Finite Group Databasebeta

← C2×#20241945 #26183613

Subgroups — C2×#20241945 #26183613

Order 12032.

1498 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689634C211yesmaximal, characteristic
2C2×#15688986C211yesmaximal, characteristic, complemented
3#20241945C218yesmaximal, direct product factor
4C2×#15688861C212yesmaximal, complemented
5C2×#15688858C212yesmaximal, complemented
6C22×C22⋊C8×C47C211yesmaximal, characteristic, complemented
7C22×#10625350C411yescharacteristic, complemented
8#15689634—22no—
9#15689634—22no—
10#15689634C414yescomplemented
11#15689634C2214yes—
12C2×#13094007—21no—
13C2×#13094003C411yescharacteristic, complemented
14C2×#13094003C2211yescharacteristic
15C2×#13093998C411yescharacteristic, complemented
16C2×#13093998C2211yescharacteristic
17C2×#13093993—21no—
18C2×C4×C22⋊D47—21no—
19C2×C4×C22⋊D47—21no—
20C2×#13093983C411yescharacteristic, complemented
21C22×C22⋊Dic47—21no—
22#15688986—22no—
23#15688986C2214yescomplemented
24C2×C47⋊#95—21no—
25C2×C47⋊#95C2212yes—
26#15688861C2218yescomplemented
27#15688858C2218yescomplemented
28C24×C4×C47C2211yescharacteristic
29C23×C8×C47—21no—
30C2×C22⋊C8×C47—21no—
31C2×C22⋊C8×C47—22no—
32C2×C22⋊C8×C47C2212yes—
33C2×C22⋊C8×C47C2214yescomplemented
34C2×#10625350—22no—
35C2×#10625350—44no—
36C2×#10625350C2×C414yescomplemented
37C22×Dic94C2×C411yescharacteristic
38C22×D188C2×C411yescharacteristic
39C22×C22⋊D47—21no—
40C22×C22⋊D47—21no—
41C22×C22⋊D47—21no—
42C23×Dic47—22no—
43C22×C4×D47—21no—
44#13094007—24no—
45#13094007—24no—
46#13094003—22no—
47#13094003—21no—
48#13094003—21no—
49#13094003C2×C412yes—
50#13094003C2×C414yescomplemented
51#13094003C2312yes—
52#13093998—22no—
53#13093998—21no—
54#13093998—21no—
55#13093998C2×C412yes—
56#13093998C2×C414yescomplemented
57#13093998C2312yes—
58#13093993—24no—
59#13093993—24no—
60C4×C22⋊D47—24no—
61C4×C22⋊D47—24no—
62C4×C22⋊D47—24no—
63C4×C22⋊D47—24no—
64#13093983—22no—
65#13093983C2×C414yescomplemented
66C2×C4⋊Dic47C2×C411yescharacteristic
67C2×C4⋊Dic47C2×C411yescharacteristic
68C2×Dic47⋊C4—21no—
69C2×Dic47⋊C4—21no—
70C2×C22⋊Dic47—21no—
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—21no—
73C2×C22⋊Dic47—24no—
74C2×C22⋊Dic47—44no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×C4×Dic47—21no—
78C47⋊#95—24no—
79C47⋊#95—22no—
80C47⋊#95C2314yes—
81C25×C47D411yescharacteristic
82C23×C4×C47—44no—
83C23×C4×C47C2×C411yescharacteristic
84C23×C4×C47C2×C411yescharacteristic
85C23×C4×C47C2×C412yes—
86C23×C4×C47D411yescharacteristic
87C23×C4×C47D411yescharacteristic
88C23×C4×C47D412yes—
89C23×C4×C47D411yescharacteristic
90C23×C4×C47D412yes—
91C23×C4×C47C2311yescharacteristic
92C23×C4×C47C2312yes—
93C22×C8×C47—22no—
94C22×C8×C47—24no—
95C22×C8×C47—44no—
96C22⋊C8×C47—22no—
97C22⋊C8×C47—24no—
98C22⋊C8×C47C2314yes—
99C24×C47—24no—
100C24×C47—24no—
101C24×C47—42no—
102C24×C47C22⋊C411yescharacteristic
103C24×C47C22⋊C412yes—
104C24×C47C22⋊C411yescharacteristic
105C24×C47C2×D412yes—
106C24×C47C2×D411yescharacteristic
107C23×D47—21no—
108C22×C4×C47—24no—
109C22×C4×C47—24no—
110C22×C4×C47—42no—
111C22×C4×C47—44no—
112C22×C4×C47—42no—
113C22×C4×C47—44no—
114C22×C4×C47—42no—
115C22×C4×C47—44no—
116C22×C4×C47—42no—
117C22×C4×C47—44no—
118C22×C4×C47C22⋊C411yescharacteristic
119C22×C4×C47C22⋊C411yescharacteristic
120C22×C4×C47C22⋊C412yes—
121C22×C4×C47C22⋊C412yes—
122C22×C4×C47C22⋊C412yes—
123C22×C4×C47C22⋊C412yes—
124C22×C4×C47C22⋊C412yes—
125C22×C4×C47C22×C411yescharacteristic
126C22×C4×C47C22×C412yes—
127C22×C4×C47C22×C411yescharacteristic
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2411yescharacteristic
136C2×C22⋊D47—22no—
137C2×C22⋊D47—24no—
138C2×C22⋊D47—24no—
139C2×C22⋊D47—22no—
140C2×C22⋊D47—44no—
141C2×C22⋊D47—44no—
142C2×C22⋊D47—44no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—24no—
147C22×Dic47—44no—
148C22×Dic47—44no—
149C22×Dic47—44no—
150#10625350—48no—
151#10625350—48no—
152C2×D188—21no—
153C2×D188—42no—
154C2×D188C22×C412yes—
155C2×C4×D47—24no—
156C2×C4×D47—44no—
157C2×Dic94—21no—
158C2×Dic94—42no—
159C2×Dic94C22×C412yes—
160C2×C8×C47—24no—
161C2×C8×C47—44no—
162C2×C8×C47—48no—
163C22⋊Dic47—24no—
164C22⋊Dic47—22no—
165C22⋊Dic47—24no—
166C22⋊Dic47—22no—
167C22⋊Dic47—48no—
168(C2×D47)⋊C4—22no—
169(C2×D47)⋊C4—24no—
170(C2×D47)⋊C4—22no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47C22×C412yes—
174C4⋊Dic47C22×C412yes—
175Dic47⋊C4—22no—
176Dic47⋊C4—24no—
177Dic47⋊C4—22no—
178C4×Dic47—24no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—24no—
186C23×C47—22no—
187C23×C47—24no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—416no—
193C23×C47#9914yes—
194C23×C47C2×C22⋊C412yes—
195C23×C47C2×C22⋊C412yes—
196C23×C47C2×C22⋊C411yescharacteristic
197C23×C47C2×C22⋊C411yescharacteristic
198C23×C47#13514yes—
199C23×C47C22×D411yescharacteristic
200C22×D47—22no—
201C22×D47—42no—
202C22×D47—42no—
203C22×D47—42no—
204C2×C4×C47—24no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—42no—
217C2×C4×C47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—41no—
220C2×C4×C47—42no—
221C2×C4×C47—41no—
222C2×C4×C47C2×C22⋊C412yes—
223C2×C4×C47C2×C22⋊C411yescharacteristic
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47C2×C22⋊C411yescharacteristic
226C2×C4×C47C23×C411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C22⋊D47—48no—
229C22⋊D47—44no—
230C22⋊D47—48no—
231C22⋊D47—44no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—44no—
239C2×Dic47—44no—
240C2×Dic47—44no—
241C2×Dic47—88no—
242D188—42no—
243D188—42no—
244C4×D47—48no—
245Dic94—42no—
246Dic94—42no—
247C8×C47—44no—
248C2×#1516—471nomaximal, Sylow
249C22×C47—21no—
250C22×C47—21no—
251C22×C47—22no—
252C22×C47—21no—
253C22×C47—22no—
254C22×C47—21no—
255C22×C47—28no—
256C22×C47—28no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—48no—
261C22×C47—48no—
262C22×C47—44no—
263C22×C47C2×#9914yes—
264C22×C47C2×#9912yes—
265C22×C47#41612yes—
266C22×C47#41612yes—
267C22×C47#42112yes—
268C22×C47C22×C22⋊C411yescharacteristic
269C22×C47C2×#13512yes—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—84no—
274C4×C47—21no—
275C4×C47—21no—
276C4×C47—21no—
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—21no—
280Dic47—42no—
281Dic47—42no—
282Dic47—42no—
283Dic47—84no—
284C2×#525—471no—
285C22×C22⋊C8D4711yescharacteristic, complemented
286C2×#381—471no—
287#1516—478no—
288C2×#328—472no—
289C2×#327—472no—
290C2×C47—22no—
291C2×C47—24no—
292C2×C47—22no—
293C2×C47—42no—
294C2×C47#151614yescomplemented
295C2×C47C22×#9911yescharacteristic
296C2×C47C2×#41611yescharacteristic
297C2×C47C2×#42111yescharacteristic
298D47—82no—
299C22×#41—471no—
300C24×C4C2×D4711yescharacteristic
301C23×C8—21no—
302#525—474no—
303#525—474no—
304#525—942no—
305#525—942no—
306C2×#123—941no—
307C2×C22⋊Q8—471no—
308C2×C22⋊Q8—471no—
309C2×#121—471no—
310C2×#121—471no—
311C2×#120—941no—
312C2×C4×D4—941no—
313C2×C4×D4—941no—
314C2×#117—471no—
315C22×C22⋊C4—941no—
316C2×C22⋊C8—22no—
317C2×C22⋊C8—21no—
318C2×C22⋊C8C2×D4712yes—
319C2×C22⋊C8C2×D4714yescomplemented
320#381—474no—
321#381—942no—
322C2×#95—472no—
323C2×#95—941no—
324#328—478no—
325#327—478no—
326C47C2×#151611yescharacteristic, complemented, Sylow
327C25D18811yescharacteristic
328C2×#41—474no—
329C2×#41—942no—
330C2×#41—1884no—
331C22×Q8—471no—
332C22×D4—471no—
333C22×D4—941no—
334C22×D4—941no—
335C22×D4—941no—
336C23×C4—44no—
337C23×C4—941no—
338C23×C4—942no—
339C23×C4C4×D4711yescharacteristic
340C23×C4C4×D4711yescharacteristic
341C23×C4C4×D4712yes—
342C23×C4D18811yescharacteristic
343C23×C4D18812yes—
344C23×C4C22⋊D4711yescharacteristic
345C23×C4C22⋊D4711yescharacteristic
346C23×C4C22⋊D4712yes—
347C23×C4C22×D4711yescharacteristic
348C23×C4C22×D4712yes—
349C22×C8—22no—
350C22×C8—24no—
351C22×C8—44no—
352#123—944no—
353#123—944no—
354C22⋊Q8—472no—
355C22⋊Q8—474no—
356C22⋊Q8—472no—
357C22⋊Q8—942no—
358C22⋊Q8—941no—
359C22⋊Q8—941no—
360#121—472no—
361#121—474no—
362#121—472no—
363#121—942no—
364#121—941no—
365#121—941no—
366#120—944no—
367#120—944no—
368C4×D4—944no—
369C4×D4—944no—
370C4×D4—944no—
371C4×D4—944no—
372#117—474no—
373#117—942no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—944no—
384C2×C22⋊C4—1884no—
385C2×C42—941no—
386C22⋊C8—22no—
387C22⋊C8—24no—
388C22⋊C8C22×D4714yes—
389#95—474no—
390#95—942no—
391#95—944no—
392C24—24no—
393C24—24no—
394C24—42no—
395C24—941no—
396C24(C2×D47)⋊C411yescharacteristic
397C24(C2×D47)⋊C412yes—
398C24(C2×D47)⋊C411yescharacteristic
399C24C2×D18812yes—
400C24C2×D18811yescharacteristic
401#41—1888no—
402#41—1888no—
403C2×Q8—472no—
404C2×Q8—941no—
405C2×Q8—1882no—
406C2×D4—472no—
407C2×D4—944no—
408C2×D4—942no—
409C2×D4—944no—
410C2×D4—942no—
411C2×D4—941no—
412C2×D4—1882no—
413C2×D4—1884no—
414C2×D4—1884no—
415C2×D4—1884no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—944no—
430C22×C4—944no—
431C22×C4—1884no—
432C22×C4—1884no—
433C22×C4—1884no—
434C22×C4—1884no—
435C22×C4(C2×D47)⋊C412yes—
436C22×C4(C2×D47)⋊C412yes—
437C22×C4(C2×D47)⋊C411yescharacteristic
438C22×C4(C2×D47)⋊C411yescharacteristic
439C22×C4(C2×D47)⋊C412yes—
440C22×C4(C2×D47)⋊C412yes—
441C22×C4(C2×D47)⋊C412yes—
442C22×C4C2×C4×D4712yes—
443C22×C4C2×C4×D4711yescharacteristic
444C22×C4C2×C4×D4712yes—
445C22×C4C2×C4×D4711yescharacteristic
446C22×C4C2×D18812yes—
447C22×C4C2×D18811yescharacteristic
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C2×C22⋊D4712yes—
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C23×D4711yescharacteristic
453C2×C8—24no—
454C2×C8—44no—
455C2×C8—48no—
456C4⋊C4—472no—
457C4⋊C4—472no—
458C4⋊C4—942no—
459C4⋊C4—944no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—942no—
463C22⋊C4—944no—
464C22⋊C4—944no—
465C22⋊C4—942no—
466C22⋊C4—942no—
467C22⋊C4—942no—
468C22⋊C4—944no—
469C22⋊C4—942no—
470C22⋊C4—1888no—
471C42—944no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—416no—
486C23—942no—
487C23—1882no—
488C23—1882no—
489C23—1882no—
490C23#1309393814yes—
491C23C2×(C2×D47)⋊C412yes—
492C23C2×(C2×D47)⋊C412yes—
493C23C2×(C2×D47)⋊C411yescharacteristic
494C23C2×(C2×D47)⋊C411yescharacteristic
495C23#1309404614yes—
496C23C22×D18811yescharacteristic
497Q8—1882no—
498Q8—1882no—
499D4—1882no—
500D4—1882no—
501D4—1888no—
502D4—1888no—
503D4—1884no—
504D4—1884no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—3768no—
527C2×C4—1888no—
528C2×C4—1884no—
529C2×C4—1884no—
530C2×C4—1884no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4C2×(C2×D47)⋊C412yes—
535C2×C4C2×(C2×D47)⋊C411yescharacteristic
536C2×C4C2×(C2×D47)⋊C412yes—
537C2×C4C2×(C2×D47)⋊C411yescharacteristic
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4C22×C22⋊D4711yescharacteristic
540C8—44no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—28no—
545C22—28no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—48no—
553C22—48no—
554C22—44no—
555C22—3764no—
556C22—1882no—
557C22—1882no—
558C22—1882no—
559C22C2×#1309393814yes—
560C22C2×#1309393812yes—
561C22#1568912012yes—
562C22#1568912012yes—
563C22#1568913912yes—
564C22C22×(C2×D47)⋊C411yescharacteristic
565C22C2×#1309404612yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—3764no—
573C4—1882no—
574C4—1882no—
575C4—1882no—
576C2—22no—
577C2—24no—
578C2—22no—
579C2—42no—
580C2—3762no—
581C2#2024194514yesdirect product factor
582C2C22×#1309393811yescharacteristic
583C2C2×#1568912011yescharacteristic
584C2C2×#1568913911yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C22⋊C8×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689634C2
cosocleC22×C4×C47C24
hypercenterC22×C22⋊C8D47
hypocenterC47C2×#1516
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15688986C2semidirect
3. #20241945C2direct (N × H)
4. C2×#15688861C2semidirect
5. C2×#15688858C2semidirect
6. C22×C22⋊C8×C47C2semidirect
7. C22×#10625350C4semidirect
10. #15689634C4semidirect
13. C2×#13094003C4semidirect
15. C2×#13093998C4semidirect
20. C2×#13093983C4semidirect
23. #15688986C22semidirect
26. #15688861C22semidirect
27. #15688858C22semidirect
33. C2×C22⋊C8×C47C22semidirect
36. C2×#10625350C2×C4semidirect
50. #13094003C2×C4semidirect
56. #13093998C2×C4semidirect
65. #13093983C2×C4semidirect
285. C22×C22⋊C8D47semidirect
294. C2×C47#1516semidirect
319. C2×C22⋊C8C2×D47semidirect
326. C47C2×#1516semidirect
581. C2#20241945direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896342
C2×#156889862
#202419452
C2×#156888612
C2×#156888582
C22×C22⋊C8×C472
C2×#151647