Finite Group Databasebeta

← #24992105

Subgroups — #24992105

Order 11008.

546 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 135 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#19143676C211yesmaximal, characteristic, complemented
2#19137302C211yesmaximal, characteristic
3#19132985C212yesmaximal
4#19132423C212yesmaximal, complemented
5C43×#1461C211yesmaximal, characteristic, complemented
6C2×#13063314C411yescharacteristic
7C2×C4×D172C2211yescharacteristic
8Q8×C43⋊C8—24no—
9#14582952—22no—
10#14582952C2212yes—
11#14582806C411yescharacteristic
12#14582800C411yescharacteristic
13#14582790—22no—
14#14582770—22no—
15#14582768C411yescharacteristic
16#14582745C2211yescharacteristic
17#14582626—24no—
18C2×C4×Q8×C43C2211yescharacteristic
19C43×#431C2212yes—
20C43×#324—22no—
21C22×D172C2×C411yescharacteristic
22C22×C4×D43—21no—
23#13063314—22no—
24#13063314C814yescomplemented
25C4×D172—22no—
26C4×D172—22no—
27C2×C4⋊Dic43C2×C411yescharacteristic
28C2×Dic43⋊C4—22no—
29C2×(C2×D43)⋊C4—22no—
30C2×(C2×D43)⋊C4—21no—
31C2×(C2×D43)⋊C4—22no—
32C2×C4×Dic43C2×C411yescharacteristic
33C2×C4×Dic43C2×C411yescharacteristic
34C4⋊(C43⋊C8)—24no—
35C4⋊(C43⋊C8)—22no—
36C22⋊(C43⋊C8)—22no—
37C4×C43⋊C8—24no—
38C4×C43⋊C8D414yes—
39C43⋊#95—22no—
40C43⋊#95—21no—
41C43⋊#95—21no—
42C22×Q8×C43C2×C411yescharacteristic
43C4×Q8×C43—22no—
44C4×Q8×C43—22no—
45C2×C4⋊C4×C43C2×C411yescharacteristic
46C2×C4⋊C4×C43D412yes—
47C2×C42×C43D412yes—
48C2×C42×C43C2311yescharacteristic
49C4⋊C8×C43—22no—
50C22⋊C8×C43—22no—
51C4×C8×C43—22no—
52C23×D43—21no—
53C2×Q8×C43—22no—
54C2×Q8×C43—22no—
55C2×Q8×C43C2×C812yes—
56C2×Q8×C43M1612yes—
57C22×C4×C43C22⋊C412yes—
58C22×C4×C43C22⋊C412yes—
59C22×C4×C43C22×C411yescharacteristic
60C22×C4×C43C2×D411yescharacteristic
61C22×C4×C43C2×D411yescharacteristic
62C22×Dic43—21no—
63C22×Dic43—21no—
64C22×Dic43—21no—
65C2×D172—21no—
66C2×D172—22no—
67C2×D172—22no—
68C2×D172C2×C812yes—
69C2×C4×D43—42no—
70C2×C4×D43—82no—
71C2×C8×C43—42no—
72C4⋊C4×C43—22no—
73C4⋊C4×C43—24no—
74C42×C43—22no—
75C42×C43—22no—
76C42×C43C2×D412yes—
77C42×C43#4112yes—
78(C2×D43)⋊C4—28no—
79(C2×D43)⋊C4—41no—
80(C2×D43)⋊C4—41no—
81(C2×D43)⋊C4—82no—
82C4⋊Dic43—22no—
83Dic43⋊C4—82no—
84C4×Dic43—21no—
85C4×Dic43—22no—
86C4×Dic43C2×C812yes—
87C2×C43⋊C8—24no—
88C23×C43C2×C22⋊C411yescharacteristic
89C22×D43—22no—
90C22×D43—42no—
91C22×D43—42no—
92C22×D43—41no—
93C22×D43—41no—
94Q8×C43—24no—
95Q8×C43—24no—
96C2×C4×C43—22no—
97C2×C4×C43—22no—
98C2×C4×C43—22no—
99C2×C4×C43—41no—
100C2×C4×C43—41no—
101C2×C4×C43—42no—
102C2×C4×C43C22⋊C814yes—
103C2×C4×C43C4×D412yes—
104C2×C4×C43#12011yescharacteristic
105C2×C4×C43#12112yes—
106C2×C4×C43#12311yescharacteristic
107C2×C4×C43C22×C811yescharacteristic
108C2×C4×C43C2×M1611yescharacteristic
109C2×C4×C43#13112yes—
110C2×Dic43—22no—
111C2×Dic43—41no—
112C2×Dic43—41no—
113C2×Dic43—42no—
114C2×Dic43—41no—
115C2×Dic43—41no—
116C2×Dic43—41no—
117C2×Dic43—41no—
118D172—24no—
119D172—42no—
120C4×D43—84no—
121C8×C43—82no—
122C43⋊C8—44no—
123#9102—431nomaximal, Sylow
124C22×C43—21no—
125C22×C43—21no—
126C22×C43#38611yescharacteristic
127C22×C43C2×C22⋊C811yescharacteristic
128C22×C43#40811yescharacteristic
129C2×D43—44no—
130C2×D43—42no—
131C2×D43—82no—
132C2×D43—81no—
133C2×D43—81no—
134C4×C43—24no—
135C4×C43—22no—
136C4×C43—21no—
137C4×C43—21no—
138C4×C43—42no—
139C4×C43C8×D412yes—
140C4×C43#43512yes—
141Dic43—42no—
142Dic43—81no—
143Dic43—81no—
144#2982—431no—
145#2326—431no—
146#1478—432no—
147#1461D4311yescharacteristic, complemented
148#1274—432no—
149C2×C43—21no—
150C2×C43—21no—
151C2×C43#186911yescharacteristic
152C2×C43#191011yescharacteristic
153C2×C43#191211yescharacteristic
154D43—82no—
155C2×#124—431no—
156C2×C4×Q8C2×D4311yescharacteristic
157C2×C4×D4—431no—
158C8×Q8—864no—
159#431—432no—
160#431—862no—
161#431C2×D4312yes—
162#391—431no—
163#390—431no—
164#388—862no—
165#385—862no—
166#384—431no—
167#376—431no—
168D4⋊C8—864no—
169#324—22no—
170C43#910211yescharacteristic, complemented, Sylow
171C22×Q8C4×D4311yescharacteristic
172C22×D4—431no—
173C23×C4—861no—
174#124—434no—
175#124—862no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—862no—
179C4×D4—862no—
180C2×C4⋊C4—431no—
181C2×C4⋊C4—862no—
182C2×C4⋊C4C4×D4311yescharacteristic
183C2×C4⋊C4C22⋊D4312yes—
184C2×C22⋊C4—862no—
185C2×C22⋊C4—862no—
186C2×C22⋊C4—861no—
187C2×C42—431no—
188C2×C42—431no—
189C2×C42D17212yes—
190C2×C42C22×D4311yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—864no—
193C4⋊C8—862no—
194C22⋊C8—22no—
195C22⋊C8—862no—
196C4×C8—22no—
197C4×C8—434no—
198C4×C8—864no—
199#95—862no—
200#95—861no—
201#95—861no—
202C24—861no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C8×D4312yes—
206C2×Q8#1062187212yes—
207C2×D4—432no—
208C2×D4—862no—
209C2×D4—861no—
210C2×D4—862no—
211C22×C4—861no—
212C22×C4—861no—
213C22×C4—861no—
214C22×C4—3442no—
215C22×C4—1722no—
216C22×C4(C2×D43)⋊C412yes—
217C22×C4(C2×D43)⋊C412yes—
218C22×C4C2×C4×D4311yescharacteristic
219C22×C4C2×D17211yescharacteristic
220C22×C4C2×C22⋊D4311yescharacteristic
221C2×C8—42no—
222C2×C8—864no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—862no—
226C4⋊C4—3442no—
227C22⋊C4—868no—
228C22⋊C4—3442no—
229C22⋊C4—1721no—
230C22⋊C4—1721no—
231C42—22no—
232C42—22no—
233C42—432no—
234C42—861no—
235C42—862no—
236C42D4×D4312yes—
237C42#1062190012yes—
238C23—862no—
239C23—1722no—
240C23—1722no—
241C23—1721no—
242C23—1721no—
243C23C2×(C2×D43)⋊C411yescharacteristic
244Q8—24no—
245Q8—24no—
246D4—864no—
247D4—1722no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—862no—
255C2×C4—3444no—
256C2×C4—1721no—
257C2×C4—1722no—
258C2×C4—1721no—
259C2×C4—1721no—
260C2×C4—1721no—
261C2×C4—1721no—
262C2×C4—1721no—
263C2×C4(C2×D43)⋊C814yes—
264C2×C4(C22⋊D43)⋊C412yes—
265C2×C4#1306329511yescharacteristic
266C2×C4#1306329712yes—
267C2×C4#1306330911yescharacteristic
268C2×C4C2×C8×D4311yescharacteristic
269C2×C4C2×#1062187211yescharacteristic
270C2×C4#1306333312yes—
271C8—82no—
272C8—1724no—
273C22—21no—
274C22—21no—
275C22—3442no—
276C22—3441no—
277C22—3441no—
278C22—1724no—
279C22—1722no—
280C22#1458277611yescharacteristic
281C22C2×(C2×D43)⋊C811yescharacteristic
282C22#1458286711yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—24no—
287C4—42no—
288C4—3441no—
289C4—3441no—
290C4—1722no—
291C4(C22⋊D43)⋊C812yes—
292C4#1458296712yes—
293C2—21no—
294C2—21no—
295C2—3442no—
296C2#1913488511yescharacteristic
297C2#1913512011yescharacteristic
298C2#1913513011yescharacteristic
subgroupG/H
centerC22#14582776
commutatorC2×C4×C43C22×C8
FrattiniC2×C42C22×D43
FittingC43×#1461C2
socleC22×C43#386
prime radicalC22×D172C2×C4
cosocleC2×C42×C43C23
hypercenter#1461D43
hypocenterC43#9102
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #19143676C2semidirect
4. #19132423C2semidirect
5. C43×#1461C2semidirect
24. #13063314C8semidirect
147. #1461D43semidirect
170. C43#9102semidirect

Maximal subgroups

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

subgroupindex
#191436762
#191373022
#191329852
#191324232
C43×#14612
#910243