Finite Group Databasebeta

← #26104586

Subgroups — #26104586

Order 12032.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094003C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3#20241213C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic
7C2×C4×Dic94C411yescharacteristic
8C2×C22⋊(C47⋊C8)—21no—
9#15689029C2211yescharacteristic
10#15689021C2211yescharacteristic
11#15689007—21no—
12#15688996—21no—
13#15688983C411yescharacteristic
14C4×Dic47⋊C4—21no—
15C4×C4⋊Dic47C411yescharacteristic
16C4×C22⋊Dic47—21no—
17#15688878—21no—
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C4⋊C8×C47—21no—
21C2×C22⋊C8×C47—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—
38C22⋊(C47⋊C8)—42no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C22×C8×C47—22no—
48C2×C42×C47—42no—
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—42no—
54C22⋊C8×C47—42no—
55C24×C47C22⋊C411yescharacteristic
56C22×C4×C47—22no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—41no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47SD1612yes—
69C22×C4×C47Q1612yes—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C2×Dic94—21no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94M1612yes—
81C2×C8×C47—42no—
82C2×C8×C47—42no—
83C2×C8×C47—42no—
84C42×C47—22no—
85C42×C47—22no—
86C42×C47—41no—
87C42×C47—41no—
88C22⋊Dic47—24no—
89C22⋊Dic47—42no—
90C4⋊Dic47—21no—
91C4⋊Dic47—41no—
92C4⋊Dic47—41no—
93C4⋊Dic47M1612yes—
94Dic47⋊C4—24no—
95Dic47⋊C4—42no—
96Dic47⋊C4—42no—
97C4×Dic47—42no—
98C4×Dic47—42no—
99C2×C47⋊C8—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47#10312yes—
109C23×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
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—21no—
122C2×C4×C47—21no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—44no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47#9912yes—
129C2×C4×C47#10312yes—
130C2×C4×C47C2×M1611yescharacteristic
131C2×C4×C47C2×M1611yescharacteristic
132C2×C4×C47C2×SD1611yescharacteristic
133C2×C4×C47C2×Q1611yescharacteristic
134C2×C4×C47#13512yes—
135C2×Dic47—22no—
136C2×Dic47—22no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147Dic94—42no—
148Dic94—42no—
149C8×C47—82no—
150C47⋊C8—82no—
151#7648—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—22no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47#40711yescharacteristic
160C22×C47C2×#9911yescharacteristic
161C22×C47#41311yescharacteristic
162C22×C47C2×#10311yescharacteristic
163C22×C47#41611yescharacteristic
164C22×C47#42111yescharacteristic
165C22×C47#42111yescharacteristic
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—22no—
169C4×C47—22no—
170C4×C47—21no—
171C4×C47—22no—
172C4×C47—21no—
173C4×C47—41no—
174Dic47—42no—
175Dic47—42no—
176Dic47—81no—
177Dic47—81no—
178C4×C22⋊Q8—471no—
179#1817—471no—
180#1797D4711yescharacteristic
181#1272—472no—
182#1271—472no—
183C2×C47—42no—
184C2×C47#146511yescharacteristic
185C2×C47#148711yescharacteristic
186C2×C47#148811yescharacteristic
187C2×C47#151611yescharacteristic
188C2×C47#152011yescharacteristic
189C2×C47#153511yescharacteristic
190C2×C47#154211yescharacteristic
191C2×C22⋊Q8—471no—
192C2×C4×Q8—471no—
193C22×C42C2×D4711yescharacteristic
194C2×C4⋊C8—21no—
195C2×C22⋊C8—21no—
196C2×C22⋊C8—941no—
197#391—471no—
198#389—471no—
199#387—941no—
200#385—941no—
201#380—471no—
202C4×C4⋊C4—471no—
203C4×C4⋊C4—941no—
204C4×C22⋊C4—941no—
205#336—472no—
206#336—941no—
207#336C2×D4712yes—
208C47#764811yescharacteristic, complemented, Sylow
209C22×Q8—471no—
210C23×C4C4×D4711yescharacteristic
211C23×C4D18811yescharacteristic
212C23×C4C22⋊D4711yescharacteristic
213C22×C8—22no—
214C22×C8—942no—
215C22⋊Q8—942no—
216C22⋊Q8—942no—
217C4×Q8—1882no—
218C2×C4⋊C4—471no—
219C2×C4⋊C4—471no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C22⋊C4—941no—
225C2×C22⋊C4—941no—
226C2×C42—42no—
227C2×C42—941no—
228C2×C42—941no—
229C2×C42C4×D4711yescharacteristic
230C2×C42D18811yescharacteristic
231C2×C42C22⋊D4711yescharacteristic
232C2×C42C22×D4711yescharacteristic
233C4⋊C8—42no—
234C22⋊C8—42no—
235C22⋊C8—1882no—
236#95—941no—
237#95—941no—
238#95—941no—
239#95—941no—
240#95—941no—
241C24(C2×D47)⋊C411yescharacteristic
242C2×Q8—472no—
243C2×Q8—941no—
244C2×Q8—1881no—
245C2×Q8—1881no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—941no—
256C22×C4—941no—
257C22×C4—941no—
258C22×C4—941no—
259C22×C4(C2×D47)⋊C411yescharacteristic
260C22×C4(C2×D47)⋊C411yescharacteristic
261C22×C4(C2×D47)⋊C411yescharacteristic
262C22×C4#1062533512yes—
263C22×C4#1062533712yes—
264C22×C4C2×C4×D4711yescharacteristic
265C22×C4C2×D18811yescharacteristic
266C22×C4C2×C22⋊D4711yescharacteristic
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—1882no—
271C2×C8—1882no—
272C2×C8—1882no—
273C4⋊C4—472no—
274C4⋊C4—944no—
275C4⋊C4—941no—
276C4⋊C4—1881no—
277C4⋊C4—1881no—
278C4⋊C4—1882no—
279C4⋊C4—1882no—
280C22⋊C4—944no—
281C22⋊C4—1882no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—1882no—
287C42—1882no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1309394712yes—
295C23C2×(C2×D47)⋊C411yescharacteristic
296Q8—1882no—
297Q8—1882no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—942no—
317C2×C4—942no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4#1309393812yes—
329C2×C4#1309394712yes—
330C2×C4M16×D4711yescharacteristic
331C2×C4M16×D4711yescharacteristic
332C2×C4C2×#1062533511yescharacteristic
333C2×C4C2×#1062533711yescharacteristic
334C2×C4#1309404612yes—
335C8—82no—
336C8—3762no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1568908611yescharacteristic
345C22C2×#1309393811yescharacteristic
346C22#1568910511yescharacteristic
347C22C2×#1309394711yescharacteristic
348C22#1568912011yescharacteristic
349C22#1568913911yescharacteristic
350C22#1568914111yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—3761no—
360C4—3761no—
361C4—1882no—
362C4—1882no—
363C2—42no—
364C2#2024172411yescharacteristic
365C2#2024181611yescharacteristic
366C2#2024182311yescharacteristic
367C2#2024194511yescharacteristic
368C2#2024195911yescharacteristic
369C2#2024201711yescharacteristic
370C2#2024204611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1797D47
hypocenterC47#7648
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20241213C2semidirect
4. #20241210C2semidirect
208. C47#7648semidirect

Maximal subgroups

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

subgroupindex
C4×#130940032
#202434562
#202412132
#202412102
C47×#17972
#764847