Finite Group Databasebeta

← #10694008

Subgroups — #10694008

Order 768.

726 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×C4×#127C211yesmaximal, characteristic
2C2×D4⋊(C3⋊C8)C214yesmaximal, complemented
3#69392C211yesmaximal, characteristic, complemented
4C4×#803C211yesmaximal, characteristic, complemented
5#9648—31nomaximal, Sylow
6C2×C3×#127C411yescharacteristic
7C2×C3×C4×D4—22no—
8C2×C3×C4×D4C2212yes—
9C3×#390—22no—
10C3×#390C412yes—
11C3×#384C411yescharacteristic
12C3×C43C2211yescharacteristic
13C2×C4⋊(C3⋊C8)C2212yes—
14C2×#803—22no—
15C2×C4×C3⋊C8C2212yes—
16D4⋊(C3⋊C8)—28no—
17#4524—22no—
18C4×#127D311yescharacteristic
19#1831—31no—
20C4×#97—31no—
21C2×D4⋊C8—34no—
22C22×C3×D4—22no—
23C22×C3×D4C2×C412yes—
24C23×C3×C4—22no—
25C3×#127—24no—
26C3×C4×D4—24no—
27C3×C4×D4—24no—
28C3×C4×D4—48no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4C2×C412yes—
31C2×C3×C22⋊C4—24no—
32C2×C3×C22⋊C4—22no—
33C2×C3×C42C2×C411yescharacteristic
34C2×C3×C42C2×C411yescharacteristic
35C2×C3×C42D412yes—
36C2×C3×C42D412yes—
37C2×C3×C42C2311yescharacteristic
38C22×C3⋊C8—22no—
39C22×C3⋊C8—22no—
40C4⋊(C3⋊C8)—24no—
41#803—28no—
42C4×C3⋊C8—22no—
43C4×C3⋊C8D414yescomplemented
44C2×#127Dic311yescharacteristic
45C2×C4×D4—22no—
46C2×C4×D4C2×D312yes—
47C2×C4⋊C8—32no—
48C2×#97—62no—
49C2×C4×C8—32no—
50#390—22no—
51#390Dic312yes—
52#384Dic311yescharacteristic
53C43C2×D311yescharacteristic
54#336—62no—
55D4⋊C8—68no—
56C24×C3—22no—
57C2×C3×D4—28no—
58C2×C3×D4—24no—
59C2×C3×D4—22no—
60C2×C3×D4—24no—
61C2×C3×D4—44no—
62C2×C3×D4—44no—
63C2×C3×D4M1614yes—
64C22×C3×C4—22no—
65C22×C3×C4—42no—
66C22×C3×C4—42no—
67C22×C3×C4—48no—
68C22×C3×C4C22⋊C412yes—
69C22×C3×C4C22⋊C412yes—
70C22×C3×C4C22×C411yescharacteristic
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D411yescharacteristic
73C3×C4⋊C4—24no—
74C3×C4⋊C4—44no—
75C3×C22⋊C4—48no—
76C3×C22⋊C4—44no—
77C3×C42—24no—
78C3×C42—22no—
79C3×C42—22no—
80C3×C42—24no—
81C3×C42C2×D412yes—
82C3×C42#4112yes—
83C2×C3⋊C8—28no—
84C2×C3⋊C8—24no—
85C2×C3⋊C8—42no—
86C2×C3⋊C8—44no—
87C22×D4—22no—
88C22×D4C2×Dic312yes—
89C23×C4—22no—
90C22×C8—62no—
91C22×C8—62no—
92#127—24no—
93C4×D4—24no—
94C4×D4—24no—
95C4×D4—48no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4C2×Dic312yes—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—22no—
100C2×C42C2×Dic311yescharacteristic
101C2×C42C2×Dic311yescharacteristic
102C2×C42#6712yes—
103C2×C42#6712yes—
104C2×C42C22×D311yescharacteristic
105C4⋊C8—64no—
106#97—68no—
107C4×C8—34no—
108C4×C8—62no—
109C23×C3—24no—
110C23×C3—42no—
111C23×C3—42no—
112C23×C3—44no—
113C23×C3—44no—
114C23×C3C2×C22⋊C411yescharacteristic
115C3×D4—28no—
116C3×D4—416no—
117C3×D4—44no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—44no—
126C2×C3×C4—84no—
127C2×C3×C4—84no—
128C2×C3×C4#10414yes—
129C2×C3×C4C4×D412yes—
130C2×C3×C4#12112yes—
131C2×C3×C4#12411yescharacteristic
132C2×C3×C4#12711yescharacteristic
133C2×C3×C4C2×M1612yes—
134C2×C3×C4#13112yes—
135C2×C3×C4AGL(1,ℤ/8)14yes—
136C3⋊C8—28no—
137C3⋊C8—82no—
138C24—22no—
139C2×D4—28no—
140C2×D4—22no—
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4#20814yes—
146C22×C4—22no—
147C22×C4—48no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4#21712yes—
151C22×C4#21712yes—
152C22×C4C22×Dic311yescharacteristic
153C22×C4C2×#6711yescharacteristic
154C22×C4C2×#6711yescharacteristic
155C2×C8—68no—
156C2×C8—64no—
157C2×C8—124no—
158C2×C8—122no—
159C4⋊C4—24no—
160C4⋊C4—44no—
161C22⋊C4—48no—
162C22⋊C4—44no—
163C42—24no—
164C42—22no—
165C42—24no—
166C42—22no—
167C42D3×D412yes—
168C42#23712yes—
169C22×C3—44no—
170C22×C3—48no—
171C22×C3—82no—
172C22×C3—82no—
173C22×C3—84no—
174C22×C3#39011yescharacteristic
175C22×C3#40711yescharacteristic
176C22×C3#40811yescharacteristic
177C22×C3#41812yes—
178C22×C3C2×#10412yes—
179C3×C4—22no—
180C3×C4—24no—
181C3×C4—42no—
182C3×C4—82no—
183C3×C4#43514yes—
184C3×C4#49012yes—
185C3×C4#49612yes—
186C23—24no—
187C23—44no—
188C23—42no—
189C23—42no—
190C23—44no—
191C23C2×#21711yescharacteristic
192D4—28no—
193D4—416no—
194D4—44no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—84no—
204C2×C4—84no—
205C2×C4#83714yes—
206C2×C4C2×#20812yes—
207C2×C4#93214yes—
208C2×C4D4×Dic312yes—
209C2×C4#93611yescharacteristic
210C2×C4#93812yes—
211C2×C4#94011yescharacteristic
212C2×C4#94812yes—
213C8—68no—
214C8—242no—
215C2×C3—84no—
216C2×C3#148412yes—
217C2×C3#149012yes—
218C2×C3#195511yescharacteristic
219C2×C3#196111yescharacteristic
220C2×C3#196211yescharacteristic
221C22—48no—
222C22—44no—
223C22—82no—
224C22—84no—
225C22—82no—
226C22#520711yescharacteristic
227C22#521512yes—
228C22#522111yescharacteristic
229C22#523411yescharacteristic
230C22C2×#83712yes—
231C4—22no—
232C4—24no—
233C4—42no—
234C4—82no—
235C4#501114yes—
236C4#507112yes—
237C4#507412yes—
238C3#964811yescharacteristic, complemented, Sylow
239C2—84no—
240C2#7159912yes—
241C2#7161511yescharacteristic
242C2#7176511yescharacteristic
243C2#7180112yes—
244C2#7181511yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#127C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#127C4
cosocleC2×C3×C42C23
hypercenterC4×#127D3
hypocenterC3#9648
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C2×D4⋊(C3⋊C8)C2semidirect
3. #69392C2semidirect
4. C4×#803C2semidirect
43. C4×C3⋊C8D4semidirect
238. C3#9648semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1272
C2×D4⋊(C3⋊C8)2
#693922
C4×#8032
#96483