Finite Group Databasebeta

← #11797987

Subgroups — #11797987

Order 1152.

272 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 43 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#10593334C211yesmaximal, characteristic
2#10590922C211yesmaximal, characteristic, complemented
3#10590741C211yesmaximal, characteristic, complemented
4#10590179C211yesmaximal, characteristic, complemented
5#10590067C211yesmaximal, characteristic, complemented
6#10590065C211yesmaximal, characteristic, complemented
7#10590064C211yesmaximal, characteristic, complemented
8C3×#901—21no—
9#63898—21no—
10#63822C2211yescharacteristic
11C2×#3713C2211yescharacteristic
12#63773C2211yescharacteristic
13#63773C2211yescharacteristic
14#63735C412yescomplemented
15#63734C412yescomplemented
16#63642C2211yescharacteristic
17C3×#821—21no—
18#63578C2211yescharacteristic
19#63574C2211yescharacteristic
20#63568—21no—
21#5145—61no—
22#4785—61no—
23C3×#235—21no—
24C2×C3×D12—21no—
25C2×D3×Dic3—41no—
26C2×C32×C8D411yescharacteristic
27C4×C3⋊Dic3D411yescharacteristic
28C2×C3⋊(C3⋊C8)C2311yescharacteristic
29C3×C4⋊Dic3—21no—
30C3×#208—21no—
31C3×ΓL(1,𝔽25)—41no—
32C3×D3×C8—41no—
33(C2×D3)⋊Dic3—41no—
34#3718C2×C411yescharacteristic, complemented
35#3715—21no—
36#3713—21no—
37#3713C2×C412yes—
38#3713C2×C411yescharacteristic, complemented
39#3708—21no—
40#3708C2×C412yescomplemented
41AGL(1,ℤ/16)—91nomaximal, Sylow
42C3×#131—61no—
43#950—61no—
44D4×Dic3—61no—
45C2×#213—61no—
46#901—21no—
47#893—61no—
48C3×#102—61no—
49#837—61no—
50#825—61no—
51#822—61no—
52#821—21no—
53#815—61no—
54#815—61no—
55#808—61no—
56C22×C3×D3—41no—
57C2×C32×C4C2×D411yescharacteristic
58C2×C3⋊Dic3—21no—
59C3×#67—41no—
60C2×C3×Dic3—41no—
61C3×D12—22no—
62C3×D12—21no—
63C3×D12—41no—
64C3×C4×D3—41no—
65C3×Dic6—21no—
66D3×Dic3—82no—
67C32×C8—21no—
68C3⋊(C3⋊C8)C22×C411yescharacteristic
69C3⋊(C3⋊C8)#4111yescharacteristic
70C3×C3⋊C8—41no—
71#509—91no—
72#444—91no—
73#442—91no—
74#365—91no—
75#361—91no—
76#360—91no—
77#347—91no—
78C3×#41—61no—
79C2×C3×D4—61no—
80C22×Dic3—121no—
81#235—21no—
82C2×D12—21no—
83C2×C4×D3—121no—
84C3×M16—61no—
85C3×M16—121no—
86C2×C3×C8—21no—
87C2×C3×C8—21no—
88C2×C3×C8—121no—
89C3×C4⋊C4—61no—
90#217—121no—
91#216—61no—
92#215—61no—
93#214—61no—
94#213—62no—
95#213—61no—
96#213—62no—
97(C2×D3)⋊C4—121no—
98C4⋊Dic3—21no—
99C4×Dic3—61no—
100C4×Dic3—61no—
101#208—21no—
102C2×C3⋊C8—61no—
103C2×C3⋊C8—61no—
104ΓL(1,𝔽25)—41no—
105D3×C8—41no—
106C22×C32AGL(1,ℤ/8)11yescharacteristic
107C2×C3×D3—41no—
108C2×C3×D3—42no—
109C2×C3×D3—81no—
110C2×C3×D3—81no—
111C32×C4C4×D411yescharacteristic
112C32×C4AGL(1,ℤ/8)11yescharacteristic
113C3⋊Dic3—41no—
114C3×Dic3—41no—
115C3×Dic3—81no—
116#135—91no—
117C2×D8—91no—
118#131—61no—
119C4×D4—181no—
120SD32—92no—
121D16—92no—
122M32—91no—
123M32—91no—
124#108—91no—
125#106—91no—
126#104—181no—
127#102—61no—
128#97—91no—
129C23×C3—121no—
130C22×D3—41no—
131C3×Q8—61no—
132C3×D4—62no—
133C3×D4—61no—
134C3×D4—121no—
135C3×D4—121no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—121no—
139C2×C3×C4—121no—
140#67—41no—
141C2×Dic3—41no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—121no—
145C2×Dic3—242no—
146D12—22no—
147D12—21no—
148D12—41no—
149C4×D3—41no—
150C4×D3—242no—
151Dic6—21no—
152C3×C8—21no—
153C3×C8—21no—
154C3×C8—41no—
155C3×C8—121no—
156C3⋊C8—41no—
157C3⋊C8—61no—
158C3⋊C8—61no—
159C3⋊C8—61no—
160C3⋊C8—61no—
161C2×C32—21no—
162C2×C32#44211yescharacteristic
163C3×D3—81no—
164C3×D3—82no—
165#41—61no—
166C2×D4—61no—
167C22×C4—361no—
168Q16—91no—
169SD16—181no—
170D8—91no—
171D8—92no—
172D8—181no—
173M16—61no—
174M16—121no—
175C2×C8—91no—
176C2×C8—121no—
177C2×C8GO+(4,𝔽2)11yescharacteristic
178C4⋊C4—61no—
179C22⋊C4—361no—
180C42—91no—
181C16—92no—
182C16—181no—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3—121no—
186C22×C3—122no—
187C22×C3—241no—
188C22×C3—241no—
189C2×D3—41no—
190C2×D3—42no—
191C2×D3—81no—
192C2×D3—81no—
193C3×C4—21no—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—121no—
198C3×C4—241no—
199Dic3—41no—
200Dic3—81no—
201Dic3—121no—
202Dic3—121no—
203Dic3—241no—
204C32AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
205C23—121no—
206Q8—61no—
207D4—62no—
208D4—61no—
209D4—121no—
210D4—121no—
211C2×C4—121no—
212C2×C4—121no—
213C2×C4—181no—
214C2×C4—722no—
215C2×C4C2×GO+(4,𝔽2)11yescharacteristic
216C8—21no—
217C8—91no—
218C8—91no—
219C8—121no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—41no—
225C2×C3—242no—
226C2×C3—241no—
227D3—82no—
228D3—81no—
229C22—121no—
230C22—122no—
231C22—241no—
232C22—241no—
233C22#6422411yescharacteristic
234C4—121no—
235C4—241no—
236C4—361no—
237C4C4×GO+(4,𝔽2)11yescharacteristic
238C4#6423611yescharacteristic
239C3—21no—
240C3—21no—
241C2—21no—
242C2—242no—
243C2—241no—
244C2#1059341911yescharacteristic
subgroupG/H
centerC2#10593419
commutatorC3⋊(C3⋊C8)C22×C4
FrattiniC2×C4C2×GO+(4,𝔽2)
FittingC2×C32×C8D4
socleC2×C32#442
prime radical#10593334C2
cosocleC2×C3⋊(C3⋊C8)C23
hypercenterC2×C8GO+(4,𝔽2)
hypocenterC32AGL(1,ℤ/16)
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #10590922C2semidirect
3. #10590741C2semidirect
4. #10590179C2semidirect
5. #10590067C2semidirect
6. #10590065C2semidirect
7. #10590064C2semidirect
14. #63735C4semidirect
15. #63734C4semidirect
34. #3718C2×C4semidirect
38. #3713C2×C4semidirect
40. #3708C2×C4semidirect
204. C32AGL(1,ℤ/16)semidirect

Maximal subgroups

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

subgroupindex
#105933342
#105909222
#105907412
#105901792
#105900672
#105900652
#105900642
AGL(1,ℤ/16)9