Finite Group Databasebeta

← #11780287

Subgroups — #11780287

Order 1152.

702 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 143 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
1Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2#10592719C211yesmaximal, characteristic, complemented
3#10592540C211yesmaximal, characteristic
4#10592379C211yesmaximal, characteristic, complemented
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589352C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic
8#68251—34nomaximal
9C2×Q8×C3⋊D3—21no—
10C32×C4×Q8C2211yescharacteristic
11C2×#3744—21no—
12#64106—21no—
13#64106—22no—
14C4×C4⋊(C3⋊D3)—21no—
15C4×C4⋊(C3⋊D3)C2211yescharacteristic
16#64078—21no—
17#64078C2211yescharacteristic
18#64078C2211yescharacteristic
19C32×C4⋊C8C2211yescharacteristic
20C32×C4×C8C2211yescharacteristic
21#63662—21no—
22#63644—22no—
23#63643—21no—
24#63635—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26Q8×D12—34no—
27#5020—34no—
28#4841—34no—
29#4680—34no—
30C3×Q8⋊C8D314yescomplemented
31#4486—34no—
32#4459—34no—
33C2×C32×Q8D411yescharacteristic
34Q8×C3⋊D3—42no—
35Q8×C3⋊D3—42no—
36C2×C4⋊(C3⋊D3)D411yescharacteristic
37C2×C4×C3⋊D3—22no—
38C2×C4×C3⋊D3—21no—
39C2×#634—22no—
40C2×#634—21no—
41C2×#634—21no—
42C2×C32×C8—21no—
43C2×C32×C8—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4D411yescharacteristic
46C32×C42—21no—
47C32×C42C2311yescharacteristic
48#3752—22no—
49#3752—21no—
50C4⋊(C3⋊Dic3)—22no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)D411yescharacteristic
55C4⋊(C3⋊Dic3)D412yes—
56C2×C3⋊(C3⋊C8)—21no—
57#3744—42no—
58#1651—91noSylow
59C2×D3×Q8—64no—
60C3×C4×Q8C2×D314yes—
61C2×#204—64no—
62#897—68no—
63#897—64no—
64C4×D12—34no—
65C4×D12—64no—
66#869—34no—
67#869—34no—
68#869—64no—
69C3×C4⋊C8C2×D314yes—
70C3×C4×C8C2×D314yes—
71#835—64no—
72#817—68no—
73#816—64no—
74#808—64no—
75C4⋊(C3⋊C8)—34no—
76C22×C3⋊D3—21no—
77C32×Q8—21no—
78C32×Q8SD1612yescomplemented
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4C2×D411yescharacteristic
82C2×C32×C4C2×D411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C3⋊Dic3—21no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—41no—
87C4⋊(C3⋊D3)—21no—
88C4⋊(C3⋊D3)SD1612yes—
89C4×C3⋊D3—44no—
90C4×C3⋊D3—41no—
91C4×C3⋊D3—41no—
92#634—44no—
93#634—41no—
94#634—41no—
95#634—41no—
96#634—41no—
97C32×C8—22no—
98C32×C8—41no—
99C3⋊(C3⋊C8)—41no—
100D4×Q8—91no—
101#498—91no—
102#475—91no—
103#461—91no—
104#332—91no—
105Q8⋊C8C3⋊D311yescharacteristic, complemented
106D4⋊C8—91no—
107C2×C3×Q8D1214yes—
108D3×Q8—128no—
109D3×Q8—128no—
110C2×D12—34no—
111C2×C4×D3—68no—
112C2×C4×D3—64no—
113C2×Dic6—68no—
114C2×Dic6—64no—
115C2×Dic6—64no—
116C2×C3×C8—24no—
117C2×C3×C8—24no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4D1214yes—
120C3×C42—24no—
121C3×C42C22×D314yes—
122(C2×D3)⋊C4—68no—
123(C2×D3)⋊C4—64no—
124C4⋊Dic3—34no—
125C4⋊Dic3—38no—
126C4⋊Dic3—68no—
127C4⋊Dic3—64no—
128C4⋊Dic3—64no—
129C4⋊Dic3—64no—
130C2×C3⋊C8—64no—
131#204—128no—
132C22×C32#12011yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3—41no—
135C2×C3⋊D3—41no—
136C32×C4—22no—
137C32×C4—21no—
138C32×C4—41no—
139C32×C4C2×SD1611yescharacteristic
140C32×C4C2×SD1611yescharacteristic
141C32×C4AGL(1,ℤ/8)11yescharacteristic
142C32×C4#13711yescharacteristic
143C3⋊Dic3—41no—
144C3⋊Dic3—42no—
145C3⋊Dic3—81no—
146C22×Q8—181no—
147C2×SD16—181no—
148C4⋊Q8—91no—
149C4⋊Q8—91no—
150C4⋊Q8—181no—
151C22⋊Q8—182no—
152C22⋊Q8—181no—
153C4×Q8C2×C3⋊D311yescharacteristic
154C4×D4—91no—
155C4×D4—181no—
156#106—182no—
157#106—181no—
158C4⋊C8—91no—
159C4⋊C8C2×C3⋊D311yescharacteristic
160#103—181no—
161#103—181no—
162C4×C8C2×C3⋊D311yescharacteristic
163C22×D3—64no—
164C3×Q8—24no—
165C3×Q8#20418yescomplemented
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4C2×D1214yes—
169C2×C3×C4D3×D414yes—
170C2×C3×C4D3×D414yes—
171C2×Dic3—64no—
172C2×Dic3—68no—
173C2×Dic3—124no—
174D12—38no—
175D12—64no—
176C4×D3—1216no—
177C4×D3—124no—
178C4×D3—124no—
179Dic6—1216no—
180Dic6—124no—
181Dic6—124no—
182Dic6—124no—
183Dic6—124no—
184C3×C8—28no—
185C3×C8—44no—
186C3⋊C8—124no—
187C2×C32#44811yescharacteristic
188C2×C32#45011yescharacteristic
189C2×C32#45611yescharacteristic
190C3⋊D3—42no—
191C2×Q8—182no—
192C2×Q8—181no—
193C2×Q8—181no—
194C2×Q8—362no—
195C2×Q8—362no—
196C2×Q8C4⋊(C3⋊D3)11yescharacteristic
197C2×D4—91no—
198C22×C4—182no—
199C22×C4—181no—
200SD16—362no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—181no—
204C4⋊C4—21no—
205C4⋊C4—91no—
206C4⋊C4—92no—
207C4⋊C4—182no—
208C4⋊C4—181no—
209C4⋊C4—181no—
210C4⋊C4—181no—
211C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
212C22⋊C4—182no—
213C22⋊C4—181no—
214C42—21no—
215C42C22×C3⋊D311yescharacteristic
216C22×C3#88214yes—
217C2×D3—68no—
218C2×D3—124no—
219C2×D3—124no—
220C3×C4—28no—
221C3×C4—24no—
222C3×C4—44no—
223C3×C4C2×#20414yes—
224C3×C4#90814yes—
225C3×C4D3×SD1614yes—
226C3×C4#91814yes—
227Dic3—124no—
228Dic3—128no—
229Dic3—244no—
230C32#165111yescharacteristic, complemented, Sylow
231C23—181no—
232Q8—21no—
233Q8—364no—
234Q8—361no—
235Q8—361no—
236Q8—361no—
237Q8—361no—
238Q8#374412yescomplemented
239D4—92no—
240D4—181no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—182no—
244C2×C4—181no—
245C2×C4—364no—
246C2×C4—361no—
247C2×C4—361no—
248C2×C4—361no—
249C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
250C2×C4D4×C3⋊D311yescharacteristic
251C2×C4D4×C3⋊D311yescharacteristic
252C8—22no—
253C8—41no—
254C8—361no—
255C2×C3#473114yes—
256C2×C3#480614yes—
257C2×C3#482614yes—
258D3—128no—
259C22—182no—
260C22—361no—
261C22—361no—
262C22#6409111yescharacteristic
263C4—21no—
264C4—22no—
265C4—41no—
266C4—362no—
267C4—361no—
268C4—721no—
269C4C2×#374411yescharacteristic
270C4#6411711yescharacteristic
271C4SD16×C3⋊D311yescharacteristic
272C4#6412711yescharacteristic
273C3#6825114yescomplemented
274C2—362no—
275C2#1059243011yescharacteristic
276C2#1059250511yescharacteristic
277C2#1059252511yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1651
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
4. #10592379C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589352C2semidirect
30. C3×Q8⋊C8D3semidirect
78. C32×Q8SD16semidirect
105. Q8⋊C8C3⋊D3semidirect
165. C3×Q8#204semidirect
230. C32#1651semidirect
238. Q8#3744semidirect
273. C3#68251semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊(C3⋊D3)2
#105927192
#105925402
#105923792
C32×Q8⋊C82
#105893522
#105893252
#682513