Finite Group Databasebeta

← #76746

Subgroups — #76746

Order 384.

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

normal subgroups only

This group has 472 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#4947C212yesmaximal, complemented
7#4947C211yesmaximal, characteristic, complemented
8#4946C211yesmaximal, characteristic, complemented
9#4658C212yesmaximal, complemented
10#2824—31nomaximal, Sylow
11C22×C3×D4C2212yes—
12C22×C3×D4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×#217C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C3×#95C2211yescharacteristic
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32#400—32no—
33#397—32no—
34#394—32no—
35#394—32no—
36#393—32no—
37#393—31no—
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—
57C2×Dic6—42no—
58C2×Dic6—41no—
59C3×C22⋊C4—41no—
60C3×C22⋊C4—42no—
61#217—22no—
62#217—42no—
63#217—42no—
64#217—42no—
65#217—42no—
66#217—42no—
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69C22×Q8—31no—
70C22×D4C2×D312yes—
71C22×D4C2×D311yescharacteristic
72C23×C4—31no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4C2×D311yescharacteristic
82C2×C22⋊C4C2×D312yes—
83#95—32no—
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—
125Dic6—44no—
126C24C22×D312yes—
127C24C22×D311yescharacteristic
128C2×Q8—122no—
129C2×Q8—121no—
130C2×D4—24no—
131C2×D4—42no—
132C2×D4—41no—
133C2×D4—42no—
134C2×D4—42no—
135C22×C4—32no—
136C22×C4—32no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—121no—
140C22×C4—122no—
141C22×C4C22×D312yes—
142C22×C4C22×D311yescharacteristic
143C22×C4C22×D311yescharacteristic
144C4⋊C4—122no—
145C4⋊C4—122no—
146C22⋊C4—41no—
147C22⋊C4—42no—
148C22⋊C4—62no—
149C22⋊C4—122no—
150C22⋊C4—122no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C22⋊C4—122no—
154C22×C3—24no—
155C22×C3—22no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—42no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3—42no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3C2×#4111yescharacteristic
167C22×C3H(5,𝔽2)12yes—
168C22×C3H(5,𝔽2)12yes—
169C22×C3H(5,𝔽2)11yescharacteristic
170C22×C32−1+411yescharacteristic
171C3×C4—42no—
172C3×C4—41no—
173C3×C4—41no—
174Dic3—42no—
175Dic3—42no—
176Dic3—42no—
177Dic3—42no—
178C23—21no—
179C23—22no—
180C23—22no—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23#23712yes—
190C23C23×D311yescharacteristic
191Q8—124no—
192D4—48no—
193D4—44no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
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—62no—
212C2×C4—62no—
213C2×C4—124no—
214C2×C3—42no—
215C2×C3—82no—
216C2×C3#53812yes—
217C2×C3#53911yescharacteristic
218C2×C3#56011yescharacteristic
219C2×C3H(3,𝔽4)11yescharacteristic
220C2×C3#56212yes—
221C22—24no—
222C22—22no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22C2×#23711yescharacteristic
234C22#100412yes—
235C22#100411yescharacteristic
236C22#100412yes—
237C22#101011yescharacteristic
238C4—41no—
239C4—42no—
240C4—41no—
241C4—122no—
242C4—122no—
243C4—122no—
244C4—122no—
245C3#282411yescharacteristic, complemented, Sylow
246C2—42no—
247C2—82no—
248C2#559312yes—
249C2#559411yescharacteristic
250C2#560211yescharacteristic
251C2#562112yes—
252C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2824
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. #4947C2semidirect
7. #4947C2semidirect
8. #4946C2semidirect
9. #4658C2semidirect
245. C3#2824semidirect

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
#49472
#49462
#46582
#28243