Finite Group Databasebeta

← #72222654

Subgroups — #72222654

Order 2304.

448 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 116 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#11797889C211yesmaximal, characteristic
2C3×#69076C212yesmaximal, complemented
3#11772198C211yesmaximal, characteristic, complemented
4#11771667C212yesmaximal, complemented
5#11771325C211yesmaximal, characteristic, complemented
6#10707075—32nomaximal
7C3×#4901—22no—
8C3×#4700C2212yes—
9#10590629C411yescharacteristic
10C32×#346C2211yescharacteristic
11#10589327C2211yescharacteristic
12#10589321C2211yescharacteristic
13C3×#4508—22no—
14C3×#4463C2212yes—
15#10589144C411yescharacteristic
16#10589120C412yes—
17C3×#2171—32no—
18#70922—32no—
19#69076D312yescomplemented
20#67767—32no—
21#67569—32no—
22#67567—32no—
23#67512—32no—
24#67512—61no—
25C3×#904C2×C412yes—
26C3×#901—22no—
27C2×C32×C16D412yes—
28C32×C4×C8C2311yescharacteristic
29C2×C3⋊(C3⋊C16)C2×C411yescharacteristic
30C2×C3⋊(C3⋊C16)C2×C411yescharacteristic
31C4×C3⋊(C3⋊C8)—21no—
32C3×#819C2×C412yes—
33C3×#812—22no—
34C3×#805—22no—
35C3×#798—42no—
36C3×D3×C16—42no—
37#63550C811yescharacteristic, complemented
38#63547C812yescomplemented
39#63545C811yescharacteristic, complemented
40#12068—91noSylow
41C3×#504—62no—
42C3×#443—32no—
43#5194—32no—
44#4901—22no—
45#4700C2×D312yes—
46C3×#364—32no—
47C3×#350—62no—
48C3×#346—21no—
49C3×#346C2×D312yes—
50#4567—32no—
51#4508—22no—
52#4497—32no—
53#4493—32no—
54#4463C2×D312yes—
55#4461—32no—
56#4461—61no—
57#4455—32no—
58#4455—61no—
59C3×#235—22no—
60C2×C32×C8C22×C411yescharacteristic
61C2×C32×C8C2×D411yescharacteristic
62C32×C42#4111yescharacteristic
63C2×C3⋊(C3⋊C8)—41no—
64C3×#208—22no—
65C3×Dic12C2×C812yes—
66C3×D24C2×C812yes—
67C3×#204—22no—
68C3×ΓL(1,𝔽25)—42no—
69C3×D3×C8—42no—
70C32×C16—22no—
71C3⋊(C3⋊C16)—21no—
72C3⋊(C3⋊C16)C2×C811yescharacteristic
73C3⋊(C3⋊C16)C2×C811yescharacteristic
74C3×C3⋊C16—42no—
75#2179—91no—
76#2171—32no—
77#1369—91no—
78#1336—92no—
79#1313—91no—
80C3×#135—32no—
81C3×#131—62no—
82#904C4×D312yes—
83#901—22no—
84C3×M32—62no—
85C3×M32—122no—
86C2×C3×C16—22no—
87C2×C3×C16—122no—
88C2×C3×C16#6712yes—
89C2×C3×C16#6712yes—
90C3×#108—32no—
91C3×#104—62no—
92C3×C4×C8—21no—
93C3×C4×C8C22×D312yes—
94#829—32no—
95#828—32no—
96#827—32no—
97#826—32no—
98#819C4×D312yes—
99#812—22no—
100C2×C3⋊C16—32no—
101C2×C3⋊C16—32no—
102C2×C3⋊C16—61no—
103C2×C3⋊C16—61no—
104#805—22no—
105C4×C3⋊C8—62no—
106C4×C3⋊C8—62no—
107#798—42no—
108D3×C16—42no—
109C2×C32×C4—21no—
110C2×C32×C4C4×D411yescharacteristic
111C3×#67—42no—
112C3×D12—22no—
113C3×C4×D3—42no—
114C3×Dic6—22no—
115C32×C8C22×C811yescharacteristic
116C32×C8#13111yescharacteristic
117C32×C8AGL(1,ℤ/8)12yes—
118C3⋊(C3⋊C8)—81no—
119C3×C3⋊C8—42no—
120#508—91no—
121#504—62no—
122#443—32no—
123#368—91no—
124#364—32no—
125#359—92no—
126#350—62no—
127#346—91no—
128#346D3211yescharacteristic
129#322—91no—
130C3×#41—62no—
131#235—22no—
132C3×Q16—32no—
133C3×SD16—62no—
134C3×D8—32no—
135C3×M16—62no—
136C3×M16—122no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—122no—
140C2×C3×C8C2×C4×D312yes—
141C2×C3×C8C2×#6712yes—
142C3×C42—21no—
143C3×C42#23512yes—
144#208—22no—
145C2×C3⋊C8—122no—
146C2×C3⋊C8—122no—
147Dic12D3×C812yes—
148D24D3×C812yes—
149#204—22no—
150ΓL(1,𝔽25)—42no—
151D3×C8—42no—
152C3×C16—22no—
153C3×C16—22no—
154C3×C16—42no—
155C3×C16—122no—
156C3⋊C16—32no—
157C3⋊C16—32no—
158C3⋊C16—42no—
159C3⋊C16—61no—
160C3⋊C16—61no—
161C3⋊C16—62no—
162C3⋊C16—62no—
163C22×C32#44411yescharacteristic
164C2×C3×D3—42no—
165C32×C4—41no—
166C32×C4C8×D411yescharacteristic
167C32×C4#44211yescharacteristic
168C3×Dic3—42no—
169#135—32no—
170#131—62no—
171Q32—91no—
172SD32—92no—
173D16—91no—
174M32—62no—
175M32—122no—
176C2×C16—91no—
177C2×C16—91no—
178C2×C16—122no—
179C2×C16#62512yes—
180#108—32no—
181#104—62no—
182C4×C8—181no—
183C4×C8C2×D3211yescharacteristic
184C3×Q8—62no—
185C3×D4—62no—
186C3×D4—122no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—122no—
191C2×C3×C4C4×#6712yes—
192#67—42no—
193D12—22no—
194C4×D3—42no—
195Dic6—22no—
196C3×C8—21no—
197C3×C8—21no—
198C3×C8—22no—
199C3×C8—122no—
200C3×C8C2×D3×C812yes—
201C3×C8#90112yes—
202C3×C8#93212yes—
203C3×C8#94912yes—
204C3⋊C8—42no—
205C3⋊C8—242no—
206C3⋊C8—242no—
207C2×C32—21no—
208C2×C32#158011yescharacteristic
209C3×D3—82no—
210#41—62no—
211Q16—32no—
212SD16—62no—
213D8—32no—
214M16—62no—
215M16—122no—
216C2×C8—122no—
217C2×C8—361no—
218C2×C8C4×D3211yescharacteristic
219C2×C8C2×#62511yescharacteristic
220C42#379811yescharacteristic
221C16—22no—
222C16—91no—
223C16—91no—
224C16—122no—
225C16—181no—
226C22×C3—21no—
227C22×C3—122no—
228C22×C3#514512yes—
229C2×D3—42no—
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—42no—
233C3×C4—42no—
234C3×C4—122no—
235C3×C4#501412yes—
236C3×C4C8×#6712yes—
237Dic3—42no—
238C32#1206811yescharacteristic, complemented, Sylow
239Q8—62no—
240D4—62no—
241D4—122no—
242C2×C4—21no—
243C2×C4—122no—
244C2×C4C4×#62511yescharacteristic
245C8—122no—
246C8—721no—
247C8D32×C811yescharacteristic
248C8#6380311yescharacteristic
249C8#6382312yes—
250C2×C3—22no—
251C2×C3—22no—
252C2×C3—21no—
253C2×C3—242no—
254C2×C3#7084012yes—
255D3—82no—
256C22—122no—
257C22#1059092211yescharacteristic
258C4—41no—
259C4—122no—
260C4#1059034411yescharacteristic
261C4C8×#62511yescharacteristic
262C3—21no—
263C3#1070707512yescomplemented
264C2—21no—
265C2—242no—
266C2#1177735511yescharacteristic
subgroupG/H
centerC4#10590344
commutatorC32×C8C22×C8
FrattiniC4×C8C2×D32
FittingC32×#346C22
socleC2×C32#1580
prime radical#10590629C4
cosocleC32×C4×C8C23
hypercenter#346D32
hypocenterC32#12068
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ #346

Complements

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

NH ≅ G/Ntype
2. C3×#69076C2semidirect
3. #11772198C2semidirect
4. #11771667C2semidirect
5. #11771325C2semidirect
19. #69076D3semidirect
37. #63550C8semidirect
38. #63547C8semidirect
39. #63545C8semidirect
238. C32#12068semidirect
263. C3#10707075semidirect

Maximal subgroups

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

subgroupindex
#117978892
C3×#690762
#117721982
#117716672
#117713252
#107070753