Finite Group Databasebeta

← #70013

Subgroups — #70013

Order 384.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#5822C212yesmaximal, complemented
2#5748C211yesmaximal, characteristic, complemented
3C3×#408C211yesmaximal, characteristic, complemented
4#5110C212yesmaximal, complemented
5C3⋊#338C211yesmaximal, characteristic, complemented
6#2017—31nomaximal, Sylow
7C2×C3×#41C2211yescharacteristic
8C2×#235C2212yes—
9C2×C3×M16C2211yescharacteristic
10C22×C3×C8C2211yescharacteristic
11#951—22no—
12#950—24no—
13#949—22no—
14#942—22no—
15#932—22no—
16C2×#208C2212yes—
17#909—22no—
18#908—22no—
19#904—22no—
20C3×C22⋊C8—22no—
21(C2×D3)⋊C8—24no—
22#575—32no—
23#575—31no—
24#408—32no—
25#408D311yescharacteristic, complemented
26#338—31no—
27C3×#41—22no—
28C3×#41—22no—
29C2×C3×Q8D411yescharacteristic
30C2×C3×D4—21no—
31C2×C3×D4D411yescharacteristic
32C22×C3×C4—21no—
33C22×C3×C4C2311yescharacteristic
34C2×#67—22no—
35#235—22no—
36#235—22no—
37#235—22no—
38#235—22no—
39C2×D12D412yes—
40C2×C4×D3—22no—
41C2×Dic6D412yes—
42C3×M16—22no—
43C2×C3×C8—22no—
44C2×C3×C8—21no—
45C2×C3×C8D412yes—
46#216—24no—
47#215—24no—
48#214—24no—
49#213—24no—
50#208—22no—
51#208—22no—
52C2×C3⋊C8D414yes—
53Dic12—22no—
54D24—22no—
55#204—24no—
56C2×#41—32no—
57C2×#41C2×D311yescharacteristic
58#137—62no—
59#137—62no—
60#137—62no—
61AGL(1,ℤ/8)—62no—
62AGL(1,ℤ/8)—62no—
63AGL(1,ℤ/8)—62no—
64#135—64no—
65#135—62no—
66C2×M16—32no—
67C2×M16C2×D311yescharacteristic
68C22×C8C2×D311yescharacteristic
69C22⋊C8—22no—
70C22⋊C8—64no—
71C23×C3—21no—
72C23×C3#4111yescharacteristic
73C22×D3—22no—
74C3×Q8—22no—
75C3×D4—22no—
76C3×D4—42no—
77C2×C3×C4—21no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4C2×D412yes—
82C2×C3×C4C2×D412yes—
83C2×C3×C4C2×D411yescharacteristic
84#67—42no—
85#67—42no—
86C2×Dic3—22no—
87D12—22no—
88D12—22no—
89C4×D3—42no—
90C4×D3—42no—
91Dic6—22no—
92Dic6—22no—
93C3×C8—22no—
94C3×C8—41no—
95C3⋊C8—24no—
96#41—22no—
97#41—22no—
98#41—62no—
99#41—62no—
100#41—62no—
101#41—62no—
102C2×Q8—32no—
103C2×Q8D1211yescharacteristic
104C2×D4—21no—
105C2×D4—32no—
106C2×D4—62no—
107C2×D4D1211yescharacteristic
108C22×C4—21no—
109C22×C4—62no—
110C22×C4C22×D311yescharacteristic
111Q16—62no—
112Q16—64no—
113SD16—64no—
114SD16—64no—
115SD16—64no—
116D8—62no—
117D8—64no—
118M16—22no—
119M16—62no—
120M16—62no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—34no—
124C2×C8#6712yes—
125C22×C3—41no—
126C22×C3—42no—
127C22×C3—41no—
128C22×C3#12112yes—
129C22×C3#12111yescharacteristic
130C2×D3—44no—
131C2×D3—42no—
132C3×C4—41no—
133C3×C4#12012yes—
134C3×C4#12011yescharacteristic
135C3×C4#12711yescharacteristic
136Dic3—42no—
137C23—21no—
138C23—62no—
139C23#23511yescharacteristic
140Q8—22no—
141Q8—62no—
142Q8—62no—
143D4—22no—
144D4—42no—
145D4—62no—
146D4—62no—
147D4—122no—
148D4—122no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—62no—
153C2×C4—122no—
154C2×C4—122no—
155C2×C4C2×D1211yescharacteristic
156C2×C4D3×D412yes—
157C2×C4D3×D412yes—
158C2×C4C2×#6711yescharacteristic
159C8—22no—
160C8—41no—
161C8—64no—
162C2×C3—22no—
163C2×C3—21no—
164C2×C3—81no—
165C2×C3#39211yescharacteristic
166D3—82no—
167C22—41no—
168C22—42no—
169C22—41no—
170C22—124no—
171C22—122no—
172C22#88412yes—
173C22#93011yescharacteristic
174C4—41no—
175C4—122no—
176C4#88212yes—
177C4#93711yescharacteristic
178C4#94011yescharacteristic
179C3#201711yescharacteristic, complemented, Sylow
180C2—21no—
181C2—22no—
182C2—81no—
183C2—242no—
184C2#496011yescharacteristic
subgroupG/H
centerC4#940
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#408C2
socleC2×C3#392
hypercenter#408D3
hypocenterC3#2017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #5822C2semidirect
2. #5748C2semidirect
3. C3×#408C2semidirect
4. #5110C2semidirect
5. C3⋊#338C2semidirect
25. #408D3semidirect
179. C3#2017semidirect

Maximal subgroups

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

subgroupindex
#58222
#57482
C3×#4082
#51102
C3⋊#3382
#20173