Finite Group Databasebeta

← #26104459

Subgroups — #26104459

Order 12032.

550 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094003C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3#20241213C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1728C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7C2×C4×Dic94C411yescharacteristic
8C2×C4⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)—21no—
10#15689029C2211yescharacteristic
11#15689021C2211yescharacteristic
12#15689007—21no—
13#15688996—21no—
14#15688983C411yescharacteristic
15C4×Dic47⋊C4—21no—
16C4×C4⋊Dic47C411yescharacteristic
17C4×C22⋊Dic47—21no—
18#15688878C2212yes—
19C22×M16×C47—21no—
20C22×C42×C47C2211yescharacteristic
21C47×#336—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—22no—
25#13094003—22no—
26#13094003—22no—
27C4×Dic94—42no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47—21no—
38C4⋊(C47⋊C8)—42no—
39C22⋊(C47⋊C8)—42no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C2×M16×C47—24no—
49C2×M16×C47—42no—
50C2×M16×C47—42no—
51C22×C8×C47—22no—
52C2×C42×C47—42no—
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47C2311yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47SD1612yes—
72C22×C4×C47Q1612yes—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C2×Dic94—21no—
81C2×Dic94—41no—
82C2×Dic94—41no—
83C2×Dic94M1612yescomplemented
84M16×C47—48no—
85C2×C8×C47—24no—
86C2×C8×C47—42no—
87C2×C8×C47—42no—
88C42×C47—22no—
89C42×C47—22no—
90C42×C47—41no—
91C42×C47—41no—
92C22⋊Dic47—24no—
93C22⋊Dic47—42no—
94C4⋊Dic47—21no—
95C4⋊Dic47—41no—
96C4⋊Dic47—41no—
97C4⋊Dic47M1612yescomplemented
98Dic47⋊C4—24no—
99Dic47⋊C4—42no—
100Dic47⋊C4—41no—
101Dic47⋊C4—41no—
102C4×Dic47—42no—
103C4×Dic47—42no—
104C2×C47⋊C8—42no—
105C2×C47⋊C8—42no—
106C2×C47⋊C8—42no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47#10312yes—
114C23×C47C2×C22⋊C411yescharacteristic
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—24no—
122C2×C4×C47—24no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47#9912yes—
134C2×C4×C47#10312yes—
135C2×C4×C47#10412yes—
136C2×C4×C47C2×M1611yescharacteristic
137C2×C4×C47C2×M1611yescharacteristic
138C2×C4×C47C2×SD1611yescharacteristic
139C2×C4×C47C2×Q1611yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C47#13711yescharacteristic
142C2×Dic47—22no—
143C2×Dic47—22no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154Dic94—42no—
155Dic94—42no—
156C8×C47—44no—
157C47⋊C8—82no—
158#7617—471nomaximal, Sylow
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—22no—
162C22×C47—41no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47#40711yescharacteristic
167C22×C47C2×#9911yescharacteristic
168C22×C47#41311yescharacteristic
169C22×C47C2×#10311yescharacteristic
170C22×C47#41711yescharacteristic
171C22×C47C2×#10411yescharacteristic
172C22×C47#42111yescharacteristic
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—41no—
180C4×C47—41no—
181C4×C47#32814yes—
182Dic47—42no—
183Dic47—42no—
184Dic47—81no—
185Dic47—81no—
186C4×C22⋊Q8—471no—
187#1797—471no—
188#1728D4711yescharacteristic
189#1272—472no—
190#1271—472no—
191C2×C47—42no—
192C2×C47#146511yescharacteristic
193C2×C47#148711yescharacteristic
194C2×C47#148911yescharacteristic
195C2×C47C2×#32811yescharacteristic
196C2×C47#151311yescharacteristic
197C2×C47#152911yescharacteristic
198C2×C47#155011yescharacteristic
199C22×M16—21no—
200C2×C22⋊Q8—471no—
201C2×C4×Q8—471no—
202C22×C42C2×D4711yescharacteristic
203C2×C4⋊C8—941no—
204C2×C22⋊C8—941no—
205#391—471no—
206#389—471no—
207#387—941no—
208#385—941no—
209#380—471no—
210C4×C4⋊C4—471no—
211C4×C4⋊C4—941no—
212C4×C22⋊C4—941no—
213#336—21no—
214#336—472no—
215#336C2×D4712yes—
216C47#761711yescharacteristic, complemented, Sylow
217C22×Q8—471no—
218C23×C4C4×D4711yescharacteristic
219C23×C4D18811yescharacteristic
220C23×C4C22⋊D4711yescharacteristic
221C2×M16—24no—
222C2×M16—42no—
223C2×M16—42no—
224C22×C8—22no—
225C22×C8—942no—
226C22⋊Q8—942no—
227C22⋊Q8—942no—
228C4×Q8—1882no—
229C2×C4⋊C4—471no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4—471no—
232C2×C4⋊C4—941no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C22⋊C4—941no—
236C2×C22⋊C4—941no—
237C2×C42—42no—
238C2×C42—941no—
239C2×C42—941no—
240C2×C42C4×D4711yescharacteristic
241C2×C42D18811yescharacteristic
242C2×C42C22⋊D4711yescharacteristic
243C2×C42C22×D4711yescharacteristic
244C4⋊C8—1882no—
245C22⋊C8—1882no—
246#95—941no—
247#95—941no—
248#95—941no—
249#95—941no—
250#95—941no—
251C24(C2×D47)⋊C411yescharacteristic
252C2×Q8—472no—
253C2×Q8—941no—
254C2×Q8—1881no—
255C2×Q8—1881no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—941no—
267C22×C4—941no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4(C2×D47)⋊C411yescharacteristic
271C22×C4(C2×D47)⋊C411yescharacteristic
272C22×C4(C2×D47)⋊C411yescharacteristic
273C22×C4#1062533512yes—
274C22×C4#1062533712yes—
275C22×C4C2×C4×D4711yescharacteristic
276C22×C4C2×D18811yescharacteristic
277C22×C4C2×C22⋊D4711yescharacteristic
278M16—48no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—1882no—
283C2×C8—1882no—
284C2×C8—1882no—
285C4⋊C4—472no—
286C4⋊C4—944no—
287C4⋊C4—941no—
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C4⋊C4—1882no—
291C4⋊C4—1881no—
292C4⋊C4—1881no—
293C22⋊C4—944no—
294C22⋊C4—1882no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—1882no—
300C42—1882no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#1309394712yes—
308C23C2×(C2×D47)⋊C411yescharacteristic
309Q8—1882no—
310Q8—1882no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—942no—
330C2×C4—942no—
331C2×C4—1881no—
332C2×C4—1881no—
333C2×C4—1881no—
334C2×C4—1881no—
335C2×C4—1881no—
336C2×C4—1881no—
337C2×C4—1881no—
338C2×C4—1881no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4#1309393812yes—
342C2×C4#1309394712yes—
343C2×C4#1309395212yes—
344C2×C4M16×D4711yescharacteristic
345C2×C4M16×D4711yescharacteristic
346C2×C4C2×#1062533511yescharacteristic
347C2×C4C2×#1062533711yescharacteristic
348C2×C4#1309405211yescharacteristic
349C2×C4#1309405711yescharacteristic
350C8—44no—
351C8—3762no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1568908611yescharacteristic
360C22C2×#1309393811yescharacteristic
361C22#1568910511yescharacteristic
362C22C2×#1309394711yescharacteristic
363C22#1568912511yescharacteristic
364C22C2×#1309395211yescharacteristic
365C22#1568913911yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—3761no—
375C4—3761no—
376C4—1882no—
377C4—1882no—
378C4#1568885914yes—
379C2—42no—
380C2#2024172411yescharacteristic
381C2#2024181611yescharacteristic
382C2#2024182811yescharacteristic
383C2C2×#1568885911yescharacteristic
384C2#2024193211yescharacteristic
385C2#2024199311yescharacteristic
386C2#2024207911yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1728C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1728D47
hypocenterC47#7617
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
3. #20241213C2semidirect
4. #20241210C2semidirect
83. C2×Dic94M16semidirect
97. C4⋊Dic47M16semidirect
216. C47#7617semidirect

Maximal subgroups

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

subgroupindex
C4×#130940032
#202433682
#202412132
#202412102
C47×#17282
#761747