Finite Group Databasebeta

← #26108016

Subgroups — #26108016

Order 12032.

706 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 159 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
1#20243384C211yesmaximal, characteristic
2#20242060C212yesmaximal
3#20241817C212yesmaximal
4#20241712C211yesmaximal, characteristic
5C2×C47×#557C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093943—21no—
9#15688892—21no—
10#15688869—24no—
11Q8⋊(C47⋊C8)—24no—
12#15688849—22no—
13C47×#557—24no—
14C47×#557—24no—
15C2×C4⋊Q8×C47—21no—
16C2×C4⋊Q8×C47C2211yescharacteristic
17C2×#125×C47—22no—
18C2×#125×C47C412yes—
19C2×C4×Q8×C47—21no—
20C2×C4×Q8×C47C412yes—
21C2×C4×Q8×C47C2211yescharacteristic
22C2×C4×Q8×C47C2211yescharacteristic
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27#13093943—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C22×Q8×C47—21no—
31C22×Q8×C47C2×C411yescharacteristic
32C22×Q8×C47D411yescharacteristic
33C4⋊Q8×C47—28no—
34C4⋊Q8×C47—22no—
35C4⋊Q8×C47D414yes—
36#125×C47—216no—
37#125×C47—24no—
38#125×C47—24no—
39C4×Q8×C47—28no—
40C4×Q8×C47—24no—
41C4×Q8×C47—24no—
42C4×Q8×C47—22no—
43C4×Q8×C47—22no—
44C4×Q8×C47—22no—
45C4×Q8×C47D414yes—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47—21no—
51C2×C4⋊C4×C47C2×C412yes—
52C2×C4⋊C4×C47C2×C411yescharacteristic
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C4⋊C4×C47D412yes—
55C2×C42×C47—22no—
56C2×C42×C47—21no—
57C2×C42×C47C2×C412yes—
58C2×C42×C47C2311yescharacteristic
59C2×Q8×C47—24no—
60C2×Q8×C47—21no—
61C2×Q8×C47—22no—
62C2×Q8×C47—24no—
63C2×Q8×C47—22no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C2×Q8×C47—42no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47C22⋊C412yes—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—22no—
73C22×C4×C47C22⋊C412yes—
74C22×C4×C47C22⋊C412yes—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—28no—
79C4⋊C4×C47—28no—
80C4⋊C4×C47—28no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—24no—
84C4⋊C4×C47—24no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47C22⋊C414yes—
88C4⋊C4×C47C22⋊C412yes—
89C42×C47—28no—
90C42×C47—22no—
91C42×C47—22no—
92C42×C47—24no—
93C42×C47C2×D412yes—
94C42×C47C2×D412yes—
95#10625329—42no—
96#10625329—42no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—42no—
99C23×C47C2×C22⋊C411yescharacteristic
100Q8×C47—24no—
101Q8×C47—22no—
102Q8×C47—42no—
103Q8×C47—44no—
104Q8×C47—42no—
105Q8×C47—44no—
106Q8×C47#10114yescomplemented
107C2×C4×C47—24no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—24no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—41no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47#12012yes—
124C2×C4×C47#12012yes—
125C2×C4×C47#13512yes—
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C47#13711yescharacteristic
128C47⋊C8—82no—
129C47⋊C8—82no—
130#8390—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47#37911yescharacteristic
134C22×C47#41611yescharacteristic
135C22×C47#41711yescharacteristic
136C4×C47—22no—
137C4×C47—24no—
138C4×C47—21no—
139C4×C47—42no—
140C4×C47—42no—
141C4×C47—42no—
142C4×C47—44no—
143C4×C47C2×#10112yes—
144C4×C47#44912yes—
145C4×C47#45212yes—
146C2×#557D4711yescharacteristic
147#1801—471no—
148#1546—472no—
149#1487—472no—
150#1461—471no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#178411yescharacteristic
154C2×C47#178811yescharacteristic
155C2×C47#179611yescharacteristic
156#557—24no—
157#557—24no—
158C2×C4⋊Q8—21no—
159C2×C4⋊Q8C2×D4711yescharacteristic
160C2×#125—22no—
161C2×#125Dic4712yes—
162C2×C4×Q8—21no—
163C2×C4×Q8Dic4712yes—
164C2×C4×Q8C2×D4711yescharacteristic
165C2×C4×Q8C2×D4711yescharacteristic
166#431—472no—
167#423—472no—
168C2×#101—941no—
169#343—941no—
170#332—944no—
171Q8⋊C8—944no—
172#324—942no—
173C47#839011yescharacteristic, complemented, Sylow
174C22×Q8—21no—
175C22×Q8C2×Dic4711yescharacteristic
176C22×Q8C22⋊D4711yescharacteristic
177C2×M16—942no—
178C4⋊Q8—28no—
179C4⋊Q8—22no—
180C4⋊Q8C22⋊D4714yes—
181#125—216no—
182#125—24no—
183#125—24no—
184C4×Q8—28no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8C22⋊D4714yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×Dic4712yes—
197C2×C4⋊C4C2×Dic4711yescharacteristic
198C2×C4⋊C4C22⋊D4711yescharacteristic
199C2×C4⋊C4C22⋊D4712yes—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42C2×Dic4712yes—
203C2×C42C22×D4711yescharacteristic
204C4⋊C8—942no—
205C4⋊C8—942no—
206C4⋊C8—942no—
207#101—1882no—
208C22⋊C8—942no—
209C4×C8—942no—
210C2×Q8—24no—
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C22⋊Dic4712yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4C22⋊Dic4712yes—
225C22×C4C22⋊Dic4712yes—
226C22×C4C22×Dic4711yescharacteristic
227C22×C4C2×C22⋊D4711yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229M16—1882no—
230M16—1882no—
231C2×C8—1882no—
232C2×C8—1882no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4C22⋊Dic4714yes—
243C4⋊C4C22⋊Dic4712yes—
244C42—28no—
245C42—22no—
246C42—22no—
247C42—24no—
248C42C2×C22⋊D4712yes—
249C42C2×C22⋊D4712yes—
250C23C2×C22⋊Dic4711yescharacteristic
251Q8—24no—
252Q8—22no—
253Q8—42no—
254Q8—44no—
255Q8—42no—
256Q8—44no—
257Q8#1309394314yescomplemented
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4C2×C22⋊Dic4711yescharacteristic
273C2×C4C2×C22⋊Dic4711yescharacteristic
274C2×C4#1309399312yes—
275C2×C4#1309399312yes—
276C2×C4#1309404412yes—
277C2×C4#1309404911yescharacteristic
278C2×C4#1309405411yescharacteristic
279C8—3762no—
280C8—3762no—
281C22—21no—
282C22—21no—
283C22#1568897911yescharacteristic
284C22#1568911911yescharacteristic
285C22#1568912411yescharacteristic
286C4—24no—
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—42no—
292C4—42no—
293C4C2×#1309394312yes—
294C4#1568926812yes—
295C4#1568928512yes—
296C2—21no—
297C2—21no—
298C2#2024331211yescharacteristic
299C2#2024332811yescharacteristic
300C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#557C2
socleC22×C47#379
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×#557D47
hypocenterC47#8390
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
106. Q8×C47#101semidirect
173. C47#8390semidirect
257. Q8#13093943semidirect

Maximal subgroups

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

subgroupindex
#202433842
#202420602
#202418172
#202417122
C2×C47×#5572
#839047