Finite Group Databasebeta

← #68310

Subgroups — #68310

Order 384.

228 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 63 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#4719C211yesmaximal, characteristic
3#4648C211yesmaximal, characteristic
4C3×#328C211yesmaximal, characteristic
5#4538C211yesmaximal, characteristic
6#4467C211yesmaximal, characteristic
7#4452C211yesmaximal, characteristic
8#1506—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10Q8×Dic3C411yescharacteristic, complemented
11#944C411yescharacteristic, complemented
12#924C2211yescharacteristic
13C2×C4×Dic3—21no—
14C2×C4×Dic3C2211yescharacteristic
15#889—21no—
16#888C411yescharacteristic, complemented
17#887—21no—
18#887C411yescharacteristic, complemented
19#878—21no—
20#877—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26#815—21no—
27Dic3⋊C8—21no—
28#533—31no—
29#432—31no—
30#376—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—21no—
51Dic3⋊C4—21no—
52Dic3⋊C4—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55C4×Dic3—24no—
56C4×Dic3—21no—
57C4×Dic3—21no—
58C4×Dic3C2×C411yescharacteristic
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3D411yescharacteristic
61C4×Dic3D411yescharacteristic
62C2×C3⋊C8—21no—
63C4⋊Q8—31no—
64C4⋊Q8—31no—
65#125—61no—
66C22⋊Q8—31no—
67C22⋊Q8—61no—
68C22⋊Q8C2×D311yescharacteristic
69C4×Q8—31no—
70C4×Q8—31no—
71C4×Q8—61no—
72#117—61no—
73C2×C42—31no—
74C2×C42—61no—
75C4⋊C8—61no—
76C22⋊C8—31no—
77C22⋊C8C2×D311yescharacteristic
78#97—61no—
79#95—31no—
80#95—61no—
81#95C2×D311yescharacteristic
82C23×C3C2×D411yescharacteristic
83C3×Q8—41no—
84C2×C3×C4—22no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4C22×C411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—22no—
95C2×Dic3—22no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C411yescharacteristic
102C2×Dic3C22⋊C411yescharacteristic
103C2×Dic3C22⋊C411yescharacteristic
104Dic6—42no—
105Dic6—41no—
106C3×C8—41no—
107C3⋊C8—41no—
108C2×Q8—31no—
109C2×Q8—61no—
110C2×Q8C4×D311yescharacteristic
111C22×C4—21no—
112C22×C4—31no—
113C22×C4—31no—
114C22×C4—61no—
115C22×C4C22×D311yescharacteristic
116C2×C8—21no—
117C2×C8—61no—
118C4⋊C4—21no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4C4×D311yescharacteristic
127C22⋊C4—21no—
128C22⋊C4—61no—
129C42—31no—
130C42—31no—
131C42—31no—
132C42—31no—
133C42—61no—
134C42—61no—
135C42—64no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3C2×C22⋊C411yescharacteristic
139C22×C3#13711yescharacteristic
140C22×C3#13711yescharacteristic
141C3×C4—21no—
142C3×C4—21no—
143C3×C4—42no—
144C3×C4—41no—
145C3×C4—41no—
146Dic3—21no—
147Dic3—21no—
148Dic3—21no—
149Dic3—42no—
150Dic3—41no—
151Dic3—41no—
152Dic3#10412yes—
153C23D3×D411yescharacteristic
154Q8—41no—
155Q8—121no—
156Q8—122no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—62no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—62no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—121no—
174C2×C4—121no—
175C2×C4C2×C4×D311yescharacteristic
176C2×C4D3×D411yescharacteristic
177C8—41no—
178C8—121no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3#41011yescharacteristic
182C2×C3#41911yescharacteristic
183C2×C3C2×#10411yescharacteristic
184C22—21no—
185C22—21no—
186C22D3×C22⋊C411yescharacteristic
187C22#91511yescharacteristic
188C22#91811yescharacteristic
189C4—21no—
190C4—21no—
191C4—32no—
192C4—42no—
193C4—41no—
194C4—41no—
195C4—61no—
196C4—61no—
197C4—61no—
198C4—122no—
199C4—121no—
200C4—121no—
201C3#150611yescharacteristic, complemented, Sylow
202C2—21no—
203C2—21no—
204C2#474011yescharacteristic
205C2#480211yescharacteristic
206C2D3×#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#1506
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
16. #888C4semidirect
18. #887C4semidirect
201. C3#1506semidirect

Maximal subgroups

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

subgroupindex
#56222
#47192
#46482
C3×#3282
#45382
#44672
#44522
#15063