Finite Group Databasebeta

← #11799792

Subgroups — #11799792

Order 1152.

282 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 45 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#10591166C211yesmaximal, characteristic
2C2×#63576C211yesmaximal, characteristic, complemented
3C22⋊#3708C211yesmaximal, characteristic, complemented
4#10590068C212yesmaximal, complemented
5#10590061C212yesmaximal, complemented
6C22×C3⋊(C3⋊C8)C2211yescharacteristic
7C3×#930—21no—
8C2×C3×C4⋊Dic3—21no—
9C2×#3713C2211yescharacteristic
10C2×#3708C2212yes—
11#63576—21no—
12#63576C2212yes—
13#63576C2211yescharacteristic
14#63562—21no—
15#5038—61no—
16C2×#807—61no—
17C22×C32×C4D411yescharacteristic
18C2×C3×#67—41no—
19C22×C3×Dic3—41no—
20C2×C3×D12—21no—
21C2×C3⋊(C3⋊C8)—22no—
22C2×C3⋊(C3⋊C8)D411yescharacteristic
23C2×C3⋊(C3⋊C8)C2311yescharacteristic
24C3×(C2×D3)⋊C4—41no—
25C3×C4⋊Dic3—22no—
26C3×C4⋊Dic3—21no—
27C3×C4⋊Dic3—41no—
28#3713—22no—
29#3708—22no—
30#2232—91nomaximal, Sylow
31C3×#121—61no—
32C2×C3×C4⋊C4—61no—
33C2×#213—61no—
34#930—21no—
35C2×C4⋊Dic3—21no—
36C22×C3⋊C8—61no—
37C22×C3⋊C8—61no—
38#832—61no—
39#809—61no—
40#807—62no—
41#807—61no—
42#807—62no—
43C23×C32D811yescharacteristic
44C22×C3×D3—41no—
45C2×C32×C4—21no—
46C2×C32×C4—21no—
47C2×C32×C4D811yescharacteristic
48C2×C32×C4C2×D411yescharacteristic
49C3×#67—82no—
50C2×C3×Dic3—42no—
51C2×C3×Dic3—41no—
52C2×C3×Dic3—82no—
53C3×D12—41no—
54C3×D12—41no—
55C3⋊(C3⋊C8)—21no—
56C3⋊(C3⋊C8)#4112yes—
57#466—91no—
58C2×#107—91no—
59#366—92no—
60#357—92no—
61C22⋊C16—91no—
62C2×C3×D4—61no—
63C2×C3×D4—121no—
64C22×C3×C4—21no—
65C22×C3×C4—21no—
66C22×C3×C4—121no—
67C2×#67—41no—
68C22×Dic3—41no—
69C2×D12—21no—
70C3×C4⋊C4—62no—
71C3×C4⋊C4—61no—
72C3×C4⋊C4—121no—
73C3×C22⋊C4—121no—
74#213—122no—
75(C2×D3)⋊C4—41no—
76C4⋊Dic3—22no—
77C4⋊Dic3—21no—
78C4⋊Dic3—41no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—61no—
81C2×C3⋊C8—61no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8—64no—
84C2×C3⋊C8—122no—
85C22×C32—21no—
86C22×C32—21no—
87C22×C32D1612yes—
88C22×C32C2×D811yescharacteristic
89C2×C3×D3—81no—
90C2×C3×D3—81no—
91C2×C3×D3—81no—
92C32×C4—21no—
93C32×C4#12311yescharacteristic
94C32×C4#13711yescharacteristic
95C3×Dic3—82no—
96C3×Dic3—81no—
97C2×D8—91no—
98C22×C8—91no—
99#121—61no—
100C2×C4⋊C4—61no—
101C2×C16—92no—
102#107—92no—
103#107—91no—
104#107—181no—
105#102—181no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—121no—
109C22×D3—41no—
110C3×D4—121no—
111C3×D4—121no—
112C3×D4—242no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—121no—
122C2×C3×C4—122no—
123C2×C3×C4—242no—
124#67—82no—
125C2×Dic3—42no—
126C2×Dic3—41no—
127C2×Dic3—82no—
128D12—41no—
129D12—41no—
130C3⋊C8—62no—
131C3⋊C8—62no—
132C3⋊C8—62no—
133C3⋊C8—121no—
134C2×C32—22no—
135C2×C32#48011yescharacteristic
136C2×C32C2×D1611yescharacteristic
137C2×C32#51011yescharacteristic
138C3×D3—161no—
139C2×D4—61no—
140C2×D4—121no—
141C22×C4—121no—
142C22×C4GO+(4,𝔽2)11yescharacteristic
143D8—182no—
144C2×C8—91no—
145C2×C8—91no—
146C2×C8—182no—
147C4⋊C4—61no—
148C4⋊C4—62no—
149C4⋊C4—121no—
150C22⋊C4—121no—
151C16—182no—
152C22×C3—22no—
153C22×C3—22no—
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—22no—
157C22×C3—22no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—241no—
161C22×C3—241no—
162C22×C3—241no—
163C2×D3—81no—
164C2×D3—81no—
165C2×D3—81no—
166C3×C4—22no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—41no—
172C3×C4—242no—
173C3×C4—241no—
174Dic3—82no—
175Dic3—81no—
176C32#223211yescharacteristic, complemented, Sylow
177C23—121no—
178C23#377411yescharacteristic
179D4—121no—
180D4—121no—
181D4—242no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—121no—
185C2×C4—122no—
186C2×C4—242no—
187C2×C4#377411yescharacteristic
188C2×C4C2×GO+(4,𝔽2)11yescharacteristic
189C8—92no—
190C8—181no—
191C2×C3—24no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—42no—
199C2×C3—481no—
200D3—161no—
201C22—21no—
202C22—21no—
203C22—241no—
204C22—241no—
205C22—241no—
206C22#6373412yes—
207C22C2×#377411yescharacteristic
208C4—21no—
209C4—242no—
210C4—241no—
211C4#6423311yescharacteristic
212C4#6423911yescharacteristic
213C3—21no—
214C3—21no—
215C2—22no—
216C2—481no—
217C2C2×#6373411yescharacteristic
218C2#1059333611yescharacteristic
219C2#1059349211yescharacteristic
subgroupG/H
centerC22C2×#3774
commutator, cosocleC2×C3⋊(C3⋊C8)C23
FrattiniC2×C4C2×GO+(4,𝔽2)
FittingC22×C32×C4D4
socleC22×C32C2×D8
prime radical#10591166C2
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32#2232
LCSG ▷ C2×C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#63576C2semidirect
3. C22⋊#3708C2semidirect
4. #10590068C2semidirect
5. #10590061C2semidirect
176. C32#2232semidirect

Maximal subgroups

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

subgroupindex
#105911662
C2×#635762
C22⋊#37082
#105900682
#105900612
#22329