Finite Group Databasebeta

← #26108036

Subgroups — #26108036

Order 12032.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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, complemented
2#20242069C212yesmaximal, complemented
3#20241813C212yesmaximal
4#20241712C211yesmaximal, characteristic, complemented
5C2×C47×#556C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093943—21no—
9#15688892—21no—
10#15688872—24no—
11D4⋊(C47⋊C8)—24no—
12#15688849—22no—
13C47×#556—24no—
14C47×#556—24no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#126×C47—22no—
17C2×#124×C47C412yes—
18C2×C22⋊Q8×C47—21no—
19C2×C4×Q8×C47C412yes—
20C2×C4×Q8×C47C2211yescharacteristic
21C2×C4×D4×C47C2211yescharacteristic
22C2×#117×C47—21no—
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×C47C2×C412yes—
31C22×Q8×C47D411yescharacteristic
32C22×D4×C47C2×C411yescharacteristic
33C23×C4×C47—21no—
34C4⋊Q8×C47—22no—
35C4⋊Q8×C47D414yes—
36#126×C47—216no—
37#124×C47—24no—
38#124×C47—24no—
39C22⋊Q8×C47—28no—
40C4×Q8×C47—24no—
41C4×Q8×C47—24no—
42C4×Q8×C47—22no—
43C4×Q8×C47—22no—
44C4×D4×C47—22no—
45C4×D4×C47D414yes—
46#117×C47—28no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47—21no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D412yes—
53C2×C22⋊C4×C47—22no—
54C2×C22⋊C4×C47—22no—
55C2×C22⋊C4×C47—21no—
56C2×C42×C47—22no—
57C2×C42×C47C2×C412yes—
58C2×C42×C47C2311yescharacteristic
59C24×C47—21no—
60C2×Q8×C47—22no—
61C2×Q8×C47—22no—
62C2×Q8×C47—22no—
63C2×Q8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—44no—
66C2×Q8×C47C22⋊C414yes—
67C2×D4×C47—21no—
68C2×D4×C47—42no—
69C2×D4×C47C22⋊C412yes—
70C22×C4×C47—21no—
71C22×C4×C47—22no—
72C22×C4×C47—24no—
73C22×C4×C47—42no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47C22⋊C412yes—
77C22×C4×C47C22⋊C412yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C4⋊C4×C47—28no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—24no—
84C4⋊C4×C47—24no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47C22⋊C412yes—
88C22⋊C4×C47—28no—
89C22⋊C4×C47—28no—
90C22⋊C4×C47—24no—
91C42×C47—28no—
92C42×C47—22no—
93C42×C47—22no—
94C42×C47C2×D412yes—
95C42×C47C2×D412yes—
96#10625329—42no—
97#10625329—42no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47C2×C22⋊C411yescharacteristic
106Q8×C47—44no—
107Q8×C47—44no—
108Q8×C47—42no—
109Q8×C47—42no—
110D4×C47—24no—
111D4×C47—42no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—24no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—41no—
120C2×C4×C47—44no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—44no—
124C2×C4×C47—44no—
125C2×C4×C47—42no—
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47C2×C22⋊C411yescharacteristic
128C2×C4×C47#12012yes—
129C2×C4×C47#12012yes—
130C2×C4×C47#13512yes—
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C47#13711yescharacteristic
133C47⋊C8—82no—
134C47⋊C8—82no—
135#8394—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47—82no—
142C22×C47—82no—
143C22×C47#37911yescharacteristic
144C22×C47#41611yescharacteristic
145C22×C47#41711yescharacteristic
146C4×C47—22no—
147C4×C47—21no—
148C4×C47—42no—
149C4×C47—44no—
150C4×C47—42no—
151C4×C47—42no—
152C4×C47#41312yes—
153C4×C47#44912yes—
154C4×C47#45212yes—
155C2×#556D4711yescharacteristic
156#1801—471no—
157#1548—472no—
158#1486—472no—
159#1461—471no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—82no—
163C2×C47#178511yescharacteristic
164C2×C47#178811yescharacteristic
165C2×C47#179611yescharacteristic
166#556—24no—
167#556—24no—
168C2×C4⋊Q8C2×D4711yescharacteristic
169C2×#126—22no—
170C2×#124Dic4712yes—
171C2×C22⋊Q8—21no—
172C2×C4×Q8Dic4712yes—
173C2×C4×Q8C2×D4711yescharacteristic
174C2×C4×D4C2×D4711yescharacteristic
175C2×#117—21no—
176#431—472no—
177#423—472no—
178C2×#101—941no—
179#343—941no—
180#333—944no—
181D4⋊C8—944no—
182#324—942no—
183C47#839411yescharacteristic, complemented, Sylow
184C22×Q8C2×Dic4712yes—
185C22×Q8C22⋊D4711yescharacteristic
186C22×D4C2×Dic4711yescharacteristic
187C23×C4—21no—
188C2×M16—942no—
189C4⋊Q8—22no—
190C4⋊Q8C22⋊D4714yes—
191#126—216no—
192#124—24no—
193#124—24no—
194C22⋊Q8—28no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4C22⋊D4714yes—
201#117—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic4711yescharacteristic
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C4⋊C4C22⋊D4712yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42C2×Dic4712yes—
213C2×C42C22×D4711yescharacteristic
214C4⋊C8—942no—
215C4⋊C8—942no—
216C4⋊C8—942no—
217#101—1882no—
218C22⋊C8—942no—
219C4×C8—942no—
220C24—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—44no—
227C2×Q8C22⋊Dic4714yes—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊Dic4712yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C22⋊Dic4712yes—
238C22×C4C22⋊Dic4712yes—
239C22×C4C22×Dic4711yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242M16—1882no—
243M16—1882no—
244C2×C8—1882no—
245C2×C8—1882no—
246C4⋊C4—28no—
247C4⋊C4—21no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4C22⋊Dic4712yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42C2×C22⋊D4712yes—
260C42C2×C22⋊D4712yes—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic4711yescharacteristic
267Q8—44no—
268Q8—42no—
269Q8—42no—
270Q8—44no—
271D4—24no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4C2×C22⋊Dic4711yescharacteristic
288C2×C4C2×C22⋊Dic4711yescharacteristic
289C2×C4#1309399312yes—
290C2×C4#1309399312yes—
291C2×C4#1309404412yes—
292C2×C4#1309404911yescharacteristic
293C2×C4#1309405411yescharacteristic
294C8—3762no—
295C8—3762no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22#1568897911yescharacteristic
304C22#1568911911yescharacteristic
305C22#1568912411yescharacteristic
306C4—21no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4#1568910712yes—
313C4#1568926812yes—
314C4#1568928512yes—
315C2—21no—
316C2—21no—
317C2—82no—
318C2#2024331611yescharacteristic
319C2#2024332811yescharacteristic
320C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#556C2
socleC22×C47#379
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenterC2×#556D47
hypocenterC47#8394
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #20243384C2semidirect
2. #20242069C2semidirect
4. #20241712C2semidirect
183. C47#8394semidirect

Maximal subgroups

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

subgroupindex
#202433842
#202420692
#202418132
#202417122
C2×C47×#5562
#839447