Finite Group Databasebeta

← #26225079

Subgroups — #26225079

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689523C211yesmaximal, characteristic
2C4×#13094015C212yesmaximal
3#20244351C212yesmaximal
4#20244132C211yesmaximal, characteristic
5#20244129C211yesmaximal, characteristic
6#20244125C212yesmaximal
7#20243551C211yesmaximal, characteristic
8#20243286C212yesmaximal
9#20243285C211yesmaximal, characteristic
10#97×Dic47C211yesmaximal, characteristic
11C47×#1776C211yesmaximal, characteristic
12C2×#13094015C2212yes—
13#15689523C418yescomplemented
14C2×#13093958C2211yescharacteristic
15C2×#13093957C2212yes—
16C2×#13093956C2211yescharacteristic
17C2×#13093955C2212yes—
18C2×#13093935C2211yescharacteristic
19C2×#13093934C2211yescharacteristic
20C2×#13093934C2211yescharacteristic
21C2×C8×Dic47C2211yescharacteristic
22C2×C8×Dic47C2211yescharacteristic
23#15689022C2212yes—
24#15689019C2212yes—
25#15689010—22no—
26#15689006—22no—
27#15688988C2212yes—
28C4×Dic47⋊C4—22no—
29C4×C4⋊Dic47—22no—
30C4×C4⋊Dic47C2212yes—
31C42×Dic47C2211yescharacteristic
32#15688886C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C2×#107×C47C2211yescharacteristic
36C2×#106×C47C2211yescharacteristic
37C2×#97×C47C2211yescharacteristic
38C4×C4⋊C4×C47C2212yes—
39C47×#340C2212yes—
40C22×C47⋊C8C2311yescharacteristic
41C22×C47⋊C8C2311yescharacteristic
42#13094015—24no—
43#13094015C2×C418yes—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47C2312yes—
47C2×C4⋊Dic47C2312yes—
48C2×Dic47⋊C4—22no—
49C2×Dic47⋊C4—22no—
50C2×C4×Dic47—22no—
51C2×C4×Dic47D412yes—
52C2×C4×Dic47D412yes—
53C2×C4×Dic47C2311yescharacteristic
54C2×C4×Dic47C2311yescharacteristic
55#13093958—22no—
56#13093957C2×C418yes—
57#13093956C2×C414yes—
58#13093955—24no—
59#13093935Q814yes—
60#13093934—22no—
61#13093934—22no—
62C8×Dic47—22no—
63C8×Dic47C2×C414yes—
64C47⋊#95—22no—
65C47⋊#95—22no—
66C47⋊#95—22no—
67C47⋊#95—22no—
68C47⋊#95—22no—
69C22×C8×C47C2311yescharacteristic
70C22×C8×C47C2311yescharacteristic
71C2×C4⋊C4×C47C2312yes—
72C2×C4⋊C4×C47C2312yes—
73C2×C42×C47—22no—
74C2×C42×C47C2311yescharacteristic
75#107×C47—22no—
76#106×C47C2×C414yes—
77#97×C47Q814yes—
78#95×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2411yescharacteristic
83C22×Dic47—22no—
84C22×Dic47—22no—
85C22×Dic47C2×D411yescharacteristic
86C22×Dic47C2×D411yescharacteristic
87C22×Dic47C2×D412yes—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47C22×C412yes—
91C2×C8×C47C2×Q812yes—
92C2×C8×C47C2×Q812yes—
93C2×C8×C47#4112yes—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47C22×C414yes—
98C42×C47—44no—
99C42×C47C2×Q812yes—
100C42×C47C2×Q812yes—
101C4⋊Dic47—22no—
102C4⋊Dic47—28no—
103C4⋊Dic47—22no—
104C4⋊Dic47—22no—
105C4⋊Dic47—44no—
106C4⋊Dic47C22×C414yes—
107Dic47⋊C4—28no—
108Dic47⋊C4—44no—
109C4×Dic47—21no—
110C4×Dic47—21no—
111C4×Dic47—42no—
112C4×Dic47—42no—
113C4×Dic47—44no—
114C4×Dic47C22×C412yes—
115C4×Dic47#4112yes—
116C2×C47⋊C8—21no—
117C2×C47⋊C8—21no—
118C2×C47⋊C8C22×C412yes—
119C2×C47⋊C8C2×Q812yes—
120C2×C47⋊C8C2×Q812yes—
121C2×C47⋊C8#4112yes—
122C23×C47C22×D411yescharacteristic
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—24no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47C4⋊Q814yes—
132C2×C4×C47C23×C411yescharacteristic
133C2×C4×C47C22×Q811yescharacteristic
134C2×C4×C47C22×Q811yescharacteristic
135C2×C4×C47C2×#4111yescharacteristic
136C2×C4×C47C2×#4111yescharacteristic
137C2×C4×C47C2×#4111yescharacteristic
138C2×Dic47—22no—
139C2×Dic47—22no—
140C2×Dic47—24no—
141C2×Dic47—21no—
142C2×Dic47—21no—
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47C4×D414yes—
151C8×C47—22no—
152C8×C47C4×Q814yes—
153C47⋊C8—22no—
154C47⋊C8C4×Q814yes—
155#21728—471nomaximal, Sylow
156C22×C47C2×C4×D411yescharacteristic
157C22×C47C2×C4⋊Q811yescharacteristic
158C22×C47#53211yescharacteristic
159C22×C47#57512yes—
160C22×C47#57512yes—
161C4×C47—24no—
162C4×C47—44no—
163C4×C47—82no—
164C4×C47C2×C4×Q811yescharacteristic
165C4×C47C2×C4×Q811yescharacteristic
166C4×C47#53311yescharacteristic
167C4×C47#53311yescharacteristic
168Dic47—24no—
169Dic47—41no—
170Dic47—41no—
171Dic47—44no—
172Dic47—82no—
173C2×#499—471no—
174C4×#125—472no—
175#1991—472no—
176#1944—471no—
177#1944—471no—
178#1943—472no—
179#1846—471no—
180#1776—471no—
181#1776—472no—
182#1776D4711yescharacteristic
183C4×#97—471no—
184C2×C47C4×C4⋊Q811yescharacteristic
185C2×C47#294312yes—
186C2×C47#316612yes—
187C2×C47#318311yescharacteristic
188C2×C47#318311yescharacteristic
189C2×#125—472no—
190#499—478no—
191C2×#107—472no—
192C2×#107—471no—
193C2×#107C2×D4711yescharacteristic
194C2×#106—471no—
195C2×#106—472no—
196C2×#106C2×D4711yescharacteristic
197C2×#97—471no—
198C2×#97—471no—
199C2×#97—471no—
200C2×#97C2×D4711yescharacteristic
201C2×C4×C8—471no—
202C2×C4×C8—471no—
203#389—472no—
204#389—472no—
205#387—942no—
206#387—942no—
207#382—472no—
208C4×C4⋊C4—472no—
209C4×C4⋊C4—942no—
210C4×C4⋊C4—942no—
211C4×C4⋊C4C2×D4712yes—
212C43—471no—
213#340—472no—
214#340—472no—
215#340—472no—
216#340C2×D4712yes—
217C47#2172811yescharacteristic, complemented, Sylow
218C22×C8—471no—
219C22×C8—471no—
220C22×C8C22×D4711yescharacteristic
221C22×C8C22×D4711yescharacteristic
222#125—478no—
223#125—944no—
224C2×C4⋊C4—472no—
225C2×C4⋊C4—472no—
226C2×C4⋊C4—942no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—942no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4C22×D4712yes—
231C2×C4⋊C4C22×D4712yes—
232C2×C42—22no—
233C2×C42—471no—
234C2×C42—471no—
235C2×C42—472no—
236C2×C42—472no—
237C2×C42—942no—
238C2×C42C22×D4711yescharacteristic
239#107—22no—
240#107—478no—
241#107—942no—
242#106—474no—
243#106—944no—
244#106C4×D4714yes—
245#97—474no—
246#97—942no—
247#97—942no—
248#97Dic9414yes—
249C4×C8—474no—
250C4×C8—942no—
251#95—22no—
252#95—942no—
253#95—942no—
254#95—942no—
255#95—942no—
256#95—942no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—472no—
260C22×C4—471no—
261C22×C4—471no—
262C22×C4—942no—
263C22×C4—942no—
264C22×C4D4×D4712yes—
265C22×C4C23×D4711yescharacteristic
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—472no—
269C2×C8—472no—
270C2×C8—472no—
271C2×C8—472no—
272C2×C8—941no—
273C2×C8—941no—
274C2×C8C2×Dic9412yes—
275C2×C8C2×Dic9412yes—
276C2×C8C2×C4×D4712yes—
277C2×C8#1062535012yes—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—474no—
282C4⋊C4—948no—
283C4⋊C4—948no—
284C4⋊C4—942no—
285C4⋊C4—942no—
286C4⋊C4—942no—
287C4⋊C4—1884no—
288C4⋊C4—1884no—
289C4⋊C4C2×C4×D4714yes—
290C42—44no—
291C42—472no—
292C42—472no—
293C42—941no—
294C42—941no—
295C42—1882no—
296C42—1882no—
297C42—1884no—
298C42C2×Dic9412yes—
299C42Q8×D4712yes—
300C23C2×D4×D4711yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—474no—
310C2×C4—942no—
311C2×C4—942no—
312C2×C4—944no—
313C2×C4—942no—
314C2×C4—942no—
315C2×C4—941no—
316C2×C4—941no—
317C2×C4—1882no—
318C2×C4—1882no—
319C2×C4—1882no—
320C2×C4—1882no—
321C2×C4—1882no—
322C2×C4#1309402314yes—
323C2×C4C22×C4×D4711yescharacteristic
324C2×C4C2×Q8×D4711yescharacteristic
325C2×C4C22×Dic9411yescharacteristic
326C2×C4#41×D4711yescharacteristic
327C2×C4#41×D4711yescharacteristic
328C2×C4C2×#1062535011yescharacteristic
329C8—22no—
330C8—474no—
331C8—942no—
332C8C4×Dic9414yes—
333C22C4×D4×D4711yescharacteristic
334C22C2×#1309402311yescharacteristic
335C22#1568967011yescharacteristic
336C22#1568991912yes—
337C22#1568992212yes—
338C4—24no—
339C4—44no—
340C4—82no—
341C4—944no—
342C4—3762no—
343C4—1884no—
344C4—1881no—
345C4—1881no—
346C4C4×Q8×D4711yescharacteristic
347C4C2×C4×Dic9411yescharacteristic
348C4#1568967611yescharacteristic
349C4#1568967711yescharacteristic
350C2C4×#1309402311yescharacteristic
351C2#2025218312yes—
352C2#2025393812yes—
353C2#2025411511yescharacteristic
354C2#2025412011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1776C2
socle, prime radicalC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenter#1776D47
hypocenterC47#21728
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
13. #15689523C4semidirect
217. C47#21728semidirect

Maximal subgroups

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

subgroupindex
C2×#156895232
C4×#130940152
#202443512
#202441322
#202441292
#202441252
#202435512
#202432862
#202432852
#97×Dic472
C47×#17762
#2172847