Finite Group Databasebeta

← #26104425

Subgroups — #26104425

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689021C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic
3#20241212C214yesmaximal, complemented
4C47×#1728C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×C22⋊(C47⋊C8)—21no—
9#15689021—21no—
10#15689021—21no—
11#15689021C412yes—
12#15689021C2212yes—
13C2×C47⋊#95—21no—
14#15688878—21no—
15#15688878C2212yes—
16C22×M16×C47—21no—
17C22×C42×C47C2211yescharacteristic
18C47×#336—21no—
19C47×#336C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—24no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47C2×C412yes—
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4—42no—
32C2×Dic47⋊C4—42no—
33C22⋊(C47⋊C8)—42no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C2×M16×C47—24no—
40C2×M16×C47—42no—
41C2×M16×C47—42no—
42C22×C8×C47—22no—
43C2×C42×C47—42no—
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47C2311yescharacteristic
48C24×C47C22⋊C411yescharacteristic
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—41no—
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47D812yes—
61C22×C4×C47SD1612yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—24no—
66C22×Dic47—22no—
67C22×Dic47—22no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73M16×C47—48no—
74C2×C8×C47—24no—
75C2×C8×C47—42no—
76C2×C8×C47—42no—
77C42×C47—41no—
78C42×C47—41no—
79C42×C47—41no—
80C42×C47—41no—
81C4⋊Dic47—24no—
82C4⋊Dic47—22no—
83C4⋊Dic47—42no—
84C4⋊Dic47—42no—
85C4⋊Dic47M1614yescomplemented
86Dic47⋊C4—44no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—42no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47#9912yes—
97C23×C47#10212yes—
98C23×C47#10412yes—
99C23×C47C2×C22⋊C411yescharacteristic
100C2×C4×C47—28no—
101C2×C4×C47—24no—
102C2×C4×C47—24no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
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—44no—
117C2×C4×C47#10112yes—
118C2×C4×C47#10212yes—
119C2×C4×C47C2×M1612yes—
120C2×C4×C47C2×D811yescharacteristic
121C2×C4×C47C2×SD1611yescharacteristic
122C2×C4×C47#13712yes—
123C2×Dic47—24no—
124C2×Dic47—48no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—84no—
131C8×C47—44no—
132C47⋊C8—82no—
133#7610—471nomaximal, Sylow
134C22×C47—22no—
135C22×C47—22no—
136C22×C47—22no—
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—22no—
140C22×C47#32714yes—
141C22×C47#40711yescharacteristic
142C22×C47C2×#9911yescharacteristic
143C22×C47C2×#10111yescharacteristic
144C22×C47C2×#10211yescharacteristic
145C22×C47#41911yescharacteristic
146C22×C47C2×#10411yescharacteristic
147C22×C47#42111yescharacteristic
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—22no—
154C4×C47—41no—
155Dic47—44no—
156Dic47—82no—
157C2×#389—471no—
158#1817—471no—
159#1728D4711yescharacteristic
160#1271—474no—
161C2×C47—24no—
162C2×C47#146311yescharacteristic
163C2×C47#148611yescharacteristic
164C2×C47#149111yescharacteristic
165C2×C47C2×#32711yescharacteristic
166C2×C47#151111yescharacteristic
167C2×C47#152811yescharacteristic
168C2×C47#154611yescharacteristic
169C22×M16—21no—
170C22×C4⋊C4—471no—
171C22×C4⋊C4—471no—
172C22×C4⋊C4—941no—
173C22×C42C2×D4711yescharacteristic
174C2×C22⋊C8—941no—
175#389—472no—
176#389—472no—
177#389—941no—
178#389—941no—
179C2×#95—941no—
180#336—21no—
181#336—472no—
182#336—941no—
183#336C2×D4712yes—
184C47#761011yescharacteristic, complemented, Sylow
185C23×C4—941no—
186C23×C4—941no—
187C23×C4C4×D4711yescharacteristic
188C23×C4D18811yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190C2×M16—24no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—22no—
194C22×C8—942no—
195C2×C4⋊C4—472no—
196C2×C4⋊C4—472no—
197C2×C4⋊C4—944no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4—942no—
201C2×C4⋊C4—942no—
202C2×C4⋊C4—1882no—
203C2×C4⋊C4—1882no—
204C2×C4⋊C4—1882no—
205C2×C42—42no—
206C2×C42C4×D4711yescharacteristic
207C2×C42D18811yescharacteristic
208C2×C42C22⋊D4711yescharacteristic
209C2×C42C22×D4711yescharacteristic
210C22⋊C8—1882no—
211#95—942no—
212#95—942no—
213C24(C2×D47)⋊C411yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—944no—
223C22×C4—942no—
224C22×C4—942no—
225C22×C4—1882no—
226C22×C4—1882no—
227C22×C4—1882no—
228C22×C4—1882no—
229C22×C4—1882no—
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4#1062533412yes—
234C22×C4#1062533612yes—
235C22×C4C2×C4×D4711yescharacteristic
236C22×C4C2×D18811yescharacteristic
237C22×C4C2×C22⋊D4711yescharacteristic
238M16—48no—
239C2×C8—24no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—1882no—
243C2×C8—1882no—
244C2×C8—1882no—
245C4⋊C4—474no—
246C4⋊C4—944no—
247C4⋊C4—942no—
248C4⋊C4—1882no—
249C4⋊C4—1882no—
250C4⋊C4—1884no—
251C4⋊C4—1882no—
252C4⋊C4—1882no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1309393812yes—
262C23#1309394412yes—
263C23#1309395212yes—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—944no—
283C2×C4—3764no—
284C2×C4—1882no—
285C2×C4—1882no—
286C2×C4—1882no—
287C2×C4—1888no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4#1309394212yes—
291C2×C4#1309394412yes—
292C2×C4M16×D4712yes—
293C2×C4C2×#1062533411yescharacteristic
294C2×C4C2×#1062533611yescharacteristic
295C2×C4#1309405712yes—
296C8—44no—
297C8—3762no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1568885614yes—
305C22#1568908611yescharacteristic
306C22C2×#1309393811yescharacteristic
307C22C2×#1309394211yescharacteristic
308C22C2×#1309394411yescharacteristic
309C22#1568913311yescharacteristic
310C22C2×#1309395211yescharacteristic
311C22#1568913911yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—21no—
316C4—22no—
317C4—21no—
318C4—41no—
319C4—3762no—
320C4—1884no—
321C2—24no—
322C2#2024171711yescharacteristic
323C2#2024181211yescharacteristic
324C2#2024183611yescharacteristic
325C2C2×#1568885611yescharacteristic
326C2#2024192211yescharacteristic
327C2#2024198911yescharacteristic
328C2#2024206111yescharacteristic
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#7610
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. #20241212C2semidirect
85. C4⋊Dic47M16semidirect
184. C47#7610semidirect

Maximal subgroups

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

subgroupindex
C2×#156890212
#202434562
#202412122
C47×#17282
#761047