Finite Group Databasebeta

← #71879

Subgroups — #71879

Order 384.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#537C211yesmaximal, characteristic, complemented
2C2×#949C212yesmaximal, complemented
3C2×#837C212yesmaximal, complemented
4#4988C211yesmaximal, characteristic, complemented
5#4530C211yesmaximal, characteristic, complemented
6#2006—31nomaximal, Sylow
7C2×C3×#41C2212yes—
8C22×D12C2211yescharacteristic
9C3×C22⋊Q8—22no—
10C3×#121—22no—
11C3×#121—22no—
12C2×C3×C4⋊C4C2211yescharacteristic
13#949—24no—
14#949—24no—
15C2×#215—22no—
16C2×#213—22no—
17C2×(C2×D3)⋊C4—22no—
18C2×C4×Dic3C2211yescharacteristic
19C2×#208C2212yes—
20#837—24no—
21C2×AGL(1,ℤ/8)—32no—
22#537D311yescharacteristic, complemented
23C2×#104—32no—
24#390—31no—
25#343—31no—
26C23×D3—21no—
27C3×#41—22no—
28C3×#41—22no—
29C3×#41—22no—
30C3×#41—22no—
31C2×C3×Q8D412yes—
32C2×C3×D4—22no—
33C2×C3×D4—22no—
34C2×C3×D4D412yes—
35C22×C3×C4—22no—
36C22×C3×C4D412yes—
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—21no—
39C2×D12—22no—
40C2×D12—22no—
41C2×D12—21no—
42C2×D12D412yescomplemented
43C3×C4⋊C4—22no—
44C3×C4⋊C4—22no—
45C3×C22⋊C4—24no—
46#215—28no—
47#213—28no—
48(C2×D3)⋊C4—44no—
49C4×Dic3—22no—
50#208—22no—
51#208D414yescomplemented
52C2×C3⋊C8—22no—
53C2×#41C2×D312yes—
54C22×D4—31no—
55AGL(1,ℤ/8)—64no—
56AGL(1,ℤ/8)—64no—
57C2×SD16—62no—
58C2×D8—62no—
59C2×M16—32no—
60C22⋊Q8—22no—
61#121—22no—
62#121—22no—
63C2×C4⋊C4C2×D311yescharacteristic
64C2×C22⋊C4—62no—
65C2×C42—31no—
66#104—64no—
67C23×C3—22no—
68C23×C3C2×D411yescharacteristic
69C22×D3—22no—
70C22×D3—42no—
71C22×D3—42no—
72C22×D3—41no—
73C22×D3—41no—
74C3×Q8—22no—
75C3×Q8—22no—
76C3×D4—22no—
77C3×D4—22no—
78C3×D4—42no—
79C3×D4—42no—
80C3×D4—82no—
81C2×C3×C4—22no—
82C2×C3×C4—22no—
83C2×C3×C4—42no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D412yes—
88C2×C3×C4C2×D412yes—
89C2×C3×C4#4111yescharacteristic
90C2×Dic3—22no—
91C2×Dic3—41no—
92C2×Dic3—41no—
93D12—24no—
94D12—42no—
95C3⋊C8—24no—
96C24—61no—
97#41—22no—
98#41—22no—
99#41—22no—
100#41—22no—
101C2×Q8#6712yes—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—32no—
105C2×D4—62no—
106C2×D4—61no—
107C2×D4—62no—
108C2×D4#6712yes—
109C22×C4—22no—
110C22×C4—61no—
111C22×C4#6712yes—
112C22×C4C22×D311yescharacteristic
113SD16—68no—
114D8—68no—
115M16—34no—
116M16—62no—
117C2×C8—62no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C22⋊C4—24no—
121C22⋊C4—124no—
122C42—62no—
123C22×C3—21no—
124C22×C3—21no—
125C22×C3—42no—
126C22×C3—42no—
127C22×C3—42no—
128C22×C3#12011yescharacteristic
129C22×C3#12111yescharacteristic
130C22×C3#12711yescharacteristic
131C2×D3—44no—
132C2×D3—42no—
133C2×D3—82no—
134C2×D3—81no—
135C2×D3—81no—
136C3×C4—42no—
137C3×C4—42no—
138C3×C4#12012yes—
139C3×C4#12112yes—
140Dic3—42no—
141C23—22no—
142C23—62no—
143C23—122no—
144C23—122no—
145C23—121no—
146C23—121no—
147C23C2×#6711yescharacteristic
148Q8—22no—
149Q8—22no—
150D4—22no—
151D4—22no—
152D4—42no—
153D4—42no—
154D4—64no—
155D4—82no—
156D4—122no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—62no—
163C2×C4—121no—
164C2×C4—121no—
165C2×C4D3×D411yescharacteristic
166C2×C4D3×D412yes—
167C2×C4#23711yescharacteristic
168C2×C4C2×#6712yes—
169C8—64no—
170C2×C3—21no—
171C2×C3—21no—
172C2×C3—82no—
173C2×C3#39211yescharacteristic
174C2×C3#45311yescharacteristic
175C2×C3#45411yescharacteristic
176D3—82no—
177C22—21no—
178C22—21no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—124no—
183C22—122no—
184C22—242no—
185C22—241no—
186C22—241no—
187C22#93811yescharacteristic
188C22#94011yescharacteristic
189C22#95311yescharacteristic
190C4—42no—
191C4—42no—
192C4—122no—
193C4#93712yes—
194C4#93912yes—
195C3#200611yescharacteristic, complemented, Sylow
196C2—21no—
197C2—21no—
198C2—82no—
199C2—242no—
200C2#522311yescharacteristic
201C2#524111yescharacteristic
202C2#524511yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#537C2
socleC22×C3#120
hypercenter#537D3
hypocenterC3#2006
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C3×#537C2semidirect
2. C2×#949C2semidirect
3. C2×#837C2semidirect
4. #4988C2semidirect
5. #4530C2semidirect
22. #537D3semidirect
42. C2×D12D4semidirect
51. #208D4semidirect
195. C3#2006semidirect

Maximal subgroups

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

subgroupindex
C3×#5372
C2×#9492
C2×#8372
#49882
#45302
#20063