Finite Group Databasebeta

← #26147460

Subgroups — #26147460

Order 12032.

2430 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Dic47C211yesmaximal, characteristic
2C2×#15688986C211yesmaximal, characteristic
3C2×#15688981C212yesmaximal
4Dic47⋊#95C214yesmaximal, complemented
5#20241614C214yesmaximal, complemented
6#20241610C212yesmaximal
7C2×C47×#381C211yesmaximal, characteristic
8C24×Dic47C2211yescharacteristic
9C24×Dic47C2211yescharacteristic
10C22×Dic47⋊C4C2214yes—
11C22×C22⋊Dic47C2212yes—
12C22×C4×Dic47—28no—
13C22×C4×Dic47C418yescomplemented
14C22×C4×Dic47C2212yes—
15C22×C4×Dic47C2212yes—
16#15688986—24no—
17#15688981—28no—
18C2×C47⋊#95C2214yes—
19C2×C47⋊#95C2218yes—
20C2×C47⋊#95C2214yes—
21C24×C4×C47C2211yescharacteristic
22C22×C22⋊C4×C47C2212yes—
23C47×#381—24no—
24C2×#95×C47C2214yes—
25C23×Dic47—24no—
26C23×Dic47—28no—
27C23×Dic47—28no—
28C23×Dic47C2×C414yes—
29C23×Dic47C2×C414yes—
30C23×Dic47D418yes—
31C23×Dic47C2314yes—
32C23×Dic47C2311yescharacteristic
33C23×Dic47C2312yes—
34C23×Dic47C2311yescharacteristic
35C2×Dic47⋊C4—28no—
36C2×Dic47⋊C4—216no—
37C2×C22⋊Dic47—24no—
38C2×C22⋊Dic47—24no—
39C2×C22⋊Dic47—24no—
40C2×C4×Dic47—232no—
41C2×C4×Dic47—28no—
42C2×C4×Dic47—28no—
43C2×C4×Dic47C2×C418yes—
44C2×C4×Dic47C2×C418yes—
45C47⋊#95—28no—
46C47⋊#95—28no—
47C47⋊#95—216no—
48C25×C47C2311yescharacteristic
49C23×C4×C47—24no—
50C23×C4×C47—24no—
51C23×C4×C47C2×C414yes—
52C23×C4×C47D414yes—
53C23×C4×C47Q814yes—
54C23×C4×C47C2314yes—
55C23×C4×C47C2312yes—
56C2×C22⋊C4×C47—24no—
57C2×C22⋊C4×C47—24no—
58C2×C22⋊C4×C47—24no—
59#95×C47—28no—
60C24×C47—24no—
61C24×C47—24no—
62C24×C47C22×C412yes—
63C24×C47C2×D414yes—
64C24×C47C2×D412yes—
65C24×C47C2×Q812yes—
66C24×C47#4114yes—
67C24×C47C2411yescharacteristic
68C22×C4×C47—28no—
69C22×C4×C47—28no—
70C22×C4×C47—216no—
71C22×C4×C47—216no—
72C22×C4×C47—24no—
73C22×C4×C47—24no—
74C22×C4×C47—24no—
75C22×C4×C47—24no—
76C22×C4×C47—24no—
77C22×C4×C47—28no—
78C22×C4×C47—28no—
79C22×C4×C47C4⋊C4116yescomplemented
80C22×C4×C47C22×C414yes—
81C22×C4×C47C2×D414yes—
82C22×C4×C47C2×Q814yes—
83C22×Dic47—232no—
84C22×Dic47—28no—
85C22×Dic47—24no—
86C22×Dic47—24no—
87C22×Dic47—24no—
88C22×Dic47—216no—
89C22×Dic47—216no—
90C22×Dic47—232no—
91C22×Dic47—24no—
92C22×Dic47—24no—
93C22×Dic47—28no—
94C22×Dic47—24no—
95C22×Dic47—28no—
96C22×Dic47C22⋊C4116yes—
97C22×Dic47C22×C412yes—
98C22×Dic47C22×C414yes—
99C22×Dic47C22×C412yes—
100C22×Dic47C2×D414yes—
101C22×Dic47C2×D414yes—
102C22×Dic47#4118yes—
103C22×Dic47#4118yes—
104C22⋊C4×C47—48no—
105C22⋊Dic47—48no—
106Dic47⋊C4—416no—
107C4×Dic47—216no—
108C4×Dic47—216no—
109C23×C47—24no—
110C23×C47—24no—
111C23×C47—28no—
112C23×C47—28no—
113C23×C47—216no—
114C23×C47—28no—
115C23×C47—28no—
116C23×C47C2×C22⋊C414yes—
117C23×C47C2×C4⋊C414yes—
118C23×C47#11714yes—
119C23×C47#12118yes—
120C23×C47C22⋊Q818yes—
121C23×C47C23×C411yescharacteristic
122C23×C47C22×D412yes—
123C23×C47C22×D411yescharacteristic
124C23×C47C22×Q811yescharacteristic
125C23×C47C2×#4114yes—
126C23×C47C2×#4114yes—
127C23×C47C2×#4112yes—
128C2×C4×C47—24no—
129C2×C4×C47—24no—
130C2×C4×C47—28no—
131C2×C4×C47—28no—
132C2×C4×C47—232no—
133C2×C4×C47—44no—
134C2×C4×C47—44no—
135C2×C4×C47—44no—
136C2×C4×C47—48no—
137C2×C4×C47—48no—
138C2×C4×C47C2×C4⋊C418yes—
139C2×Dic47—232no—
140C2×Dic47—24no—
141C2×Dic47—24no—
142C2×Dic47—24no—
143C2×Dic47—28no—
144C2×Dic47—216no—
145C2×Dic47—232no—
146C2×Dic47—44no—
147C2×Dic47—44no—
148C2×Dic47—44no—
149C2×Dic47—48no—
150C2×Dic47—48no—
151C2×Dic47C2×C22⋊C414yes—
152C2×Dic47C2×C22⋊C414yes—
153C2×Dic47#11718yes—
154C2×Dic47#11718yes—
155C2×Dic47#123116yes—
156#14685—471nomaximal, Sylow
157C22×C47—216no—
158C22×C47—216no—
159C22×C47—24no—
160C22×C47—24no—
161C22×C47—28no—
162C22×C47—28no—
163C22×C47#38018yes—
164C22×C47C22×C22⋊C411yescharacteristic
165C22×C47C22×C4⋊C411yescharacteristic
166C22×C47C2×#11712yes—
167C22×C47C2×#11712yes—
168C22×C47C2×#11711yescharacteristic
169C22×C47C2×#12112yes—
170C22×C47C2×C22⋊Q812yes—
171C22×C47C2×#12314yes—
172C22×C47#52514yes—
173C22×C47#52918yes—
174C22×C47#53214yes—
175C22×C47#53314yes—
176C4×C47—28no—
177C4×C47—84no—
178Dic47—24no—
179Dic47—28no—
180Dic47—84no—
181Dic47#38118yescomplemented
182C23×C42—471no—
183C2×#381—471no—
184C2×#381D4711yescharacteristic
185C2×#380—472no—
186C4⋊#95—474no—
187#1434—474no—
188C22⋊#95—472no—
189C2×C47—28no—
190C2×C47C2×#38011yescharacteristic
191C2×C47C2×#38111yescharacteristic
192C2×C47#228011yescharacteristic
193C2×C47#228312yes—
194C2×C47#228512yes—
195C2×C47#285414yes—
196C2×C47#285514yes—
197C24×C4—471no—
198C24×C4—471no—
199C24×C4C2×D4711yescharacteristic
200C22×C4⋊C4—474no—
201C22×C22⋊C4—472no—
202C22×C22⋊C4C2×D4712yes—
203C22×C42—478no—
204C22×C42—472no—
205C22×C42—472no—
206C22×C42—948no—
207#381—24no—
208#381—944no—
209#380—948no—
210C2×#95—478no—
211C2×#95—474no—
212C2×#95—474no—
213C2×#95C2×D4714yes—
214C47#1468511yescharacteristic, complemented, Sylow
215C25C22×D4711yescharacteristic
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—471no—
219C23×C4—474no—
220C23×C4—478no—
221C23×C4—471no—
222C23×C4—472no—
223C23×C4—474no—
224C23×C4—474no—
225C23×C4—948no—
226C23×C4—948no—
227C23×C4—944no—
228C23×C4Dic9414yes—
229C23×C4C4×D4714yes—
230C23×C4C22⋊D4714yes—
231C23×C4C22×D4714yes—
232C23×C4C22×D4712yes—
233C2×C4⋊C4—9416no—
234C2×C4⋊C4—948no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—944no—
239C2×C22⋊C4—944no—
240C2×C22⋊C4—944no—
241C2×C42—478no—
242C2×C42—478no—
243C2×C42—9432no—
244C2×C42—948no—
245C2×C42—948no—
246#95—28no—
247#95—9416no—
248#95—948no—
249#95—948no—
250C24—24no—
251C24—24no—
252C24C2×Dic9412yes—
253C24C2×C4×D4712yes—
254C24D4×D4714yes—
255C24#1062535214yes—
256C24C2×C22⋊D4712yes—
257C24C23×D4711yescharacteristic
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—216no—
261C22×C4—216no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—472no—
270C22×C4—474no—
271C22×C4—478no—
272C22×C4—4716no—
273C22×C4—472no—
274C22×C4—474no—
275C22×C4—474no—
276C22×C4—478no—
277C22×C4—9416no—
278C22×C4—9432no—
279C22×C4—9416no—
280C22×C4—9432no—
281C22×C4—948no—
282C22×C4—944no—
283C22×C4—944no—
284C22×C4—944no—
285C22×C4—944no—
286C22×C4—944no—
287C22×C4—948no—
288C22×C4—948no—
289C22×C4—944no—
290C22×C4Dic47⋊C4116yescomplemented
291C22×C4C2×Dic9414yes—
292C22×C4C2×C4×D4714yes—
293C22×C4C2×C22⋊D4714yes—
294C4⋊C4—18816no—
295C22⋊C4—48no—
296C22⋊C4—1888no—
297C42—9416no—
298C42—9416no—
299C23—24no—
300C23—24no—
301C23—28no—
302C23—28no—
303C23—216no—
304C23—28no—
305C23—28no—
306C23C22⋊C4×D4714yes—
307C23C2×Dic47⋊C414yes—
308C23#1309398014yes—
309C23#1309399618yes—
310C23#1309400118yes—
311C23C22×C4×D4711yescharacteristic
312C23C2×D4×D4712yes—
313C23C22×C22⋊D4711yescharacteristic
314C23C22×Dic9411yescharacteristic
315C23#41×D4714yes—
316C23#41×D4714yes—
317C23C2×#1062535212yes—
318C2×C4—232no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—474no—
329C2×C4—478no—
330C2×C4—4716no—
331C2×C4—474no—
332C2×C4—478no—
333C2×C4—9432no—
334C2×C4—9432no—
335C2×C4—944no—
336C2×C4—944no—
337C2×C4—944no—
338C2×C4—948no—
339C2×C4—9416no—
340C2×C4—1884no—
341C2×C4—1884no—
342C2×C4—1884no—
343C2×C4—1888no—
344C2×C4—1888no—
345C2×C4C2×Dic47⋊C418yes—
346C22—24no—
347C22—24no—
348C22—28no—
349C22—28no—
350C22—216no—
351C22—216no—
352C22#1568898118yes—
353C22C2×C22⋊C4×D4711yescharacteristic
354C22C22×Dic47⋊C411yescharacteristic
355C22#117×D4712yes—
356C22#117×D4712yes—
357C22C2×#1309398011yescharacteristic
358C22C2×#1309399612yes—
359C22C2×#1309400112yes—
360C22#123×D4714yes—
361C22#1568963114yes—
362C22#1568965018yes—
363C22#1568966914yes—
364C22#1568967614yes—
365C4—28no—
366C4—84no—
367C4—478no—
368C4—944no—
369C4—948no—
370C4—3764no—
371C2—28no—
372C2C2×#1568898111yescharacteristic
373C2#381×D4711yescharacteristic
374C2#2024576811yescharacteristic
375C2#2024577912yes—
376C2#2024579112yes—
377C2#2025171714yes—
378C2#2025172514yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#381C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#381D47
hypocenterC47#14685
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
4. Dic47⋊#95C2semidirect
5. #20241614C2semidirect
13. C22×C4×Dic47C4semidirect
79. C22×C4×C47C4⋊C4semidirect
181. Dic47#381semidirect
214. C47#14685semidirect
290. C22×C4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Dic472
C2×#156889862
C2×#156889812
Dic47⋊#952
#202416142
#202416102
C2×C47×#3812
#1468547