Finite Group Databasebeta

← #26109665

Subgroups — #26109665

Order 12032.

868 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243298C211yesmaximal, characteristic
2#20241611C211yesmaximal, characteristic
3#20241266C212yesmaximal, complemented
4#20241266C212yesmaximal, complemented
5C47×#2280C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)C2212yes—
10C2×C47⋊#95—21no—
11C2×C47⋊#95C2212yes—
12C24×C4×C47C2211yescharacteristic
13C2×#117×C47C411yescharacteristic, complemented
14C2×#117×C47C411yescharacteristic, complemented
15C47×#381—21no—
16C47×#380C411yescharacteristic, complemented
17C47×#380C411yescharacteristic, complemented
18C47×#380C2211yescharacteristic
19C47×#380C2211yescharacteristic
20C47×#379—21no—
21C4×C22⋊C4×C47—21no—
22C4×C22⋊C4×C47—21no—
23C23×Dic47—22no—
24C22×C47⋊C8—22no—
25C2×C22⋊Dic47—24no—
26C2×C22⋊Dic47—42no—
27C2×C22⋊Dic47—42no—
28C22⋊(C47⋊C8)—24no—
29C22⋊(C47⋊C8)—42no—
30C22⋊(C47⋊C8)—42no—
31C47⋊#95—24no—
32C47⋊#95—42no—
33C25×C47D411yescharacteristic
34C23×C4×C47—24no—
35C23×C4×C47—42no—
36C23×C4×C47—42no—
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43#117×C47—42no—
44#117×C47—42no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C42×C47—21no—
56C2×C42×C47—21no—
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61C24×C47—22no—
62C24×C47—22no—
63C24×C47—21no—
64C24×C47—21no—
65C24×C47—22no—
66C24×C47—22no—
67C24×C47—41no—
68C24×C47—41no—
69C24×C47C22⋊C411yescharacteristic
70C24×C47C22⋊C411yescharacteristic
71C24×C47C2×D411yescharacteristic
72C22×C4×C47—22no—
73C22×C4×C47—22no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—416no—
94C22×C4×C47C22⋊C411yescharacteristic
95C22×C4×C47C22⋊C411yescharacteristic
96C22×C4×C47C22×C411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47#4114yes—
99C22×Dic47—24no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22×Dic47—42no—
103C22×Dic47—42no—
104C22×Dic47—42no—
105C22×Dic47—42no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C22⋊C4×C47—42no—
111C22⋊C4×C47—42no—
112C22⋊C4×C47—42no—
113C22⋊C4×C47—42no—
114C22⋊C4×C47—42no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47—41no—
117C42×C47—42no—
118C42×C47—42no—
119C22⋊Dic47—48no—
120C2×C47⋊C8—42no—
121C2×C47⋊C8—42no—
122C2×C47⋊C8—42no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—21no—
127C23×C47—24no—
128C23×C47—24no—
129C23×C47—21no—
130C23×C47—21no—
131C23×C47—21no—
132C23×C47—21no—
133C23×C47—21no—
134C23×C47—21no—
135C23×C47—21no—
136C23×C47—21no—
137C23×C47—21no—
138C23×C47—21no—
139C23×C47—21no—
140C23×C47—21no—
141C23×C47—44no—
142C23×C47—44no—
143C23×C47—44no—
144C23×C47—44no—
145C23×C47—42no—
146C23×C47—42no—
147C23×C47—42no—
148C23×C47—42no—
149C23×C47#9912yes—
150C23×C47#10412yes—
151C23×C47C2×C22⋊C411yescharacteristic
152C23×C47#12312yes—
153C23×C47#13512yes—
154C23×C47AGL(1,ℤ/8)11yescharacteristic
155C23×C47#13711yescharacteristic
156C2×C4×C47—28no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—44no—
167C2×C4×C47—44no—
168C2×C4×C47—44no—
169C2×C4×C47—44no—
170C2×C4×C47—44no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—41no—
180C2×C4×C47—41no—
181C2×C4×C47—41no—
182C2×C4×C47—41no—
183C2×C4×C47#9912yes—
184C2×C4×C47#11712yes—
185C2×C4×C47#12312yes—
186C2×Dic47—44no—
187C2×Dic47—44no—
188C2×Dic47—44no—
189C2×Dic47—82no—
190C2×Dic47—82no—
191C2×Dic47—82no—
192C2×Dic47—82no—
193C47⋊C8—82no—
194#8715—471nomaximal, Sylow
195C22×C47—22no—
196C22×C47—22no—
197C22×C47—22no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—22no—
203C22×C47—22no—
204C22×C47—22no—
205C22×C47—22no—
206C22×C47—22no—
207C22×C47—22no—
208C22×C47—22no—
209C22×C47—48no—
210C22×C47—48no—
211C22×C47—41no—
212C22×C47—41no—
213C22×C47—41no—
214C22×C47—41no—
215C22×C47—41no—
216C22×C47—41no—
217C22×C47—41no—
218C22×C47—41no—
219C22×C47#38111yescharacteristic
220C22×C47C2×#9911yescharacteristic
221C22×C47C2×#9911yescharacteristic
222C22×C47#41611yescharacteristic
223C22×C47#41711yescharacteristic
224C22×C47C2×#10411yescharacteristic
225C22×C47#42111yescharacteristic
226C22×C47#48212yes—
227C22×C47#48512yes—
228C4×C47—22no—
229C4×C47—22no—
230C4×C47—22no—
231C4×C47—41no—
232C4×C47—81no—
233C4×C47—81no—
234C4×C47—81no—
235C4×C47—81no—
236Dic47—84no—
237#2280D4711yescharacteristic
238#1779—471no—
239C22⋊#95—471no—
240#1294—472no—
241#1294—472no—
242C2×C47—22no—
243C2×C47—22no—
244C2×C47—42no—
245C2×C47—42no—
246C2×C47#150111yescharacteristic
247C2×C47#150411yescharacteristic
248C2×C47#151311yescharacteristic
249C2×C47#151611yescharacteristic
250C2×C47#181911yescharacteristic
251C2×C47#182511yescharacteristic
252C2×C47#182611yescharacteristic
253C24×C4C2×D4711yescharacteristic
254C2×#117Dic4711yescharacteristic, complemented
255C2×#117Dic4711yescharacteristic, complemented
256C22×C22⋊C4—941no—
257C2×C22⋊C8—472no—
258C2×C22⋊C8—941no—
259C2×C22⋊C8—941no—
260#381—21no—
261#380Dic4711yescharacteristic, complemented
262#380Dic4711yescharacteristic, complemented
263#380C2×D4711yescharacteristic
264#380C2×D4711yescharacteristic
265#379—21no—
266C4×C22⋊C4—21no—
267C4×C22⋊C4—21no—
268C2×#95—472no—
269C2×#95—941no—
270C47#871511yescharacteristic, complemented, Sylow
271C25C22⋊D4711yescharacteristic
272C23×C4—24no—
273C23×C4—42no—
274C23×C4—42no—
275C23×C4—942no—
276C23×C4C2×Dic4711yescharacteristic
277C23×C4C2×Dic4711yescharacteristic
278C23×C4C22⋊D4711yescharacteristic
279C23×C4C22⋊D4711yescharacteristic
280C23×C4C22⋊D4711yescharacteristic
281C23×C4C22×D4711yescharacteristic
282C22×C8—942no—
283#117—42no—
284#117—42no—
285C2×C4⋊C4C2×Dic4711yescharacteristic
286C2×C4⋊C4C2×Dic4711yescharacteristic
287C2×C4⋊C4C2×Dic4711yescharacteristic
288C2×C4⋊C4C2×Dic4711yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—944no—
296C2×C22⋊C4—1882no—
297C2×C22⋊C4—1882no—
298C2×C42—21no—
299C2×C42—21no—
300C22⋊C8—944no—
301C22⋊C8—1882no—
302C22⋊C8—1882no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—944no—
308#95—1882no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—41no—
316C24—41no—
317C24C22⋊Dic4711yescharacteristic
318C24C22⋊Dic4711yescharacteristic
319C24C2×C22⋊D4711yescharacteristic
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—416no—
342C22×C4—944no—
343C22×C4—1882no—
344C22×C4—1882no—
345C22×C4—1882no—
346C22×C4—1882no—
347C22×C4—1882no—
348C22×C4—1882no—
349C22×C4C22⋊Dic4711yescharacteristic
350C22×C4C22⋊Dic4711yescharacteristic
351C22×C4#1062535014yes—
352C22×C4C22×Dic4711yescharacteristic
353C22×C4C2×C22⋊D4711yescharacteristic
354C2×C8—1882no—
355C2×C8—1882no—
356C2×C8—1882no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—1888no—
369C42—42no—
370C42—42no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—24no—
376C23—24no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#1309393912yes—
398C23#1309395112yes—
399C23C2×C22⋊Dic4711yescharacteristic
400C23#1309400712yes—
401C23#1309404412yes—
402C23#1309404911yescharacteristic
403C23#1309405411yescharacteristic
404C2×C4—28no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—3762no—
432C2×C4—3762no—
433C2×C4—3762no—
434C2×C4—3762no—
435C2×C4—1884no—
436C2×C4—1884no—
437C2×C4—1884no—
438C2×C4#1309393912yes—
439C2×C4#1309398312yes—
440C2×C4#1309400712yes—
441C8—3762no—
442C22—22no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—48no—
457C22—48no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22#1568898611yescharacteristic
467C22C2×#1309393911yescharacteristic
468C22C2×#1309393911yescharacteristic
469C22#1568911911yescharacteristic
470C22#1568912411yescharacteristic
471C22C2×#1309395111yescharacteristic
472C22#1568914011yescharacteristic
473C22#1568944712yes—
474C22#1568946412yes—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—3764no—
484C2—22no—
485C2—22no—
486C2—42no—
487C2—42no—
488C2#2024187911yescharacteristic
489C2#2024189211yescharacteristic
490C2#2024193111yescharacteristic
491C2#2024194411yescharacteristic
492C2#2024346211yescharacteristic
493C2#2024348311yescharacteristic
494C2#2024348711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2280C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2280D47
hypocenterC47#8715
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
3. #20241266C2semidirect
4. #20241266C2semidirect
13. C2×#117×C47C4semidirect
14. C2×#117×C47C4semidirect
16. C47×#380C4semidirect
17. C47×#380C4semidirect
254. C2×#117Dic47semidirect
255. C2×#117Dic47semidirect
261. #380Dic47semidirect
262. #380Dic47semidirect
270. C47#8715semidirect

Maximal subgroups

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

subgroupindex
#202432982
#202416112
#202412662
C47×#22802
#871547