
485C22⋊C4—948no—
486C22⋊C4—942no—
487C22⋊C4—942no—
488C42—24no—
489C42—22no—
490C42—948no—
491C42—942no—
492C42—942no—
493C42C22×Dic4714yescomplemented
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23D4×Dic4712yes—
501C23#1309399411yescharacteristic
502C23#1309399411yescharacteristic
503C23#1309400112yes—
504C23C2×D4×D4711yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×C4⋊Dic4714yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×C4⋊Dic4714yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—944no—
526C2×C4—3762no—
527C2×C4—3762no—
528C2×C4—3762no—
529C2×C4—3768no—
530C2×C4—1884no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1884no—
535C2×C4—1884no—
536C2×C4C2×C4⋊Dic4712yes—
537C2×C4C2×C4⋊Dic4712yes—
538C2×C4D4×Dic4712yes—
539C2×C4D4×Dic4712yes—
540C2×C4D4×Dic4712yes—
541C2×C4#1309399411yescharacteristic
542C2×C4#1309399411yescharacteristic
543C2×C4#1309400112yes—
544C2×C4#1309400112yes—
545C2×C4#1309400112yes—
546C2×C4#1309400812yes—
547C2×C4#1309400812yes—
548C2×C4C23×Dic4711yescharacteristic
549C2×C4C2×D4×D4711yescharacteristic
550C2×C4C22×D18811yescharacteristic
551C2×C4C22×Dic9411yescharacteristic
552C2×C4C2×#1062535211yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C8—24no—
555C8—1882no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22#1568899812yes—
567C22#1568899812yes—
568C22#1568929712yes—
569C22#1568930212yes—
570C22C2×D4×Dic4711yescharacteristic
571C22C2×#1309399411yescharacteristic
572C22C2×#1309400111yescharacteristic
573C4—24no—
574C4—44no—
575C4—3762no—
576C4—1882no—
577C4—1884no—
578C4—1882no—
579C4#1568899812yes—
580C4#1568899812yes—
581C4C22×C4⋊Dic4711yescharacteristic
582C4C2×D4×Dic4711yescharacteristic
583C4C2×#1309400111yescharacteristic
584C4C2×#1309400811yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2#2024362314yesdirect product factor
590C2C2×#1568899811yescharacteristic
591C2C2×#1568930211yescharacteristic
592C2C2×#1568929711yescharacteristic
subgroupG/H
centerC23#13093994
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#104×C47C2
socleC23×C47#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC22×#104D47
hypocenterC47C2×#1860
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
4. C2×#15688989C2semidirect
5. #20243623C2direct (N × H)
6. C2×#15688891C2semidirect
7. C2×#15688883C2semidirect
23. #15689165C22semidirect
31. #15688989C22semidirect
33. #15688891C22semidirect
34. #15688883C22semidirect
38. C2×#104×C47C4semidirect
102. C2×M16×C47C2×C4semidirect
107. C2×C42×C47C2×C4semidirect
109. #104×C47C2×C4semidirect
167. M16×C47C22×C4semidirect
172. C42×C47C22×C4semidirect
301. C2×C47#1860semidirect
325. C2×#104Dic47semidirect
336. C47C2×#1860semidirect
360. C2×M16C2×Dic47semidirect
399. C2×C42C2×Dic47semidirect
403. #104C2×Dic47semidirect
467. M16C22×Dic47semidirect
493. C42C22×Dic47semidirect
589. C2#20243623direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#156891652
C2×#156891242
C2×#156889892
#202436232
C2×#156888912
C2×#156888832
C22×#104×C472
C2×#186047