Finite Group Databasebeta

← #26182125

Subgroups — #26182125

Order 12032.

1322 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093980C211yesmaximal, characteristic
2#20252017C212yesmaximal, complemented
3#20241719C214yesmaximal, complemented
4#20241717C212yesmaximal, complemented
5#20241715C211yesmaximal, characteristic
6#20241699C214yesmaximal, complemented
7C47×#1462C211yesmaximal, characteristic
8C24×Dic47C411yescharacteristic
9C2×#13093980—22no—
10C2×#13093980—22no—
11C2×#13093980—22no—
12C2×#13093980—22no—
13C2×#13093980C412yes—
14C2×#13093980C412yes—
15C2×#13093980C2212yes—
16C2×#13093980C2212yes—
17C22×Dic47⋊C4C411yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×C22⋊Dic47C411yescharacteristic
20C22×C4×Dic47C411yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22#15689177—22no—
23#15689177C2214yescomplemented
24#15689173—22no—
25#15689173C2214yescomplemented
26#15689146—22no—
27#15689087C2212yes—
28M16×Dic47—22no—
29#15688848C2218yescomplemented
30#15688847C2214yes—
31C22×C22⋊C4×C47C2211yescharacteristic
32C47×#407C2212yes—
33C47×#323C2214yes—
34C23×Dic47—21no—
35C23×Dic47—21no—
36C23×Dic47—21no—
37C23×Dic47—21no—
38C23×Dic47—28no—
39C23×Dic47C2×C411yescharacteristic
40C23×Dic47C2×C412yes—
41C23×Dic47C2×C411yescharacteristic
42C23×Dic47C2×C412yes—
43C23×Dic47C2×C411yescharacteristic
44C23×Dic47C2×C411yescharacteristic
45C2×#10625329—22no—
46#13093980—24no—
47#13093980—24no—
48#13093980—24no—
49#13093980—24no—
50#13093980—24no—
51#13093980—24no—
52#13093980—24no—
53#13093980—24no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—22no—
58C2×Dic47⋊C4C2×C412yes—
59C2×Dic47⋊C4C2×C412yes—
60C2×Dic47⋊C4C2×C412yes—
61C2×Dic47⋊C4C2312yes—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47C2×C412yes—
65C2×C22⋊Dic47C2×C412yes—
66C2×C4×Dic47—21no—
67C2×C4×Dic47—21no—
68C2×C4×Dic47—21no—
69C2×C4×Dic47—21no—
70C2×C4×Dic47C2×C412yes—
71C2×C4×Dic47C2×C412yes—
72C2×C4×Dic47C2×C411yescharacteristic
73C2×C4×Dic47C2×C411yescharacteristic
74C2×C4×Dic47D412yes—
75C2×C4×Dic47D412yes—
76C2×C4×Dic47D412yes—
77C2×C4×Dic47D412yes—
78C2×C4×Dic47C2311yescharacteristic
79C2×C4×Dic47C2311yescharacteristic
80Dic47⋊C8—24no—
81Dic47⋊C8—24no—
82C22⋊(C47⋊C8)—22no—
83C22⋊(C47⋊C8)C2314yes—
84#13093934—24no—
85C8×Dic47—24no—
86C25×C47C2×C411yescharacteristic
87C23×C4×C47C2×C411yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C2×M16×C47—22no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47C2×C412yes—
93C2×C22⋊C4×C47C2312yes—
94C22⋊C8×C47—22no—
95C22⋊C8×C47C2314yes—
96C24×C47—21no—
97C24×C47—21no—
98C24×C47—24no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47C22×C411yescharacteristic
102C24×C47C22×C412yes—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—42no—
106C22×C4×C47—42no—
107C22×C4×C47C22×C412yes—
108C22×C4×C47C22×C411yescharacteristic
109C22×C4×C47C2×D412yes—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2411yescharacteristic
112C22×Dic47—28no—
113C22×Dic47—28no—
114C22×Dic47—28no—
115C22×Dic47—28no—
116C22×Dic47—21no—
117C22×Dic47—21no—
118C22×Dic47—24no—
119C22×Dic47—22no—
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—24no—
124C22×Dic47—24no—
125C22×Dic47—24no—
126C22×Dic47—24no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—42no—
130C22×Dic47—42no—
131C22×Dic47C22⋊C411yescharacteristic
132C22×Dic47C22⋊C411yescharacteristic
133C22×Dic47C22⋊C411yescharacteristic
134C22×Dic47C22⋊C411yescharacteristic
135C22×Dic47C22⋊C412yes—
136C22×Dic47C22⋊C412yes—
137C22×Dic47M1618yescomplemented
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C412yes—
140C22×Dic47C22×C412yes—
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144M16×C47—44no—
145C2×C8×C47—24no—
146C22⋊C4×C47—24no—
147C22⋊C4×C47—24no—
148C22⋊Dic47—24no—
149C22⋊Dic47—24no—
150Dic47⋊C4—24no—
151Dic47⋊C4—24no—
152Dic47⋊C4—24no—
153Dic47⋊C4—24no—
154C4×Dic47—22no—
155C4×Dic47—22no—
156C4×Dic47—22no—
157C4×Dic47—22no—
158C4×Dic47C22⋊C412yes—
159C4×Dic47C22⋊C412yes—
160C4×Dic47C22⋊C412yes—
161C4×Dic47C22⋊C412yes—
162C4×Dic47C2×D412yes—
163C4×Dic47C2×D412yes—
164C4×Dic47C2×D412yes—
165C4×Dic47C2×D412yes—
166#10625329—44no—
167C2×C47⋊C8—24no—
168C23×C47—22no—
169C23×C47—22no—
170C23×C47—22no—
171C23×C47—24no—
172C23×C47—24no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—42no—
176C23×C47—42no—
177C23×C47—416no—
178C23×C47—42no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47C2×C22⋊C411yescharacteristic
183C23×C47C2×C22⋊C411yescharacteristic
184C23×C47C2×M1614yes—
185C23×C47C23×C411yescharacteristic
186C2×C4×C47—24no—
187C2×C4×C47—22no—
188C2×C4×C47—41no—
189C2×C4×C47—42no—
190C2×C4×C47—41no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—42no—
194C2×C4×C47C2×C22⋊C412yes—
195C2×C4×C47C22×D412yes—
196C2×Dic47—28no—
197C2×Dic47—28no—
198C2×Dic47—24no—
199C2×Dic47—24no—
200C2×Dic47—22no—
201C2×Dic47—22no—
202C2×Dic47—22no—
203C2×Dic47—24no—
204C2×Dic47—22no—
205C2×Dic47—22no—
206C2×Dic47—22no—
207C2×Dic47—22no—
208C2×Dic47—28no—
209C2×Dic47—28no—
210C2×Dic47—22no—
211C2×Dic47—42no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47C2×C22⋊C411yescharacteristic
218C2×Dic47C2×C22⋊C411yescharacteristic
219C2×Dic47C2×C22⋊C412yes—
220C2×Dic47C2×C22⋊C412yes—
221C2×Dic47C2×C22⋊C411yescharacteristic
222C2×Dic47C2×C22⋊C411yescharacteristic
223C2×Dic47C2×M1614yes—
224C2×Dic47C2×M1614yes—
225C8×C47—44no—
226C47⋊C8—44no—
227#17357—471nomaximal, Sylow
228C22×C47—28no—
229C22×C47—21no—
230C22×C47—21no—
231C22×C47—21no—
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—21no—
235C22×C47—24no—
236C22×C47—44no—
237C22×C47—44no—
238C22×C47—48no—
239C22×C47—48no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47#41012yes—
243C22×C47#41312yes—
244C22×C47C22×C22⋊C411yescharacteristic
245C22×C47C22×M1612yes—
246C4×C47—22no—
247C4×C47—22no—
248C4×C47—42no—
249Dic47—21no—
250Dic47—21no—
251Dic47—24no—
252Dic47—22no—
253Dic47—22no—
254Dic47—42no—
255Dic47#40712yes—
256Dic47#40712yes—
257C22×#117—471no—
258#2920—472no—
259#1464—474no—
260#1463—472no—
261#1462—471no—
262#1462D4711yescharacteristic
263#1458—474no—
264C2×C47—22no—
265C2×C47—22no—
266C2×C47—22no—
267C2×C47—44no—
268C2×C47C2×#40711yescharacteristic
269C2×C47C2×#41011yescharacteristic
270C2×C47C2×#41311yescharacteristic
271C24×C4—471no—
272C2×#117—472no—
273C2×#117—472no—
274C2×#117—472no—
275C2×#117—472no—
276C2×#117—942no—
277C2×#117—942no—
278C2×#117—942no—
279C2×#117—942no—
280C22×C4⋊C4—471no—
281C22×C4⋊C4—471no—
282C22×C22⋊C4—471no—
283C22×C22⋊C4C2×D4711yescharacteristic
284C22×C42—471no—
285C22×C42—471no—
286#432—474no—
287#432—942no—
288#431—474no—
289#431—942no—
290#423—942no—
291#407—472no—
292#407C2×D4712yes—
293C4×M16—942no—
294#324—478no—
295#323—474no—
296#323C2×D4714yes—
297C47#1735711yescharacteristic, complemented, Sylow
298C25C4×D4711yescharacteristic
299C23×C4—472no—
300C23×C4—471no—
301C23×C4—471no—
302C23×C4—471no—
303C23×C4—472no—
304C23×C4—471no—
305C23×C4—941no—
306C23×C4—941no—
307C23×C4—941no—
308C23×C4—941no—
309C23×C4—948no—
310C23×C4C4×D4711yescharacteristic
311C23×C4C22×D4711yescharacteristic
312C2×M16—22no—
313C2×M16—942no—
314#117—944no—
315#117—944no—
316#117—944no—
317#117—944no—
318#117—944no—
319#117—944no—
320#117—944no—
321#117—944no—
322C2×C4⋊C4—472no—
323C2×C4⋊C4—472no—
324C2×C4⋊C4—472no—
325C2×C4⋊C4—472no—
326C2×C4⋊C4—942no—
327C2×C4⋊C4—942no—
328C2×C4⋊C4—942no—
329C2×C4⋊C4—942no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—472no—
333C2×C22⋊C4—472no—
334C2×C22⋊C4—942no—
335C2×C22⋊C4—942no—
336C2×C22⋊C4C4×D4712yes—
337C2×C22⋊C4C22×D4712yes—
338C2×C42—471no—
339C2×C42—471no—
340C2×C42—472no—
341C2×C42—472no—
342C2×C42—472no—
343C2×C42—472no—
344C2×C42—471no—
345C2×C42—471no—
346C2×C42—472no—
347C2×C42—472no—
348C2×C42—941no—
349C2×C42—941no—
350C2×C42—941no—
351C2×C42—941no—
352C4⋊C8—944no—
353C4⋊C8—944no—
354C22⋊C8—22no—
355C22⋊C8—474no—
356C22⋊C8—942no—
357C22⋊C8C22×D4714yes—
358#97—944no—
359C4×C8—944no—
360C24—21no—
361C24—21no—
362C24—24no—
363C24—42no—
364C24—42no—
365C24C2×C4×D4711yescharacteristic
366C24C2×C4×D4712yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—471no—
372C22×C4—471no—
373C22×C4—471no—
374C22×C4—471no—
375C22×C4—471no—
376C22×C4—472no—
377C22×C4—478no—
378C22×C4—471no—
379C22×C4—472no—
380C22×C4—472no—
381C22×C4—471no—
382C22×C4—471no—
383C22×C4—472no—
384C22×C4—942no—
385C22×C4—942no—
386C22×C4—942no—
387C22×C4—942no—
388C22×C4—944no—
389C22×C4—944no—
390C22×C4—944no—
391C22×C4—944no—
392C22×C4—948no—
393C22×C4—948no—
394C22×C4—948no—
395C22×C4—948no—
396C22×C4—941no—
397C22×C4—941no—
398C22×C4—944no—
399C22×C4—941no—
400C22×C4—941no—
401C22×C4—1882no—
402C22×C4—1882no—
403C22×C4C2×C4×D4712yes—
404C22×C4C2×C4×D4711yescharacteristic
405C22×C4D4×D4712yes—
406C22×C4D4×D4712yes—
407C22×C4C23×D4711yescharacteristic
408M16—44no—
409M16—1884no—
410C2×C8—24no—
411C2×C8—944no—
412C4⋊C4—944no—
413C4⋊C4—944no—
414C4⋊C4—944no—
415C4⋊C4—944no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—944no—
419C22⋊C4—944no—
420C42—472no—
421C42—472no—
422C42—472no—
423C42—472no—
424C42—472no—
425C42—472no—
426C42—472no—
427C42—472no—
428C42—942no—
429C42—942no—
430C42—942no—
431C42—942no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—416no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23C22⋊C4×D4711yescharacteristic
447C23C22⋊C4×D4711yescharacteristic
448C23M16×D4714yes—
449C23C22×C4×D4711yescharacteristic
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—471no—
459C2×C4—471no—
460C2×C4—474no—
461C2×C4—471no—
462C2×C4—471no—
463C2×C4—472no—
464C2×C4—472no—
465C2×C4—474no—
466C2×C4—942no—
467C2×C4—942no—
468C2×C4—942no—
469C2×C4—942no—
470C2×C4—948no—
471C2×C4—948no—
472C2×C4—944no—
473C2×C4—944no—
474C2×C4—948no—
475C2×C4—948no—
476C2×C4—942no—
477C2×C4—944no—
478C2×C4—942no—
479C2×C4—942no—
480C2×C4—942no—
481C2×C4—1882no—
482C2×C4—1881no—
483C2×C4—1881no—
484C2×C4—1882no—
485C2×C4—1882no—
486C2×C4—1882no—
487C2×C4C22⋊C4×D4712yes—
488C2×C4C2×D4×D4712yes—
489C8—44no—
490C8—1884no—
491C22—28no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—24no—
499C22—44no—
500C22—48no—
501C22—48no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22#1568909512yes—
506C22#1568910412yes—
507C22C2×C22⋊C4×D4711yescharacteristic
508C22C2×M16×D4712yes—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—472no—
513C4—472no—
514C4—941no—
515C4—941no—
516C4—944no—
517C4—942no—
518C4—942no—
519C4—1882no—
520C2—22no—
521C2—22no—
522C2—22no—
523C2—44no—
524C2#407×D4711yescharacteristic
525C2C2×#1568909511yescharacteristic
526C2C2×#1568910411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1462C2
socleC22×C47C22×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenter#1462D47
hypocenterC47#17357
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #20252017C2semidirect
3. #20241719C2semidirect
4. #20241717C2semidirect
6. #20241699C2semidirect
23. #15689177C22semidirect
25. #15689173C22semidirect
29. #15688848C22semidirect
137. C22×Dic47M16semidirect
297. C47#17357semidirect

Maximal subgroups

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

subgroupindex
C22×#130939802
#202520172
#202417192
#202417172
#202417152
#202416992
C47×#14622
#1735747