Finite Group Databasebeta

← #26189403

Subgroups — #26189403

Order 12032.

958 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 187 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
1#20257198C211yesmaximal, characteristic, complemented
2#20253396C212yesmaximal
3#20242395C214yesmaximal, complemented
4#20242327C214yesmaximal
5#20241951C212yesmaximal, complemented
6#20241712C211yesmaximal, characteristic, complemented
7C47×#2536C211yesmaximal, characteristic
8Q8×Dic94—22no—
9#15689852—22no—
10#15689849—24no—
11#15689754—21no—
12#15689754C2214yes—
13#15689680—22no—
14#15689680C2212yes—
15#15689601—21no—
16#15689453C2214yes—
17#15689413—22no—
18#15689328—22no—
19#15689274C2214yes—
20C4×#10625335—24no—
21#15689176C2212yes—
22#15688859C2218yes—
23#15688849C2214yes—
24C2×C22⋊Q8×C47—21no—
25C2×C4×Q8×C47C2211yescharacteristic
26C2×C4×D4×C47C2211yescharacteristic
27C47×#398C2212yes—
28C47×#394C2212yes—
29C47×#388—22no—
30C47×#382C2211yescharacteristic
31C47×#380—21no—
32C2×#10625335—24no—
33#13094024—22no—
34#13094024—22no—
35#13094024D412yes—
36#13094024D412yes—
37#13094023—24no—
38#13094023—28no—
39#13094016—24no—
40#13094016—22no—
41#13094015—24no—
42#13094015—28no—
43#13094003—24no—
44#13094003—22no—
45#13094003C2314yes—
46Q8×Dic47—24no—
47Q8×Dic47—24no—
48C4×Dic94—24no—
49C4×Dic94—24no—
50C4×Dic94—22no—
51C4×Dic94D414yes—
52#13093983—22no—
53#13093983—24no—
54#13093955—24no—
55C4⋊(C47⋊C8)—22no—
56#13093947—24no—
57#13093945—24no—
58C22⋊(C47⋊C8)C2314yes—
59C4×C47⋊C8—22no—
60C22×Q8×C47D411yescharacteristic
61C22×D4×C47C2311yescharacteristic
62C23×C4×C47—21no—
63C22⋊Q8×C47—44no—
64C4×Q8×C47—22no—
65C4×Q8×C47—22no—
66C4×D4×C47—22no—
67C4×D4×C47—22no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47D411yescharacteristic
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C4⋊C4×C47C2312yes—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47—22no—
75C2×C22⋊C4×C47—22no—
76C2×C42×C47D412yes—
77C2×C42×C47C2311yescharacteristic
78#95×C47—22no—
79#95×C47—21no—
80#95×C47C2312yes—
81C24×C47—21no—
82C2×Q8×C47—22no—
83C2×Q8×C47—22no—
84C2×Q8×C47SD1614yescomplemented
85C2×D4×C47—22no—
86C2×D4×C47—42no—
87C22×C4×C47—22no—
88C22×C4×C47—21no—
89C22×C4×C47—42no—
90C22×C4×C47—42no—
91C22×C4×C47—42no—
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2411yescharacteristic
97C2×Dic94—22no—
98C2×Dic94—24no—
99C2×Dic94C2×D414yes—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—24no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—42no—
104C4⋊C4×C47—44no—
105C22⋊C4×C47—44no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—44no—
109C42×C47—22no—
110C42×C47—22no—
111C42×C47C2×D412yes—
112C42×C47C2×D412yes—
113C22⋊Dic47—24no—
114#10625335—48no—
115C4⋊Dic47—24no—
116C4⋊Dic47—24no—
117C4⋊Dic47—24no—
118C4⋊Dic47—22no—
119C4⋊Dic47—22no—
120C4⋊Dic47—22no—
121C4⋊Dic47C2×D414yes—
122Dic47⋊C4—28no—
123Dic47⋊C4—24no—
124Dic47⋊C4—24no—
125C4×Dic47—28no—
126C4×Dic47—24no—
127C2×C47⋊C8—24no—
128C23×C47—22no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47—41no—
135C23×C47C22×D411yescharacteristic
136Q8×C47—28no—
137D4×C47—24no—
138D4×C47—42no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—41no—
148C2×C4×C47—42no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—84no—
152C2×C4×C47#12011yescharacteristic
153C2×C4×C47#12011yescharacteristic
154C2×C4×C47#12012yes—
155C2×C4×C47C2×SD1614yes—
156C2×C4×C47C2×SD1612yes—
157C2×C4×C47#13712yes—
158C2×C4×C47C22×D412yes—
159C2×Dic47—24no—
160C2×Dic47—24no—
161Dic94—44no—
162Dic94—44no—
163Dic94—84no—
164C47⋊C8—44no—
165#18220—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47C2×#12011yescharacteristic
176C22×C47C22×SD1611yescharacteristic
177C22×C47C2×#13711yescharacteristic
178C4×C47—24no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—22no—
182C4×C47—42no—
183C4×C47—82no—
184C4×C47—81no—
185C4×C47#44814yes—
186Dic47—44no—
187Dic47—44no—
188#3502—471no—
189#3101—472no—
190#2536D4711yescharacteristic
191#1611—474no—
192#1599—474no—
193#1518—472no—
194#1461—471no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—82no—
198C2×C47C2×#44811yescharacteristic
199C2×C47#302011yescharacteristic
200C2×C47#302511yescharacteristic
201Q82—942no—
202#557—944no—
203#557—942no—
204#544—474no—
205#544—941no—
206#533—472no—
207#533—942no—
208C2×C22⋊Q8—21no—
209#519—941no—
210C2×C4×Q8C2×D4711yescharacteristic
211C2×C4×D4C2×D4711yescharacteristic
212#483—474no—
213#476—942no—
214#461—942no—
215#450—474no—
216C4×SD16—944no—
217#431—472no—
218#398C2×D4712yes—
219#394C2×D4712yes—
220#388—22no—
221#382C2×D4711yescharacteristic
222#380—21no—
223#328—478no—
224#324—474no—
225C47#1822011yescharacteristic, complemented, Sylow
226C22×Q8C22⋊D4711yescharacteristic
227C22×D4C22×D4711yescharacteristic
228C23×C4—21no—
229C2×SD16—944no—
230C4⋊Q8—472no—
231C4⋊Q8—472no—
232C4⋊Q8—942no—
233C4⋊Q8—942no—
234C4⋊Q8—948no—
235C4⋊Q8—944no—
236#125—944no—
237#125—948no—
238#125—944no—
239#125—942no—
240C22⋊Q8—44no—
241C22⋊Q8—474no—
242C22⋊Q8—942no—
243C22⋊Q8—944no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—474no—
247C4×Q8—944no—
248C4×Q8—944no—
249C4×Q8—944no—
250C4×Q8—944no—
251C4×Q8—942no—
252C4×D4—22no—
253C4×D4—22no—
254#117—942no—
255#117—944no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C22⋊D4711yescharacteristic
259C2×C4⋊C4C22×D4711yescharacteristic
260C2×C4⋊C4C22×D4712yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42C22⋊D4712yes—
265C2×C42C22×D4711yescharacteristic
266#106—944no—
267C4⋊C8—942no—
268#103—944no—
269#102—944no—
270C22⋊C8—474no—
271C4×C8—942no—
272#95—22no—
273#95—21no—
274#95C22×D4712yes—
275C24—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—474no—
279C2×Q8—944no—
280C2×Q8—942no—
281C2×Q8#1062533514yescomplemented
282C2×D4—22no—
283C2×D4—42no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C22⋊D4712yes—
290C22×C4C2×C22⋊D4712yes—
291C22×C4C2×C22⋊D4711yescharacteristic
292C22×C4C2×C22⋊D4711yescharacteristic
293C22×C4C23×D4711yescharacteristic
294SD16—1888no—
295C2×C8—944no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—474no—
302C4⋊C4—948no—
303C4⋊C4—944no—
304C4⋊C4—944no—
305C4⋊C4—944no—
306C4⋊C4—944no—
307C4⋊C4—942no—
308C4⋊C4—942no—
309C4⋊C4—942no—
310C4⋊C4—944no—
311C22⋊C4—44no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—44no—
315C22⋊C4—944no—
316C42—22no—
317C42—22no—
318C42—948no—
319C42—944no—
320C42D4×D4712yes—
321C42D4×D4712yes—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C22×C22⋊D4711yescharacteristic
330Q8—28no—
331Q8—3764no—
332Q8—1884no—
333Q8—1884no—
334D4—24no—
335D4—42no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—84no—
349C2×C4—944no—
350C2×C4—944no—
351C2×C4#1309399211yescharacteristic
352C2×C4#1309399211yescharacteristic
353C2×C4#1309399212yes—
354C2×C4C2×#1062533514yes—
355C2×C4C2×#1062533512yes—
356C2×C4#1309405412yes—
357C2×C4C2×D4×D4712yes—
358C8—1884no—
359C22—21no—
360C22—21no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#1309399211yescharacteristic
369C22C22×#1062533511yescharacteristic
370C22C2×#1309405411yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—24no—
375C4—42no—
376C4—82no—
377C4—81no—
378C4—1884no—
379C4—1884no—
380C4#1568926114yes—
381C2—21no—
382C2—21no—
383C2—82no—
384C2C2×#1568926111yescharacteristic
385C2#2025275111yescharacteristic
386C2#2025277911yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2536C2
socleC22×C47C2×#120
prime radicalC22×D4×C47C23
hypercenter#2536D47
hypocenterC47#18220
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #20257198C2semidirect
3. #20242395C2semidirect
5. #20241951C2semidirect
6. #20241712C2semidirect
84. C2×Q8×C47SD16semidirect
225. C47#18220semidirect
281. C2×Q8#10625335semidirect

Maximal subgroups

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

subgroupindex
#202571982
#202533962
#202423952
#202423272
#202419512
#202417122
C47×#25362
#1822047