Finite Group Databasebeta

← #71245

Subgroups — #71245

Order 384.

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

normal subgroups only

This group has 208 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#5684C211yesmaximal, characteristic
2C3×#489C211yesmaximal, characteristic, complemented
3#5086C211yesmaximal, characteristic, complemented
4#5081C211yesmaximal, characteristic, complemented
5#4540C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1661—31nomaximal, Sylow
9C3×C4⋊Q8C2211yescharacteristic
10C3×#124C2211yescharacteristic
11#946—21no—
12C2×#216—21no—
13C2×#215—21no—
14#939—21no—
15#886—21no—
16#884—21no—
17#883—21no—
18#878—21no—
19#875C2211yescharacteristic
20#870C2211yescharacteristic
21C3×#103—21no—
22C3×#102—21no—
23C3×#97C2211yescharacteristic
24#808—21no—
25#807—21no—
26#803C2211yescharacteristic
27#803C2211yescharacteristic
28#562—31no—
29#501—31no—
30#497—31no—
31#489D311yescharacteristic, complemented
32#330—31no—
33#329—31no—
34#329—31no—
35C2×C3×Q8—21no—
36C2×C3×Q8D411yescharacteristic
37C2×C3×D4D411yescharacteristic
38C2×#67—21no—
39C2×#67—21no—
40C22×Dic3—21no—
41C2×D12D411yescharacteristic
42C2×C4×D3—21no—
43C2×Dic6D411yescharacteristic
44C2×C3×C8—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4—21no—
49C3×C42C2311yescharacteristic
50#217—21no—
51#216—42no—
52#215—42no—
53(C2×D3)⋊C4—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56C4⋊Dic3—21no—
57C4⋊Dic3D411yescharacteristic
58C4⋊Dic3D411yescharacteristic
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64C2×Q16—61no—
65C2×SD16—61no—
66C4⋊Q8C2×D311yescharacteristic
67#125—31no—
68#124—31no—
69#124C2×D311yescharacteristic
70#123—61no—
71#123—61no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74#121—61no—
75#121—61no—
76#107—61no—
77#106—61no—
78#103—21no—
79#102—21no—
80#97—31no—
81#97—31no—
82#97C2×D311yescharacteristic
83C23×C3—21no—
84C22×D3—21no—
85C3×Q8—41no—
86C3×Q8—41no—
87C3×Q8—41no—
88C3×D4—41no—
89C2×C3×C4—21no—
90C2×C3×C4—41no—
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94#67—81no—
95#67—81no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—81no—
100D12—41no—
101C4×D3—81no—
102Dic6—41no—
103C3×C8—41no—
104C3⋊C8—22no—
105C3⋊C8—22no—
106C2×Q8—21no—
107C2×Q8—31no—
108C2×Q8#6711yescharacteristic
109C2×D4—31no—
110C2×D4—61no—
111C2×D4—61no—
112C2×D4#6711yescharacteristic
113C22×C4—61no—
114C22×C4—61no—
115Q16—122no—
116SD16—122no—
117C2×C8—21no—
118C2×C8—61no—
119C2×C8—61no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—31no—
123C4⋊C4—31no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C42C22×D311yescharacteristic
135C22×C3—41no—
136C22×C3—41no—
137C22×C3—41no—
138C22×C3#12011yescharacteristic
139C2×D3—41no—
140C2×D3—41no—
141C2×D3—41no—
142C3×C4—21no—
143C3×C4—21no—
144C3×C4—21no—
145C3×C4—41no—
146C3×C4—81no—
147Dic3—41no—
148Dic3—41no—
149Dic3—41no—
150C23—21no—
151C23—61no—
152Q8—41no—
153Q8—41no—
154Q8—41no—
155Q8—121no—
156D4—41no—
157D4—121no—
158D4—241no—
159D4—241no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—241no—
166C2×C4—241no—
167C2×C4D3×D411yescharacteristic
168C2×C4D3×D411yescharacteristic
169C2×C4C2×#6711yescharacteristic
170C8—41no—
171C8—62no—
172C8—62no—
173C2×C3—81no—
174C2×C3#45311yescharacteristic
175C2×C3#45511yescharacteristic
176C2×C3#45611yescharacteristic
177D3—81no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—121no—
182C22—121no—
183C22—121no—
184C22#93711yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—41no—
189C4—81no—
190C4—121no—
191C4—121no—
192C4—121no—
193C3#166111yescharacteristic, complemented, Sylow
194C2—81no—
195C2—241no—
196C2#506111yescharacteristic
197C2#519811yescharacteristic
198C2#520511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#489C2
socleC22×C3#120
prime radical#5684C2
hypercenter#489D3
hypocenterC3#1661
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. C3×#489C2semidirect
3. #5086C2semidirect
4. #5081C2semidirect
5. #4540C2semidirect
6. #4465C2semidirect
7. #4464C2semidirect
31. #489D3semidirect
193. C3#1661semidirect

Maximal subgroups

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

subgroupindex
#56842
C3×#4892
#50862
#50812
#45402
#44652
#44642
#16613