Finite Group Databasebeta

← #71205

Subgroups — #71205

Order 384.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#5727C211yesmaximal, characteristic, complemented
2C3×#486C211yesmaximal, characteristic, complemented
3#5075C211yesmaximal, characteristic, complemented
4#5072C211yesmaximal, characteristic
5#4547C211yesmaximal, characteristic, complemented
6#4459C211yesmaximal, characteristic, complemented
7#4456C211yesmaximal, characteristic, complemented
8#1627—31nomaximal, Sylow
9C2×D3×Q8—21no—
10C3×C4⋊Q8C2211yescharacteristic
11C3×#127C2211yescharacteristic
12#946—21no—
13C2×#214—21no—
14C2×#213—21no—
15#896—22no—
16#893—21no—
17#888—21no—
18C4×D12C2211yescharacteristic
19C4×Dic6C2211yescharacteristic
20C3×#102—22no—
21C3×C4×C8C2211yescharacteristic
22#832—21no—
23#832—21no—
24C4⋊(C3⋊C8)C2211yescharacteristic
25C4⋊(C3⋊C8)C2211yescharacteristic
26D4×Q8—31no—
27#486D311yescharacteristic, complemented
28#460—31no—
29#459—31no—
30#332—31no—
31Q8⋊C8—31no—
32D4⋊C8—31no—
33C2×C3×Q8—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37D3×Q8—42no—
38D3×Q8—42no—
39C2×D12D411yescharacteristic
40C2×C4×D3—22no—
41C2×C4×D3—21no—
42C2×Dic6—22no—
43C2×Dic6D411yescharacteristic
44C2×C3×C8—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4D412yes—
47C3×C42C2311yescharacteristic
48#214—42no—
49#213—42no—
50(C2×D3)⋊C4—22no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3D411yescharacteristic
53C4⋊Dic3D411yescharacteristic
54Dic3⋊C4—22no—
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C2×C3⋊C8—21no—
58C2×C3⋊C8—21no—
59C22×Q8—61no—
60C2×SD16—61no—
61C2×D8—61no—
62C4⋊Q8—61no—
63C4⋊Q8C2×D311yescharacteristic
64#127C2×D311yescharacteristic
65C22⋊Q8—61no—
66C22⋊Q8—62no—
67C4×Q8—31no—
68C4×D4—31no—
69C4×D4—61no—
70C4⋊C8—31no—
71C4⋊C8—31no—
72#102—22no—
73#102—61no—
74#102—61no—
75C4×C8C2×D311yescharacteristic
76C23×C3—41no—
77C22×D3—21no—
78C3×Q8—41no—
79C3×Q8—41no—
80C3×D4—42no—
81C3×D4—41no—
82C3×D4—41no—
83C3×D4—41no—
84C3×D4—41no—
85C2×C3×C4—22no—
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C2×Dic3—21no—
90C2×Dic3—21no—
91C2×Dic3—21no—
92D12—21no—
93D12SD1612yescomplemented
94C4×D3—44no—
95C4×D3—41no—
96C4×D3—41no—
97Dic6—21no—
98Dic6—44no—
99Dic6D812yescomplemented
100C3×C8—41no—
101C3⋊C8—41no—
102C3⋊C8—41no—
103C2×Q8—21no—
104C2×Q8—31no—
105C2×Q8—62no—
106C2×Q8—122no—
107C2×Q8—122no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—31no—
112C22×C4—62no—
113C22×C4—61no—
114SD16—122no—
115D8—122no—
116C2×C8—21no—
117C2×C8—61no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—31no—
121C4⋊C4—31no—
122C4⋊C4—62no—
123C4⋊C4—61no—
124C4⋊C4#6712yes—
125C22⋊C4—62no—
126C22⋊C4—61no—
127C42—61no—
128C42C22×D311yescharacteristic
129C22×C3—81no—
130C22×C3—81no—
131C22×C3—81no—
132C22×C3#12011yescharacteristic
133C2×D3—22no—
134C2×D3—41no—
135C2×D3—41no—
136C3×C4—21no—
137C3×C4—42no—
138C3×C4C2×D811yescharacteristic
139C3×C4C2×SD1611yescharacteristic
140C3×C4AGL(1,ℤ/8)11yescharacteristic
141C3×C4#13711yescharacteristic
142Dic3—22no—
143Dic3—41no—
144Dic3—41no—
145C23—41no—
146C23—61no—
147Q8—32no—
148Q8—41no—
149Q8—41no—
150Q8—61no—
151Q8—124no—
152D4—32no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—61no—
159C2×C4—22no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—124no—
164C2×C4—121no—
165C2×C4—121no—
166C2×C4D3×D411yescharacteristic
167C2×C4D3×D411yescharacteristic
168C2×C4C2×#6711yescharacteristic
169C8—41no—
170C8—121no—
171C8—121no—
172C2×C3—161no—
173C2×C3#44711yescharacteristic
174C2×C3#44811yescharacteristic
175C2×C3#45411yescharacteristic
176D3—42no—
177C22—62no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—121no—
182C22—121no—
183C22#93711yescharacteristic
184C4—21no—
185C4—42no—
186C4—62no—
187C4—121no—
188C4—121no—
189C4D3×D811yescharacteristic
190C4#91111yescharacteristic
191C4D3×SD1611yescharacteristic
192C4#91511yescharacteristic
193C3#162711yescharacteristic, complemented, Sylow
194C2—122no—
195C2—161no—
196C2#506211yescharacteristic
197C2#515611yescharacteristic
198C2#517011yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#486C2
socleC22×C3#120
prime radical#5072C2
hypercenter#486D3
hypocenterC3#1627
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5727C2semidirect
2. C3×#486C2semidirect
3. #5075C2semidirect
5. #4547C2semidirect
6. #4459C2semidirect
7. #4456C2semidirect
27. #486D3semidirect
93. D12SD16semidirect
99. Dic6D8semidirect
193. C3#1627semidirect

Maximal subgroups

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

subgroupindex
#57272
C3×#4862
#50752
#50722
#45472
#44592
#44562
#16273