Finite Group Databasebeta

← #26134566

Subgroups — #26134566

Order 12032.

840 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688910C214yesmaximal, complemented
2#20243376C211yesmaximal, characteristic, complemented
3C4×#13093941C211yesmaximal, characteristic, complemented
4C47×#2408C211yesmaximal, characteristic
5C2×#13093941C2212yes—
6C2×#13093939C2212yes—
7C4×#10625329—21no—
8#15688983—21no—
9#15688910—28no—
10#15688892—21no—
11#15688890—21no—
12C23×D4×C47C411yescharacteristic, complemented
13C2×#121×C47—21no—
14C2×#120×C47C2212yes—
15C2×C4×D4×C47C2211yescharacteristic
16C47×#394—21no—
17C47×#392—21no—
18C47×#390—21no—
19C47×#380C411yescharacteristic, complemented
20C47×#379C412yescomplemented
21C2×#10625329—22no—
22C2×C4⋊Dic47—41no—
23C2×C22⋊Dic47—22no—
24#13093941—22no—
25#13093941D414yescomplemented
26#13093939—24no—
27#13093935—41no—
28C4×C47⋊C8—41no—
29C47⋊#95—41no—
30C25×C47C2×C412yes—
31C22×D4×C47—22no—
32C22×D4×C47—21no—
33C22×D4×C47—21no—
34C22×D4×C47—41no—
35C22×D4×C47—41no—
36C22×D4×C47—44no—
37C22×D4×C47C2311yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40#121×C47—44no—
41#120×C47—24no—
42#120×C47—24no—
43C4×D4×C47—41no—
44C4×D4×C47—41no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47C2×C412yes—
51C2×C22⋊C4×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C42×C47D411yescharacteristic
54#95×C47—21no—
55C24×C47—22no—
56C24×C47—24no—
57C24×C47—42no—
58C24×C47—44no—
59C24×C47—42no—
60C24×C47—82no—
61C24×C47C22×C411yescharacteristic
62C24×C47C2×D411yescharacteristic
63C2×D4×C47—24no—
64C2×D4×C47—41no—
65C2×D4×C47—41no—
66C2×D4×C47—42no—
67C2×D4×C47—42no—
68C2×D4×C47—44no—
69C2×D4×C47—48no—
70C2×D4×C47—44no—
71C2×D4×C47—44no—
72C2×D4×C47—44no—
73C2×D4×C47—42no—
74C2×D4×C47—42no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—82no—
78C2×D4×C47C2×D412yes—
79C2×D4×C47#4112yes—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47C22⋊C411yescharacteristic
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22⋊C411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×Dic47—42no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—42no—
95C22⋊C4×C47—24no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—44no—
99C22⋊C4×C47—44no—
100C22⋊C4×C47—82no—
101C42×C47—21no—
102C42×C47—21no—
103C22⋊Dic47—42no—
104C22⋊Dic47—42no—
105C4⋊Dic47—82no—
106#10625329—24no—
107#10625329—42no—
108C2×C47⋊C8—42no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—24no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—44no—
122C23×C47—44no—
123C23×C47—42no—
124C23×C47—42no—
125C23×C47—42no—
126C23×C47—84no—
127C23×C47—88no—
128C23×C47—88no—
129C23×C47—84no—
130C23×C47C2×C22⋊C411yescharacteristic
131C23×C47C4×D412yes—
132C23×C47#12411yescharacteristic
133C23×C47#12711yescharacteristic
134D4×C47—42no—
135D4×C47—42no—
136D4×C47—44no—
137D4×C47—48no—
138D4×C47—44no—
139D4×C47—84no—
140D4×C47—84no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—22no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47—42no—
156C2×C4×C47—81no—
157C2×C4×C47#9911yescharacteristic
158C2×C4×C47#9911yescharacteristic
159C2×C4×C47#12112yes—
160C2×Dic47—82no—
161C2×Dic47—82no—
162C2×Dic47—82no—
163C47⋊C8—44no—
164#13218—471nomaximal, Sylow
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—22no—
169C22×C47—21no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47—42no—
173C22×C47—42no—
174C22×C47—44no—
175C22×C47—44no—
176C22×C47—41no—
177C22×C47—41no—
178C22×C47—41no—
179C22×C47—41no—
180C22×C47—44no—
181C22×C47—81no—
182C22×C47—82no—
183C22×C47—82no—
184C22×C47—88no—
185C22×C47—88no—
186C22×C47—82no—
187C22×C47—82no—
188C22×C47#39011yescharacteristic
189C22×C47C2×#9911yescharacteristic
190C22×C47#41011yescharacteristic
191C4×C47—21no—
192C4×C47—22no—
193C4×C47—41no—
194C4×C47—41no—
195C4×C47—41no—
196C4×C47—82no—
197C4×C47AΣL(1,𝔽16)12yescomplemented
198Dic47—162no—
199#2408D4711yescharacteristic
200C2×AΣL(1,𝔽16)—474no—
201#1799—471no—
202C4×#100—471no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—41no—
206C2×C47—42no—
207C2×C47—41no—
208C2×C47—84no—
209C2×C47#196411yescharacteristic
210C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
211C2×C47#212211yescharacteristic
212C23×D4Dic4711yescharacteristic, complemented
213C2×#121—21no—
214C2×#120C2×D4712yes—
215C2×C4×D4C2×D4711yescharacteristic
216C2×#100—472no—
217C2×#99—472no—
218C4×M16—941no—
219#394—21no—
220#392—21no—
221#390—21no—
222#380—941no—
223#380Dic4711yescharacteristic, complemented
224#379Dic4712yescomplemented
225AΣL(1,𝔽16)—948no—
226#343—941no—
227#342—941no—
228C47#1321811yescharacteristic, complemented, Sylow
229C25C2×Dic4712yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—41no—
234C22×D4—41no—
235C22×D4—44no—
236C22×D4C22×D4711yescharacteristic
237C23×C4C2×Dic4711yescharacteristic
238C23×C4C22⋊D4711yescharacteristic
239C2×M16—942no—
240#121—44no—
241#120—24no—
242#120—24no—
243C4×D4—41no—
244C4×D4—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1881no—
247C2×C4⋊C4C22⋊D4711yescharacteristic
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—942no—
252C2×C22⋊C4C2×Dic4712yes—
253C2×C22⋊C4C2×Dic4711yescharacteristic
254C2×C22⋊C4C22⋊D4711yescharacteristic
255C2×C42C22⋊D4711yescharacteristic
256#100—474no—
257#100—942no—
258#99—944no—
259#97—1881no—
260C4×C8—1881no—
261#95—21no—
262#95—1881no—
263C24—22no—
264C24—24no—
265C24—42no—
266C24—44no—
267C24—42no—
268C24—82no—
269C24C22×Dic4711yescharacteristic
270C24C2×C22⋊D4711yescharacteristic
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—82no—
286C2×D4D4×D4712yes—
287C2×D4#1062535212yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—1882no—
296C22×C4C22⋊Dic4711yescharacteristic
297C22×C4C22⋊Dic4711yescharacteristic
298C22×C4C22⋊Dic4711yescharacteristic
299C22×C4C22⋊Dic4711yescharacteristic
300C22×C4C2×C22⋊D4711yescharacteristic
301M16—944no—
302M16—1882no—
303C2×C8—1882no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—3762no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—1882no—
314C22⋊C4—1882no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—84no—
335C23—88no—
336C23—88no—
337C23—84no—
338C23C2×C22⋊Dic4711yescharacteristic
339C23D4×Dic4712yes—
340C23#1309401111yescharacteristic
341C23#1309402011yescharacteristic
342D4—42no—
343D4—42no—
344D4—44no—
345D4—48no—
346D4—44no—
347D4—84no—
348D4—84no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—3762no—
366C2×C4—3762no—
367C2×C4—3762no—
368C2×C4#1309393911yescharacteristic
369C2×C4#1309393911yescharacteristic
370C2×C4#1309399512yes—
371C8—1884no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—44no—
382C22—44no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—44no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—88no—
392C22—88no—
393C22—82no—
394C22—82no—
395C22#1568902611yescharacteristic
396C22C2×#1309393911yescharacteristic
397C22#1568909711yescharacteristic
398C4—22no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—7522no—
405C4#1568891012yescomplemented
406C2—21no—
407C2—21no—
408C2—41no—
409C2—42no—
410C2—41no—
411C2—84no—
412C2#2024422511yescharacteristic
413C2C2×#1568891011yescharacteristic
414C2#2024503811yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
Fitting, prime radicalC47×#2408C2
socleC22×C47C2×#99
cosocleC22×D4×C47C23
hypercenter#2408D47
hypocenterC47#13218
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#15688910C2semidirect
2. #20243376C2semidirect
3. C4×#13093941C2semidirect
12. C23×D4×C47C4semidirect
19. C47×#380C4semidirect
20. C47×#379C4semidirect
25. #13093941D4semidirect
197. C4×C47AΣL(1,𝔽16)semidirect
212. C23×D4Dic47semidirect
223. #380Dic47semidirect
224. #379Dic47semidirect
228. C47#13218semidirect
405. C4#15688910semidirect

Maximal subgroups

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

subgroupindex
C2×#156889102
#202433762
C4×#130939412
C47×#24082
#1321847