Finite Group Databasebeta

← #11779861

Subgroups — #11779861

Order 1152.

750 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 141 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#10596048C211yesmaximal, characteristic
2#10592719C211yesmaximal, characteristic, complemented
3#10592540C211yesmaximal, characteristic, complemented
4#10592393C211yesmaximal, characteristic, complemented
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589351C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic, complemented
8#68248—34nomaximal
9C2×#3832—21no—
10C32×C4×Q8C2211yescharacteristic
11C2×#3744—21no—
12#64104—22no—
13#64104—21no—
14#64083—21no—
15#64083C2211yescharacteristic
16C4×C4⋊(C3⋊D3)—21no—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18#64078C2211yescharacteristic
19C32×C4⋊C8C2211yescharacteristic
20C32×C4×C8C2211yescharacteristic
21#63662—21no—
22#63648—22no—
23#63643—21no—
24#63635—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5577—34no—
27#5020—34no—
28#4841—34no—
29#4694—34no—
30C3×Q8⋊C8D314yescomplemented
31#4485—34no—
32#4459—34no—
33C2×C32×Q8D411yescharacteristic
34#3832—42no—
35#3832—42no—
36C2×C4⋊(C3⋊D3)—22no—
37C2×C4⋊(C3⋊D3)—21no—
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)D411yescharacteristic
41C2×C4⋊(C3⋊D3)D412yes—
42C2×C4×C3⋊D3—22no—
43C2×C4×C3⋊D3—21no—
44C2×#634—21no—
45C2×C32×C8—21no—
46C2×C32×C8—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4D411yescharacteristic
49C32×C42—21no—
50C32×C42C2311yescharacteristic
51#3752—22no—
52#3752—21no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)D411yescharacteristic
56C2×C3⋊(C3⋊C8)—21no—
57#3744—42no—
58#1642—91noSylow
59C2×#239—64no—
60C3×C4×Q8C2×D314yes—
61C2×#204—64no—
62#895—68no—
63#895—64no—
64#874—34no—
65#874—64no—
66C4×D12—34no—
67C4×D12—64no—
68#869—34no—
69C3×C4⋊C8C2×D314yes—
70C3×C4×C8C2×D314yes—
71#835—64no—
72#821—68no—
73#816—64no—
74#808—64no—
75C4⋊(C3⋊C8)—34no—
76C22×C3⋊D3—21no—
77C22×C3⋊D3—22no—
78C32×Q8—21no—
79C32×Q8SD1612yescomplemented
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4C2×D411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C3⋊Dic3—21no—
86C2×C3⋊Dic3—41no—
87C4⋊(C3⋊D3)—21no—
88C4⋊(C3⋊D3)—21no—
89C4⋊(C3⋊D3)—42no—
90C4⋊(C3⋊D3)—44no—
91C4⋊(C3⋊D3)—41no—
92C4⋊(C3⋊D3)—41no—
93C4⋊(C3⋊D3)—41no—
94C4⋊(C3⋊D3)—41no—
95C4⋊(C3⋊D3)—82no—
96C4×C3⋊D3—44no—
97C4×C3⋊D3—41no—
98C4×C3⋊D3—41no—
99#634—41no—
100#634—41no—
101C32×C8—41no—
102C32×C8—41no—
103C3⋊(C3⋊C8)—41no—
104#550—91no—
105#486—91no—
106#475—91no—
107#461—91no—
108#331—91no—
109Q8⋊C8C3⋊D311yescharacteristic, complemented
110D4⋊C8—91no—
111C2×C3×Q8D1214yes—
112#239—128no—
113#239—128no—
114C2×D12—34no—
115C2×D12—38no—
116C2×D12—68no—
117C2×D12—64no—
118C2×D12—64no—
119C2×D12—68no—
120C2×C4×D3—68no—
121C2×C4×D3—64no—
122C2×Dic6—64no—
123C2×C3×C8—24no—
124C2×C3×C8—24no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4D1214yes—
127C3×C42—24no—
128C3×C42C22×D314yes—
129(C2×D3)⋊C4—68no—
130(C2×D3)⋊C4—64no—
131C4⋊Dic3—34no—
132C4⋊Dic3—64no—
133C4⋊Dic3—64no—
134C2×C3⋊C8—64no—
135#204—128no—
136C22×C32#12011yescharacteristic
137C2×C3⋊D3—41no—
138C2×C3⋊D3—42no—
139C2×C3⋊D3—42no—
140C2×C3⋊D3—41no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—41no—
143C32×C4—22no—
144C32×C4—21no—
145C32×C4—41no—
146C32×C4C2×SD1611yescharacteristic
147C32×C4#13511yescharacteristic
148C32×C4AGL(1,ℤ/8)11yescharacteristic
149C32×C4AGL(1,ℤ/8)11yescharacteristic
150C3⋊Dic3—41no—
151C3⋊Dic3—81no—
152C2×#41—181no—
153C2×SD16—181no—
154C4⋊Q8—91no—
155#127—91no—
156#127—181no—
157#121—182no—
158#121—181no—
159C4×Q8C2×C3⋊D311yescharacteristic
160C4×D4—91no—
161C4×D4—181no—
162#106—181no—
163C4⋊C8—91no—
164C4⋊C8C2×C3⋊D311yescharacteristic
165#103—181no—
166#103—181no—
167#102—182no—
168C4×C8C2×C3⋊D311yescharacteristic
169C22×D3—64no—
170C22×D3—68no—
171C3×Q8—24no—
172C3×Q8#20418yescomplemented
173C2×C3×C4—24no—
174C2×C3×C4—24no—
175C2×C3×C4C2×D1214yes—
176C2×C3×C4D3×D414yes—
177C2×C3×C4D3×D414yes—
178C2×Dic3—64no—
179C2×Dic3—124no—
180D12—64no—
181D12—64no—
182D12—128no—
183D12—1216no—
184D12—124no—
185D12—124no—
186D12—124no—
187D12—124no—
188D12—248no—
189C4×D3—1216no—
190C4×D3—124no—
191C4×D3—124no—
192Dic6—124no—
193Dic6—124no—
194C3×C8—44no—
195C3×C8—44no—
196C3⋊C8—124no—
197C2×C32#44911yescharacteristic
198C2×C32#45011yescharacteristic
199C2×C32#45511yescharacteristic
200C3⋊D3—81no—
201C3⋊D3—82no—
202#41—362no—
203#41—362no—
204C2×Q8—181no—
205C2×Q8C4⋊(C3⋊D3)11yescharacteristic
206C2×D4—91no—
207C2×D4—92no—
208C2×D4—182no—
209C2×D4—182no—
210C2×D4—181no—
211C2×D4—181no—
212C22×C4—182no—
213C22×C4—181no—
214SD16—362no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—181no—
218C4⋊C4—21no—
219C4⋊C4—91no—
220C4⋊C4—181no—
221C4⋊C4—181no—
222C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
223C22⋊C4—182no—
224C22⋊C4—181no—
225C42—21no—
226C42C22×C3⋊D311yescharacteristic
227C22×C3#88214yes—
228C2×D3—128no—
229C2×D3—124no—
230C2×D3—124no—
231C2×D3—128no—
232C2×D3—128no—
233C2×D3—124no—
234C3×C4—28no—
235C3×C4—24no—
236C3×C4—44no—
237C3×C4C2×#20414yes—
238C3×C4#90814yes—
239C3×C4#91414yes—
240C3×C4#91914yes—
241Dic3—124no—
242Dic3—244no—
243C32#164211yescharacteristic, complemented, Sylow
244C23—182no—
245C23—181no—
246Q8—21no—
247Q8—361no—
248Q8—361no—
249Q8#374412yescomplemented
250D4—181no—
251D4—181no—
252D4—362no—
253D4—364no—
254D4—361no—
255D4—361no—
256D4—361no—
257D4—361no—
258D4—722no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—181no—
262C2×C4—364no—
263C2×C4—361no—
264C2×C4—361no—
265C2×C4—361no—
266C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
267C2×C4D4×C3⋊D311yescharacteristic
268C2×C4D4×C3⋊D311yescharacteristic
269C8—41no—
270C8—41no—
271C8—361no—
272C2×C3#473114yes—
273C2×C3#481014yes—
274C2×C3#482514yes—
275D3—244no—
276D3—248no—
277C22—362no—
278C22—361no—
279C22—362no—
280C22—361no—
281C22—361no—
282C22—362no—
283C22#6409111yescharacteristic
284C4—21no—
285C4—22no—
286C4—41no—
287C4—361no—
288C4—721no—
289C4C2×#374411yescharacteristic
290C4#6411711yescharacteristic
291C4#6412311yescharacteristic
292C4#6412811yescharacteristic
293C3#6824814yescomplemented
294C2—722no—
295C2—721no—
296C2#1059243011yescharacteristic
297C2#1059250911yescharacteristic
298C2#1059252411yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radical#10596048C2
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1642
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #10592719C2semidirect
3. #10592540C2semidirect
4. #10592393C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589351C2semidirect
7. #10589325C2semidirect
30. C3×Q8⋊C8D3semidirect
79. C32×Q8SD16semidirect
109. Q8⋊C8C3⋊D3semidirect
172. C3×Q8#204semidirect
243. C32#1642semidirect
249. Q8#3744semidirect
293. C3#68248semidirect

Maximal subgroups

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

subgroupindex
#105960482
#105927192
#105925402
#105923932
C32×Q8⋊C82
#105893512
#105893252
#682483