Finite Group Databasebeta

← #26122797

Subgroups — #26122797

Order 12032.

1630 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689634C211yesmaximal, characteristic
2C22×#13093952C212yesmaximal, complemented
3C2×#15689087C211yesmaximal, characteristic, complemented
4C2×#15688883C212yesmaximal, complemented
5C2×C4×C22⋊C4×C47C211yesmaximal, characteristic, complemented
6C22×#10625350C2211yescharacteristic
7C22×#10625329C2212yes—
8#15689634—24no—
9#15689634—24no—
10C2×#13094007—21no—
11C2×#13094003C412yescomplemented
12C2×#13093998C412yescomplemented
13C2×#13093993—21no—
14C2×C4×C22⋊D47—22no—
15C2×#13093983C2211yescharacteristic
16C2×#13093952—24no—
17C2×#13093952—24no—
18C2×#13093952—24no—
19#15689087—24no—
20C2×C22⋊(C47⋊C8)—22no—
21#15688883—24no—
22C24×C4×C47C2211yescharacteristic
23C22×C22⋊C4×C47—21no—
24C22×C42×C47C2212yes—
25C4×C22⋊C4×C47—24no—
26C2×#95×C47—21no—
27C2×#10625350—22no—
28C2×#10625350—22no—
29C2×#10625350—22no—
30C2×#10625350—22no—
31C2×#10625350—22no—
32C2×#10625350D414yes—
33C22×Dic94C2×C411yescharacteristic
34C22×D188C2×C411yescharacteristic
35C22×C22⋊D47—22no—
36C22×C22⋊D47—21no—
37C22×C4×D47—21no—
38C2×#10625329—22no—
39C2×#10625329—22no—
40C2×#10625329—22no—
41C2×#10625329—22no—
42C2×#10625329—22no—
43C2×#10625329D414yescomplemented
44C22×C47⋊C8—22no—
45#13094007—28no—
46#13094003—24no—
47#13094003—24no—
48#13093998—24no—
49#13093998—24no—
50#13093993—28no—
51C4×C22⋊D47—216no—
52#13093983—22no—
53#13093983D414yes—
54C2×C4⋊Dic47C2×C412yes—
55C2×Dic47⋊C4—22no—
56C2×C22⋊Dic47—22no—
57C2×C22⋊Dic47—21no—
58C2×(C2×D47)⋊C4—22no—
59C2×C4×Dic47—21no—
60#13093952—216no—
61C22⋊(C47⋊C8)—28no—
62C25×C47D411yescharacteristic
63C23×C4×C47—22no—
64C23×C4×C47—22no—
65C23×C4×C47—22no—
66C23×C4×C47—22no—
67C23×C4×C47—22no—
68C23×C4×C47—22no—
69C23×C4×C47—22no—
70C23×C4×C47C2×C412yes—
71C23×C4×C47D411yescharacteristic
72C23×C4×C47D412yes—
73C23×C4×C47C2311yescharacteristic
74C2×C22⋊C4×C47—28no—
75C2×C22⋊C4×C47—24no—
76C2×C42×C47—22no—
77C2×C42×C47—22no—
78C2×C42×C47—22no—
79C2×C42×C47—22no—
80C2×C42×C47—22no—
81C2×C42×C47D414yescomplemented
82#95×C47—24no—
83C24×C47—24no—
84C24×C47—21no—
85C24×C47—21no—
86C24×C47—22no—
87C24×C47—22no—
88C24×C47—42no—
89C24×C47C22⋊C412yes—
90C24×C47C2×D411yescharacteristic
91C23×D47—21no—
92C22×C4×C47—28no—
93C22×C4×C47—24no—
94C22×C4×C47—22no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—24no—
101C22×C4×C47—24no—
102C22×C4×C47—22no—
103C22×C4×C47—24no—
104C22×C4×C47—24no—
105C22×C4×C47—22no—
106C22×C4×C47—44no—
107C22×C4×C47—42no—
108C22×C4×C47—42no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—42no—
112C22×C4×C47—42no—
113C22×C4×C47—42no—
114C22×C4×C47—42no—
115C22×C4×C47—42no—
116C22×C4×C47—42no—
117C22×C4×C47C22⋊C412yes—
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C2×D412yes—
120C22×C4×C47C2×D412yes—
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C2×C22⋊D47—28no—
127C2×C22⋊D47—24no—
128C2×C22⋊D47—44no—
129C2×C22⋊D47—44no—
130C2×C22⋊D47—41no—
131C2×C22⋊D47—42no—
132C2×C22⋊D47—41no—
133C22×Dic47—22no—
134C22×Dic47—21no—
135#10625350—24no—
136#10625350—24no—
137#10625350—24no—
138#10625350—24no—
139#10625350—42no—
140#10625350—42no—
141#10625350—42no—
142#10625350—42no—
143C2×D188—22no—
144C2×D188—22no—
145C2×D188—21no—
146C2×D188C22⋊C412yescomplemented
147C2×C4×D47—24no—
148C2×C4×D47—41no—
149C2×C4×D47—42no—
150C2×C4×D47—41no—
151C2×Dic94—22no—
152C2×Dic94—22no—
153C2×Dic94—21no—
154C2×Dic94C22⋊C412yescomplemented
155C22⋊C4×C47—216no—
156C42×C47—24no—
157C42×C47—24no—
158C42×C47—42no—
159C42×C47—42no—
160C22⋊Dic47—28no—
161C22⋊Dic47—24no—
162(C2×D47)⋊C4—28no—
163C4⋊Dic47—22no—
164C4⋊Dic47C22⋊C414yescomplemented
165Dic47⋊C4—28no—
166C4×Dic47—24no—
167#10625329—24no—
168#10625329—24no—
169#10625329—42no—
170#10625329—42no—
171C2×C47⋊C8—24no—
172C2×C47⋊C8—42no—
173C2×C47⋊C8—42no—
174C23×C47—22no—
175C23×C47—22no—
176C23×C47—24no—
177C23×C47—24no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—48no—
185C23×C47—48no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47#10414yescomplemented
191C23×C47C2×C22⋊C411yescharacteristic
192C23×C47#12012yes—
193C23×C47#12011yescharacteristic
194C23×C47#12112yes—
195C23×C47#12311yescharacteristic
196C22×D47—22no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47—42no—
200C22×D47—42no—
201C2×C4×C47—28no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—48no—
206C2×C4×C47—44no—
207C2×C4×C47—48no—
208C2×C4×C47—44no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—44no—
212C2×C4×C47—44no—
213C2×C4×C47—44no—
214C2×C4×C47—44no—
215C2×C4×C47C2×C22⋊C411yescharacteristic
216C2×C4×C47C2×C22⋊C411yescharacteristic
217C2×C4×C47C4×D412yes—
218C2×C4×C47C4×D412yes—
219C2×C4×C47#12012yes—
220C2×C4×C47#12011yescharacteristic
221C2×C4×C47#12012yes—
222C2×C4×C47#12112yes—
223C2×C4×C47#12112yes—
224C2×C4×C47#12112yes—
225C2×C4×C47#12112yes—
226C2×C4×C47#12112yes—
227C2×C4×C47#12312yes—
228C2×C4×C47#12311yescharacteristic
229C2×C4×C47#12412yes—
230C2×C4×C47#12712yes—
231C22⋊D47—416no—
232C22⋊D47—44no—
233C22⋊D47—44no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240D188—24no—
241D188—42no—
242C4×D47—44no—
243C4×D47—44no—
244Dic94—24no—
245Dic94—42no—
246C47⋊C8—44no—
247#10971—471nomaximal, Sylow
248C22×C47—24no—
249C22×C47—21no—
250C22×C47—21no—
251C22×C47—21no—
252C22×C47—21no—
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—24no—
256C22×C47—24no—
257C22×C47—48no—
258C22×C47—48no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47—42no—
262C22×C47—42no—
263C22×C47#37911yescharacteristic
264C22×C47#38611yescharacteristic
265C22×C47#38611yescharacteristic
266C22×C47#39212yes—
267C22×C47#39412yes—
268C22×C47C2×#10412yes—
269C22×C47#42112yes—
270C2×D47—42no—
271C2×D47—44no—
272C2×D47—82no—
273C2×D47—81no—
274C2×D47—81no—
275C4×C47—48no—
276C4×C47—42no—
277C4×C47#38612yes—
278C4×C47#39012yes—
279C4×C47#39212yes—
280C4×C47#39412yes—
281Dic47—42no—
282Dic47—44no—
283C2×#525—471no—
284C22×#104—472no—
285C2×#407—471no—
286C2×C4×C22⋊C4D4711yescharacteristic, complemented
287C2×#339—472no—
288C2×C47—22no—
289C2×C47—22no—
290C2×C47—44no—
291C2×C47#143811yescharacteristic
292C2×C47#178912yes—
293C2×C47#187112yes—
294C2×C47#189611yescharacteristic
295C2×C47#189711yescharacteristic
296D47—82no—
297C22×#41—471no—
298C24×C4C2×D4711yescharacteristic
299C22×M16—472no—
300#525—944no—
301#525—944no—
302C2×#123—941no—
303C2×C22⋊Q8—472no—
304C2×#121—472no—
305C2×#120—941no—
306C2×C4×D4—942no—
307C2×#117—471no—
308C22×C22⋊C4—21no—
309C22×C42C2×D4712yes—
310C2×#104—944no—
311C2×#104—944no—
312C2×#104—944no—
313#407—944no—
314C2×C22⋊C8—942no—
315C4×C22⋊C4—24no—
316C2×#95—21no—
317#339—944no—
318C47#1097111yescharacteristic, complemented, Sylow
319C25C22⋊D4711yescharacteristic
320C2×#41—474no—
321C2×#41—942no—
322C2×#41—942no—
323C2×#41—942no—
324C2×#41—942no—
325C2×#41—942no—
326C22×Q8—471no—
327C22×D4—471no—
328C22×D4—942no—
329C22×D4—941no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—941no—
338C23×C4C4×D4712yes—
339C23×C4D18812yes—
340C23×C4C22⋊D4711yescharacteristic
341C23×C4C22×D4711yescharacteristic
342C2×M16—474no—
343C2×M16—942no—
344C2×M16—942no—
345C2×M16—942no—
346C2×M16—942no—
347C2×M16—942no—
348C22×C8—942no—
349#123—948no—
350C22⋊Q8—944no—
351C22⋊Q8—944no—
352#121—944no—
353#121—944no—
354#120—948no—
355C4×D4—9416no—
356#117—474no—
357#117—942no—
358C2×C4⋊C4—472no—
359C2×C4⋊C4—942no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—942no—
363C2×C22⋊C4—941no—
364C2×C22⋊C4—942no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—941no—
371C2×C42C22⋊D4714yescomplemented
372#104—9416no—
373C22⋊C8—948no—
374#95—24no—
375C24—24no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—22no—
380C24—42no—
381C24—941no—
382C24(C2×D47)⋊C412yes—
383C24C2×C22⋊D4711yescharacteristic
384#41—944no—
385#41—944no—
386#41—944no—
387#41—944no—
388#41—1882no—
389#41—1882no—
390#41—1882no—
391#41—1882no—
392C2×Q8—472no—
393C2×Q8—941no—
394C2×Q8—942no—
395C2×Q8—942no—
396C2×D4—472no—
397C2×D4—948no—
398C2×D4—944no—
399C2×D4—941no—
400C2×D4—942no—
401C2×D4—942no—
402C2×D4—1884no—
403C2×D4—1884no—
404C2×D4—1881no—
405C2×D4—1882no—
406C2×D4—1881no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—942no—
433C22×C4—941no—
434C22×C4—944no—
435C22×C4—1881no—
436C22×C4—1882no—
437C22×C4—1881no—
438C22×C4(C2×D47)⋊C412yes—
439C22×C4C2×C4×D4711yescharacteristic
440C22×C4C2×D18811yescharacteristic
441C22×C4#1062535012yes—
442C22×C4D4×D4712yes—
443C22×C4D4×D4712yes—
444C22×C4D4×D4712yes—
445C22×C4#1062535212yes—
446C22×C4C2×C22⋊D4712yes—
447M16—944no—
448M16—944no—
449M16—1882no—
450M16—1882no—
451C2×C8—944no—
452C2×C8—1882no—
453C2×C8—1882no—
454C4⋊C4—474no—
455C4⋊C4—948no—
456C4⋊C4—942no—
457C22⋊C4—216no—
458C22⋊C4—948no—
459C22⋊C4—948no—
460C22⋊C4—944no—
461C42—24no—
462C42—24no—
463C42—42no—
464C42—42no—
465C42—944no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—48no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—942no—
483C23—1881no—
484C23—1881no—
485C23—1882no—
486C23—1882no—
487C23#1309395214yescomplemented
488C23C2×(C2×D47)⋊C411yescharacteristic
489C23#1309399212yes—
490C23#1309399311yescharacteristic
491C23#1309399612yes—
492C23#1309400711yescharacteristic
493Q8—944no—
494Q8—1882no—
495D4—944no—
496D4—18816no—
497D4—1884no—
498D4—1884no—
499D4—1882no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—944no—
515C2×C4—942no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1884no—
521C2×C4—1884no—
522C2×C4C22⋊C4×D4711yescharacteristic
523C2×C4C22⋊C4×D4711yescharacteristic
524C2×C4(C22⋊D47)⋊C412yes—
525C2×C4C4×C22⋊D4712yes—
526C2×C4#1309399212yes—
527C2×C4#1309399411yescharacteristic
528C2×C4#1309399412yes—
529C2×C4#1309399512yes—
530C2×C4#1309399612yes—
531C2×C4#1309399712yes—
532C2×C4#1309399712yes—
533C2×C4#1309399812yes—
534C2×C4#1309400612yes—
535C2×C4#1309400811yescharacteristic
536C2×C4#1309401012yes—
537C2×C4#1309402012yes—
538C8—1884no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—48no—
549C22—48no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—3761no—
555C22—3761no—
556C22—3762no—
557C22—1884no—
558C22—1882no—
559C22#1568897811yescharacteristic
560C22#1568900011yescharacteristic
561C22#1568900211yescharacteristic
562C22#1568903112yes—
563C22#1568903812yes—
564C22C2×#1309395212yes—
565C22#1568914112yes—
566C4—48no—
567C4—42no—
568C4—1884no—
569C4—1882no—
570C4#1568899912yes—
571C4#1568902312yes—
572C4#1568903212yes—
573C4#1568903912yes—
574C2—22no—
575C2—22no—
576C2—44no—
577C2—3762no—
578C2#2024162711yescharacteristic
579C2#2024333312yes—
580C2#2024368812yes—
581C2#2024383111yescharacteristic
582C2#2024383611yescharacteristic
subgroupG/H
centerC22×C4C2×D188
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C4×C22⋊C4×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#15689634C2
cosocleC23×C4×C47C23
hypercenterC2×C4×C22⋊C4D47
hypocenterC47#10971
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#13093952C2semidirect
3. C2×#15689087C2semidirect
4. C2×#15688883C2semidirect
5. C2×C4×C22⋊C4×C47C2semidirect
11. C2×#13094003C4semidirect
12. C2×#13093998C4semidirect
43. C2×#10625329D4semidirect
81. C2×C42×C47D4semidirect
146. C2×D188C22⋊C4semidirect
154. C2×Dic94C22⋊C4semidirect
164. C4⋊Dic47C22⋊C4semidirect
190. C23×C47#104semidirect
286. C2×C4×C22⋊C4D47semidirect
318. C47#10971semidirect
371. C2×C42C22⋊D47semidirect
487. C23#13093952semidirect

Maximal subgroups

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

subgroupindex
C2×#156896342
C22×#130939522
C2×#156890872
C2×#156888832
C2×C4×C22⋊C4×C472
#1097147