Finite Group Databasebeta

← #82192

Subgroups — #82192

Order 384.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#235C211yesmaximal, characteristic
2C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#929C212yesmaximal, complemented
4#5213C212yesmaximal, complemented
5#5211C211yesmaximal, characteristic, complemented
6#4991C212yesmaximal, complemented
7#4989C214yesmaximal, complemented
8#4979C212yesmaximal, complemented
9#2406—31nomaximal, Sylow
10C23×C3×C4C2211yescharacteristic
11C23×C3×C4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C22×#67C2212yes—
14C2×#235—24no—
15C2×#235—28no—
16C22×D12C2211yescharacteristic
17C22×C4×D3C2212yes—
18C22×Dic6C2211yescharacteristic
19C2×C3×C4⋊C4—24no—
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C4⋊C4C2212yes—
22C2×#217C2212yes—
23#929—28no—
24C2×(C2×D3)⋊C4C2214yes—
25C2×(C2×D3)⋊C4C2214yes—
26C2×Dic3⋊C4C2214yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C22×#41—31no—
31C2×#123—32no—
32C22×C4⋊C4D311yescharacteristic, complemented
33#395—32no—
34#394—32no—
35#386—32no—
36#385—34no—
37#381—31no—
38C24×C3C2311yescharacteristic
39C23×D3C2312yes—
40C22×C3×C4—22no—
41C22×C3×C4—24no—
42C22×C3×C4—24no—
43C22×C3×C4—24no—
44C22×C3×C4—22no—
45C22×C3×C4D414yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C2×#67—24no—
51C2×#67—24no—
52C2×#67D418yes—
53C22×Dic3C2314yes—
54C22×Dic3C2312yes—
55#235—232no—
56C2×D12—24no—
57C2×D12—22no—
58C2×C4×D3—24no—
59C2×C4×D3—24no—
60C2×C4×D3—24no—
61C2×Dic6—22no—
62C2×Dic6—24no—
63C3×C4⋊C4—24no—
64C3×C4⋊C4—24no—
65#217—24no—
66(C2×D3)⋊C4—28no—
67(C2×D3)⋊C4—28no—
68Dic3⋊C4—28no—
69C2×#41—68no—
70C2×#41—64no—
71C22×Q8—31no—
72C22×D4—32no—
73C22×D4—31no—
74C23×C4—32no—
75C23×C4C2×D311yescharacteristic
76C23×C4C2×D311yescharacteristic
77C23×C4C2×D311yescharacteristic
78#123—68no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—34no—
81C2×C4⋊C4C2×D312yes—
82C2×C4⋊C4C2×D312yes—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—34no—
85C2×C22⋊C4—34no—
86#95—32no—
87#95—32no—
88#95—34no—
89C23×C3—22no—
90C23×C3—22no—
91C23×C3C2×D414yes—
92C23×C3C2×D412yes—
93C23×C3C2411yescharacteristic
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—24no—
97C22×D3—24no—
98C22×D3C2×D414yes—
99C2×C3×C4—28no—
100C2×C3×C4—28no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—24no—
106C2×C3×C4—22no—
107C2×C3×C4—44no—
108C2×C3×C4C2×D412yes—
109C2×C3×C4C2×D412yes—
110C2×C3×C4#4114yescomplemented
111#67—216no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—24no—
115C2×Dic3—24no—
116C2×Dic3—24no—
117C2×Dic3C2×D414yes—
118D12—28no—
119C4×D3—216no—
120Dic6—28no—
121C24—32no—
122C24C22×D311yescharacteristic
123#41—632no—
124C2×Q8—64no—
125C2×Q8—62no—
126C2×D4—38no—
127C2×D4—64no—
128C2×D4—64no—
129C2×D4—64no—
130C2×D4—62no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—32no—
137C22×C4—34no—
138C22×C4—64no—
139C22×C4—64no—
140C22×C4—64no—
141C22×C4#6714yes—
142C22×C4C22×D312yes—
143C22×C4C22×D312yes—
144C22×C4C22×D311yescharacteristic
145C22×C4C22×D311yescharacteristic
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—68no—
149C22⋊C4—68no—
150C22⋊C4—64no—
151C22⋊C4—68no—
152C22×C3—22no—
153C22×C3—22no—
154C22×C3—28no—
155C22×C3#12014yes—
156C22×C3C22×D412yes—
157C22×C3C22×D411yescharacteristic
158C22×C3C2×#4112yes—
159C22×C32−1+412yes—
160C2×D3—24no—
161C2×D3—28no—
162C2×D3—42no—
163C2×D3—42no—
164C2×D3—44no—
165C3×C4—22no—
166C3×C4—22no—
167C3×C4—24no—
168C3×C4—42no—
169Dic3—24no—
170Dic3—44no—
171C23—22no—
172C23—22no—
173C23—34no—
174C23—64no—
175C23—62no—
176C23—64no—
177C23—62no—
178C23D3×D414yes—
179C23C2×#6712yes—
180C23C23×D311yescharacteristic
181Q8—68no—
182D4—68no—
183D4—616no—
184C2×C4—28no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—34no—
193C2×C4—44no—
194C2×C4—616no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—64no—
198C2×C4—64no—
199C2×C4—64no—
200C2×C4#23514yescomplemented
201C2×C4C2×#6712yes—
202C2×C4C2×#6712yes—
203C2×C3—24no—
204C2×C3C2×#12011yescharacteristic
205C2×C3#52511yescharacteristic
206C2×C3#53611yescharacteristic
207C2×C3#54714yes—
208D3—44no—
209C22—28no—
210C22—22no—
211C22—22no—
212C22—68no—
213C22—64no—
214C22—124no—
215C22—122no—
216C22—122no—
217C22#93714yes—
218C22C2×#23512yes—
219C22C2×D3×D412yes—
220C22#100712yes—
221C22C22×#6711yescharacteristic
222C4—24no—
223C4—22no—
224C4—22no—
225C4—42no—
226C4—64no—
227C4—124no—
228C3#240611yescharacteristic, complemented, Sylow
229C2—24no—
230C2—124no—
231C2#550714yes—
232C2#579111yescharacteristic
233C2C2×#93711yescharacteristic
234C2#581611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC22×#235C2
hypercenterC22×C4⋊C4D3
hypocenterC3#2406
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4⋊C4C2semidirect
3. C2×#929C2semidirect
4. #5213C2semidirect
5. #5211C2semidirect
6. #4991C2semidirect
7. #4989C2semidirect
8. #4979C2semidirect
32. C22×C4⋊C4D3semidirect
110. C2×C3×C4#41semidirect
200. C2×C4#235semidirect
228. C3#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C22×C3×C4⋊C42
C2×#9292
#52132
#52112
#49912
#49892
#49792
#24063