Finite Group Databasebeta

← #76748

Subgroups — #76748

Order 384.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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×#392C211yesmaximal, characteristic
2#5222C212yesmaximal, complemented
3#5222C212yesmaximal, complemented
4#5220C212yesmaximal, complemented
5#4952C212yesmaximal, complemented
6#4949C211yesmaximal, characteristic, complemented
7#4948C212yesmaximal, complemented
8#4946C211yesmaximal, characteristic, complemented
9#4658C212yesmaximal, complemented
10#2826—31nomaximal, Sylow
11C22×C3×D4C2212yes—
12C22×C3×D4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×#217C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C3×#95C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#399—31no—
34#397—32no—
35#396—32no—
36#394—32no—
37#394—32no—
38#392D311yescharacteristic
39#386—32no—
40#385—31no—
41C24×C3C2312yes—
42C24×C3C2311yescharacteristic
43C2×C3×D4—24no—
44C2×C3×D4—42no—
45C2×C3×D4—42no—
46C2×C3×D4—42no—
47C2×C3×D4—41no—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×Dic3—42no—
52C22×Dic3—41no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C3×C22⋊C4—42no—
58C3×C22⋊C4—41no—
59#217—22no—
60#217—42no—
61#217—42no—
62#217—42no—
63#217—42no—
64#217—42no—
65C4⋊Dic3—41no—
66Dic3⋊C4—42no—
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69C22×D4C2×D312yes—
70C22×D4C2×D311yescharacteristic
71C23×C4—31no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4C2×D311yescharacteristic
83C2×C22⋊C4C2×D312yes—
84#95—32no—
85#95—32no—
86#95—32no—
87#95—32no—
88#95—31no—
89#95C2×D311yescharacteristic
90C23×C3—22no—
91C23×C3—21no—
92C23×C3—22no—
93C23×C3—21no—
94C23×C3—22no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3#4112yes—
102C23×C3C2411yescharacteristic
103C3×D4—48no—
104C3×D4—44no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×Dic3—22no—
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—
125C24C22×D312yes—
126C24C22×D311yescharacteristic
127C2×D4—24no—
128C2×D4—42no—
129C2×D4—41no—
130C2×D4—42no—
131C2×D4—42no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—121no—
137C22×C4—122no—
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C22×C4C22×D311yescharacteristic
141C4⋊C4—122no—
142C4⋊C4—122no—
143C4⋊C4—122no—
144C4⋊C4—121no—
145C22⋊C4—41no—
146C22⋊C4—42no—
147C22⋊C4—62no—
148C22⋊C4—122no—
149C22⋊C4—122no—
150C22⋊C4—122no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C22×C3—24no—
154C22×C3—22no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—42no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3—42no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3C2×#4111yescharacteristic
166C22×C3H(5,𝔽2)12yes—
167C22×C3H(5,𝔽2)12yes—
168C22×C3H(5,𝔽2)11yescharacteristic
169C22×C32−1+411yescharacteristic
170C3×C4—42no—
171C3×C4—41no—
172C3×C4—41no—
173Dic3—42no—
174Dic3—42no—
175Dic3—42no—
176Dic3—42no—
177C23—21no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23#23712yes—
189C23C23×D311yescharacteristic
190D4—48no—
191D4—44no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
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—62no—
211C2×C4—124no—
212C2×C3—42no—
213C2×C3—82no—
214C2×C3#53812yes—
215C2×C3#53911yescharacteristic
216C2×C3#56011yescharacteristic
217C2×C3#56011yescharacteristic
218C2×C3#56212yes—
219C22—24no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22C2×#23711yescharacteristic
232C22#100412yes—
233C22#100411yescharacteristic
234C22#100412yes—
235C22#101011yescharacteristic
236C4—41no—
237C4—42no—
238C4—41no—
239C4—122no—
240C4—122no—
241C4—122no—
242C4—122no—
243C3#282611yescharacteristic, complemented, Sylow
244C2—42no—
245C2—82no—
246C2#559312yes—
247C2#559511yescharacteristic
248C2#560211yescharacteristic
249C2#562112yes—
250C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2826
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
2. #5222C2semidirect
3. #5222C2semidirect
4. #5220C2semidirect
5. #4952C2semidirect
6. #4949C2semidirect
7. #4948C2semidirect
8. #4946C2semidirect
9. #4658C2semidirect
243. C3#2826semidirect

Maximal subgroups

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

subgroupindex
C3×#3922
#52222
#52202
#49522
#49492
#49482
#49462
#46582
#28263