Finite Group Databasebeta

← #26134579

Subgroups — #26134579

Order 12032.

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

normal subgroups only

This group has 844 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#20243380C211yesmaximal, characteristic, complemented
3C4×#13093939C211yesmaximal, characteristic, complemented
4C47×#2408C211yesmaximal, characteristic
5#15689147—21no—
6C2×#13093941C2212yes—
7C2×#13093939C2212yes—
8C4×C22⋊Dic47—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×C22⋊Dic47—22no—
23C2×C4×Dic47—41no—
24C4⋊(C47⋊C8)—41no—
25C4⋊(C47⋊C8)—41no—
26#13093941—24no—
27#13093939—22no—
28#13093939D414yescomplemented
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—44no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47—82no—
77C2×D4×C47C2×D412yes—
78C2×D4×C47#4112yes—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—22no—
84C22×C4×C47—42no—
85C22×C4×C47—42no—
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47C22⋊C411yescharacteristic
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×Dic47—42no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—42no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—44no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—82no—
101C42×C47—21no—
102C42×C47—21no—
103C22⋊Dic47—24no—
104C22⋊Dic47—42no—
105C4×Dic47—82no—
106#10625329—42no—
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#12112yes—
133D4×C47—42no—
134D4×C47—42no—
135D4×C47—44no—
136D4×C47—48no—
137D4×C47—44no—
138D4×C47—88no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—22no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—81no—
155C2×C4×C47#9911yescharacteristic
156C2×C4×C47#9911yescharacteristic
157C2×C4×C47#12411yescharacteristic
158C2×C4×C47#12711yescharacteristic
159C2×Dic47—44no—
160C2×Dic47—82no—
161C2×Dic47—82no—
162C47⋊C8—82no—
163#13221—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—44no—
172C22×C47—44no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47—41no—
177C22×C47—44no—
178C22×C47—81no—
179C22×C47—82no—
180C22×C47—82no—
181C22×C47—88no—
182C22×C47—88no—
183C22×C47—82no—
184C22×C47—82no—
185C22×C47#39011yescharacteristic
186C22×C47C2×#9911yescharacteristic
187C22×C47#41011yescharacteristic
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—41no—
192C4×C47—41no—
193C4×C47—41no—
194C4×C47—82no—
195C4×C47AΣL(1,𝔽16)12yescomplemented
196Dic47—84no—
197#2408D4711yescharacteristic
198C2×AΣL(1,𝔽16)—474no—
199#1800—471no—
200C4×#99—471no—
201C2×C47—21no—
202C2×C47—21no—
203C2×C47—42no—
204C2×C47—42no—
205C2×C47—84no—
206C2×C47#196411yescharacteristic
207C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
208C2×C47#212211yescharacteristic
209C23×D4Dic4711yescharacteristic, complemented
210C2×#121—21no—
211C2×#120C2×D4712yes—
212C2×C4×D4C2×D4711yescharacteristic
213#423—941no—
214C2×#100—472no—
215C2×#99—472no—
216#394—21no—
217#392—21no—
218#390—21no—
219#380Dic4711yescharacteristic, complemented
220#379Dic4712yescomplemented
221C4×C22⋊C4—941no—
222AΣL(1,𝔽16)—948no—
223#343—941no—
224#342—941no—
225C47#1322111yescharacteristic, complemented, Sylow
226C25C2×Dic4712yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—41no—
231C22×D4—41no—
232C22×D4—44no—
233C22×D4C22×D4711yescharacteristic
234C23×C4C2×Dic4711yescharacteristic
235C23×C4C22⋊D4711yescharacteristic
236C2×M16—942no—
237#121—44no—
238#120—24no—
239#120—24no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C22⋊D4711yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—942no—
248C2×C22⋊C4C2×Dic4712yes—
249C2×C22⋊C4C2×Dic4711yescharacteristic
250C2×C22⋊C4C22⋊D4711yescharacteristic
251C2×C42—1881no—
252C2×C42C22⋊D4711yescharacteristic
253C4⋊C8—1881no—
254C4⋊C8—1881no—
255#100—944no—
256#99—474no—
257#99—942no—
258#95—21no—
259#95—1881no—
260C24—22no—
261C24—24no—
262C24—42no—
263C24—44no—
264C24—42no—
265C24—82no—
266C24C22×Dic4711yescharacteristic
267C24C2×C22⋊D4711yescharacteristic
268C2×D4—24no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—82no—
282C2×D4D4×D4712yes—
283C2×D4#1062535212yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—1882no—
292C22×C4C22⋊Dic4711yescharacteristic
293C22×C4C22⋊Dic4711yescharacteristic
294C22×C4C22⋊Dic4711yescharacteristic
295C22×C4C22⋊Dic4711yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297M16—1882no—
298M16—1882no—
299C2×C8—1882no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C22⋊C4—24no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—82no—
309C22⋊C4—944no—
310C22⋊C4—1882no—
311C42—21no—
312C42—21no—
313C42—3762no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—84no—
332C23—88no—
333C23—88no—
334C23—84no—
335C23C2×C22⋊Dic4711yescharacteristic
336C23D4×Dic4712yes—
337C23#1309399512yes—
338D4—42no—
339D4—42no—
340D4—44no—
341D4—48no—
342D4—44no—
343D4—88no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—3762no—
361C2×C4—3762no—
362C2×C4—1884no—
363C2×C4#1309393911yescharacteristic
364C2×C4#1309393911yescharacteristic
365C2×C4#1309401111yescharacteristic
366C2×C4#1309402011yescharacteristic
367C8—3762no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—44no—
376C22—44no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—44no—
382C22—81no—
383C22—82no—
384C22—82no—
385C22—88no—
386C22—88no—
387C22—82no—
388C22—82no—
389C22#1568902611yescharacteristic
390C22C2×#1309393911yescharacteristic
391C22#1568909711yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—82no—
399C4—3764no—
400C4#1568891012yescomplemented
401C2—21no—
402C2—21no—
403C2—42no—
404C2—42no—
405C2—84no—
406C2#2024422511yescharacteristic
407C2C2×#1568891011yescharacteristic
408C2#2024503811yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
Fitting, prime radicalC47×#2408C2
socleC22×C47C2×#99
cosocleC22×D4×C47C23
hypercenter#2408D47
hypocenterC47#13221
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. #20243380C2semidirect
3. C4×#13093939C2semidirect
12. C23×D4×C47C4semidirect
19. C47×#380C4semidirect
20. C47×#379C4semidirect
28. #13093939D4semidirect
195. C4×C47AΣL(1,𝔽16)semidirect
209. C23×D4Dic47semidirect
219. #380Dic47semidirect
220. #379Dic47semidirect
225. C47#13221semidirect
400. C4#15688910semidirect

Maximal subgroups

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

subgroupindex
C2×#156889102
#202433802
C4×#130939392
C47×#24082
#1322147