Finite Group Databasebeta

← #11778093

Subgroups — #11778093

Order 1152.

930 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 129 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#10595981C211yesmaximal, characteristic, complemented
2C32×#394C211yesmaximal, characteristic, complemented
3#10592916C212yesmaximal, complemented
4#10589394C211yesmaximal, characteristic, complemented
5#10589317C212yesmaximal, complemented
6#69703—34nomaximal
7C22×C32×D4C2211yescharacteristic
8C2×#3828—21no—
9C2×C32×C4⋊C4C2211yescharacteristic
10C2×C32×C22⋊C4—21no—
11C2×C32×C22⋊C4—21no—
12C2×#3753—22no—
13#64139—21no—
14#64139C2212yes—
15#64106—22no—
16#64104—22no—
17C32×#95C2211yescharacteristic
18#63659—22no—
19#63657C2212yes—
20#5510—34no—
21C3×#394D314yescomplemented
22#5217—38no—
23#4528—34no—
24#4451—38no—
25C24×C32—21no—
26C2×C32×D4—22no—
27C2×C32×D4—42no—
28C22×C32×C4—21no—
29C22×C32×C4D411yescharacteristic
30C22×C32×C4D411yescharacteristic
31C22×C32×C4C2311yescharacteristic
32C2×#638—22no—
33#3828—42no—
34#3828—42no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)—21no—
37C2×C4⋊(C3⋊D3)D412yes—
38C2×C4×C3⋊D3—22no—
39C2×#634—21no—
40C32×C4⋊C4—22no—
41C32×C22⋊C4—42no—
42C32×C22⋊C4—42no—
43#3753—44no—
44#3752—22no—
45#3752—22no—
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)D412yes—
48C2×C3⋊(C3⋊C8)—22no—
49#1598—91noSylow
50C22×C3×D4C2×D314yes—
51C2×#235—64no—
52C2×C3×C4⋊C4C2×D314yes—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—24no—
55C2×#213—68no—
56#930—38no—
57#930—64no—
58#897—68no—
59#895—68no—
60C3×#95C2×D314yes—
61#832—68no—
62#830—38no—
63C23×C32—22no—
64C23×C32—41no—
65C23×C32—41no—
66C23×C32—41no—
67C23×C32—41no—
68C23×C32—41no—
69C23×C32—41no—
70C23×C32C2×D411yescharacteristic
71C22×C3⋊D3—22no—
72C32×D4—24no—
73C32×D4—42no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4—41no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4—41no—
82C2×C32×C4D812yescomplemented
83C2×C32×C4C2×D412yes—
84#638—42no—
85#638—42no—
86C2×C3⋊Dic3—22no—
87C4⋊(C3⋊D3)—42no—
88C4⋊(C3⋊D3)—42no—
89C4⋊(C3⋊D3)—81no—
90C4×C3⋊D3—44no—
91#634—42no—
92C3⋊(C3⋊C8)—42no—
93#537—91no—
94#447—92no—
95#394C3⋊D311yescharacteristic, complemented
96#327—92no—
97#324—91no—
98C24×C3—24no—
99C2×C3×D4—28no—
100C2×C3×D4—48no—
101C22×C3×C4—24no—
102C22×C3×C4#6714yes—
103C22×C3×C4#6714yes—
104C22×C3×C4C22×D314yes—
105C2×#67—68no—
106#235—128no—
107#235—128no—
108C2×D12—38no—
109C2×D12—64no—
110C2×D12—64no—
111C2×C4×D3—68no—
112C2×Dic6—64no—
113C3×C4⋊C4—28no—
114C3×C22⋊C4—48no—
115C3×C22⋊C4—48no—
116#213—1216no—
117(C2×D3)⋊C4—68no—
118(C2×D3)⋊C4—68no—
119C4⋊Dic3—38no—
120C4⋊Dic3—64no—
121C2×C3⋊C8—68no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—42no—
125C22×C32—42no—
126C22×C32—42no—
127C22×C32—81no—
128C22×C32—81no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32#12011yescharacteristic
132C22×C32C2×D811yescharacteristic
133C22×C32AGL(1,ℤ/8)11yescharacteristic
134C2×C3⋊D3—42no—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—42no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—41no—
140C32×C4—81no—
141C3⋊Dic3—42no—
142C2×#41—181no—
143C22×D4C2×C3⋊D311yescharacteristic
144C2×D8—182no—
145C22⋊Q8—182no—
146#121—92no—
147#121—182no—
148#121—181no—
149C2×C4⋊C4C2×C3⋊D311yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152#102—182no—
153C22⋊C8—92no—
154#95C2×C3⋊D311yescharacteristic
155C23×C3—28no—
156C23×C3—44no—
157C23×C3—44no—
158C23×C3—44no—
159C23×C3—44no—
160C23×C3—44no—
161C23×C3—44no—
162C23×C3C2×#6714yes—
163C22×D3—68no—
164C3×D4—216no—
165C3×D4—48no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—24no—
169C2×C3×C4—44no—
170C2×C3×C4—44no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4—44no—
174C2×C3×C4#21318yescomplemented
175C2×C3×C4D3×D418yes—
176#67—128no—
177#67—128no—
178C2×Dic3—68no—
179D12—128no—
180D12—128no—
181D12—244no—
182C4×D3—1216no—
183Dic6—128no—
184C3⋊C8—128no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—82no—
188C2×C32#44711yescharacteristic
189C2×C32#45411yescharacteristic
190C2×C32B(4,𝔽2)11yescharacteristic
191C3⋊D3—82no—
192C24—21no—
193#41—362no—
194#41—362no—
195C2×Q8—181no—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—92no—
199C2×D4—181no—
200C2×D4—181no—
201C2×D4—182no—
202C22×C4—21no—
203C22×C4—182no—
204C22×C4#63811yescharacteristic
205C22×C4#63811yescharacteristic
206C22×C4C22×C3⋊D311yescharacteristic
207D8—364no—
208C2×C8—182no—
209C4⋊C4—22no—
210C4⋊C4—92no—
211C4⋊C4—181no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—182no—
215C22⋊C4—182no—
216C22×C3—24no—
217C22×C3—24no—
218C22×C3—48no—
219C22×C3—48no—
220C22×C3—48no—
221C22×C3—84no—
222C22×C3—84no—
223C22×C3—84no—
224C22×C3—84no—
225C22×C3C2×#21314yes—
226C22×C3#93714yes—
227C22×C3#94914yes—
228C2×D3—128no—
229C2×D3—128no—
230C2×D3—128no—
231C3×C4—28no—
232C3×C4—28no—
233C3×C4—44no—
234C3×C4—84no—
235Dic3—128no—
236C32#159811yescharacteristic, complemented, Sylow
237C23—22no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—182no—
245C23C2×#63811yescharacteristic
246Q8—362no—
247D4—24no—
248D4—42no—
249D4—362no—
250D4—362no—
251D4—362no—
252D4—362no—
253D4—721no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—182no—
263C2×C4—364no—
264C2×C4#375312yescomplemented
265C2×C4D4×C3⋊D312yes—
266C8—362no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—88no—
270C2×C3#503214yes—
271C2×C3#503614yes—
272C2×C3#506214yes—
273D3—248no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—362no—
284C22—362no—
285C22—362no—
286C22C2×#375311yescharacteristic
287C22#6414611yescharacteristic
288C22#6415811yescharacteristic
289C4—22no—
290C4—22no—
291C4—41no—
292C4—81no—
293C4—362no—
294C3#6970314yescomplemented
295C2—21no—
296C2—21no—
297C2—82no—
298C2—722no—
299C2#1059273111yescharacteristic
300C2#1059273511yescharacteristic
301C2#1059276111yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#394C2
socleC22×C32#120
hypercenter#394C3⋊D3
hypocenterC32#1598
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #10595981C2semidirect
2. C32×#394C2semidirect
3. #10592916C2semidirect
4. #10589394C2semidirect
5. #10589317C2semidirect
21. C3×#394D3semidirect
82. C2×C32×C4D8semidirect
95. #394C3⋊D3semidirect
174. C2×C3×C4#213semidirect
236. C32#1598semidirect
264. C2×C4#3753semidirect
294. C3#69703semidirect

Maximal subgroups

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

subgroupindex
#105959812
C32×#3942
#105929162
#105893942
#105893172
#697033