Finite Group Databasebeta

← #74839

Subgroups — #74839

Order 384.

464 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C3×#386C211yesmaximal, characteristic
2#5221C211yesmaximal, characteristic
3#5220C211yesmaximal, characteristic
4#5210C212yesmaximal, complemented
5#4949C212yesmaximal, complemented
6#4948C212yesmaximal, complemented
7#4942C212yesmaximal, complemented
8#4657C212yesmaximal, complemented
9#4655C212yesmaximal, complemented
10#2748—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19C2×#217C2212yes—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27C3×#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#399—32no—
33#398—32no—
34#396—32no—
35#390—31no—
36#387—32no—
37#386—31no—
38#386D311yescharacteristic
39#385—32no—
40#380—32no—
41C24×C3C2311yescharacteristic
42C24×C3C2311yescharacteristic
43C2×C3×D4—41no—
44C2×C3×D4—41no—
45C2×C3×D4Q814yes—
46C22×C3×C4—41no—
47C22×C3×C4—42no—
48C22×C3×C4C2312yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3—24no—
52C22×Dic3—41no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C3×C22⋊C4—42no—
58C3×C22⋊C4—41no—
59#217—24no—
60#217—42no—
61#217—42no—
62#217—41no—
63C4⋊Dic3—42no—
64C4⋊Dic3—41no—
65Dic3⋊C4—24no—
66Dic3⋊C4—42no—
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69C4×Dic3—41no—
70C22×D4C2×D311yescharacteristic
71C23×C4—31no—
72C23×C4C2×D311yescharacteristic
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4C2×D312yes—
84C2×C22⋊C4C2×D311yescharacteristic
85C2×C42—31no—
86#95—32no—
87#95—32no—
88#95—32no—
89#95—32no—
90#95C2×D312yes—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—22no—
95C23×C3—21no—
96C23×C3—21no—
97C23×C3—21no—
98C23×C3—21no—
99C23×C3C2×Q812yes—
100C23×C3C2×Q812yes—
101C23×C3C2411yescharacteristic
102C3×D4—44no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—44no—
112C2×C3×C4C2×Q812yes—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—44no—
125C2×Dic3#4114yes—
126C24C22×D311yescharacteristic
127C24C22×D311yescharacteristic
128C2×D4—41no—
129C2×D4—41no—
130C2×D4Dic614yes—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—64no—
138C22×C4—121no—
139C22×C4C22×D312yes—
140C22×C4C22×D312yes—
141C22×C4C22×D311yescharacteristic
142C4⋊C4—64no—
143C4⋊C4—122no—
144C4⋊C4—122no—
145C4⋊C4—122no—
146C4⋊C4—121no—
147C4⋊C4—122no—
148C22⋊C4—41no—
149C22⋊C4—42no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C22⋊C4—121no—
154C42—121no—
155C22×C3—22no—
156C22×C3—22no—
157C22×C3—24no—
158C22×C3—24no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3C22×Q811yescharacteristic
166C22×C3C2×#4112yes—
167C22×C3H(5,𝔽2)11yescharacteristic
168C22×C3H(5,𝔽2)12yes—
169C22×C3H(5,𝔽2)11yescharacteristic
170C3×C4—22no—
171C3×C4—42no—
172C3×C4—42no—
173Dic3—24no—
174Dic3—42no—
175Dic3—42no—
176Dic3—42no—
177C23—22no—
178C23—21no—
179C23—22no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23—22no—
185C23C2×Dic612yes—
186C23C2×Dic612yes—
187C23C23×D311yescharacteristic
188D4—44no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—34no—
198C2×C4—44no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—124no—
211C2×C4C2×Dic612yes—
212C2×C3—42no—
213C2×C3—42no—
214C2×C3#54012yes—
215C2×C3#54311yescharacteristic
216C2×C3#55111yescharacteristic
217C2×C3#55412yes—
218C2×C3#56011yescharacteristic
219C22—22no—
220C22—22no—
221C22—24no—
222C22—24no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—42no—
229C22C22×Dic611yescharacteristic
230C22#100411yescharacteristic
231C22#100411yescharacteristic
232C22#100412yes—
233C22D3×#4112yes—
234C4—22no—
235C4—42no—
236C4—42no—
237C4—64no—
238C4—122no—
239C4—122no—
240C4—122no—
241C3#274811yescharacteristic, complemented, Sylow
242C2—42no—
243C2—42no—
244C2#548311yescharacteristic
245C2#553912yes—
246C2#559511yescharacteristic
247C2#561512yes—
248C2#565911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC22×C3×D4C22
hypercenter#386D3
hypocenterC3#2748
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #5210C2semidirect
5. #4949C2semidirect
6. #4948C2semidirect
7. #4942C2semidirect
8. #4657C2semidirect
9. #4655C2semidirect
241. C3#2748semidirect

Maximal subgroups

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

subgroupindex
C3×#3862
#52212
#52202
#52102
#49492
#49482
#49422
#46572
#46552
#27483