Finite Group Databasebeta

← #76607

Subgroups — #76607

Order 384.

504 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 177 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×#944C211yesmaximal, characteristic
2C2×#888C211yesmaximal, characteristic
3C3×#391C211yesmaximal, characteristic
4#5229C211yesmaximal, characteristic
5#4973C211yesmaximal, characteristic
6C42×Dic3C211yesmaximal, characteristic
7#4929C211yesmaximal, characteristic
8#4650C214yesmaximal
9#4646C214yesmaximal
10#2862—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×Dic6C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16#944—24no—
17C2×C4⋊Dic3C2211yescharacteristic
18C2×Dic3⋊C4C2212yes—
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×C4×Dic3C2212yes—
22C2×C4×Dic3C2211yescharacteristic
23C2×C4×Dic3C2211yescharacteristic
24C2×C4×Dic3C2212yes—
25#888—24no—
26C3×#95C2214yes—
27C3⋊#95C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2214yes—
30C2×C4⋊Q8—31no—
31C2×C4⋊Q8—31no—
32#391—31no—
33#391—31no—
34#391—31no—
35#391—34no—
36#391D311yescharacteristic
37#382—34no—
38C43—31no—
39C2×C3×Q8—22no—
40C2×C3×Q8—22no—
41C2×C3×Q8—22no—
42C22×C3×C4C2312yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×Dic3C2312yes—
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2311yescharacteristic
50C22×Dic3C2311yescharacteristic
51C2×Dic6—22no—
52C2×Dic6—22no—
53C2×Dic6—22no—
54C2×Dic6—24no—
55C2×Dic6—28no—
56C3×C4⋊C4—22no—
57C3×C42—22no—
58C4⋊Dic3—22no—
59Dic3⋊C4—24no—
60Dic3⋊C4—24no—
61Dic3⋊C4—24no—
62C4×Dic3—22no—
63C4×Dic3—24no—
64C4×Dic3D414yes—
65C4×Dic3Q818yescomplemented
66C22×Q8—32no—
67C22×Q8—31no—
68C22×Q8C2×D311yescharacteristic
69C4⋊Q8—64no—
70C4⋊Q8—64no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4C2×D311yescharacteristic
76C2×C42—31no—
77C2×C42—32no—
78C2×C42—32no—
79C2×C42—31no—
80C2×C42C2×D311yescharacteristic
81#95—34no—
82#95—34no—
83#95—34no—
84#95C2×D314yes—
85C23×C3C2411yescharacteristic
86C3×Q8—44no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—22no—
91C2×C3×C4—24no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4C2×D412yes—
95C2×C3×C4C2×Q814yes—
96C2×C3×C4#4112yes—
97C2×C3×C4#4112yes—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—24no—
105C2×Dic3C2×D412yes—
106C2×Dic3C2×D412yes—
107C2×Dic3C2×Q814yes—
108C2×Dic3C2×Q814yes—
109C2×Dic3#4114yes—
110Dic6—44no—
111Dic6—48no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—64no—
116C2×Q8—68no—
117C2×Q8—62no—
118C2×Q8—62no—
119C2×Q8—62no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4C22×D312yes—
126C22×C4C22×D312yes—
127C22×C4C22×D312yes—
128C22×C4C22×D311yescharacteristic
129C4⋊C4—22no—
130C4⋊C4—64no—
131C4⋊C4—64no—
132C4⋊C4—64no—
133C4⋊C4—62no—
134C42—22no—
135C42—34no—
136C42—38no—
137C42—64no—
138C42—62no—
139C22×C3C22×D411yescharacteristic
140C22×C3C22×Q812yes—
141C22×C3C2×#4111yescharacteristic
142C22×C3C2×#4111yescharacteristic
143C22×C3C2×#4112yes—
144C3×C4—24no—
145C3×C4—42no—
146C3×C4—42no—
147C3×C4#12414yes—
148Dic3—24no—
149Dic3—42no—
150Dic3—42no—
151Dic3#12414yes—
152Dic3C4⋊Q814yes—
153C23C23×D311yescharacteristic
154Q8—44no—
155Q8—128no—
156Q8—124no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—34no—
167C2×C4—34no—
168C2×C4—34no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—64no—
176C2×C4#23512yes—
177C2×C4D3×D412yes—
178C2×C4#23712yes—
179C2×C4D3×Q814yes—
180C2×C3C2×#12411yescharacteristic
181C2×C3C2×#12411yescharacteristic
182C2×C3C2×C4⋊Q811yescharacteristic
183C2×C3#53312yes—
184C2×C3#53312yes—
185C22C2×#23511yescharacteristic
186C22C2×D3×D411yescharacteristic
187C22C2×#23711yescharacteristic
188C22C2×D3×Q812yes—
189C22D3×#4112yes—
190C4—24no—
191C4—34no—
192C4—34no—
193C4—42no—
194C4—42no—
195C4—64no—
196C4—122no—
197C4—122no—
198C4#88514yes—
199C3#286211yescharacteristic, complemented, Sylow
200C2C2×#88511yescharacteristic
201C2#557812yes—
202C2#562212yes—
203C2D3×#12411yescharacteristic
204C2D3×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC3×#391C2
hypercenter#391D3
hypocenterC3#2862
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
65. C4×Dic3Q8semidirect
199. C3#2862semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8882
C3×#3912
#52292
#49732
C42×Dic32
#49292
#46502
#46462
#28623