Finite Group Databasebeta

← #69520

Subgroups — #69520

Order 384.

256 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 83 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×#380C211yesmaximal, characteristic
2#5212C211yesmaximal, characteristic
3C2×#830C211yesmaximal, characteristic
4C2×#808C211yesmaximal, characteristic, complemented
5#4529C211yesmaximal, characteristic, complemented
6#4529C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#2077—31nomaximal, Sylow
9C23×C3×C4C2211yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4—21no—
13C2×#217—21no—
14C2×C4⋊Dic3C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C22×C3⋊C8C2211yescharacteristic
17C22×C3⋊C8C2211yescharacteristic
18C3×#95—21no—
19C3⋊#95—21no—
20#830—22no—
21#808—22no—
22C2×#106—31no—
23C2×C22⋊C8—31no—
24#380—31no—
25#380D311yescharacteristic
26#340—31no—
27#340—31no—
28#340—31no—
29C24×C3D411yescharacteristic
30C22×C3×C4—21no—
31C22×C3×C4—21no—
32C22×C3×C4—21no—
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4D411yescharacteristic
39C22×C3×C4C2311yescharacteristic
40C22×Dic3—21no—
41C22×Dic3—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46C3×C22⋊C4—42no—
47#217—42no—
48C4⋊Dic3—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3—21no—
52C2×C3⋊C8—21no—
53C2×C3⋊C8—21no—
54C2×C3⋊C8—21no—
55C2×C3⋊C8—21no—
56C2×C3⋊C8—21no—
57C2×C3⋊C8D412yes—
58C23×C4C2×D311yescharacteristic
59C22×C8—31no—
60C22×C8—31no—
61C2×C4⋊C4—31no—
62C2×C4⋊C4—31no—
63C2×C4⋊C4C2×D311yescharacteristic
64C2×C4⋊C4C2×D311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—61no—
67#106—62no—
68C22⋊C8—62no—
69#95—21no—
70#95—61no—
71C23×C3—21no—
72C23×C3—21no—
73C23×C3—21no—
74C23×C3—21no—
75C23×C3—21no—
76C23×C3—21no—
77C23×C3SD1612yes—
78C23×C3C2×D411yescharacteristic
79C2×C3×C4—21no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4#4111yescharacteristic
94C2×C3×C4#4111yescharacteristic
95C2×C3×C4#4111yescharacteristic
96C2×C3×C4#4111yescharacteristic
97C2×C3×C4#4111yescharacteristic
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C3⋊C8—22no—
105C3⋊C8—41no—
106C24#6711yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—61no—
116C22×C4—61no—
117C22×C4#6711yescharacteristic
118C22×C4C22×D311yescharacteristic
119C2×C8—32no—
120C2×C8—61no—
121C2×C8—61no—
122C2×C8—61no—
123C2×C8—61no—
124C2×C8—61no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C22⋊C4—42no—
134C22⋊C4—122no—
135C22×C3—22no—
136C22×C3—22no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3#12111yescharacteristic
143C22×C3#12311yescharacteristic
144C22×C3#12311yescharacteristic
145C22×C3C2×SD1611yescharacteristic
146C22×C3#13511yescharacteristic
147C22×C3AGL(1,ℤ/8)11yescharacteristic
148C22×C3#13711yescharacteristic
149C3×C4—41no—
150C3×C4—81no—
151C3×C4—81no—
152C3×C4#12311yescharacteristic
153C3×C4#12411yescharacteristic
154C3×C4#12611yescharacteristic
155C3×C4#12611yescharacteristic
156Dic3—81no—
157Dic3—81no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23#21412yes—
165C23C2×#6711yescharacteristic
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—121no—
180C2×C4—121no—
181C2×C4—121no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—121no—
185C2×C4#23511yescharacteristic
186C2×C4#23511yescharacteristic
187C2×C4D3×D411yescharacteristic
188C2×C4#23711yescharacteristic
189C2×C4#23711yescharacteristic
190C2×C4#23711yescharacteristic
191C8—62no—
192C8—121no—
193C2×C3—42no—
194C2×C3#39711yescharacteristic
195C2×C3#46511yescharacteristic
196C2×C3#46811yescharacteristic
197C2×C3#48111yescharacteristic
198C2×C3#48211yescharacteristic
199C2×C3#48311yescharacteristic
200C2×C3#48511yescharacteristic
201C22—22no—
202C22—22no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22#92911yescharacteristic
209C22C2×#21411yescharacteristic
210C22#93511yescharacteristic
211C22#93911yescharacteristic
212C22#94211yescharacteristic
213C22#94911yescharacteristic
214C22#95011yescharacteristic
215C4—41no—
216C4—81no—
217C4—81no—
218C4—241no—
219C4—241no—
220C4#87911yescharacteristic
221C4#87911yescharacteristic
222C4#88311yescharacteristic
223C4#88511yescharacteristic
224C3#207711yescharacteristic, complemented, Sylow
225C2—42no—
226C2#495211yescharacteristic
227C2#497111yescharacteristic
228C2#500311yescharacteristic
229C2#503411yescharacteristic
230C2#504111yescharacteristic
231C2#504511yescharacteristic
232C2#504911yescharacteristic
subgroupG/H
centerC23C2×#67
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#380C2
socleC23×C3C2×D4
prime radicalC24×C3D4
hypercenter#380D3
hypocenterC3#2077
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
4. C2×#808C2semidirect
5. #4529C2semidirect
6. #4529C2semidirect
7. #4529C2semidirect
224. C3#2077semidirect

Maximal subgroups

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

subgroupindex
C3×#3802
#52122
C2×#8302
C2×#8082
#45292
#20773