Finite Group Databasebeta

← #68309

Subgroups — #68309

Order 384.

240 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 71 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#5622C211yesmaximal, characteristic
2#4718C211yesmaximal, characteristic
3#4649C211yesmaximal, characteristic
4C3×#328C211yesmaximal, characteristic
5#4538C211yesmaximal, characteristic
6#4467C211yesmaximal, characteristic
7#4452C211yesmaximal, characteristic
8#1503—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10Q8×Dic3C411yescharacteristic, complemented
11#944C411yescharacteristic, complemented
12#924C2211yescharacteristic
13C2×Dic3⋊C4—21no—
14C2×C4×Dic3—21no—
15C2×C4×Dic3C2211yescharacteristic
16#889—21no—
17#888C411yescharacteristic, complemented
18#887—21no—
19#887C411yescharacteristic, complemented
20#878—21no—
21#877—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28#533—31no—
29#431—31no—
30C4×C4⋊C4—31no—
31#328—31no—
32#328—31no—
33#328—31no—
34#328D311yescharacteristic
35C2×C3×Q8C2×C411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—21no—
39C22×Dic3D411yescharacteristic
40C22×Dic3D411yescharacteristic
41C2×Dic6—21no—
42C2×Dic6C2×C411yescharacteristic
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4C2×C411yescharacteristic
46C3×C22⋊C4—21no—
47#217—21no—
48C4⋊Dic3—21no—
49C4⋊Dic3C2×C411yescharacteristic
50Dic3⋊C4—24no—
51Dic3⋊C4—21no—
52Dic3⋊C4—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C4×Dic3—21no—
58C4×Dic3—42no—
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3C2×C411yescharacteristic
61C4×Dic3D411yescharacteristic
62C4×Dic3D411yescharacteristic
63C2×C3⋊C8—21no—
64C4⋊Q8—31no—
65C4⋊Q8—31no—
66#125—61no—
67C22⋊Q8—31no—
68C22⋊Q8—61no—
69C22⋊Q8C2×D311yescharacteristic
70C4×Q8—31no—
71C4×Q8—31no—
72C4×Q8—61no—
73#117—61no—
74C2×C4⋊C4—61no—
75C2×C42—31no—
76C2×C42—61no—
77C4⋊C8—61no—
78C22⋊C8—31no—
79C22⋊C8C2×D311yescharacteristic
80C4×C8—61no—
81#95—31no—
82#95C2×D311yescharacteristic
83C23×C3C2×D411yescharacteristic
84C3×Q8—41no—
85C2×C3×C4—22no—
86C2×C3×C4—21no—
87C2×C3×C4—21no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4C22×C411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×Dic3—22no—
94C2×Dic3—22no—
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3C22⋊C411yescharacteristic
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C411yescharacteristic
102C2×Dic3C22⋊C411yescharacteristic
103C2×Dic3SD1612yes—
104C2×Dic3Q1612yes—
105Dic6—42no—
106Dic6—41no—
107C3×C8—41no—
108C3⋊C8—41no—
109C2×Q8—31no—
110C2×Q8—61no—
111C2×Q8C4×D311yescharacteristic
112C22×C4—21no—
113C22×C4—31no—
114C22×C4—31no—
115C22×C4—61no—
116C22×C4C22×D311yescharacteristic
117C2×C8—21no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—31no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—64no—
127C4⋊C4—61no—
128C4⋊C4C4×D311yescharacteristic
129C22⋊C4—21no—
130C22⋊C4—61no—
131C42—31no—
132C42—31no—
133C42—31no—
134C42—31no—
135C42—61no—
136C42—61no—
137C42—122no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3C2×C22⋊C411yescharacteristic
141C22×C3C2×SD1611yescharacteristic
142C22×C3C2×Q1611yescharacteristic
143C3×C4—21no—
144C3×C4—21no—
145C3×C4—42no—
146C3×C4—41no—
147C3×C4—41no—
148Dic3—21no—
149Dic3—42no—
150Dic3—41no—
151Dic3—41no—
152Dic3#10312yes—
153Dic3#10312yes—
154Dic3#10412yes—
155C23D3×D411yescharacteristic
156Q8—41no—
157Q8—121no—
158Q8—122no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—32no—
166C2×C4—32no—
167C2×C4—31no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—121no—
176C2×C4—121no—
177C2×C4C2×C4×D311yescharacteristic
178C2×C4D3×D411yescharacteristic
179C8—41no—
180C8—121no—
181C2×C3—21no—
182C2×C3—21no—
183C2×C3#41011yescharacteristic
184C2×C3C2×#10311yescharacteristic
185C2×C3C2×#10411yescharacteristic
186C22—21no—
187C22—21no—
188C22D3×C22⋊C411yescharacteristic
189C22D3×SD1611yescharacteristic
190C22D3×Q1611yescharacteristic
191C4—21no—
192C4—21no—
193C4—32no—
194C4—32no—
195C4—32no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—61no—
200C4—122no—
201C4—121no—
202C4—121no—
203C3#150311yescharacteristic, complemented, Sylow
204C2—21no—
205C2—21no—
206C2#474011yescharacteristic
207C2D3×#10311yescharacteristic
208C2D3×#10411yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3C2×C22⋊C4
prime radicalC23×C3C2×D4
cosocleC22×C3×C4C23
hypercenter#328D3
hypocenterC3#1503
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
10. Q8×Dic3C4semidirect
11. #944C4semidirect
17. #888C4semidirect
19. #887C4semidirect
203. C3#1503semidirect

Maximal subgroups

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

subgroupindex
#56222
#47182
#46492
C3×#3282
#45382
#44672
#44522
#15033