Finite Group Databasebeta

← #69986

Subgroups — #69986

Order 384.

338 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#924C211yesmaximal, characteristic
2C22×#208C211yesmaximal, characteristic
3C3×#407C211yesmaximal, characteristic
4C2×#824C212yesmaximal, complemented
5#4554C212yesmaximal, complemented
6#1901—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C22×Dic6C2211yescharacteristic
9C2×C3×M16C2212yes—
10C2×#217C412yes—
11C2×C4⋊Dic3C2211yescharacteristic
12#924—21no—
13#924—21no—
14#924—21no—
15#924—21no—
16C2×Dic3⋊C4C412yes—
17C2×#208—21no—
18C2×#208—24no—
19C2×#208C2212yes—
20C22×C3⋊C8—21no—
21C3×C22⋊C8—21no—
22C3×C22⋊C8—21no—
23#824—24no—
24C22×M16—31no—
25C2×C22⋊Q8—31no—
26C2×#101—32no—
27#407D311yescharacteristic
28#343—32no—
29C24×C3C2×C411yescharacteristic
30C22×C3×C4—22no—
31C22×C3×C4—22no—
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C22×Dic3C2×C412yes—
39C22×Dic3C2×C412yes—
40C2×Dic6—22no—
41C2×Dic6—21no—
42C2×Dic6—21no—
43C2×Dic6—22no—
44C3×M16—22no—
45C3×M16—22no—
46C2×C3×C8—22no—
47#217—22no—
48#217—22no—
49C4⋊Dic3—21no—
50C4⋊Dic3—21no—
51Dic3⋊C4—22no—
52Dic3⋊C4—22no—
53#208—24no—
54#208—22no—
55#208D414yes—
56C2×C3⋊C8—24no—
57C2×C3⋊C8—22no—
58C22×Q8—31no—
59C23×C4C2×D311yescharacteristic
60C2×M16—32no—
61C2×M16—64no—
62C2×M16—61no—
63C2×M16C2×D312yes—
64C22×C8—61no—
65C22⋊Q8—61no—
66C22⋊Q8—61no—
67C22⋊Q8—61no—
68C22⋊Q8—61no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—31no—
71C2×C22⋊C4—32no—
72#101—64no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C23×C3—22no—
76C23×C3—22no—
77C23×C3—41no—
78C23×C3C22⋊C411yescharacteristic
79C23×C3C22⋊C411yescharacteristic
80C23×C3C22×C411yescharacteristic
81C2×C3×C4—22no—
82C2×C3×C4—21no—
83C2×C3×C4—21no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—21no—
87C2×C3×C4—21no—
88C2×C3×C4C22⋊C411yescharacteristic
89C2×C3×C4C22⋊C411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91C2×C3×C4C2×D412yes—
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4#4112yes—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—24no—
97C2×Dic3—42no—
98Dic6—41no—
99Dic6—42no—
100Dic6—41no—
101C3×C8—42no—
102C3⋊C8—24no—
103C24C4×D311yescharacteristic
104C2×Q8—61no—
105C2×Q8—62no—
106C2×Q8—62no—
107C2×Q8—61no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—32no—
111C22×C4—32no—
112C22×C4C4×D311yescharacteristic
113C22×C4D1211yescharacteristic
114C22×C4D1211yescharacteristic
115C22×C4#6711yescharacteristic
116C22×C4#6711yescharacteristic
117C22×C4C22×D311yescharacteristic
118M16—22no—
119M16—22no—
120M16—34no—
121M16—64no—
122M16—62no—
123C2×C8—22no—
124C2×C8—64no—
125C2×C8—62no—
126C4⋊C4—62no—
127C4⋊C4—62no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C22⋊C4—62no—
131C22⋊C4—62no—
132C22×C3—21no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3—21no—
139C22×C3—44no—
140C22×C3#10112yescomplemented
141C22×C3C2×C22⋊C411yescharacteristic
142C22×C3C4×D412yes—
143C3×C4—21no—
144C3×C4—21no—
145C3×C4#12011yescharacteristic
146C3×C4#12112yes—
147C3×C4#12311yescharacteristic
148Dic3—42no—
149Dic3—42no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23(C2×D3)⋊C411yescharacteristic
154C23(C2×D3)⋊C411yescharacteristic
155C23C2×C4×D311yescharacteristic
156Q8—121no—
157Q8—121no—
158Q8—122no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—64no—
167C2×C4—62no—
168C2×C4—62no—
169C2×C4—122no—
170C2×C4(C2×D3)⋊C411yescharacteristic
171C2×C4(C2×D3)⋊C411yescharacteristic
172C2×C4C2×D1211yescharacteristic
173C2×C4D3×D412yes—
174C2×C4#23712yes—
175C2×C4C2×#6711yescharacteristic
176C8—42no—
177C8—64no—
178C2×C3—22no—
179C2×C3—22no—
180C2×C3—41no—
181C2×C3#38611yescharacteristic
182C2×C3C2×#10111yescharacteristic
183C2×C3#41311yescharacteristic
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—44no—
192C22#82412yescomplemented
193C22#67⋊C412yes—
194C22C2×(C2×D3)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—122no—
198C4—122no—
199C4#88211yescharacteristic
200C4#88611yescharacteristic
201C4#93912yes—
202C3#190111yescharacteristic, complemented, Sylow
203C2—22no—
204C2—22no—
205C2—41no—
206C2#495811yescharacteristic
207C2#513211yescharacteristic
208C2C2×#82411yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3C2×C4
cosocleC22×C3×C4C23
hypercenter#407D3
hypocenterC3#1901
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
4. C2×#824C2semidirect
5. #4554C2semidirect
140. C22×C3#101semidirect
192. C22#824semidirect
202. C3#1901semidirect

Maximal subgroups

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

subgroupindex
C2×#9242
C22×#2082
C3×#4072
C2×#8242
#45542
#19013