Finite Group Databasebeta

← #24994047

Subgroups — #24994047

Order 11008.

644 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 111 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#19140075C211yesmaximal, characteristic
2#19133110C212yesmaximal
3#19132925C211yesmaximal, characteristic, complemented
4#19132481C212yesmaximal, complemented
5C43×#2274C211yesmaximal, characteristic, complemented
6C2×#13063299C2211yescharacteristic
7#14582863C2212yes—
8#14582838C412yescomplemented
9#14582816C412yescomplemented
10#14582789—22no—
11#14582778—21no—
12#14582778—21no—
13C4×C22⋊Dic43C2211yescharacteristic
14#14582632—24no—
15#14582625—22no—
16C2×#117×C43—21no—
17C22×C4⋊C4×C43C2211yescharacteristic
18C4×C4⋊C4×C43—24no—
19C4×C22⋊C4×C43—21no—
20C4×C22⋊C4×C43C2212yes—
21C22×D172C2×C411yescharacteristic
22C22×C22⋊D43—21no—
23C2×#10621877—22no—
24#13063299—22no—
25#13063299—22no—
26C2×C4⋊Dic43C2×C411yescharacteristic
27C2×C22⋊Dic43—21no—
28C2×(C2×D43)⋊C4—22no—
29C2×(C2×D43)⋊C4—21no—
30C2×(C2×D43)⋊C4—22no—
31C2×C4×Dic43—21no—
32C22⋊(C43⋊C8)—22no—
33C22⋊(C43⋊C8)—22no—
34C43⋊#95—22no—
35C43⋊#95—22no—
36C43⋊#95C2×C412yes—
37C23×C4×C43D411yescharacteristic
38C23×C4×C43D411yescharacteristic
39C23×C4×C43C2311yescharacteristic
40#117×C43—28no—
41C2×C4⋊C4×C43—22no—
42C2×C4⋊C4×C43—22no—
43C2×C4⋊C4×C43—22no—
44C2×C4⋊C4×C43C2×C412yes—
45C2×C4⋊C4×C43D412yes—
46C2×C22⋊C4×C43—22no—
47C2×C42×C43—22no—
48C2×C42×C43—22no—
49C2×C42×C43—22no—
50#95×C43—22no—
51#95×C43D414yes—
52C24×C43C2×D411yescharacteristic
53C23×D43—21no—
54C22×C4×C43—24no—
55C22×C4×C43—21no—
56C22×C4×C43—21no—
57C22×C4×C43—24no—
58C22×C4×C43—21no—
59C22×C4×C43—21no—
60C22×C4×C43—81no—
61C22×C4×C43—81no—
62C22×C4×C43C22⋊C412yes—
63C22×C4×C43C22⋊C412yes—
64C22×C4×C43C22×C411yescharacteristic
65C22×C4×C43C2×D411yescharacteristic
66C22×C4×C43C2×D412yes—
67C22×C4×C43#4112yes—
68C2×C22⋊D43—44no—
69C2×C22⋊D43—41no—
70C2×C22⋊D43—41no—
71C22×Dic43—21no—
72C22×Dic43—22no—
73C2×D172—22no—
74C2×D172—81no—
75C4⋊C4×C43—28no—
76C4⋊C4×C43—24no—
77C4⋊C4×C43—24no—
78C22⋊C4×C43—28no—
79C42×C43—28no—
80C42×C43—82no—
81C42×C43—82no—
82C22⋊Dic43—42no—
83(C2×D43)⋊C4—44no—
84(C2×D43)⋊C4—42no—
85(C2×D43)⋊C4—82no—
86C4⋊Dic43—22no—
87C4×Dic43—81no—
88#10621877—82no—
89C2×C43⋊C8—42no—
90C23×C43—22no—
91C23×C43—22no—
92C23×C43—41no—
93C23×C43C2×C22⋊C411yescharacteristic
94C23×C43#12011yescharacteristic
95C23×C43#12311yescharacteristic
96C22×D43—42no—
97C22×D43—42no—
98C22×D43—41no—
99C22×D43—42no—
100C2×C4×C43—22no—
101C2×C4×C43—22no—
102C2×C4×C43—24no—
103C2×C4×C43—22no—
104C2×C4×C43—22no—
105C2×C4×C43—28no—
106C2×C4×C43—22no—
107C2×C4×C43—42no—
108C2×C4×C43—41no—
109C2×C4×C43—42no—
110C2×C4×C43—41no—
111C2×C4×C43—44no—
112C2×C4×C43—42no—
113C2×C4×C43—44no—
114C2×C4×C43—42no—
115C2×C4×C43—81no—
116C2×C4×C43#10414yes—
117C2×C4×C43C4×D412yes—
118C2×C4×C43#12112yes—
119C22⋊D43—82no—
120C22⋊D43—82no—
121C2×Dic43—42no—
122C2×Dic43—42no—
123C2×Dic43—42no—
124C2×Dic43—41no—
125C2×Dic43—42no—
126D172—82no—
127C43⋊C8—82no—
128#9410—431nomaximal, Sylow
129C22×C43—21no—
130C22×C43—21no—
131C22×C43—21no—
132C22×C43—21no—
133C22×C43—21no—
134C22×C43—21no—
135C22×C43—21no—
136C22×C43—44no—
137C22×C43#38611yescharacteristic
138C22×C43C2×#10412yes—
139C22×C43#45412yes—
140C2×D43—84no—
141C2×D43—81no—
142C2×D43—81no—
143C2×D43—81no—
144C4×C43—24no—
145C4×C43—22no—
146C4×C43—22no—
147C4×C43—48no—
148C4×C43—42no—
149Dic43—82no—
150Dic43—81no—
151#2591—431no—
152#2274D4311yescharacteristic, complemented
153#1507—432no—
154#1463—431no—
155#1297—432no—
156C2×C43—22no—
157C2×C43—22no—
158C2×C43—41no—
159C2×C43#188811yescharacteristic
160C2×C43#189611yescharacteristic
161C2×C43#191311yescharacteristic
162D43—161no—
163C2×#121—431no—
164C2×#117—21no—
165C22×C4⋊C4C2×D4311yescharacteristic
166#407—432no—
167#400—432no—
168#394—432no—
169#388—862no—
170#386—861no—
171#386—861no—
172C4×C4⋊C4—24no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—431no—
175C4×C22⋊C4C2×D4312yes—
176#327—864no—
177#324—862no—
178C43#941011yescharacteristic, complemented, Sylow
179C22×D4—431no—
180C22×D4—861no—
181C23×C4C22⋊D4311yescharacteristic
182C23×C4C22⋊D4311yescharacteristic
183C23×C4C22×D4311yescharacteristic
184C2×M16—862no—
185#121—862no—
186#121—862no—
187#117—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—431no—
192C2×C4⋊C4C4×D4312yes—
193C2×C4⋊C4D17212yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—862no—
196C2×C22⋊C4—861no—
197C2×C22⋊C4—862no—
198C2×C22⋊C4—861no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—861no—
203C22⋊C8—862no—
204C22⋊C8—862no—
205#95—22no—
206#95—432no—
207#95—862no—
208#95—862no—
209#95C22⋊D4314yes—
210C24—861no—
211C24C2×C22⋊D4311yescharacteristic
212C2×D4—862no—
213C2×D4—3441no—
214C2×D4—1724no—
215C2×D4—1721no—
216C2×D4—1721no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—862no—
226C22×C4—861no—
227C22×C4(C2×D43)⋊C412yes—
228C22×C4(C2×D43)⋊C412yes—
229C22×C4C2×C4×D4311yescharacteristic
230C22×C4C2×D17211yescharacteristic
231C22×C4#1062189812yes—
232C22×C4C2×C22⋊D4312yes—
233M16—3442no—
234C2×C8—1722no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—862no—
239C22⋊C4—28no—
240C22⋊C4—3442no—
241C22⋊C4—1722no—
242C22⋊C4—1724no—
243C22⋊C4—1722no—
244C42—28no—
245C42—82no—
246C42—82no—
247C42—3441no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—1722no—
252C23—1722no—
253C23—1722no—
254C23—1721no—
255C23C2×(C2×D43)⋊C411yescharacteristic
256C23#1306329411yescharacteristic
257C23#1306330811yescharacteristic
258D4—3442no—
259D4—3442no—
260D4—3442no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—1722no—
278C2×C4—1722no—
279C2×C4—1722no—
280C2×C4—1722no—
281C2×C4—1721no—
282C2×C4#1306325314yes—
283C2×C4C4×C22⋊D4312yes—
284C2×C4#1306329912yes—
285C8—3442no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—3444no—
295C22—3441no—
296C22—3441no—
297C22—3441no—
298C22#1458277811yescharacteristic
299C22C2×#1306325312yes—
300C22#1458307212yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—48no—
305C4—42no—
306C4—3442no—
307C4—3441no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—6881no—
312C2#1913499811yescharacteristic
313C2#1913503811yescharacteristic
314C2#1913513711yescharacteristic
subgroupG/H
centerC22#14582778
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#2274C2
socleC22×C43#386
prime radicalC2×#13063299C22
cosocleC23×C4×C43C23
hypercenter#2274D43
hypocenterC43#9410
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #19132925C2semidirect
4. #19132481C2semidirect
5. C43×#2274C2semidirect
8. #14582838C4semidirect
9. #14582816C4semidirect
152. #2274D43semidirect
178. C43#9410semidirect

Maximal subgroups

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

subgroupindex
#191400752
#191331102
#191329252
#191324812
C43×#22742
#941043