Finite Group Databasebeta

← #26243102

Subgroups — #26243102

Order 12032.

1950 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625337C211yesmaximal, characteristic, complemented
2C2×#15689694C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4C2×#15689131C211yesmaximal, characteristic, complemented
5C2×#15689090C211yesmaximal, characteristic
6#20244396C218yesmaximal, complemented
7#20243104C211yesmaximal, characteristic, complemented
8C2×C47×#536C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C22×#10625337—24no—
11C22×#10625337—24no—
12C22×#10625337—24no—
13C22×#10625337C2214yes—
14C22×#10625329C2211yescharacteristic
15C23×C47⋊C8C2211yescharacteristic
16#15689694—24no—
17#15689694—24no—
18C2×#13094022—21no—
19C2×#13094011—21no—
20C2×#13094005—22no—
21C2×#13094003C2214yes—
22C2×#13093983C2211yescharacteristic
23#15689132—24no—
24#15689131—24no—
25C2×#13093948C2214yes—
26C2×#13093947C2214yes—
27#15689090—24no—
28C2×C22⋊(C47⋊C8)C2214yes—
29#15688885C2214yes—
30C22×#41×C47C2211yescharacteristic
31C23×Q8×C47C2211yescharacteristic
32C47×#536—24no—
33C47×#536—24no—
34C2×C4⋊Q8×C47—21no—
35C2×#124×C47—21no—
36C2×#123×C47—22no—
37C2×C22⋊Q8×C47C2214yes—
38C2×#117×C47C2211yescharacteristic
39C22×Dic94—22no—
40C22×Dic94—22no—
41C22×Dic94—22no—
42C22×Dic94—22no—
43C22×Dic94—21no—
44C22×Dic94—22no—
45C22×Dic94D414yes—
46C22×Dic94C2312yes—
47C23×Dic47—21no—
48C2×#10625337—232no—
49C2×#10625337—28no—
50C2×#10625337—28no—
51C2×#10625337—28no—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C2×#10625329—22no—
55C22×C47⋊C8—22no—
56C22×C47⋊C8—22no—
57C22×C47⋊C8D414yes—
58C22×C47⋊C8C2312yes—
59C22×C47⋊C8C2312yes—
60#13094022—28no—
61#13094011—28no—
62#13094005—216no—
63#13094003—28no—
64#13094003—28no—
65#13093983—22no—
66#13093983D414yes—
67C2×C4⋊Dic47C2312yes—
68C2×Dic47⋊C4—24no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47—24no—
71C2×C4×Dic47—21no—
72#13093948—28no—
73#13093947—28no—
74C22⋊(C47⋊C8)—28no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47—22no—
78C2×#41×C47—22no—
79C2×#41×C47—22no—
80C2×#41×C47—22no—
81C22×Q8×C47—21no—
82C22×Q8×C47—22no—
83C22×Q8×C47—22no—
84C22×Q8×C47—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47—22no—
87C22×Q8×C47D414yes—
88C22×Q8×C47D411yescharacteristic
89C22×Q8×C47C2312yes—
90C22×D4×C47D412yes—
91C22×D4×C47D411yescharacteristic
92C23×C4×C47—21no—
93C23×C4×C47C2312yes—
94C23×C4×C47C2311yescharacteristic
95C4⋊Q8×C47—28no—
96#124×C47—28no—
97#123×C47—216no—
98C22⋊Q8×C47—28no—
99C22⋊Q8×C47—28no—
100#117×C47—22no—
101#117×C47D414yes—
102C2×C4⋊C4×C47—24no—
103C2×C4⋊C4×C47C2312yes—
104C2×C22⋊C4×C47—21no—
105C2×C22⋊C4×C47—24no—
106C2×C42×C47—21no—
107C24×C47C2×D412yes—
108C24×C47C2×D411yescharacteristic
109#41×C47—44no—
110#41×C47—44no—
111#41×C47—44no—
112#41×C47—44no—
113C2×Q8×C47—24no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47—22no—
118C2×Q8×C47—28no—
119C2×Q8×C47—28no—
120C2×Q8×C47—22no—
121C2×Q8×C47—22no—
122C2×Q8×C47—22no—
123C2×Q8×C47—22no—
124C2×Q8×C47—42no—
125C2×Q8×C47—42no—
126C2×Q8×C47—44no—
127C2×Q8×C47—44no—
128C2×Q8×C47C2×D414yes—
129C2×D4×C47—24no—
130C2×D4×C47—24no—
131C2×D4×C47—24no—
132C2×D4×C47—22no—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C22×C4×C47—24no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—42no—
145C22×C4×C47—42no—
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D412yes—
152C22×C4×C47C2×D412yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—42no—
159C22×Dic47—42no—
160C2×Dic94—28no—
161C2×Dic94—28no—
162C2×Dic94—22no—
163C2×Dic94—22no—
164C2×Dic94—22no—
165C2×Dic94—22no—
166C2×Dic94—24no—
167C2×Dic94—22no—
168C2×Dic94—42no—
169C2×Dic94—44no—
170C2×Dic94—42no—
171C2×Dic94—44no—
172C2×Dic94C2×D414yes—
173C4⋊C4×C47—216no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47C2×D414yes—
176C22⋊C4×C47—216no—
177C22⋊C4×C47—24no—
178C42×C47—24no—
179C22⋊Dic47—216no—
180C22⋊Dic47—24no—
181#10625337—232no—
182C4⋊Dic47—22no—
183C4⋊Dic47C2×D414yes—
184Dic47⋊C4—216no—
185C4×Dic47—24no—
186#10625329—44no—
187C2×C47⋊C8—28no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8C2×D414yes—
194C23×C47—24no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—28no—
199C23×C47#12012yes—
200C23×C47#12012yes—
201C23×C47#12112yes—
202C23×C47C22×D411yescharacteristic
203Q8×C47—24no—
204Q8×C47—24no—
205Q8×C47—42no—
206Q8×C47—42no—
207Q8×C47—48no—
208Q8×C47—44no—
209D4×C47—48no—
210D4×C47—44no—
211C2×C4×C47—22no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—44no—
219C2×C4×C47—48no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47#12014yes—
225C2×C4×C47#12012yes—
226C2×C4×C47#12012yes—
227C2×C4×C47#12114yes—
228C2×C4×C47#12114yes—
229C2×C4×C47#12112yes—
230C2×C4×C47#12714yes—
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47—44no—
242C2×Dic47—48no—
243Dic94—24no—
244Dic94—24no—
245Dic94—42no—
246Dic94—42no—
247Dic94—48no—
248C47⋊C8—24no—
249C47⋊C8—42no—
250#23389—471nomaximal, Sylow
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—42no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—44no—
258C22×C47#39214yes—
259C22×C47C2×#12011yescharacteristic
260C22×C47C2×#12011yescharacteristic
261C22×C47C2×#12111yescharacteristic
262C22×C47#57814yes—
263C4×C47—42no—
264C4×C47—44no—
265C4×C47—44no—
266C4×C47#39214yes—
267C4×C47C2×#12011yescharacteristic
268C4×C47C2×#12111yescharacteristic
269C4×C47C2×#12111yescharacteristic
270C4×C47C2×#12711yescharacteristic
271Dic47—44no—
272Dic47—44no—
273C23×Q16—471no—
274C2×#536—471no—
275C2×#536D4711yescharacteristic
276C2×#419—471no—
277C2×#419—471no—
278C2×#408—471no—
279#2001—478no—
280#1735—471no—
281C2×C47—24no—
282C2×C47—82no—
283C2×C47C2×#39211yescharacteristic
284C2×C47#301112yes—
285C2×C47#301112yes—
286C2×C47#305912yes—
287C22×#41C2×D4711yescharacteristic
288C23×Q8—471no—
289C23×Q8C2×D4711yescharacteristic
290C22×Q16—474no—
291C22×Q16—944no—
292C22×Q16—944no—
293C22×Q16—944no—
294C22×M16—471no—
295C23×C8—471no—
296#536—24no—
297#536—24no—
298#536—944no—
299#536—944no—
300C2×C4⋊Q8—21no—
301C2×C4⋊Q8—941no—
302C2×#124—21no—
303C2×#124—941no—
304C2×#123—22no—
305C2×#123—942no—
306C2×C22⋊Q8—474no—
307C2×C22⋊Q8C2×D4714yes—
308C2×#117—471no—
309C2×#117C2×D4711yescharacteristic
310#419—944no—
311#419—944no—
312C2×#103—474no—
313C2×#103—474no—
314#408—944no—
315C2×C22⋊C8—474no—
316#340—474no—
317C47#2338911yescharacteristic, complemented, Sylow
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C22×Q8—21no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—474no—
331C22×Q8—472no—
332C22×Q8—942no—
333C22×Q8—941no—
334C22×Q8—942no—
335C22×Q8—942no—
336C22×Q8—942no—
337C22×Q8—942no—
338C22×Q8C22⋊D4714yes—
339C22×Q8C22⋊D4711yescharacteristic
340C22×Q8C22×D4712yes—
341C22×D4C22⋊D4712yes—
342C22×D4C22⋊D4711yescharacteristic
343C23×C4—21no—
344C23×C4—941no—
345C23×C4C22×D4712yes—
346C23×C4C22×D4711yescharacteristic
347C2×Q16—948no—
348C2×Q16—948no—
349C2×Q16—9432no—
350C2×Q16—948no—
351C2×M16—942no—
352C2×M16—942no—
353C2×M16—942no—
354C22×C8—474no—
355C22×C8—472no—
356C22×C8—472no—
357C22×C8—942no—
358C22×C8—942no—
359C4⋊Q8—28no—
360C4⋊Q8—948no—
361#124—28no—
362#124—948no—
363#123—216no—
364#123—9416no—
365C22⋊Q8—28no—
366C22⋊Q8—28no—
367C22⋊Q8—948no—
368C22⋊Q8—948no—
369#117—22no—
370#117—474no—
371#117—942no—
372#117C22⋊D4714yes—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—944no—
376C2×C4⋊C4C22×D4712yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—944no—
381C2×C42—21no—
382C2×C42—941no—
383#103—948no—
384#103—948no—
385C22⋊C8—948no—
386C24C2×C22⋊D4712yes—
387C24C2×C22⋊D4711yescharacteristic
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—474no—
408C2×Q8—944no—
409C2×Q8—942no—
410C2×Q8—948no—
411C2×Q8—948no—
412C2×Q8—942no—
413C2×Q8—942no—
414C2×Q8—942no—
415C2×Q8—942no—
416C2×Q8—1882no—
417C2×Q8—1884no—
418C2×Q8—1882no—
419C2×Q8—1884no—
420C2×Q8C2×C22⋊D4714yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—944no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—1882no—
442C22×C4—1882no—
443C22×C4D4×D4712yes—
444C22×C4D4×D4712yes—
445C22×C4D4×D4712yes—
446C22×C4#1062535212yes—
447C22×C4C2×C22⋊D4712yes—
448C22×C4C2×C22⋊D4711yescharacteristic
449C22×C4C2×C22⋊D4712yes—
450C22×C4C2×C22⋊D4712yes—
451C22×C4C23×D4711yescharacteristic
452Q16—9432no—
453M16—1884no—
454C2×C8—474no—
455C2×C8—948no—
456C2×C8—942no—
457C2×C8—942no—
458C2×C8—942no—
459C2×C8—942no—
460C2×C8—942no—
461C4⋊C4—216no—
462C4⋊C4—22no—
463C4⋊C4—474no—
464C4⋊C4—9416no—
465C4⋊C4—942no—
466C4⋊C4C2×C22⋊D4714yes—
467C22⋊C4—216no—
468C22⋊C4—24no—
469C22⋊C4—9416no—
470C22⋊C4—944no—
471C42—24no—
472C42—944no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—28no—
478C23#1309399212yes—
479C23#1309399312yes—
480C23#1309399612yes—
481C23C22×C22⋊D4711yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—48no—
485Q8—44no—
486Q8—42no—
487Q8—42no—
488Q8—944no—
489Q8—944no—
490Q8—1888no—
491Q8—1882no—
492Q8—1882no—
493D4—48no—
494D4—44no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—944no—
509C2×C4—944no—
510C2×C4—1888no—
511C2×C4—1882no—
512C2×C4—1882no—
513C2×C4—1884no—
514C2×C4#1309399212yes—
515C2×C4#1309399214yes—
516C2×C4#1309399312yes—
517C2×C4#1309399514yes—
518C2×C4#1309399612yes—
519C2×C4#1309399614yes—
520C2×C4#1309402014yes—
521C2×C4C2×D4×D4711yescharacteristic
522C2×C4C2×D4×D4711yescharacteristic
523C2×C4C2×D4×D4711yescharacteristic
524C2×C4C22×C22⋊D4711yescharacteristic
525C2×C4C22×C22⋊D4711yescharacteristic
526C2×C4C2×#1062535211yescharacteristic
527C8—944no—
528C8—1882no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—44no—
533C22—44no—
534C22—42no—
535C22—44no—
536C22#1568903114yes—
537C22C2×#1309399211yescharacteristic
538C22C2×#1309399311yescharacteristic
539C22C2×#1309399611yescharacteristic
540C22#1568993814yes—
541C4—44no—
542C4—42no—
543C4—44no—
544C4—1884no—
545C4—1884no—
546C4#1568903114yes—
547C4C2×#1309399211yescharacteristic
548C4C2×#1309399511yescharacteristic
549C4C2×#1309399611yescharacteristic
550C4C2×#1309402011yescharacteristic
551C2—24no—
552C2—82no—
553C2C2×#1568903111yescharacteristic
554C2#2025268112yes—
555C2#2025268312yes—
556C2#2025304012yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#536C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
hypercenterC2×#536D47
hypocenterC47#23389
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C23×#10625337C2semidirect
3. C2×#15689132C2semidirect
4. C2×#15689131C2semidirect
6. #20244396C2semidirect
7. #20243104C2semidirect
317. C47#23389semidirect

Maximal subgroups

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

subgroupindex
C23×#106253372
C2×#156896942
C2×#156891322
C2×#156891312
C2×#156890902
#202443962
#202431042
C2×C47×#5362
#2338947