Finite Group Databasebeta

← #11779166

Subgroups — #11779166

Order 1152.

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

normal subgroups only

This group has 630 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#10596041C211yesmaximal, characteristic, complemented
2#10592726C211yesmaximal, characteristic
3#10592547C211yesmaximal, characteristic
4#10592393C211yesmaximal, characteristic
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589348C211yesmaximal, characteristic, complemented
7#10589322C211yesmaximal, characteristic, complemented
8#68252—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10Q8×C3⋊Dic3—21no—
11C2×#3755—21no—
12C2×#3744—21no—
13#64099—21no—
14#64099—22no—
15#64083C2211yescharacteristic
16#64078—21no—
17#64078C2211yescharacteristic
18C4×#634—21no—
19C4×#634C2211yescharacteristic
20C32×C4⋊C8C2211yescharacteristic
21C32×C4×C8C2211yescharacteristic
22#63648—21no—
23#63648—22no—
24#63636—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5570—34no—
27#5027—34no—
28#4848—34no—
29#4694—34no—
30C3×Q8⋊C8D314yescomplemented
31#4482—34no—
32#4456—34no—
33C2×C32×Q8D411yescharacteristic
34C2×C4⋊(C3⋊D3)—21no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)—21no—
37C2×#634—21no—
38C2×#634D411yescharacteristic
39C2×C32×C8—21no—
40C2×C32×C8—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4D411yescharacteristic
43C32×C42—21no—
44C32×C42C2311yescharacteristic
45#3755—42no—
46C4⋊(C3⋊Dic3)—22no—
47C4⋊(C3⋊Dic3)—22no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)D411yescharacteristic
50C4⋊(C3⋊Dic3)D412yes—
51#3750—22no—
52#3750—21no—
53C4×C3⋊Dic3—22no—
54C4×C3⋊Dic3—21no—
55C2×C3⋊(C3⋊C8)—21no—
56#3744—42no—
57#1626—91noSylow
58C3×C4×Q8C2×D314yes—
59Q8×Dic3—64no—
60C2×#215—64no—
61C2×#204—64no—
62#890—68no—
63#890—64no—
64#874—34no—
65#869—34no—
66#869—64no—
67C4×Dic6—34no—
68C4×Dic6—64no—
69C3×C4⋊C8C2×D314yes—
70C3×C4×C8C2×D314yes—
71#821—68no—
72#821—64no—
73#809—64no—
74C4⋊(C3⋊C8)—34no—
75C22×C3⋊D3—41no—
76C32×Q8—21no—
77C32×Q8SD1612yescomplemented
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4C2×D411yescharacteristic
81C2×C32×C4C2×D411yescharacteristic
82C2×C32×C4C2×D411yescharacteristic
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3—21no—
85C2×C3⋊Dic3—22no—
86C4⋊(C3⋊D3)—42no—
87C4⋊(C3⋊D3)—41no—
88C4⋊(C3⋊D3)—41no—
89C4⋊(C3⋊D3)—41no—
90C4⋊(C3⋊D3)—41no—
91#634—21no—
92#634—21no—
93#634—41no—
94#634—41no—
95C32×C8—41no—
96C32×C8—41no—
97C3⋊(C3⋊C8)—41no—
98#557—91no—
99#486—91no—
100#460—91no—
101#460—91no—
102#332—91no—
103Q8⋊C8—91no—
104Q8⋊C8C3⋊D311yescharacteristic, complemented
105C2×C3×Q8D1214yes—
106C2×D12—64no—
107C2×D12—64no—
108C2×D12—64no—
109C2×Dic6—34no—
110C2×Dic6—64no—
111C2×C3×C8—24no—
112C2×C3×C8—24no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4D1214yes—
115C3×C42—24no—
116C3×C42C22×D314yes—
117#215—128no—
118C4⋊Dic3—34no—
119C4⋊Dic3—38no—
120C4⋊Dic3—68no—
121C4⋊Dic3—68no—
122C4⋊Dic3—64no—
123Dic3⋊C4—68no—
124Dic3⋊C4—64no—
125C4×Dic3—68no—
126C4×Dic3—64no—
127C2×C3⋊C8—64no—
128#204—128no—
129C22×C32#12011yescharacteristic
130C2×C3⋊D3—81no—
131C2×C3⋊D3—81no—
132C2×C3⋊D3—81no—
133C32×C4—22no—
134C32×C4—21no—
135C32×C4—41no—
136C32×C4C2×SD1611yescharacteristic
137C32×C4#13511yescharacteristic
138C32×C4AGL(1,ℤ/8)11yescharacteristic
139C32×C4#13711yescharacteristic
140C3⋊Dic3—41no—
141C3⋊Dic3—42no—
142C3⋊Dic3—41no—
143C2×SD16—181no—
144C2×SD16—181no—
145C4⋊Q8—91no—
146C4⋊Q8—181no—
147#127—91no—
148#125—182no—
149#125—181no—
150C4×Q8—91no—
151C4×Q8—181no—
152C4×Q8—181no—
153C4×Q8C2×C3⋊D311yescharacteristic
154C4⋊C8—91no—
155C4⋊C8C2×C3⋊D311yescharacteristic
156#102—181no—
157#102—182no—
158#102—181no—
159C4×C8C2×C3⋊D311yescharacteristic
160C22×D3—124no—
161C3×Q8—24no—
162C3×Q8#20418yescomplemented
163C2×C3×C4—24no—
164C2×C3×C4—24no—
165C2×C3×C4C2×D1214yes—
166C2×C3×C4D3×D414yes—
167C2×C3×C4D3×D414yes—
168C2×Dic3—64no—
169C2×Dic3—68no—
170C2×Dic3—64no—
171D12—128no—
172D12—124no—
173D12—124no—
174D12—124no—
175D12—124no—
176Dic6—64no—
177Dic6—64no—
178Dic6—124no—
179Dic6—124no—
180C3×C8—44no—
181C3×C8—44no—
182C3⋊C8—124no—
183C2×C32#44811yescharacteristic
184C2×C32#44911yescharacteristic
185C2×C32#45411yescharacteristic
186C3⋊D3—161no—
187C2×Q8—91no—
188C2×Q8—181no—
189C2×Q8C4⋊(C3⋊D3)11yescharacteristic
190C2×D4—181no—
191C2×D4—181no—
192C2×D4—181no—
193SD16—362no—
194SD16—362no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—181no—
198C4⋊C4—21no—
199C4⋊C4—91no—
200C4⋊C4—92no—
201C4⋊C4—182no—
202C4⋊C4—182no—
203C4⋊C4—182no—
204C4⋊C4—181no—
205C4⋊C4—181no—
206C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
207C42—21no—
208C42—182no—
209C42—181no—
210C42C22×C3⋊D311yescharacteristic
211C22×C3#88214yes—
212C2×D3—244no—
213C2×D3—244no—
214C2×D3—244no—
215C3×C4—28no—
216C3×C4—24no—
217C3×C4—44no—
218C3×C4C2×#20414yes—
219C3×C4#90914yes—
220C3×C4#91414yes—
221C3×C4#91914yes—
222Dic3—124no—
223Dic3—128no—
224Dic3—124no—
225C32#162611yescharacteristic, complemented, Sylow
226C23—361no—
227Q8—21no—
228Q8—181no—
229Q8—181no—
230Q8—361no—
231Q8—361no—
232Q8#374412yescomplemented
233D4—362no—
234D4—361no—
235D4—361no—
236D4—361no—
237D4—361no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—181no—
241C2×C4—182no—
242C2×C4—181no—
243C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
244C2×C4D4×C3⋊D311yescharacteristic
245C2×C4D4×C3⋊D311yescharacteristic
246C8—41no—
247C8—41no—
248C8—361no—
249C2×C3#473814yes—
250C2×C3#481014yes—
251C2×C3#482714yes—
252D3—484no—
253C22—721no—
254C22—721no—
255C22—721no—
256C22#6409111yescharacteristic
257C4—21no—
258C4—22no—
259C4—41no—
260C4—361no—
261C4—361no—
262C4—362no—
263C4C2×#374411yescharacteristic
264C4#6411811yescharacteristic
265C4#6412311yescharacteristic
266C4#6412811yescharacteristic
267C3#6825214yescomplemented
268C2—1441no—
269C2#1059243711yescharacteristic
270C2#1059250911yescharacteristic
271C2#1059252611yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radical#64083C22
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1626
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
1. #10596041C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589348C2semidirect
7. #10589322C2semidirect
30. C3×Q8⋊C8D3semidirect
77. C32×Q8SD16semidirect
104. Q8⋊C8C3⋊D3semidirect
162. C3×Q8#204semidirect
225. C32#1626semidirect
232. Q8#3744semidirect
267. C3#68252semidirect

Maximal subgroups

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

subgroupindex
#105960412
#105927262
#105925472
#105923932
C32×Q8⋊C82
#105893482
#105893222
#682523