Finite Group Databasebeta

← #11779262

Subgroups — #11779262

Order 1152.

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

normal subgroups only

This group has 648 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#10596055C211yesmaximal, characteristic, complemented
2#10592726C211yesmaximal, characteristic, complemented
3#10592542C211yesmaximal, characteristic
4#10592393C211yesmaximal, characteristic, complemented
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589348C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic, complemented
8#68245—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10#64155—21no—
11C2×#3755—21no—
12C2×#3745—21no—
13#64107—22no—
14#64101—21no—
15#64084—21no—
16#64083C2211yescharacteristic
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18#64078C2211yescharacteristic
19C4×#634—21no—
20C32×C4⋊C8C2211yescharacteristic
21C32×C4×C8C2211yescharacteristic
22#63648—21no—
23#63648—22no—
24#63636—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5584—34no—
27#5027—34no—
28#4843—34no—
29#4694—34no—
30C3×Q8⋊C8D314yescomplemented
31#4482—34no—
32#4459—34no—
33C2×C32×Q8D411yescharacteristic
34C2×C4⋊(C3⋊D3)—21no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)—21no—
37C2×C4⋊(C3⋊D3)D411yescharacteristic
38C2×C4×C3⋊D3—21no—
39C2×#634—21no—
40C2×C32×C8—21no—
41C2×C32×C8—21no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4D411yescharacteristic
44C32×C42—21no—
45C32×C42C2311yescharacteristic
46#3755—42no—
47#3752—22no—
48#3752—22no—
49#3752—21no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)D411yescharacteristic
52C4⋊(C3⋊Dic3)D412yes—
53#3750—22no—
54C4×C3⋊Dic3—22no—
55C2×C3⋊(C3⋊C8)—21no—
56#3745—42no—
57#1628—91noSylow
58C3×C4×Q8C2×D314yes—
59#946—64no—
60C2×#215—64no—
61C2×D24—64no—
62#898—68no—
63#892—64no—
64#875—64no—
65#874—34no—
66C4×D12—34no—
67#869—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—21no—
76C22×C3⋊D3—41no—
77C32×Q8—21no—
78C32×Q8—21no—
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—
86C4⋊(C3⋊D3)—21no—
87C4⋊(C3⋊D3)—21no—
88C4⋊(C3⋊D3)—42no—
89C4⋊(C3⋊D3)—41no—
90C4⋊(C3⋊D3)—41no—
91C4⋊(C3⋊D3)—41no—
92C4⋊(C3⋊D3)—41no—
93C4×C3⋊D3—41no—
94C4×C3⋊D3—41no—
95#634—41no—
96#634—41no—
97C32×C8—41no—
98C32×C8—41no—
99C3⋊(C3⋊C8)—41no—
100#556—91no—
101#486—91no—
102#460—91no—
103#459—91no—
104#332—91no—
105Q8⋊C8C3⋊D311yescharacteristic, complemented
106D4⋊C8—91no—
107C2×C3×Q8D1214yes—
108C2×D12—34no—
109C2×D12—64no—
110C2×D12—64no—
111C2×D12—64no—
112C2×C4×D3—64no—
113C2×Dic6—64no—
114C2×C3×C8—24no—
115C2×C3×C8—24no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4D1214yes—
118C3×C42—24no—
119C3×C42C22×D314yes—
120#215—128no—
121(C2×D3)⋊C4—68no—
122(C2×D3)⋊C4—68no—
123(C2×D3)⋊C4—64no—
124C4⋊Dic3—34no—
125C4⋊Dic3—38no—
126C4⋊Dic3—64no—
127Dic3⋊C4—68no—
128C4×Dic3—68no—
129C2×C3⋊C8—64no—
130D24—128no—
131C22×C32#12011yescharacteristic
132C2×C3⋊D3—41no—
133C2×C3⋊D3—41no—
134C2×C3⋊D3—41no—
135C2×C3⋊D3—81no—
136C2×C3⋊D3—81no—
137C2×C3⋊D3—81no—
138C32×C4—21no—
139C32×C4—41no—
140C32×C4—41no—
141C32×C4#13511yescharacteristic
142C32×C4#13511yescharacteristic
143C32×C4AGL(1,ℤ/8)11yescharacteristic
144C32×C4AGL(1,ℤ/8)11yescharacteristic
145C3⋊Dic3—42no—
146C3⋊Dic3—41no—
147C2×SD16—181no—
148C2×D8—181no—
149C4⋊Q8—91no—
150#127—91no—
151#126—182no—
152#124—181no—
153C22⋊Q8—181no—
154C4×Q8—181no—
155C4×Q8C2×C3⋊D311yescharacteristic
156C4×D4—91no—
157#117—181no—
158C4⋊C8—91no—
159C4⋊C8C2×C3⋊D311yescharacteristic
160#102—181no—
161#102—182no—
162#102—181no—
163C4×C8C2×C3⋊D311yescharacteristic
164C22×D3—64no—
165C22×D3—124no—
166C3×Q8—24no—
167C3×Q8—24no—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4C2×D1214yes—
171C2×C3×C4D3×D414yes—
172C2×C3×C4D3×D414yes—
173C2×Dic3—64no—
174C2×Dic3—68no—
175D12—64no—
176D12—64no—
177D12—128no—
178D12—124no—
179D12—124no—
180D12—124no—
181D12—124no—
182C4×D3—124no—
183C4×D3—124no—
184Dic6—124no—
185Dic6—124no—
186C3×C8—44no—
187C3×C8—44no—
188C3⋊C8—124no—
189C2×C32#44911yescharacteristic
190C2×C32#44911yescharacteristic
191C2×C32#45411yescharacteristic
192C3⋊D3—81no—
193C3⋊D3—161no—
194C2×Q8—181no—
195C2×Q8C4⋊(C3⋊D3)11yescharacteristic
196C2×D4—91no—
197C2×D4—181no—
198C2×D4—181no—
199C2×D4—181no—
200C22×C4—181no—
201SD16—362no—
202D8—362no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—181no—
206C4⋊C4—21no—
207C4⋊C4—91no—
208C4⋊C4—92no—
209C4⋊C4—182no—
210C4⋊C4—181no—
211C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
212C22⋊C4—182no—
213C22⋊C4—182no—
214C22⋊C4—181no—
215C42—21no—
216C42—182no—
217C42C22×C3⋊D311yescharacteristic
218C22×C3#88214yes—
219C2×D3—124no—
220C2×D3—124no—
221C2×D3—124no—
222C2×D3—244no—
223C2×D3—244no—
224C2×D3—244no—
225C3×C4—24no—
226C3×C4—44no—
227C3×C4—44no—
228C3×C4#90414yes—
229C3×C4#90814yes—
230C3×C4#91414yes—
231C3×C4#91914yes—
232Dic3—124no—
233Dic3—128no—
234C32#162811yescharacteristic, complemented, Sylow
235C23—181no—
236C23—361no—
237Q8—21no—
238Q8—21no—
239Q8—361no—
240Q8—361no—
241D4—181no—
242D4—181no—
243D4—362no—
244D4—361no—
245D4—361no—
246D4—361no—
247D4—361no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—181no—
251C2×C4—182no—
252C2×C4—361no—
253C2×C4—361no—
254C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
255C2×C4D4×C3⋊D311yescharacteristic
256C2×C4D4×C3⋊D311yescharacteristic
257C8—41no—
258C8—41no—
259C8—361no—
260C2×C3#473414yes—
261C2×C3#481014yes—
262C2×C3#482714yes—
263D3—244no—
264D3—484no—
265C22—361no—
266C22—361no—
267C22—361no—
268C22—721no—
269C22—721no—
270C22—721no—
271C22#6409111yescharacteristic
272C4—21no—
273C4—41no—
274C4—41no—
275C4—361no—
276C4—362no—
277C4#6411311yescharacteristic
278C4#6411711yescharacteristic
279C4#6412311yescharacteristic
280C4#6412811yescharacteristic
281C3#6824514yescomplemented
282C2—721no—
283C2—1441no—
284C2#1059243311yescharacteristic
285C2#1059250911yescharacteristic
286C2#1059252611yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radical#10592542C2
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1628
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. #10596055C2semidirect
2. #10592726C2semidirect
4. #10592393C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589348C2semidirect
7. #10589325C2semidirect
30. C3×Q8⋊C8D3semidirect
105. Q8⋊C8C3⋊D3semidirect
234. C32#1628semidirect
281. C3#68245semidirect

Maximal subgroups

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

subgroupindex
#105960552
#105927262
#105925422
#105923932
C32×Q8⋊C82
#105893482
#105893252
#682453