Finite Group Databasebeta

← #24994309

Subgroups — #24994309

Order 11008.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#19143680C211yesmaximal, characteristic, complemented
2#19139594C211yesmaximal, characteristic
3#19133308C212yesmaximal
4#19132450C212yesmaximal, complemented
5C43×#1467C211yesmaximal, characteristic, complemented
6C2×#13063322C2211yescharacteristic
7C2×#13063314C412yes—
8#14583030—24no—
9#14582956—21no—
10#14582956C2212yes—
11#14582952—21no—
12#14582816—21no—
13#14582816—21no—
14#14582801—21no—
15#14582801—21no—
16#14582800C412yes—
17C4×C4⋊Dic43C2211yescharacteristic
18#14582642—24no—
19C2×C4×Q8×C43C2211yescharacteristic
20C43×#432C2212yes—
21C43×#324—21no—
22C43×#324—21no—
23C22×D172—21no—
24C22×D172—21no—
25C22×D172C2×C412yes—
26#13063322—22no—
27#13063322—22no—
28#13063314—44no—
29C2×C4⋊Dic43—21no—
30C2×(C2×D43)⋊C4—22no—
31C2×(C2×D43)⋊C4—22no—
32C2×(C2×D43)⋊C4—22no—
33C2×(C2×D43)⋊C4—22no—
34C2×C4×Dic43C2×C412yes—
35C4⋊(C43⋊C8)—22no—
36C4⋊(C43⋊C8)—22no—
37C4⋊(C43⋊C8)D414yes—
38C22⋊(C43⋊C8)—22no—
39#13063236—22no—
40#13063236—22no—
41C4×C43⋊C8—22no—
42C43⋊#95—21no—
43C22×Q8×C43C2×C411yescharacteristic
44C4×Q8×C43—22no—
45C4×Q8×C43—22no—
46C2×C4⋊C4×C43C2×C411yescharacteristic
47C2×C4⋊C4×C43D411yescharacteristic
48C2×C4⋊C4×C43D411yescharacteristic
49C2×C42×C43D411yescharacteristic
50C2×C42×C43D411yescharacteristic
51C2×C42×C43C2311yescharacteristic
52C4⋊C8×C43—22no—
53C22⋊C8×C43—22no—
54#97×C43—22no—
55C23×D43—22no—
56C2×Q8×C43—22no—
57C2×Q8×C43—22no—
58C2×Q8×C43—41no—
59C22×C4×C43C22⋊C411yescharacteristic
60C22×C4×C43C22⋊C411yescharacteristic
61C22×C4×C43C22⋊C411yescharacteristic
62C22×C4×C43C22⋊C411yescharacteristic
63C22×C4×C43C22×C411yescharacteristic
64C22×C4×C43C2×D411yescharacteristic
65C22×C4×C43C2×D411yescharacteristic
66C22×Dic43—22no—
67C2×D172—24no—
68C2×D172—42no—
69C2×D172—42no—
70C2×D172—41no—
71C2×D172—42no—
72C2×D172—41no—
73C2×D172—42no—
74C2×D172—42no—
75C2×D172—82no—
76C2×C8×C43—42no—
77C4⋊C4×C43—22no—
78C4⋊C4×C43—22no—
79C4⋊C4×C43—22no—
80C42×C43—21no—
81C42×C43—21no—
82C42×C43—21no—
83C42×C43—21no—
84C42×C43C2×D412yes—
85C42×C43#4112yes—
86(C2×D43)⋊C4—44no—
87(C2×D43)⋊C4—44no—
88(C2×D43)⋊C4—82no—
89(C2×D43)⋊C4—82no—
90C4⋊Dic43—42no—
91C4×Dic43—42no—
92C2×C43⋊C8—24no—
93C23×C43C2×C22⋊C411yescharacteristic
94C22×D43—42no—
95C22×D43—42no—
96C22×D43—42no—
97C22×D43—42no—
98C22×D43—42no—
99C22×D43—42no—
100C22×D43—42no—
101Q8×C43—44no—
102C2×C4×C43—21no—
103C2×C4×C43—21no—
104C2×C4×C43—21no—
105C2×C4×C43—21no—
106C2×C4×C43—22no—
107C2×C4×C43—21no—
108C2×C4×C43—21no—
109C2×C4×C43—21no—
110C2×C4×C43—21no—
111C2×C4×C43—21no—
112C2×C4×C43—41no—
113C2×C4×C43—41no—
114C2×C4×C43—41no—
115C2×C4×C43#10012yes—
116C2×C4×C43C4×D412yes—
117C2×C4×C43#12011yescharacteristic
118C2×C4×C43#12112yes—
119C2×C4×C43#12311yescharacteristic
120C2×Dic43—42no—
121C2×Dic43—42no—
122C2×Dic43—42no—
123D172—84no—
124D172—82no—
125D172—84no—
126D172—82no—
127C8×C43—82no—
128C43⋊C8—44no—
129#9458—431nomaximal, Sylow
130C22×C43—21no—
131C22×C43—21no—
132C22×C43#38611yescharacteristic
133C22×C43C2×#10011yescharacteristic
134C22×C43#41311yescharacteristic
135C2×D43—82no—
136C2×D43—82no—
137C2×D43—82no—
138C2×D43—82no—
139C2×D43—82no—
140C2×D43—82no—
141C2×D43—82no—
142C4×C43—21no—
143C4×C43—22no—
144C4×C43—22no—
145C4×C43—22no—
146C4×C43—41no—
147C4×C43—41no—
148C4×C43—41no—
149C4×C43#45412yes—
150Dic43—82no—
151#2983—431no—
152#2548—431no—
153#1555—432no—
154#1467D4311yescharacteristic, complemented
155#1285—432no—
156C2×C43—21no—
157C2×C43—21no—
158C2×C43#188811yescharacteristic
159C2×C43#190011yescharacteristic
160C2×C43#191011yescharacteristic
161D43—162no—
162C2×#127—431no—
163C2×#124—432no—
164C2×C4×Q8C2×D4311yescharacteristic
165#446—864no—
166#432—432no—
167#432—861no—
168#432C2×D4312yes—
169#431—861no—
170#394—861no—
171#394—861no—
172#390—432no—
173#390—861no—
174#390—861no—
175C4×C4⋊C4—431no—
176#331—864no—
177#324—21no—
178#324—21no—
179C43#945811yescharacteristic, complemented, Sylow
180C22×Q8C4×D4311yescharacteristic
181C22×D4—432no—
182C22×D4—861no—
183C22×D4—861no—
184#127—862no—
185#127—862no—
186#124—1724no—
187C4×Q8—22no—
188C4×Q8—22no—
189C2×C4⋊C4—861no—
190C2×C4⋊C4C4×D4311yescharacteristic
191C2×C4⋊C4C22⋊D4311yescharacteristic
192C2×C4⋊C4C22⋊D4311yescharacteristic
193C2×C22⋊C4—862no—
194C2×C22⋊C4—862no—
195C2×C22⋊C4—862no—
196C2×C22⋊C4—862no—
197C2×C42—432no—
198C2×C42D17211yescharacteristic
199C2×C42D17211yescharacteristic
200C2×C42C22×D4311yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—434no—
203C4⋊C8—862no—
204C4⋊C8—862no—
205C22⋊C8—22no—
206C22⋊C8—862no—
207#97—22no—
208#97—862no—
209#97—862no—
210C4×C8—862no—
211#95—861no—
212C24—862no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—864no—
217C2×D4—3442no—
218C2×D4—1722no—
219C2×D4—1722no—
220C2×D4—1722no—
221C2×D4—1722no—
222C2×D4—1721no—
223C2×D4—1722no—
224C2×D4—1721no—
225C22×C4—862no—
226C22×C4(C2×D43)⋊C411yescharacteristic
227C22×C4(C2×D43)⋊C411yescharacteristic
228C22×C4(C2×D43)⋊C411yescharacteristic
229C22×C4(C2×D43)⋊C411yescharacteristic
230C22×C4C2×C4×D4311yescharacteristic
231C22×C4C2×D17211yescharacteristic
232C22×C4C2×C22⋊D4311yescharacteristic
233C2×C8—42no—
234C2×C8—864no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—1722no—
239C22⋊C4—3442no—
240C22⋊C4—3442no—
241C22⋊C4—1724no—
242C22⋊C4—1724no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—1722no—
248C42D4×D4312yes—
249C42#1062190012yes—
250C23—1722no—
251C23—1722no—
252C23—1722no—
253C23—1722no—
254C23—1722no—
255C23—1722no—
256C23—1722no—
257C23C2×(C2×D43)⋊C411yescharacteristic
258Q8—44no—
259D4—3444no—
260D4—3444no—
261D4—3442no—
262D4—3442no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—1722no—
277C2×C4—1722no—
278C2×C4—1722no—
279C2×C4#1306324112yes—
280C2×C4(C22⋊D43)⋊C412yes—
281C2×C4#1306329511yescharacteristic
282C2×C4#1306329712yes—
283C2×C4#1306330911yescharacteristic
284C8—82no—
285C8—1724no—
286C22—21no—
287C22—21no—
288C22—3442no—
289C22—3442no—
290C22—3442no—
291C22—3442no—
292C22—3442no—
293C22—3442no—
294C22—3442no—
295C22#1458277611yescharacteristic
296C22C2×#1306324111yescharacteristic
297C22#1458288111yescharacteristic
298C4—21no—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—3442no—
306C4#1458307412yes—
307C2—21no—
308C2—21no—
309C2—6882no—
310C2#1913499611yescharacteristic
311C2#1913505611yescharacteristic
312C2#1913512011yescharacteristic
subgroupG/H
centerC22#14582776
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1467C2
socleC22×C43#386
prime radicalC2×#13063322C22
cosocleC2×C42×C43C23
hypercenter#1467D43
hypocenterC43#9458
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #19143680C2semidirect
4. #19132450C2semidirect
5. C43×#1467C2semidirect
154. #1467D43semidirect
179. C43#9458semidirect

Maximal subgroups

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

subgroupindex
#191436802
#191395942
#191333082
#191324502
C43×#14672
#945843