Finite Group Databasebeta

← #26218372

Subgroups — #26218372

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic47⋊C8C211yesmaximal, characteristic
2C2×#15689078C211yesmaximal, characteristic
3#20245872C211yesmaximal, characteristic
4#20243953C214yesmaximal, complemented
5#20243946C214yesmaximal, complemented
6#20243086C211yesmaximal, characteristic
7#20243085C211yesmaximal, characteristic
8#20243081C211yesmaximal, characteristic
9C47×#1729C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C22×C4×Dic47C2211yescharacteristic
14C2×Dic47⋊C8—24no—
15C2×Dic47⋊C8C2212yes—
16C2×Dic47⋊C8C2212yes—
17#15689078—24no—
18C2×#13093934C2212yes—
19C2×C8×Dic47C2212yes—
20#15689022C412yes—
21#15689022C412yes—
22#15688998C412yes—
23#15688998C412yes—
24C4×C4⋊Dic47C2214yes—
25#15688878C2214yes—
26#15688877C2214yes—
27#15688877C2214yes—
28C22×M16×C47C2211yescharacteristic
29C23×C8×C47C2211yescharacteristic
30C2×#117×C47C2211yescharacteristic
31C47×#336C2212yes—
32C47×#336C2212yes—
33C23×Dic47C2×C411yescharacteristic
34C23×Dic47C2×C411yescharacteristic
35C2×#10625329—22no—
36C2×#10625329—22no—
37C2×#10625329Q814yes—
38C22×C47⋊C8—22no—
39C22×C47⋊C8—22no—
40C22×C47⋊C8—22no—
41C22×C47⋊C8C2312yes—
42C22×C47⋊C8C2312yes—
43#13093983—22no—
44#13093983—22no—
45C2×C4⋊Dic47C2×C414yes—
46C2×C4⋊Dic47C2×C411yescharacteristic
47C2×C4⋊Dic47C2×C411yescharacteristic
48C2×C4⋊Dic47C2×C414yes—
49C2×C22⋊Dic47C2×C412yes—
50C2×C4×Dic47—22no—
51C2×C4×Dic47—22no—
52C2×C4×Dic47C2×C412yes—
53C2×C4×Dic47C2312yes—
54C2×C4×Dic47C2312yes—
55Dic47⋊C8—24no—
56Dic47⋊C8—24no—
57#13093934—24no—
58C8×Dic47—24no—
59C47⋊#95C2×C414yes—
60C47⋊#95C2×C414yes—
61C23×C4×C47C2311yescharacteristic
62C2×M16×C47—22no—
63C2×M16×C47—22no—
64C2×M16×C47—22no—
65C22×C8×C47—22no—
66C22×C8×C47—22no—
67C22×C8×C47D414yes—
68C22×C8×C47C2312yes—
69C22×C8×C47C2311yescharacteristic
70C22×C8×C47C2311yescharacteristic
71#117×C47—22no—
72#117×C47—22no—
73C2×C4⋊C4×C47C2×C412yes—
74C2×C22⋊C4×C47C2×C412yes—
75C2×C42×C47C2312yes—
76C24×C47C22×C411yescharacteristic
77C22×C4×C47C22×C414yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D412yes—
80C22×C4×C47C2×Q812yes—
81C22×C4×C47#4112yes—
82C22×C4×C47#4112yes—
83C22×C4×C47#4112yes—
84C22×C4×C47#4112yes—
85C22×C4×C47C2411yescharacteristic
86C22×Dic47—22no—
87C22×Dic47—24no—
88C22×Dic47—22no—
89C22×Dic47—42no—
90C22×Dic47C22×C412yes—
91C22×Dic47C22×C412yes—
92C22×Dic47C22×C412yes—
93C22×Dic47C22×C412yes—
94M16×C47—44no—
95C2×C8×C47—28no—
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C2×C8×C47—22no—
101C2×C8×C47—22no—
102C2×C8×C47—22no—
103C2×C8×C47C2×D412yes—
104C2×C8×C47C2×D412yes—
105C4⋊C4×C47—24no—
106C22⋊C4×C47—24no—
107C42×C47—22no—
108C42×C47—22no—
109C22⋊Dic47—24no—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112C4⋊Dic47—28no—
113C4⋊Dic47—44no—
114C4×Dic47—24no—
115C4×Dic47—22no—
116C4×Dic47—22no—
117C4×Dic47—22no—
118C4×Dic47—22no—
119#10625329—28no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—44no—
126C2×C47⋊C8C2×Q814yes—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47#11712yes—
130C23×C47C4×D412yes—
131C23×C47C4×Q812yes—
132C23×C47C23×C411yescharacteristic
133C2×C4×C47—24no—
134C2×C4×C47—24no—
135C2×C4×C47—24no—
136C2×C4×C47#11712yes—
137C2×C4×C47C4×D412yes—
138C2×C4×C47C4×Q812yes—
139C2×C4×C47C22⋊Q814yes—
140C2×C4×C47#12314yes—
141C2×C4×C47#12514yes—
142C2×C4×C47#12614yes—
143C2×C4×C47C22×D411yescharacteristic
144C2×C4×C47C22×Q811yescharacteristic
145C2×C4×C47C2×#4111yescharacteristic
146C2×C4×C47C2×#4111yescharacteristic
147C2×C4×C47C2×#4111yescharacteristic
148C2×C4×C47C2×#4111yescharacteristic
149C2×Dic47—28no—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—22no—
153C2×Dic47—22no—
154C2×Dic47—22no—
155C2×Dic47—24no—
156C2×Dic47—22no—
157C2×Dic47—24no—
158C2×Dic47—22no—
159C2×Dic47—44no—
160C2×Dic47#13114yes—
161C8×C47—22no—
162C8×C47—22no—
163C8×C47—42no—
164C47⋊C8—24no—
165C47⋊C8—42no—
166#21024—471nomaximal, Sylow
167C22×C47—24no—
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—24no—
171C22×C47#38714yes—
172C22×C47C2×#11711yescharacteristic
173C22×C47C2×C4×D411yescharacteristic
174C22×C47C2×C4×Q811yescharacteristic
175C22×C47C2×#13112yes—
176C22×C47#56812yes—
177C4×C47—44no—
178C4×C47#38714yes—
179C4×C47C2×C22⋊Q811yescharacteristic
180C4×C47C2×#12311yescharacteristic
181C4×C47C2×#12511yescharacteristic
182C4×C47C2×#12611yescharacteristic
183Dic47—24no—
184Dic47—42no—
185Dic47—42no—
186Dic47—42no—
187C22×C4⋊C8—471no—
188C2×#405—471no—
189#2296—471no—
190#1917—474no—
191#1916—474no—
192#1729—471no—
193#1729—471no—
194#1729D4711yescharacteristic
195#1728—471no—
196C2×C47—24no—
197C2×C47C2×#38711yescharacteristic
198C2×C47#292512yes—
199C2×C47#292911yescharacteristic
200C2×C47#293011yescharacteristic
201C2×C47#296212yes—
202C22×M16—471no—
203C22×M16C2×D4711yescharacteristic
204C23×C8—471no—
205C23×C8C2×D4711yescharacteristic
206C2×#117—471no—
207C2×#117C2×D4711yescharacteristic
208C22×C42—471no—
209C2×C4⋊C8—472no—
210C2×C4⋊C8—472no—
211C2×C4⋊C8—944no—
212#405—944no—
213C2×#97—472no—
214C2×C4×C8—472no—
215#389—472no—
216#389—472no—
217#385—472no—
218#385—472no—
219C4×C4⋊C4—474no—
220#336—474no—
221#336—474no—
222#336—474no—
223#336C2×D4712yes—
224#336C2×D4712yes—
225C47#2102411yescharacteristic, complemented, Sylow
226C23×C4—471no—
227C23×C4—471no—
228C23×C4C22×D4711yescharacteristic
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—474no—
233C2×M16—942no—
234C2×M16—942no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—472no—
238C22×C8—472no—
239C22×C8—942no—
240C22×C8—942no—
241C22×C8—942no—
242C22×C8D18814yes—
243C22×C8C22×D4712yes—
244C22×C8C22×D4711yescharacteristic
245C22×C8C22×D4711yescharacteristic
246#117—22no—
247#117—22no—
248#117—942no—
249#117—942no—
250C2×C4⋊C4—471no—
251C2×C4⋊C4—471no—
252C2×C4⋊C4—474no—
253C2×C4⋊C4—474no—
254C2×C4⋊C4C4×D4712yes—
255C2×C22⋊C4—472no—
256C2×C22⋊C4C4×D4712yes—
257C2×C42—472no—
258C2×C42—472no—
259C2×C42—472no—
260C2×C42—942no—
261C2×C42—942no—
262C2×C42C22×D4712yes—
263C4⋊C8—944no—
264C4⋊C8—944no—
265#97—944no—
266C4×C8—944no—
267#95—474no—
268#95—474no—
269C24C2×C4×D4711yescharacteristic
270C22×C4—472no—
271C22×C4—472no—
272C22×C4—472no—
273C22×C4—472no—
274C22×C4—944no—
275C22×C4—942no—
276C22×C4—942no—
277C22×C4—1882no—
278C22×C4C2×C4×D4714yes—
279C22×C4C2×C4×D4711yescharacteristic
280C22×C4C2×D18812yes—
281C22×C4#1062535012yes—
282C22×C4#1062535212yes—
283C22×C4#1062535212yes—
284C22×C4#1062535212yes—
285C22×C4Q8×D4712yes—
286C22×C4C23×D4711yescharacteristic
287M16—44no—
288M16—948no—
289C2×C8—28no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—474no—
298C2×C8—942no—
299C2×C8—942no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8—942no—
303C2×C8—1884no—
304C2×C8C2×D18812yes—
305C2×C8C2×D18812yes—
306C4⋊C4—24no—
307C4⋊C4—948no—
308C4⋊C4—942no—
309C4⋊C4—942no—
310C4⋊C4—1884no—
311C22⋊C4—24no—
312C22⋊C4—944no—
313C42—22no—
314C42—22no—
315C42—944no—
316C42—942no—
317C42—942no—
318C42—942no—
319C42—942no—
320C23—22no—
321C23—22no—
322C23#1309398012yes—
323C23C4×D18812yes—
324C23#1309398912yes—
325C23C22×C4×D4711yescharacteristic
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—474no—
330C2×C4—948no—
331C2×C4—942no—
332C2×C4—942no—
333C2×C4—942no—
334C2×C4—942no—
335C2×C4—942no—
336C2×C4—944no—
337C2×C4—942no—
338C2×C4—944no—
339C2×C4—942no—
340C2×C4—1884no—
341C2×C4#1309398012yes—
342C2×C4C4×D18812yes—
343C2×C4#1309398912yes—
344C2×C4#1309400414yes—
345C2×C4#1309400814yes—
346C2×C4#1309401414yes—
347C2×C4#1309401714yes—
348C2×C4C22×D18811yescharacteristic
349C2×C4C2×Q8×D4711yescharacteristic
350C2×C4C2×#1062535211yescharacteristic
351C2×C4C2×#1062535211yescharacteristic
352C2×C4C2×#1062535211yescharacteristic
353C2×C4C2×#1062535011yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—944no—
358C8—1882no—
359C22—24no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22#1568900814yes—
364C22C2×#1309398011yescharacteristic
365C22C2×C4×D18811yescharacteristic
366C22C2×#1309398911yescharacteristic
367C22#131×D4712yes—
368C22#1568988812yes—
369C4—44no—
370C4—944no—
371C4—1882no—
372C4—1882no—
373C4—1882no—
374C4#1568900814yes—
375C4C2×#1309400411yescharacteristic
376C4C2×#1309400811yescharacteristic
377C4C2×#1309401411yescharacteristic
378C4C2×#1309401711yescharacteristic
379C2—24no—
380C2C2×#1568900811yescharacteristic
381C2#2025205012yes—
382C2#2025208411yescharacteristic
383C2#2025209311yescharacteristic
384C2#2025233412yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1729C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#1729D47
hypocenterC47#21024
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. #20243953C2semidirect
5. #20243946C2semidirect
225. C47#21024semidirect

Maximal subgroups

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

subgroupindex
C22×Dic47⋊C82
C2×#156890782
#202458722
#202439532
#202439462
#202430862
#202430852
#202430812
C47×#17292
#2102447