Finite Group Databasebeta

← #11798424

Subgroups — #11798424

Order 1152.

324 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 62 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#10589859C211yesmaximal, characteristic
2C2×C9×SD32C211yesmaximal, characteristic, complemented
3#10588872C211yesmaximal, characteristic, complemented
4#10588859C211yesmaximal, characteristic, complemented
5#10588240C211yesmaximal, characteristic, complemented
6#10588187C211yesmaximal, characteristic, complemented
7#10588185C211yesmaximal, characteristic, complemented
8#71490—31nomaximal
9#63710—21no—
10C2×C9×Q16C2211yescharacteristic, complemented
11C2×C9×D8C2211yescharacteristic
12#63505—21no—
13#63495—21no—
14#63474—21no—
15#63473—22no—
16D8×D9—21no—
17#63467C2211yescharacteristic
18#63464C2211yescharacteristic
19C9×SD32—22no—
20C2×C9×C16C2211yescharacteristic, complemented
21#63394—21no—
22#63388—21no—
23#63387—21no—
24#63372C2211yescharacteristic, complemented
25#63371C2211yescharacteristic, complemented
26#5757—31no—
27C2×C3×SD32D311yescharacteristic
28#5192—31no—
29#5179—31no—
30#4563—31no—
31#4510—31no—
32#4508—31no—
33C2×Q8×C9—21no—
34C2×D4×C9—21no—
35C2×#611—42no—
36#3699—22no—
37#3699—41no—
38D4×D9—22no—
39D4×D9—41no—
40#3697—21no—
41#3697D411yescharacteristic
42C9×Q16—21no—
43C9×Q16—21no—
44C9×D8—21no—
45C9×D8D412yes—
46C2×C8×C9C2311yescharacteristic
47#3675—41no—
48#3674—41no—
49#3673—22no—
50#3672—22no—
51#3667—21no—
52#3667D411yescharacteristic
53D72D411yescharacteristic
54#3664—21no—
55#3663—21no—
56C8×D9—21no—
57Dic36D411yescharacteristic
58C9×C16—21no—
59C9⋊C16—21no—
60#2194—91noSylow
61#1004—61no—
62C2×C3×Q16C2×D311yescharacteristic
63C2×C3×D8C2×D311yescharacteristic
64#942—61no—
65#932—61no—
66#912—61no—
67#911—62no—
68D3×D8—61no—
69#904—31no—
70#901—31no—
71C3×SD32—22no—
72C2×C3×C16C2×D311yescharacteristic
73#836—61no—
74#829—61no—
75#828—61no—
76#819—31no—
77#812—31no—
78C23×C9—41no—
79C22×D9—42no—
80Q8×C9—41no—
81Q8×C9—41no—
82D4×C9—22no—
83D4×C9—41no—
84C2×C4×C9—41no—
85C2×C4×C9C2×D411yescharacteristic
86#611—21no—
87#611—42no—
88#611—42no—
89#611—41no—
90C2×Dic9—42no—
91D36—21no—
92D36D811yescharacteristic
93C4×D9—21no—
94C4×D9—41no—
95Dic18—21no—
96Dic18D811yescharacteristic
97C8×C9C2×D411yescharacteristic
98C8×C9C2×D411yescharacteristic
99C9⋊C8—21no—
100C9⋊C8—41no—
101#576—91no—
102#510—91no—
103C2×SD32D911yescharacteristic, complemented
104#473—91no—
105#361—91no—
106#359—91no—
107#350—91no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×#67—122no—
111#237—62no—
112#237—121no—
113D3×D4—62no—
114D3×D4—121no—
115#235—31no—
116#235—61no—
117C3×Q16—21no—
118C3×Q16—21no—
119C3×D8—21no—
120C3×D8#6712yes—
121C2×C3×C8C22×D311yescharacteristic
122#216—121no—
123#215—121no—
124#214—62no—
125#213—62no—
126#208—31no—
127#208—61no—
128Dic12—31no—
129D24—31no—
130#204—61no—
131ΓL(1,𝔽25)—61no—
132D3×C8—61no—
133C3×C16—21no—
134C3⋊C16—61no—
135C22×C9—42no—
136C22×C9—161no—
137C22×C9AGL(1,ℤ/8)11yescharacteristic
138C2×D9—21no—
139C2×D9—41no—
140C2×D9—162no—
141C4×C9—81no—
142C4×C9#12011yescharacteristic
143C4×C9C2×D811yescharacteristic
144Dic9—21no—
145Dic9—41no—
146H(5,𝔽2)—181no—
147#137—181no—
148AGL(1,ℤ/8)—181no—
149AGL(1,ℤ/8)—182no—
150#135—91no—
151#135—181no—
152C2×Q16C2×D911yescharacteristic, complemented
153C2×D8—181no—
154C2×D8C2×D911yescharacteristic
155#131—91no—
156Q32—181no—
157SD32—22no—
158SD32—181no—
159M32—91no—
160C2×C16C2×D911yescharacteristic, complemented
161#108—91no—
162#101—181no—
163C23×C3—41no—
164C22×D3—122no—
165C3×Q8—41no—
166C3×Q8—41no—
167C3×D4—22no—
168C3×D4—41no—
169C2×C3×C4—41no—
170C2×C3×C4D3×D411yescharacteristic
171#67—61no—
172#67—122no—
173#67—122no—
174#67—121no—
175C2×Dic3—122no—
176D12—31no—
177D12—61no—
178C4×D3—61no—
179C4×D3—121no—
180Dic6—31no—
181Dic6—61no—
182C3×C8D3×D411yescharacteristic
183C3×C8C2×#6711yescharacteristic
184C3⋊C8—61no—
185C3⋊C8—121no—
186C2×C9—21no—
187C2×C9—82no—
188C2×C9#44711yescharacteristic
189D9—41no—
190D9—81no—
191#41—91no—
192#41—182no—
193#41—181no—
194#41—361no—
195C2×Q8—21no—
196C2×D4—21no—
197C2×D4—182no—
198C2×D4—361no—
199C2×D4—362no—
200Q16—21no—
201Q16—21no—
202Q16—91no—
203Q16—361no—
204SD16—182no—
205SD16—181no—
206SD16—361no—
207D8—21no—
208D8—91no—
209D8—182no—
210D8#61112yes—
211M16—91no—
212M16—181no—
213M16—181no—
214C2×C8—181no—
215C2×C8C22×D911yescharacteristic
216C16—21no—
217C16—181no—
218C22×C3—42no—
219C22×C3—161no—
220C22×C3#91111yescharacteristic
221C2×D3—61no—
222C2×D3—121no—
223C2×D3—482no—
224C3×C4—81no—
225C3×C4D3×D811yescharacteristic
226C3×C4#93711yescharacteristic
227Dic3—61no—
228Dic3—121no—
229C9#219411yescharacteristic, complemented, Sylow
230C23—41no—
231C23—362no—
232Q8—41no—
233Q8—41no—
234Q8—91no—
235Q8—181no—
236D4—22no—
237D4—41no—
238D4—91no—
239D4—181no—
240D4—181no—
241D4—362no—
242D4—362no—
243D4—361no—
244C2×C4—41no—
245C2×C4—181no—
246C2×C4—362no—
247C2×C4—361no—
248C2×C4D4×D911yescharacteristic
249C8—181no—
250C8—361no—
251C8D4×D911yescharacteristic
252C8C2×#61111yescharacteristic
253C2×C3—21no—
254C2×C3—82no—
255C2×C3#515611yescharacteristic
256D3—121no—
257D3—241no—
258C22—42no—
259C22—161no—
260C22—181no—
261C22—361no—
262C22—1442no—
263C22#6347311yescharacteristic
264C4—81no—
265C4—181no—
266C4—361no—
267C4D8×D911yescharacteristic
268C4#6349911yescharacteristic
269C3#7149011yescharacteristic
270C2—21no—
271C2—82no—
272C2—361no—
273C2—721no—
274C2#1058883611yescharacteristic
subgroupG/H
centerC2#10588836
commutator, cosocleC2×C8×C9C23
FrattiniC2×C3×C8C22×D3
FittingC2×C9×SD32C2
socleC2×C3#5156
prime radical#10589859C2
hypercenterC2×SD32D9
hypocenterC9#2194
LCSG ▷ C2×C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. C2×C9×SD32C2semidirect
3. #10588872C2semidirect
4. #10588859C2semidirect
5. #10588240C2semidirect
6. #10588187C2semidirect
7. #10588185C2semidirect
10. C2×C9×Q16C22semidirect
20. C2×C9×C16C22semidirect
24. #63372C22semidirect
25. #63371C22semidirect
103. C2×SD32D9semidirect
152. C2×Q16C2×D9semidirect
160. C2×C16C2×D9semidirect
229. C9#2194semidirect

Maximal subgroups

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

subgroupindex
#105898592
C2×C9×SD322
#105888722
#105888592
#105882402
#105881872
#105881852
#714903