Finite Group Databasebeta

← #84090

Subgroups — #84090

Order 384.

864 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 185 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×C3×#41C211yesmaximal, characteristic
2C2×#935C212yesmaximal, complemented
3C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
4#5230C212yesmaximal, complemented
5#5222C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#5211C211yesmaximal, characteristic, complemented
8#5210C214yesmaximal, complemented
9#2406—31nomaximal, Sylow
10C2×C3×#41—28no—
11C2×C3×#41—24no—
12C22×C3×Q8C2211yescharacteristic
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×C3×C4C2212yes—
16C23×C3×C4C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C2×#217C2214yes—
20C2×#217C2214yes—
21C2×#217C2212yes—
22#935—28no—
23C2×Dic3⋊C4—24no—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2214yes—
26C2×Dic3⋊C4C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2214yes—
29C3⋊#95C2212yes—
30C22×#41D311yescharacteristic
31C2×#123—32no—
32C22×C4⋊C4—31no—
33#395—32no—
34#394—32no—
35#386—32no—
36#385—34no—
37#381—31no—
38C24×C3C2312yes—
39C24×C3C2311yescharacteristic
40C3×#41—232no—
41C2×C3×Q8—24no—
42C2×C3×Q8—22no—
43C2×C3×D4—24no—
44C2×C3×D4—24no—
45C2×C3×D4—24no—
46C2×C3×D4—22no—
47C2×C3×D4D418yes—
48C22×C3×C4—24no—
49C22×C3×C4—24no—
50C22×C3×C4—24no—
51C22×C3×C4—24no—
52C22×C3×C4D414yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×Dic3—22no—
57C22×Dic3—24no—
58C22×Dic3—22no—
59C22×Dic3—24no—
60C22×Dic3C2314yes—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63#217—24no—
64#217—28no—
65#217—28no—
66Dic3⋊C4—24no—
67Dic3⋊C4—24no—
68Dic3⋊C4—28no—
69C2×#41—28no—
70C2×#41—24no—
71C22×Q8C2×D311yescharacteristic
72C22×D4C2×D311yescharacteristic
73C22×D4C2×D312yes—
74C23×C4—31no—
75C23×C4—31no—
76C23×C4C2×D312yes—
77C23×C4C2×D311yescharacteristic
78#123—68no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—34no—
82C2×C4⋊C4—64no—
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×C3—22no—
92C23×C3—22no—
93C23×C3—24no—
94C23×C3—24no—
95C23×C3C2×D414yes—
96C23×C3C2×D414yes—
97C23×C3C2×D412yes—
98C23×C3C2411yescharacteristic
99C3×Q8—28no—
100C3×D4—28no—
101C3×D4—216no—
102C2×C3×C4—216no—
103C2×C3×C4—28no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4C2×D412yes—
109C2×C3×C4C2×D414yes—
110C2×C3×C4C2×D412yes—
111C2×Dic3—28no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—24no—
116C2×Dic3—24no—
117C2×Dic3—24no—
118C2×Dic3—24no—
119C2×Dic3—44no—
120C2×Dic3#4114yescomplemented
121C24C22×D312yes—
122C24C22×D311yescharacteristic
123#41—232no—
124C2×Q8—24no—
125C2×Q8—22no—
126C2×D4—24no—
127C2×D4—24no—
128C2×D4—24no—
129C2×D4—22no—
130C2×D4#6718yes—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—32no—
136C22×C4—34no—
137C22×C4—32no—
138C22×C4—64no—
139C22×C4—62no—
140C22×C4—62no—
141C22×C4—64no—
142C22×C4#6714yes—
143C22×C4C22×D311yescharacteristic
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C4⋊C4—64no—
147C4⋊C4—64no—
148C4⋊C4—68no—
149C22⋊C4—68no—
150C22⋊C4—64no—
151C22⋊C4—68no—
152C22×C3—22no—
153C22×C3—22no—
154C22×C3—28no—
155C22×C3—24no—
156C22×C3—28no—
157C22×C3—42no—
158C22×C3—42no—
159C22×C3—44no—
160C22×C3#12014yes—
161C22×C3C22×D411yescharacteristic
162C22×C3C22×D412yes—
163C22×C3C2×#4112yes—
164C22×C32−1+412yes—
165C3×C4—22no—
166C3×C4—24no—
167C3×C4—22no—
168Dic3—24no—
169Dic3—42no—
170Dic3—44no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—24no—
175C23—22no—
176C23—22no—
177C23C2×#6714yes—
178C23C2×#6714yes—
179C23C2×#6712yes—
180C23C23×D311yescharacteristic
181Q8—28no—
182D4—28no—
183D4—216no—
184C2×C4—216no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—34no—
191C2×C4—68no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—64no—
195C2×C4—64no—
196C2×C4—64no—
197C2×C4—64no—
198C2×C4—62no—
199C2×C4—124no—
200C2×C4C2×#6712yes—
201C2×C4C2×#6714yes—
202C2×C4C2×#6712yes—
203C2×C3—24no—
204C2×C3—44no—
205C2×C3C2×#12011yescharacteristic
206C2×C3#52511yescharacteristic
207C2×C3#53611yescharacteristic
208C2×C3#54714yes—
209C22—28no—
210C22—22no—
211C22—22no—
212C22—28no—
213C22—24no—
214C22—44no—
215C22—42no—
216C22—42no—
217C22#95314yes—
218C22D3×#4112yes—
219C22#101012yes—
220C22C22×#6712yes—
221C22C22×#6711yescharacteristic
222C4—22no—
223C4—24no—
224C4—22no—
225C4—64no—
226C4—124no—
227C4—122no—
228C3#240611yescharacteristic, complemented, Sylow
229C2—24no—
230C2—44no—
231C2#582414yes—
232C2#582511yescharacteristic
233C2#582811yescharacteristic
234C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC22×C3×#41C2
hypercenterC22×#41D3
hypocenterC3#2406
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. C22×Dic3⋊C4C2semidirect
4. #5230C2semidirect
5. #5222C2semidirect
6. #5220C2semidirect
7. #5211C2semidirect
8. #5210C2semidirect
120. C2×Dic3#41semidirect
228. C3#2406semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×#9352
C22×Dic3⋊C42
#52302
#52222
#52202
#52112
#52102
#24063