Finite Group Databasebeta

← #78208

Subgroups — #78208

Order 384.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#896C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4#4985C211yesmaximal, characteristic, complemented
5#4981C211yesmaximal, characteristic, complemented
6#4675C212yesmaximal
7#4674C212yesmaximal
8#4667C211yesmaximal, characteristic
9#4661C211yesmaximal, characteristic, complemented
10#4647C212yesmaximal, complemented
11#4646C212yesmaximal, complemented
12#2625—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×(C2×D3)⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2212yes—
28#896—24no—
29#892—24no—
30C3×#95C2212yes—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C2×#117—31no—
40#400—31no—
41#400D311yescharacteristic, complemented
42#398—31no—
43#398—31no—
44#397—32no—
45#397—32no—
46#391—32no—
47#389—32no—
48#380—31no—
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—24no—
61C2×C4×D3D414yes—
62C2×Dic6—22no—
63C2×Dic6—24no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—22no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—42no—
70C4⋊Dic3—22no—
71C4⋊Dic3—42no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4—24no—
75Dic3⋊C4—42no—
76C4×Dic3—24no—
77C22×Q8—31no—
78C23×C4—31no—
79C22⋊Q8—64no—
80#117—64no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C22⋊C4—32no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C42—32no—
94#95—31no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95—32no—
99#95C2×D311yescharacteristic
100#95C2×D312yes—
101#95C2×D312yes—
102C23×C3C2411yescharacteristic
103C22×D3—22no—
104C22×D3—21no—
105C22×D3—21no—
106C22×D3—22no—
107C22×D3C2×D412yes—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—24no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D412yes—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3C2×D412yes—
132C2×Dic3#4114yes—
133C4×D3—28no—
134Dic6—44no—
135C24—31no—
136C2×Q8—62no—
137C2×Q8—64no—
138C22×C4—31no—
139C22×C4—34no—
140C22×C4—32no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—31no—
144C22×C4—64no—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D312yes—
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—62no—
153C4⋊C4—64no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—122no—
157C4⋊C4—122no—
158C22⋊C4—64no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—122no—
162C42—64no—
163C22×C3C22×D411yescharacteristic
164C22×C3C2×#4112yes—
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C32−1+411yescharacteristic
167C22×C32−1+412yes—
168C2×D3—22no—
169C2×D3—22no—
170C2×D3—22no—
171C2×D3—42no—
172C2×D3—42no—
173C3×C4—22no—
174C3×C4—41no—
175C3×C4—42no—
176C3×C4—41no—
177C3×C4—42no—
178Dic3—41no—
179Dic3—42no—
180Dic3—41no—
181Dic3—42no—
182Dic3#12314yes—
183C23—32no—
184C23—61no—
185C23—62no—
186C23—61no—
187C23—62no—
188C23C23×D311yescharacteristic
189Q8—124no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—32no—
202C2×C4—34no—
203C2×C4—68no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4D3×D412yes—
216C2×C3C2×#12311yescharacteristic
217C2×C3#53611yescharacteristic
218C2×C3#53711yescharacteristic
219C2×C3#54112yes—
220C2×C3#54212yes—
221D3—42no—
222C22—62no—
223C22—62no—
224C22—62no—
225C22—122no—
226C22—122no—
227C22C2×D3×D411yescharacteristic
228C22#100711yescharacteristic
229C22D3×#4112yes—
230C22#100911yescharacteristic
231C22#101012yes—
232C4—22no—
233C4—34no—
234C4—41no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—121no—
239C4—122no—
240C4—122no—
241C4—121no—
242C3#262511yescharacteristic, complemented, Sylow
243C2—122no—
244C2#562911yescharacteristic
245C2D3×#12311yescharacteristic
246C2#567511yescharacteristic
247C2#569612yes—
248C2#571012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC23×D3C23
hypercenter#400D3
hypocenterC3#2625
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C3×#400C2semidirect
4. #4985C2semidirect
5. #4981C2semidirect
9. #4661C2semidirect
10. #4647C2semidirect
11. #4646C2semidirect
41. #400D3semidirect
242. C3#2625semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8922
C3×#4002
#49852
#49812
#46752
#46742
#46672
#46612
#46472
#46462
#26253