Finite Group Databasebeta

← #25010390

Subgroups — #25010390

Order 11008.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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
1Q16×D172C211yesmaximal, characteristic
2#19136824C211yesmaximal, characteristic, complemented
3#19133981C211yesmaximal, characteristic
4#19132645C211yesmaximal, characteristic, complemented
5#19132559C211yesmaximal, characteristic, complemented
6#19132549C211yesmaximal, characteristic
7C43×#2225C211yesmaximal, characteristic, complemented
8C2×Q16×D43—21no—
9Q8×D172—21no—
10#14583279C2211yescharacteristic
11#14583141—21no—
12#14583109—21no—
13#14583101—21no—
14#14583056—22no—
15C8×D172C2211yescharacteristic
16#14582702—22no—
17C4×C43⋊C16C2211yescharacteristic
18#14582652C2211yescharacteristic
19#14582645—21no—
20C43×#500C2211yescharacteristic
21C4×Q16×C43C2211yescharacteristic
22C43×#367—21no—
23C43×C4⋊C16C2211yescharacteristic
24C43×#358—21no—
25C2×Q8×D43—22no—
26C2×Dic172—21no—
27C2×Dic172D412yes—
28C2×#10621885—22no—
29Q16×D43—42no—
30Q16×D43—42no—
31C2×#10621884—41no—
32C2×C8×D43—21no—
33#13063325—21no—
34#13063325—41no—
35#13063305—22no—
36#13063305—42no—
37C4×D172—41no—
38C4×D172D411yescharacteristic
39C2×C43⋊C16—21no—
40C4⋊(C43⋊C8)—21no—
41C4⋊(C43⋊C8)D411yescharacteristic
42#13063249—41no—
43#13063248—22no—
44(C2×D43)⋊C8—21no—
45C2×Q16×C43D411yescharacteristic
46C4⋊Q8×C43—21no—
47C4×Q8×C43—21no—
48C2×C16×C43—21no—
49#107×C43—21no—
50#107×C43—21no—
51#107×C43D411yescharacteristic
52#103×C43—21no—
53C4×C8×C43C2311yescharacteristic
54C2×Q8×C43—21no—
55C2×Q8×C43—41no—
56Q8×D43—44no—
57Q8×D43—82no—
58C2×D172D811yescharacteristic
59C2×C4×D43—21no—
60C2×C4×D43—42no—
61C2×Dic86—22no—
62C2×Dic86—42no—
63Q16×C43—21no—
64Q16×C43SD1612yescomplemented
65C2×C8×C43C2×D411yescharacteristic
66C2×C8×C43C2×D411yescharacteristic
67C4⋊C4×C43—21no—
68C4⋊C4×C43—41no—
69C4⋊C4×C43—41no—
70C4⋊C4×C43—41no—
71C42×C43—41no—
72C42×C43C2×D411yescharacteristic
73#10621885—44no—
74#10621884—82no—
75(C2×D43)⋊C4—21no—
76(C2×D43)⋊C4—42no—
77C4⋊Dic43—42no—
78C4⋊Dic43—41no—
79C4⋊Dic43—41no—
80C4⋊Dic43D811yescharacteristic
81C2×C43⋊C8—21no—
82C2×C43⋊C8—41no—
83Dic172—42no—
84Dic172—41no—
85Dic172—41no—
86C8×D43—41no—
87C8×D43—41no—
88C16×C43—41no—
89C43⋊C16—41no—
90C22×D43—21no—
91Q8×C43—22no—
92Q8×C43—41no—
93Q8×C43—81no—
94Q8×C43—81no—
95C2×C4×C43—41no—
96C2×C4×C43—41no—
97C2×C4×C43—81no—
98C2×C4×C43#12011yescharacteristic
99C2×C4×C43C2×D811yescharacteristic
100C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
101C2×Dic43—21no—
102C2×Dic43—42no—
103D172—21no—
104D172SD3212yes—
105C4×D43—41no—
106C4×D43—41no—
107C4×D43—84no—
108Dic86—42no—
109Dic86—42no—
110Dic86—84no—
111C8×C43—21no—
112C8×C43C2×SD1611yescharacteristic
113C8×C43AGL(1,ℤ/8)11yescharacteristic
114C43⋊C8—41no—
115C43⋊C8—81no—
116#12370—431nomaximal, Sylow
117C22×C43#44711yescharacteristic
118C2×D43—22no—
119C2×D43—41no—
120C2×D43—41no—
121C4×C43—21no—
122C4×C43—42no—
123C4×C43—81no—
124C4×C43—161no—
125C4×C43#45011yescharacteristic
126C4×C43#45511yescharacteristic
127C4×C43C2×SD3211yescharacteristic
128C4×C43#51011yescharacteristic
129Dic43—41no—
130Dic43—82no—
131D4×Q16—431no—
132#2241—431no—
133#2225D4311yescharacteristic, complemented
134#1673—431no—
135#1362—431no—
136#1332—431no—
137D4⋊C16—431no—
138C2×C43#163911yescharacteristic
139C2×C43#218511yescharacteristic
140C2×C43#219511yescharacteristic
141D43—42no—
142C22×Q16—861no—
143D4×Q8—861no—
144#500C2×D4311yescharacteristic
145#494—431no—
146#467—861no—
147#462—861no—
148#461—861no—
149#451—862no—
150C4×Q16C2×D4311yescharacteristic
151C8×D4—431no—
152#367—21no—
153C4⋊C16C2×D4311yescharacteristic
154#358—21no—
155#358—862no—
156C4×C16—431no—
157#335—431no—
158#332—861no—
159C43#1237011yescharacteristic, complemented, Sylow
160C22×Q8—862no—
161C2×Q16—432no—
162C2×Q16—862no—
163C2×Q16—861no—
164C2×Q16—1722no—
165C2×Q16—1722no—
166C2×Q16C22⋊D4311yescharacteristic
167C2×SD16—1721no—
168C22×C8—861no—
169C4⋊Q8—21no—
170C4⋊Q8—861no—
171C4⋊Q8—1721no—
172C22⋊Q8—862no—
173C22⋊Q8—1722no—
174C4×Q8—21no—
175C4×D4—431no—
176C4×D4—1721no—
177C2×C16—21no—
178C2×C16—861no—
179#107—21no—
180#107—21no—
181#107C22⋊D4311yescharacteristic
182C4⋊C8—431no—
183C4⋊C8—861no—
184#103—21no—
185#103—862no—
186#103—1721no—
187C22⋊C8—861no—
188C4×C8C22×D4311yescharacteristic
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—862no—
192C2×Q8—3442no—
193C2×Q8—1722no—
194C2×Q8—1724no—
195C2×D4—431no—
196C22×C4—861no—
197C22×C4—1722no—
198Q16—21no—
199Q16—1722no—
200Q16—1724no—
201Q16—1721no—
202Q16—1721no—
203Q16#1062188412yescomplemented
204SD16—3442no—
205C2×C8—861no—
206C2×C8—1721no—
207C2×C8—1721no—
208C2×C8—1721no—
209C2×C8D4×D4311yescharacteristic
210C2×C8C2×C22⋊D4311yescharacteristic
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—431no—
216C4⋊C4—1722no—
217C4⋊C4—1721no—
218C4⋊C4—1721no—
219C22⋊C4—861no—
220C22⋊C4—1722no—
221C42—41no—
222C42D4×D4311yescharacteristic
223C16—41no—
224C16—1721no—
225C23—861no—
226Q8—22no—
227Q8—41no—
228Q8—81no—
229Q8—81no—
230Q8—3444no—
231Q8—1722no—
232Q8—1722no—
233D4—432no—
234D4—861no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—861no—
239C2×C4—3444no—
240C2×C4—1722no—
241C2×C4—1721no—
242C2×C4—1721no—
243C2×C4#1306329311yescharacteristic
244C2×C4D8×D4311yescharacteristic
245C2×C4#1306334911yescharacteristic
246C8—21no—
247C8—3441no—
248C8—1721no—
249C8C2×#1062188411yescharacteristic
250C8#1306335011yescharacteristic
251C22—862no—
252C22—1721no—
253C22—1721no—
254C22#1458303111yescharacteristic
255C4—21no—
256C4—42no—
257C4—81no—
258C4—161no—
259C4—3442no—
260C4—1721no—
261C4#1458304911yescharacteristic
262C4#1458307511yescharacteristic
263C4SD32×D4311yescharacteristic
264C4#1458333811yescharacteristic
265C2—1722no—
266C2#1913377311yescharacteristic
267C2#1913651011yescharacteristic
268C2#1913657011yescharacteristic
subgroupG/H
centerC22#14583031
commutator, cosocleC4×C8×C43C23
FrattiniC4×C8C22×D43
FittingC43×#2225C2
socleC22×C43#447
prime radicalC2×D172D8
hypercenter#2225D43
hypocenterC43#12370
LCSG ▷ C4×C8×C43 ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
2. #19136824C2semidirect
4. #19132645C2semidirect
5. #19132559C2semidirect
7. C43×#2225C2semidirect
64. Q16×C43SD16semidirect
133. #2225D43semidirect
159. C43#12370semidirect
203. Q16#10621884semidirect

Maximal subgroups

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

subgroupindex
Q16×D1722
#191368242
#191339812
#191326452
#191325592
#191325492
C43×#22252
#1237043