Finite Group Databasebeta

← #68327

Subgroups — #68327

Order 384.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#5623C211yesmaximal, characteristic
2#5050C211yesmaximal, characteristic
3#4739C211yesmaximal, characteristic
4#4656C211yesmaximal, characteristic
5C3×#328C211yesmaximal, characteristic
6#4469C211yesmaximal, characteristic
7#4452C211yesmaximal, characteristic
8#1605—31nomaximal, Sylow
9C22×Dic6C2211yescharacteristic
10C3×C22⋊Q8C2211yescharacteristic
11#944—21no—
12C2×#216—21no—
13C2×C4⋊Dic3—21no—
14#924C2211yescharacteristic
15C2×Dic3⋊C4C2211yescharacteristic
16C2×Dic12—21no—
17#889—21no—
18#889—21no—
19#888—21no—
20#878—21no—
21#877—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95—21no—
25#830C2211yescharacteristic
26#816—21no—
27#810—21no—
28#544—31no—
29#451—31no—
30#451—31no—
31#395—31no—
32#328—31no—
33#328D311yescharacteristic
34#324—31no—
35C2×C3×Q8D411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4C2311yescharacteristic
38C22×Dic3—21no—
39C22×Dic3D411yescharacteristic
40C22×Dic3D411yescharacteristic
41C2×Dic6—21no—
42C2×Dic6—21no—
43C2×Dic6—21no—
44C2×Dic6—42no—
45C2×Dic6D411yescharacteristic
46C2×C3×C8—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4D411yescharacteristic
49C3×C22⋊C4—21no—
50#217—21no—
51#216—42no—
52C4⋊Dic3—21no—
53C4⋊Dic3—42no—
54C4⋊Dic3D411yescharacteristic
55Dic3⋊C4—21no—
56Dic3⋊C4—21no—
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62C4×Dic3—21no—
63C4×Dic3—21no—
64C2×C3⋊C8—21no—
65Dic12—42no—
66C22×Q8—31no—
67C2×Q16—61no—
68C2×Q16—61no—
69C4⋊Q8—61no—
70C4⋊Q8—61no—
71#125—61no—
72#125—61no—
73C22⋊Q8—31no—
74C22⋊Q8—61no—
75C22⋊Q8C2×D311yescharacteristic
76#117—61no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—61no—
79#103—61no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8C2×D311yescharacteristic
83#95—61no—
84#95C2×D311yescharacteristic
85C23×C3C2×D411yescharacteristic
86C3×Q8—41no—
87C2×C3×C4—22no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—22no—
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—21no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3Q1612yescomplemented
105Dic6—22no—
106Dic6—22no—
107Dic6—41no—
108Dic6—41no—
109Dic6—41no—
110Dic6—81no—
111C3×C8—41no—
112C3⋊C8—41no—
113C2×Q8—31no—
114C2×Q8—61no—
115C2×Q8—61no—
116C2×Q8—61no—
117C2×Q8—122no—
118C2×Q8D1211yescharacteristic
119C22×C4—21no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—61no—
123C22×C4C22×D311yescharacteristic
124Q16—122no—
125Q16—122no—
126C2×C8—21no—
127C2×C8—61no—
128C4⋊C4—21no—
129C4⋊C4—31no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—122no—
139C4⋊C4D1211yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—61no—
142C42—61no—
143C42—61no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3#12011yescharacteristic
147C22×C3C2×Q1611yescharacteristic
148C22×C3#13711yescharacteristic
149C3×C4—21no—
150C3×C4—21no—
151C3×C4—42no—
152C3×C4—41no—
153C3×C4—41no—
154Dic3—22no—
155Dic3—41no—
156Dic3—42no—
157Dic3—41no—
158Dic3—41no—
159C23D3×D411yescharacteristic
160Q8—41no—
161Q8—62no—
162Q8—62no—
163Q8—121no—
164Q8—121no—
165Q8—121no—
166Q8—241no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—32no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—62no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—121no—
181C2×C4—121no—
182C2×C4—121no—
183C2×C4C2×D1211yescharacteristic
184C2×C4D3×D411yescharacteristic
185C8—41no—
186C8—121no—
187C2×C3—21no—
188C2×C3—21no—
189C2×C3#45111yescharacteristic
190C2×C3#45611yescharacteristic
191C2×C3B(4,𝔽2)11yescharacteristic
192C22—21no—
193C22—21no—
194C22#88211yescharacteristic
195C22#91511yescharacteristic
196C22D3×Q1611yescharacteristic
197C4—21no—
198C4—21no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—62no—
203C4—122no—
204C4—121no—
205C4—121no—
206C4—121no—
207C3#160511yescharacteristic, complemented, Sylow
208C2—21no—
209C2—21no—
210C2#474111yescharacteristic
211C2#480911yescharacteristic
212C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radicalC23×C3C2×D4
hypercenter#328D3
hypocenterC3#1605
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
104. C2×Dic3Q16semidirect
207. C3#1605semidirect

Maximal subgroups

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

subgroupindex
#56232
#50502
#47392
#46562
C3×#3282
#44692
#44522
#16053