Finite Group Databasebeta

← C2×#20253308 #27084053

Subgroups — C2×#20253308 #27084053

Order 12032.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic47C211yesmaximal, characteristic, complemented
2#20253308C2116yesmaximal, direct product factor
3C2×#15689456C212yesmaximal, complemented
4C2×#15689439C212yesmaximal, complemented
5C2×#15689451C212yesmaximal, complemented
6C2×#15689434C212yesmaximal, complemented
7C2×#15689166C212yesmaximal, complemented
8C2×#15689124C212yesmaximal, complemented
9C2×#15689087C211yesmaximal, characteristic, complemented
10C2×C47×#525C211yesmaximal, characteristic
11C24×Dic47—21no—
12C22×#10625329C2212yes—
13C22×C4⋊Dic47—21no—
14C22×C4⋊Dic47—21no—
15C22×C4⋊Dic47—21no—
16C22×C4⋊Dic47—24no—
17C22×C4⋊Dic47C2212yes—
18C22×C4⋊Dic47C2212yes—
19C22×C4⋊Dic47C2212yes—
20C22×C4⋊Dic47C2218yescomplemented
21#15689456C22116yescomplemented
22#15689451C22116yescomplemented
23#15689439C22116yescomplemented
24#15689434C22116yescomplemented
25#15689166C22116yescomplemented
26C2×#13093957C2214yescomplemented
27C2×#13093955C2214yescomplemented
28#15689124C22116yescomplemented
29C2×#13093948C2214yescomplemented
30C2×#13093945C2214yescomplemented
31#15689087C2218yescomplemented
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×C22⋊(C47⋊C8)C2212yes—
34C22×#41×C47C2211yescharacteristic
35C24×C4×C47C2211yescharacteristic
36C47×#525—24no—
37C47×#525C2218yes—
38C2×#123×C47—21no—
39C2×C22⋊Q8×C47C2211yescharacteristic
40C2×C22⋊Q8×C47C2211yescharacteristic
41C2×#121×C47C2211yescharacteristic
42C2×#121×C47C2211yescharacteristic
43C2×#120×C47—21no—
44C2×C4×D4×C47—21no—
45C2×C4×D4×C47—21no—
46C2×#117×C47C2211yescharacteristic
47C23×Dic47—22no—
48C23×Dic47—22no—
49C23×Dic47—22no—
50C23×Dic47—28no—
51C23×Dic47—48no—
52C2×#10625329—28no—
53C2×#10625329C2318yes—
54C22×C47⋊C8C2314yes—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—24no—
58C2×C4⋊Dic47—24no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47—24no—
61C2×C4⋊Dic47—28no—
62C2×C4⋊Dic47—28no—
63C2×C4⋊Dic47—28no—
64C2×C4⋊Dic47C2314yes—
65C2×C4⋊Dic47C2318yes—
66C2×C4⋊Dic47C2318yes—
67C2×C4⋊Dic47C2318yes—
68#13093957C23116yescomplemented
69#13093955C23116yescomplemented
70#13093948C23116yescomplemented
71#13093945C23116yescomplemented
72C22⋊(C47⋊C8)C2318yes—
73C22⋊(C47⋊C8)C2318yes—
74C25×C47D411yescharacteristic
75C2×#41×C47—22no—
76C2×#41×C47—28no—
77C2×#41×C47C2314yes—
78C22×Q8×C47C2311yescharacteristic
79C22×D4×C47—21no—
80C22×D4×C47—21no—
81C22×D4×C47—21no—
82C22×D4×C47C2311yescharacteristic
83C23×C4×C47—28no—
84C23×C4×C47—21no—
85C23×C4×C47D411yescharacteristic
86C23×C4×C47D411yescharacteristic
87C23×C4×C47D411yescharacteristic
88C23×C4×C47D414yes—
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2311yescharacteristic
91C23×C4×C47C2314yes—
92C23×C4×C47C2311yescharacteristic
93#123×C47—28no—
94C22⋊Q8×C47—22no—
95C22⋊Q8×C47—22no—
96C22⋊Q8×C47C2314yes—
97C22⋊Q8×C47C2314yes—
98#121×C47—22no—
99#121×C47—22no—
100#121×C47C2314yes—
101#121×C47C2314yes—
102#120×C47—28no—
103C4×D4×C47—28no—
104C4×D4×C47—28no—
105#117×C47—22no—
106#117×C47C2314yes—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47—21no—
116C2×C42×C47—21no—
117C24×C47—24no—
118C24×C47—24no—
119C24×C47—24no—
120C24×C47—21no—
121C24×C47C2×D414yes—
122C24×C47C2×D411yescharacteristic
123C24×C47C2×D411yescharacteristic
124C24×C47C2×D411yescharacteristic
125#41×C47—216no—
126#41×C47—48no—
127C2×Q8×C47—21no—
128C2×Q8×C47—24no—
129C2×Q8×C47C2412yes—
130C2×D4×C47—24no—
131C2×D4×C47—24no—
132C2×D4×C47—24no—
133C2×D4×C47—21no—
134C2×D4×C47—24no—
135C2×D4×C47—44no—
136C2×D4×C47—44no—
137C2×D4×C47—44no—
138C2×D4×C47C2412yes—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—216no—
143C22×C4×C47—24no—
144C22×C4×C47—24no—
145C22×C4×C47—216no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—24no—
150C22×C4×C47—44no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D414yes—
154C22×C4×C47C2×D414yes—
155C22×C4×C47C2×D412yes—
156C22×C4×C47C2×D412yes—
157C22×C4×C47C2×D414yes—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D412yes—
160C22×C4×C47#4118yes—
161C22×C4×C47C2412yes—
162C22×C4×C47C2412yes—
163C22×C4×C47C2411yescharacteristic
164C22×C4×C47C2412yes—
165C22×Dic47—28no—
166C22×Dic47—28no—
167C22×Dic47—28no—
168C22×Dic47—24no—
169C22×Dic47—48no—
170C22×Dic47—48no—
171C22×Dic47—432no—
172C22×Dic47—48no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—24no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47C2412yes—
178C4⋊C4×C47C2412yes—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—24no—
184C42×C47—24no—
185C4⋊Dic47—28no—
186C4⋊Dic47—28no—
187C4⋊Dic47—28no—
188C4⋊Dic47—24no—
189C4⋊Dic47C2418yes—
190#10625329—216no—
191C2×C47⋊C8—28no—
192C2×C47⋊C8C2418yes—
193C23×C47—232no—
194C23×C47—28no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—28no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47—28no—
201C23×C47—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47#12314yes—
208C23×C47AGL(1,ℤ/8)14yes—
209C23×C47#13714yes—
210C23×C47C22×D412yes—
211C23×C47C22×D412yes—
212C23×C47C22×D412yes—
213C23×C47C22×D411yescharacteristic
214Q8×C47—24no—
215Q8×C47—42no—
216D4×C47—24no—
217D4×C47—48no—
218D4×C47—48no—
219D4×C47—48no—
220D4×C47—42no—
221C2×C4×C47—28no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—28no—
225C2×C4×C47—28no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—48no—
237C2×C4×C47#12314yes—
238C2×C4×C47#12318yes—
239C2×C4×C47C22×D412yes—
240C2×C4×C47C22×D412yes—
241C2×C4×C47C22×D412yes—
242C2×C4×C47C22×D411yescharacteristic
243C2×C4×C47C2×#4114yes—
244C2×C4×C47C2×#4118yes—
245C2×C4×C47C2511yescharacteristic
246C2×Dic47—28no—
247C2×Dic47—416no—
248C2×Dic47—416no—
249C2×Dic47—416no—
250C2×Dic47—48no—
251C47⋊C8—28no—
252C2×#3092—471nomaximal, Sylow
253C22×C47—232no—
254C22×C47—28no—
255C22×C47—28no—
256C22×C47—28no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47—84no—
267C22×C47C2×#12314yes—
268C22×C47C2×#12312yes—
269C22×C47C2×AGL(1,ℤ/8)14yes—
270C22×C47C2×AGL(1,ℤ/8)12yes—
271C22×C47C2×#13714yes—
272C22×C47C2×#13712yes—
273C22×C47C23×D411yescharacteristic
274C4×C47—22no—
275C4×C47—22no—
276C4×C47—42no—
277C4×C47—42no—
278C4×C47—42no—
279C4×C47C2×#12314yes—
280C4×C47C2×#12312yes—
281C4×C47C22×#4112yes—
282Dic47—48no—
283C2×#525D4711yescharacteristic
284C23×C4⋊C4—471no—
285#3092—4716no—
286C2×#484—472no—
287C2×#481—472no—
288C2×#483—472no—
289C2×#480—472no—
290C2×#428—472no—
291C2×#417—472no—
292C2×#407—471no—
293C2×C47—24no—
294C2×C47—24no—
295C2×C47—24no—
296C2×C47—82no—
297C2×C47#309214yescomplemented
298C2×C47C22×#12311yescharacteristic
299C2×C47C22×AGL(1,ℤ/8)11yescharacteristic
300C2×C47C22×#13711yescharacteristic
301C22×#41C2×D4711yescharacteristic
302C24×C4—941no—
303C24×C4C2×D4711yescharacteristic
304C22×M16—472no—
305#525—24no—
306#525C2×D4718yes—
307C2×#123—21no—
308C2×C22⋊Q8C2×D4711yescharacteristic
309C2×C22⋊Q8C2×D4711yescharacteristic
310C2×#121C2×D4711yescharacteristic
311C2×#121C2×D4711yescharacteristic
312C2×#120—21no—
313C2×C4×D4—21no—
314C2×C4×D4—21no—
315C2×#117C2×D4711yescharacteristic
316C22×C4⋊C4—472no—
317C22×C4⋊C4—472no—
318C22×C4⋊C4—478no—
319C22×C4⋊C4—472no—
320C22×C4⋊C4—941no—
321C22×C4⋊C4—941no—
322C22×C4⋊C4—941no—
323C22×C4⋊C4—944no—
324#484—4716no—
325#483—4716no—
326#481—4716no—
327#480—4716no—
328#428—4716no—
329C2×#107—474no—
330C2×#106—474no—
331#417—4716no—
332C2×#103—474no—
333C2×#102—474no—
334#407—478no—
335C2×C22⋊C8—472no—
336C2×C22⋊C8—472no—
337C47C2×#309211yescharacteristic, complemented, Sylow
338C25C22⋊D4711yescharacteristic
339C2×#41—22no—
340C2×#41—28no—
341C2×#41C22×D4714yes—
342C22×Q8C22×D4711yescharacteristic
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4C22×D4711yescharacteristic
347C23×C4—28no—
348C23×C4—21no—
349C23×C4—948no—
350C23×C4—942no—
351C23×C4—942no—
352C23×C4—942no—
353C23×C4—1888no—
354C23×C4C22⋊D4711yescharacteristic
355C23×C4C22⋊D4711yescharacteristic
356C23×C4C22⋊D4711yescharacteristic
357C23×C4C22⋊D4714yes—
358C23×C4C22×D4711yescharacteristic
359C23×C4C22×D4711yescharacteristic
360C23×C4C22×D4714yes—
361C23×C4C22×D4711yescharacteristic
362C2×M16—478no—
363C2×M16—948no—
364C22×C8—474no—
365#123—28no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8C22×D4714yes—
369C22⋊Q8C22×D4714yes—
370#121—22no—
371#121—22no—
372#121C22×D4714yes—
373#121C22×D4714yes—
374#120—28no—
375C4×D4—28no—
376C4×D4—28no—
377#117—22no—
378#117C22×D4714yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—478no—
383C2×C4⋊C4—478no—
384C2×C4⋊C4—478no—
385C2×C4⋊C4—948no—
386C2×C4⋊C4—948no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—948no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—942no—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42—21no—
402#107—4716no—
403#106—4716no—
404#103—4716no—
405#102—4716no—
406C22⋊C8—478no—
407C22⋊C8—478no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—21no—
412C24C2×C22⋊D4714yes—
413C24C2×C22⋊D4711yescharacteristic
414C24C2×C22⋊D4711yescharacteristic
415C24C2×C22⋊D4711yescharacteristic
416#41—216no—
417#41—48no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8C23×D4712yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4C23×D4712yes—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—948no—
443C22×C4—948no—
444C22×C4—948no—
445C22×C4—944no—
446C22×C4—18832no—
447C22×C4—1888no—
448C22×C4—1888no—
449C22×C4—1888no—
450C22×C4#1062535218yes—
451C22×C4C2×C22⋊D4714yes—
452C22×C4C2×C22⋊D4714yes—
453C22×C4C2×C22⋊D4712yes—
454C22×C4C2×C22⋊D4712yes—
455C22×C4C2×C22⋊D4714yes—
456C22×C4C2×C22⋊D4711yescharacteristic
457C22×C4C2×C22⋊D4711yescharacteristic
458C22×C4C2×C22⋊D4712yes—
459C22×C4C2×C22⋊D4711yescharacteristic
460C22×C4C23×D4712yes—
461C22×C4C23×D4712yes—
462C22×C4C23×D4712yes—
463C22×C4C23×D4711yescharacteristic
464M16—9416no—
465C2×C8—478no—
466C2×C8—948no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—478no—
472C4⋊C4—948no—
473C4⋊C4—948no—
474C4⋊C4—948no—
475C4⋊C4—944no—
476C4⋊C4C23×D4712yes—
477C4⋊C4C23×D4712yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C42—24no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23#1309400514yes—
499C23#1309404914yes—
500C23#1309405414yes—
501C23C22×C22⋊D4712yes—
502C23C22×C22⋊D4712yes—
503C23C22×C22⋊D4712yes—
504C23C22×C22⋊D4711yescharacteristic
505Q8—24no—
506Q8—42no—
507D4—24no—
508D4—48no—
509D4—48no—
510D4—48no—
511D4—42no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—48no—
528C2×C4—948no—
529C2×C4—18816no—
530C2×C4—18816no—
531C2×C4—18816no—
532C2×C4—1888no—
533C2×C4#1309400514yes—
534C2×C4#1309400518yes—
535C2×C4C22×C22⋊D4712yes—
536C2×C4C22×C22⋊D4712yes—
537C2×C4C22×C22⋊D4712yes—
538C2×C4C22×C22⋊D4711yescharacteristic
539C2×C4C2×#1062535214yes—
540C2×C4C2×#1062535218yes—
541C2×C4C24×D4711yescharacteristic
542C8—948no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—84no—
557C22C2×#1309400514yes—
558C22C2×#1309400512yes—
559C22C2×#1309404914yes—
560C22C2×#1309404912yes—
561C22C2×#1309405414yes—
562C22C2×#1309405412yes—
563C22C23×C22⋊D4711yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—1888no—
570C4C2×#1309400514yes—
571C4C2×#1309400512yes—
572C4C22×#1062535212yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—82no—
577C2#2025330814yesdirect product factor
578C2C22×#1309400511yescharacteristic
579C2C22×#1309404911yescharacteristic
580C2C22×#1309405411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
Fitting, prime radicalC2×C47×#525C2
socleC23×C47C22×D4
hypercenterC2×#525D47
hypocenterC47C2×#3092
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊Dic47C2semidirect
2. #20253308C2direct (N × H)
3. C2×#15689456C2semidirect
4. C2×#15689439C2semidirect
5. C2×#15689451C2semidirect
6. C2×#15689434C2semidirect
7. C2×#15689166C2semidirect
8. C2×#15689124C2semidirect
9. C2×#15689087C2semidirect
20. C22×C4⋊Dic47C22semidirect
21. #15689456C22semidirect
22. #15689451C22semidirect
23. #15689439C22semidirect
24. #15689434C22semidirect
25. #15689166C22semidirect
26. C2×#13093957C22semidirect
27. C2×#13093955C22semidirect
28. #15689124C22semidirect
29. C2×#13093948C22semidirect
30. C2×#13093945C22semidirect
31. #15689087C22semidirect
68. #13093957C23semidirect
69. #13093955C23semidirect
70. #13093948C23semidirect
71. #13093945C23semidirect
297. C2×C47#3092semidirect
337. C47C2×#3092semidirect
577. C2#20253308direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic472
#202533082
C2×#156894562
C2×#156894392
C2×#156894512
C2×#156894342
C2×#156891662
C2×#156891242
C2×#156890872
C2×C47×#5252
C2×#309247