Finite Group Databasebeta

← #11873280

Subgroups — #11873280

Order 1152.

1074 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#519C211yesmaximal, characteristic
2#10595989C211yesmaximal, characteristic
3#10592927C212yesmaximal
4#10592730C212yesmaximal, complemented
5#10592729C212yesmaximal, complemented
6C4×#3756C212yesmaximal, complemented
7#10592721C212yesmaximal, complemented
8#10592699C211yesmaximal, characteristic
9#10592695C212yesmaximal
10#82555—34nomaximal
11C22×C32×Q8C2211yescharacteristic
12C32×C4×Q8—22no—
13C32×C4×Q8C2212yes—
14C32×C4×Q8C2212yes—
15C32×#117—22no—
16C32×#117C2211yescharacteristic
17C32×#117C2211yescharacteristic
18C2×#3756C2212yes—
19C2×#3756C2212yes—
20#64133C2212yes—
21#64098—22no—
22#64088—22no—
23#64079C2211yescharacteristic
24#64078C2211yescharacteristic
25C4×#634C2212yes—
26#63662C2212yes—
27#63662C2212yes—
28#63657C2212yes—
29#63637C2212yes—
30#63637C2212yes—
31#63635C2212yes—
32#63634C2212yes—
33C4⋊(C3⋊(C3⋊C8))C2212yes—
34#63630C2211yescharacteristic
35C4×C3⋊(C3⋊C8)C2211yescharacteristic
36C3×#519D314yes—
37#5518—34no—
38#5228—38no—
39#5031—38no—
40#5030—38no—
41C4×#216—38no—
42#5022—38no—
43#5000—34no—
44#4996—38no—
45C2×C32×Q8—24no—
46C2×C32×Q8—21no—
47C2×C32×Q8C2312yes—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×#634C2312yes—
51C32×C4⋊C4—22no—
52C32×C4⋊C4—22no—
53C32×C4⋊C4D412yes—
54C32×C4⋊C4C2312yes—
55C32×C22⋊C4—21no—
56C32×C22⋊C4D412yes—
57C32×C42—22no—
58C32×C42—22no—
59C32×C42—21no—
60C32×C42C2311yescharacteristic
61C32×C42C2311yescharacteristic
62#3757—22no—
63#3756—24no—
64#3756—24no—
65C4⋊(C3⋊Dic3)—22no—
66C4⋊(C3⋊Dic3)C2312yes—
67#3750—22no—
68#3750—22no—
69#3750—22no—
70C4×C3⋊Dic3—22no—
71C2×C3⋊(C3⋊C8)C2312yes—
72C2×C3⋊(C3⋊C8)C2312yes—
73#3127—91noSylow
74C22×C3×Q8C2×D314yes—
75C3×C4×Q8—28no—
76C3×C4×Q8C2×D318yes—
77C3×C4×Q8C2×D318yes—
78C3×#117—28no—
79C3×#117C2×D314yes—
80C3×#117C2×D314yes—
81C2×#216—38no—
82C2×#216—38no—
83#924—38no—
84#889—68no—
85#879—68no—
86#870—34no—
87#869—34no—
88C4×Dic6—38no—
89#835—38no—
90#835—38no—
91#830—38no—
92#810—38no—
93#810—38no—
94#808—38no—
95#807—38no—
96C4⋊(C3⋊C8)—38no—
97#803—34no—
98C4×C3⋊C8—34no—
99C23×C32C2×D411yescharacteristic
100C32×Q8—24no—
101C32×Q8—22no—
102C32×Q8#4114yes—
103C2×C32×C4—24no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4C2×D412yes—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D412yes—
110C2×C32×C4C2411yescharacteristic
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113#634—22no—
114#634—22no—
115C3⋊(C3⋊C8)—22no—
116C3⋊(C3⋊C8)—22no—
117#541—91no—
118#519C3⋊D311yescharacteristic
119#485—92no—
120#479—92no—
121#462—92no—
122#451—92no—
123#441—92no—
124C4×Q16—92no—
125#433—91no—
126C2×C3×Q8—216no—
127C2×C3×Q8—24no—
128C2×C3×Q8C22×D318yes—
129C22×C3×C4—24no—
130C22×C3×C4C22×D314yes—
131C2×Dic6—38no—
132C3×C4⋊C4—28no—
133C3×C4⋊C4—28no—
134C3×C4⋊C4#6718yes—
135C3×C4⋊C4C22×D318yes—
136C3×C22⋊C4—24no—
137C3×C22⋊C4#6718yes—
138C3×C42—28no—
139C3×C42—28no—
140C3×C42—24no—
141C3×C42C22×D314yes—
142C3×C42C22×D314yes—
143#217—68no—
144#216—616no—
145#216—616no—
146C4⋊Dic3—38no—
147C4⋊Dic3—68no—
148Dic3⋊C4—68no—
149Dic3⋊C4—68no—
150Dic3⋊C4—68no—
151C4×Dic3—68no—
152C2×C3⋊C8—38no—
153C2×C3⋊C8—38no—
154C22×C32—21no—
155C22×C32—22no—
156C22×C32C22×D411yescharacteristic
157C32×C4—24no—
158C32×C4—21no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—42no—
162C32×C4C2×#4112yes—
163C3⋊Dic3—42no—
164C3⋊Dic3—42no—
165C22×Q8C2×C3⋊D311yescharacteristic
166C2×Q16—92no—
167C2×Q16—92no—
168C4⋊Q8—91no—
169#126—182no—
170#125—91no—
171#125—182no—
172C22⋊Q8—92no—
173C4×Q8—22no—
174C4×Q8—92no—
175C4×Q8C2×C3⋊D312yes—
176C4×Q8C2×C3⋊D312yes—
177#117—22no—
178#117C2×C3⋊D311yescharacteristic
179#117C2×C3⋊D311yescharacteristic
180#107—92no—
181#106—92no—
182C4⋊C8—92no—
183#103—92no—
184#103—92no—
185#103—92no—
186#103—92no—
187C22⋊C8—92no—
188#97—91no—
189C4×C8—91no—
190C23×C3C2×#6714yes—
191C3×Q8—216no—
192C3×Q8—28no—
193C3×Q8#235116yes—
194C2×C3×C4—216no—
195C2×C3×C4—28no—
196C2×C3×C4—28no—
197C2×C3×C4—28no—
198C2×C3×C4C2×#6718yes—
199C2×C3×C4C2×#6714yes—
200C2×C3×C4C2×#6718yes—
201C2×C3×C4C23×D314yes—
202C2×Dic3—68no—
203C2×Dic3—68no—
204Dic6—68no—
205Dic6—68no—
206C3⋊C8—68no—
207C3⋊C8—68no—
208C2×C32—41no—
209C2×C32#52511yescharacteristic
210C2×C32#57812yes—
211C2×Q8—24no—
212C2×Q8—21no—
213C2×Q8—92no—
214C2×Q8C22×C3⋊D312yes—
215C22×C4—21no—
216C22×C4C22×C3⋊D311yescharacteristic
217Q16—184no—
218Q16—184no—
219C2×C8—92no—
220C2×C8—92no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—92no—
224C4⋊C4—182no—
225C4⋊C4—182no—
226C4⋊C4—182no—
227C4⋊C4—182no—
228C4⋊C4#63812yes—
229C4⋊C4C22×C3⋊D312yes—
230C22⋊C4—21no—
231C22⋊C4—182no—
232C22⋊C4#63812yes—
233C42—22no—
234C42—22no—
235C42—21no—
236C42—182no—
237C42C22×C3⋊D311yescharacteristic
238C42C22×C3⋊D311yescharacteristic
239C22×C3—24no—
240C22×C3—28no—
241C22×C3C22×#6714yes—
242C3×C4—216no—
243C3×C4—24no—
244C3×C4—28no—
245C3×C4—28no—
246C3×C4—48no—
247C3×C4C2×#23518yes—
248Dic3—128no—
249Dic3—128no—
250C32#312711yescharacteristic, complemented, Sylow
251C23C2×#63811yescharacteristic
252Q8—24no—
253Q8—22no—
254Q8—182no—
255Q8—182no—
256Q8#382814yes—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—182no—
262C2×C4—182no—
263C2×C4C2×#63812yes—
264C2×C4C2×#63812yes—
265C2×C4C2×#63811yescharacteristic
266C2×C4C23×C3⋊D311yescharacteristic
267C8—182no—
268C8—182no—
269C2×C3—44no—
270C2×C3#579114yes—
271C2×C3#583818yes—
272C22—21no—
273C22—22no—
274C22C22×#63811yescharacteristic
275C4—24no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—362no—
281C4—362no—
282C4C2×#382812yes—
283C3#8255514yescomplemented
284C2—41no—
285C2#1059626211yescharacteristic
286C2#1059630912yes—
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#519C2
socleC22×C32C22×D4
prime radicalC23×C32C2×D4
hypercenter#519C3⋊D3
hypocenterC32#3127
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
4. #10592730C2semidirect
5. #10592729C2semidirect
6. C4×#3756C2semidirect
7. #10592721C2semidirect
250. C32#3127semidirect
283. C3#82555semidirect

Maximal subgroups

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

subgroupindex
C32×#5192
#105959892
#105929272
#105927302
#105927292
C4×#37562
#105927212
#105926992
#105926952
#825553