Finite Group Databasebeta

← #71886

Subgroups — #71886

Order 384.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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×#539C211yesmaximal, characteristic
2#5043C211yesmaximal, characteristic, complemented
3#5033C211yesmaximal, characteristic
4#4981C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4537C211yesmaximal, characteristic
7#4528C211yesmaximal, characteristic, complemented
8#1609—31nomaximal, Sylow
9C3×#126—21no—
10C3×#125—21no—
11C3×#123—21no—
12C3×#123—21no—
13C3×C22⋊Q8—21no—
14C3×C22⋊Q8C2211yescharacteristic
15C3×#121C2211yescharacteristic
16C3×C4×D4—21no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4—21no—
20#835—21no—
21#832—21no—
22C3⋊#95—21no—
23C3⋊#95C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26#808—21no—
27#807—21no—
28#539D311yescharacteristic
29#483—31no—
30#480—31no—
31#398—31no—
32#328—31no—
33#327—31no—
34#324—31no—
35C2×C3×Q8D411yescharacteristic
36C2×C3×D4—21no—
37C2×C3×D4D411yescharacteristic
38C22×C3×C4—21no—
39C22×C3×C4D411yescharacteristic
40C22×C3×C4D411yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×Dic3—21no—
43C22×Dic3—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4D411yescharacteristic
51C3×C4⋊C4D411yescharacteristic
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C42—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—21no—
60Dic3⋊C4—42no—
61C2×C3⋊C8—21no—
62C2×C3⋊C8—21no—
63#126—21no—
64#125—21no—
65#123—21no—
66#123—21no—
67C22⋊Q8—21no—
68C22⋊Q8C2×D311yescharacteristic
69#121C2×D311yescharacteristic
70C4×D4—21no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—61no—
73C2×C4⋊C4C2×D311yescharacteristic
74#107—61no—
75#106—61no—
76#103—61no—
77#102—61no—
78C22⋊C8—31no—
79C22⋊C8—31no—
80#95—31no—
81#95—61no—
82C23×C3—21no—
83C23×C3C2×D411yescharacteristic
84C3×Q8—41no—
85C3×D4—41no—
86C3×D4—41no—
87C3×D4—41no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—81no—
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D411yescharacteristic
100C2×Dic3—22no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—41no—
106C3⋊C8—41no—
107C3⋊C8—41no—
108C2×Q8#6711yescharacteristic
109C2×D4—21no—
110C2×D4#6711yescharacteristic
111C22×C4—21no—
112C22×C4—61no—
113C22×C4—61no—
114C22×C4#6711yescharacteristic
115C22×C4#6711yescharacteristic
116C22×C4C22×D311yescharacteristic
117C2×C8—61no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—122no—
128C4⋊C4#6711yescharacteristic
129C4⋊C4#6711yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C42—21no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—41no—
141C22×C3#12011yescharacteristic
142C22×C3#13511yescharacteristic
143C22×C3AGL(1,ℤ/8)11yescharacteristic
144C3×C4—21no—
145C3×C4—21no—
146C3×C4—41no—
147C3×C4—41no—
148C3×C4—41no—
149C3×C4—41no—
150C3×C4—41no—
151Dic3—42no—
152Dic3—81no—
153C23—21no—
154C23C2×#6711yescharacteristic
155Q8—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—62no—
169C2×C4—81no—
170C2×C4—121no—
171C2×C4—121no—
172C2×C4—121no—
173C2×C4—121no—
174C2×C4—121no—
175C2×C4C2×#6711yescharacteristic
176C2×C4C2×#6711yescharacteristic
177C8—121no—
178C8—121no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3—81no—
182C2×C3#44911yescharacteristic
183C2×C3#45611yescharacteristic
184C2×C3#45811yescharacteristic
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22#94911yescharacteristic
191C22#95011yescharacteristic
192C22#95311yescharacteristic
193C4—21no—
194C4—21no—
195C4—41no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—122no—
201C4—241no—
202C3#160911yescharacteristic, complemented, Sylow
203C2—21no—
204C2—21no—
205C2—81no—
206C2#523811yescharacteristic
207C2#524311yescharacteristic
208C2#524611yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#539C2
socleC22×C3#120
prime radicalC3×#121C22
hypercenter#539D3
hypocenterC3#1609
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
2. #5043C2semidirect
4. #4981C2semidirect
5. #4538C2semidirect
7. #4528C2semidirect
202. C3#1609semidirect

Maximal subgroups

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

subgroupindex
C3×#5392
#50432
#50332
#49812
#45382
#45372
#45282
#16093