Finite Group Databasebeta

← #26131801

Subgroups — #26131801

Order 12032.

962 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688922C214yesmaximal, complemented
2#20243316C212yesmaximal
3C2×C47×#536C211yesmaximal, characteristic
4C2×#13093943C2214yes—
5C2×#13093941—22no—
6#15689087—22no—
7#15688922—28no—
8C22×#41×C47C2211yescharacteristic
9C23×Q8×C47C411yescharacteristic
10C47×#536—28no—
11C2×C4⋊Q8×C47C2212yes—
12C2×#124×C47C412yes—
13C2×#123×C47—22no—
14C2×C22⋊Q8×C47—22no—
15C2×#117×C47C411yescharacteristic
16C2×#10625329—24no—
17#13093943—28no—
18#13093941—44no—
19C22⋊(C47⋊C8)—44no—
20C2×#41×C47—24no—
21C2×#41×C47—42no—
22C2×#41×C47—42no—
23C22×Q8×C47—24no—
24C22×Q8×C47—21no—
25C22×Q8×C47—42no—
26C22×Q8×C47—42no—
27C22×Q8×C47C2×C412yes—
28C22×Q8×C47C2311yescharacteristic
29C22×D4×C47C2×C411yescharacteristic
30C22×D4×C47D412yes—
31C23×C4×C47—21no—
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47D412yes—
34C4⋊Q8×C47—24no—
35C4⋊Q8×C47D418yes—
36#124×C47—24no—
37#124×C47—24no—
38#123×C47—216no—
39C22⋊Q8×C47—216no—
40#117×C47—24no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—24no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—24no—
45C2×C42×C47C2×C412yes—
46C24×C47C22⋊C412yes—
47C24×C47C22⋊C411yescharacteristic
48#41×C47—416no—
49C2×Q8×C47—24no—
50C2×Q8×C47—22no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47—416no—
54C2×Q8×C47—42no—
55C2×Q8×C47—42no—
56C2×Q8×C47—42no—
57C2×Q8×C47—42no—
58C2×Q8×C47C22⋊C414yes—
59C2×Q8×C47C2×D414yes—
60C2×D4×C47—24no—
61C2×D4×C47—22no—
62C2×D4×C47—24no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C2×D4×C47—42no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—24no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—24no—
73C22×C4×C47—24no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47—42no—
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D412yes—
80C4⋊C4×C47—24no—
81C4⋊C4×C47—216no—
82C22⋊C4×C47—24no—
83C22⋊C4×C47—216no—
84C42×C47—22no—
85C42×C47C22⋊C414yes—
86#10625329—44no—
87#10625329—44no—
88C2×C47⋊C8—44no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—42no—
95C23×C47—44no—
96C23×C47#9912yes—
97C23×C47C2×C22⋊C411yescharacteristic
98Q8×C47—44no—
99Q8×C47—44no—
100Q8×C47—48no—
101Q8×C47—44no—
102D4×C47—48no—
103D4×C47—44no—
104C2×C4×C47—28no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—24no—
112C2×C4×C47—24no—
113C2×C4×C47—48no—
114C2×C4×C47—44no—
115C2×C4×C47—48no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47#9912yes—
119C2×C4×C47C2×C22⋊C412yes—
120C2×C4×C47#12014yes—
121C47⋊C8—84no—
122#12704—471nomaximal, Sylow
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—42no—
126C22×C47—42no—
127C22×C47—44no—
128C22×C47—42no—
129C22×C47—44no—
130C22×C47—42no—
131C22×C47—82no—
132C22×C47#37911yescharacteristic
133C22×C47C2×#9911yescharacteristic
134C22×C47C2×#9911yescharacteristic
135C4×C47—24no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—44no—
139C4×C47—44no—
140C2×#536D4711yescharacteristic
141C2×#356—474no—
142#1785—472no—
143C2×C47—21no—
144C2×C47—21no—
145C2×C47—42no—
146C2×C47—82no—
147C2×C47#178111yescharacteristic
148C2×C47#213512yes—
149C22×#41C2×D4711yescharacteristic
150C23×Q8Dic4711yescharacteristic
151#536—28no—
152C2×C4⋊Q8C2×D4712yes—
153C2×#124Dic4712yes—
154C2×#123—22no—
155C2×C22⋊Q8—22no—
156C2×#117Dic4711yescharacteristic
157C2×#101—474no—
158C2×#100—942no—
159#407—942no—
160#356—948no—
161C47#1270411yescharacteristic, complemented, Sylow
162C2×#41—24no—
163C2×#41—42no—
164C2×#41—42no—
165C22×Q8—24no—
166C22×Q8—21no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8C2×Dic4712yes—
170C22×Q8C22×D4711yescharacteristic
171C22×D4C2×Dic4711yescharacteristic
172C22×D4C22⋊D4712yes—
173C23×C4—21no—
174C23×C4C2×Dic4711yescharacteristic
175C23×C4C22⋊D4712yes—
176C2×M16—944no—
177C4⋊Q8—24no—
178C4⋊Q8C22⋊D4718yes—
179#124—24no—
180#124—24no—
181#123—216no—
182C22⋊Q8—216no—
183#117—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—24no—
188C2×C42C2×Dic4712yes—
189#101—948no—
190#100—1884no—
191C22⋊C8—1884no—
192C24C22⋊Dic4712yes—
193C24C22⋊Dic4711yescharacteristic
194#41—416no—
195C2×Q8—24no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—416no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8C22⋊Dic4714yes—
205C2×Q8C2×C22⋊D4714yes—
206C2×D4—24no—
207C2×D4—22no—
208C2×D4—24no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—42no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C22⋊Dic4711yescharacteristic
224C22×C4C22×Dic4711yescharacteristic
225C22×C4C2×C22⋊D4712yes—
226M16—1884no—
227M16—1884no—
228C2×C8—1884no—
229C4⋊C4—24no—
230C4⋊C4—216no—
231C22⋊C4—24no—
232C22⋊C4—216no—
233C42—22no—
234C42C22⋊Dic4714yes—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—42no—
241C23—44no—
242C23#1309393912yes—
243C23C2×C22⋊Dic4711yescharacteristic
244Q8—44no—
245Q8—44no—
246Q8—48no—
247Q8—44no—
248D4—48no—
249D4—44no—
250C2×C4—28no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4#1309393912yes—
265C2×C4C2×C22⋊Dic4712yes—
266C2×C4#1309399314yes—
267C8—3764no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—42no—
272C22—44no—
273C22—42no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22#1568897911yescharacteristic
278C22C2×#1309393911yescharacteristic
279C22C2×#1309393911yescharacteristic
280C4—24no—
281C4—22no—
282C4—22no—
283C4—44no—
284C4—44no—
285C2—21no—
286C2—21no—
287C2—42no—
288C2—82no—
289C2#2024330311yescharacteristic
290C2#2024508312yes—
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#536C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×Q8×C47C23
hypercenterC2×#536D47
hypocenterC47#12704
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#15688922C2semidirect
161. C47#12704semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
#202433162
C2×C47×#5362
#1270447