Finite Group Databasebeta

← #11898296

Subgroups — #11898296

Order 1152.

606 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 186 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#10589679C211yesmaximal, characteristic
2#10589672C211yesmaximal, characteristic, complemented
3C8×Q8×C9C211yesmaximal, characteristic, complemented
4#10588525C213yesmaximal, complemented
5#10588512C213yesmaximal, complemented
6#10588375C213yesmaximal, complemented
7C4×#3664C213yesmaximal, complemented
8#79911—31nomaximal
9C2×#3701—21no—
10C4×Q8×C9C2211yescharacteristic
11Q8×Dic9—21no—
12C2×#3664C2213yes—
13C2×#3664C2216yes—
14#63457—23no—
15#63452—23no—
16#63436C2213yes—
17C4×D36C2213yes—
18#63431C2213yes—
19C4×Dic18C2213yes—
20C9×C4⋊C8C2213yes—
21C4×C8×C9C2213yes—
22#63380C2213yes—
23#63377C2211yescharacteristic
24#63376C2213yes—
25#5577—31no—
26#5570—31no—
27C3×C8×Q8D311yescharacteristic
28#4848—33no—
29#4841—33no—
30#4693—33no—
31C4×#204—33no—
32C2×Q8×C9D411yescharacteristic
33#3701—44no—
34C2×D36—23no—
35C2×D36—23no—
36C2×D36C2313yes—
37C2×C4×D9—23no—
38C2×Dic18C2313yes—
39C2×C8×C9C2313yes—
40C2×C8×C9C2311yescharacteristic
41C9×C4⋊C4D413yes—
42C42×C9C2313yes—
43(C2×D9)⋊C4—23no—
44C4⋊Dic9—23no—
45C4⋊Dic9—23no—
46C4⋊Dic9C2311yescharacteristic
47C4⋊Dic9C2311yescharacteristic
48Dic9⋊C4—23no—
49C4×Dic9—23no—
50#3664—26no—
51#3664—46no—
52#3392—91noSylow
53C2×#239—61no—
54C3×C4×Q8C2×D311yescharacteristic
55Q8×Dic3—61no—
56C2×#204—33no—
57C2×#204—36no—
58#895—63no—
59#890—63no—
60#874—33no—
61C4×D12—33no—
62#869—33no—
63C4×Dic6—33no—
64C3×C4⋊C8C2×D313yes—
65C3×C4×C8C2×D313yes—
66#821—33no—
67#817—31no—
68#816—33no—
69C22×D9—23no—
70Q8×C9SD1614yescomplemented
71C2×C4×C9C2×D413yes—
72C2×C4×C9C2×D413yes—
73C2×C4×C9C2411yescharacteristic
74C2×Dic9—21no—
75C2×Dic9—23no—
76C2×Dic9—21no—
77D36—23no—
78D36—23no—
79D36—46no—
80D36—83no—
81C4×D9—46no—
82Dic18—23no—
83Dic18—23no—
84C8×C9—23no—
85C8×C9#4112yescomplemented
86#557—91no—
87#550—91no—
88#492—93no—
89#461—93no—
90#460—93no—
91C8×Q8D911yescharacteristic, complemented
92C4×SD16—93no—
93C2×C3×Q8D1211yescharacteristic
94#239—124no—
95C2×D12—33no—
96C2×D12—63no—
97C2×D12—63no—
98C2×C4×D3—63no—
99C2×Dic6—33no—
100C2×C3×C8C22×D313yes—
101C2×C3×C8C22×D311yescharacteristic
102C3×C4⋊C4D1213yes—
103C3×C42C22×D313yes—
104(C2×D3)⋊C4—63no—
105C4⋊Dic3—31no—
106C4⋊Dic3—31no—
107C4⋊Dic3—63no—
108C4⋊Dic3—63no—
109Dic3⋊C4—63no—
110C4×Dic3—63no—
111#204—66no—
112#204—126no—
113C22×C9C22×D411yescharacteristic
114C2×D9—43no—
115C2×D9—43no—
116C2×D9—43no—
117C4×C9—23no—
118C4×C9C2×SD1616yes—
119C4×C9C2×#4111yescharacteristic
120C4×C9H(5,𝔽2)11yescharacteristic
121Dic9—43no—
122Dic9—41no—
123Dic9—41no—
124C2×#41—181no—
125C2×SD16—93no—
126C2×SD16—96no—
127C4⋊Q8—93no—
128#127—93no—
129#125—183no—
130#121—183no—
131C4×Q8—93no—
132C4×Q8—181no—
133C4×Q8C2×D911yescharacteristic
134C4×D4—93no—
135#106—91no—
136C4⋊C8C2×D913yes—
137#103—93no—
138#102—93no—
139C4×C8C2×D913yes—
140C22×D3—63no—
141C3×Q8#20414yes—
142C2×C3×C4C2×D1213yes—
143C2×C3×C4C2×D1213yes—
144C2×C3×C4C23×D311yescharacteristic
145C2×Dic3—61no—
146C2×Dic3—63no—
147C2×Dic3—61no—
148D12—63no—
149D12—63no—
150D12—126no—
151D12—243no—
152C4×D3—126no—
153Dic6—63no—
154Dic6—63no—
155C3×C8—23no—
156C3×C8#23912yes—
157C2×C9#55011yescharacteristic
158C2×C9C22×SD1611yescharacteristic
159C2×C9#57711yescharacteristic
160D9—83no—
161#41—364no—
162C2×Q8—93no—
163C2×Q8D3611yescharacteristic
164C2×D4—93no—
165C2×D4—183no—
166C2×D4—183no—
167C22×C4—183no—
168SD16—186no—
169SD16—366no—
170C2×C8C22×D913yes—
171C2×C8C22×D911yescharacteristic
172C4⋊C4—91no—
173C4⋊C4—91no—
174C4⋊C4—183no—
175C4⋊C4—183no—
176C4⋊C4—183no—
177C4⋊C4D3613yes—
178C22⋊C4—183no—
179C42—183no—
180C42C22×D913yes—
181C22×C3C22×D1211yescharacteristic
182C2×D3—123no—
183C2×D3—123no—
184C2×D3—123no—
185C3×C4—23no—
186C3×C4C2×#20416yes—
187C3×C4C2×#23911yescharacteristic
188C3×C4#100911yescharacteristic
189Dic3—121no—
190Dic3—121no—
191Dic3—123no—
192C9#339211yescharacteristic, complemented, Sylow
193C23—183no—
194Q8—183no—
195Q8—183no—
196Q8#366414yescomplemented
197D4—183no—
198D4—183no—
199D4—366no—
200D4—723no—
201C2×C4—181no—
202C2×C4—183no—
203C2×C4—181no—
204C2×C4—366no—
205C2×C4C2×D3613yes—
206C2×C4C2×D3613yes—
207C2×C4C23×D911yescharacteristic
208C8—23no—
209C8#370112yescomplemented
210C2×C3#557711yescharacteristic
211C2×C3C22×#20411yescharacteristic
212C2×C3#575111yescharacteristic
213D3—243no—
214C22—363no—
215C22—363no—
216C22—363no—
217C22C22×D3611yescharacteristic
218C4—23no—
219C4—361no—
220C4—361no—
221C4—363no—
222C4C2×#366416yes—
223C4C2×#370111yescharacteristic
224C4#6371511yescharacteristic
225C3#7991111yescharacteristic
226C2—723no—
227C2#1058967911yescharacteristic
228C2C22×#366411yescharacteristic
229C2#1058985311yescharacteristic
subgroupG/H
centerC22C22×D36
commutator, cosocleC2×C4×C9C24
FrattiniC2×C3×C4C23×D3
FittingC8×Q8×C9C2
socleC22×C3C22×D12
prime radical#10589679C2
hypercenterC8×Q8D9
hypocenterC9#3392
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
2. #10589672C2semidirect
3. C8×Q8×C9C2semidirect
4. #10588525C2semidirect
5. #10588512C2semidirect
6. #10588375C2semidirect
7. C4×#3664C2semidirect
70. Q8×C9SD16semidirect
85. C8×C9#41semidirect
91. C8×Q8D9semidirect
192. C9#3392semidirect
196. Q8#3664semidirect
209. C8#3701semidirect

Maximal subgroups

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

subgroupindex
#105896792
#105896722
C8×Q8×C92
#105885252
#105885122
#105883752
C4×#36642
#799113