Finite Group Databasebeta

← #74866

Subgroups — #74866

Order 384.

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

normal subgroups only

This group has 480 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#4947C212yesmaximal, complemented
6#4947C212yesmaximal, complemented
7#4946C212yesmaximal, complemented
8#4660C212yesmaximal, complemented
9#4646C212yesmaximal, complemented
10#2804—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C22×Dic6C2212yes—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2211yescharacteristic
20C2×#217C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C3×#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#400—32no—
33#393—32no—
34#393—32no—
35#391—32no—
36#390—31no—
37#386—31no—
38#386D311yescharacteristic
39#385—32no—
40#385—32no—
41C24×C3C2311yescharacteristic
42C24×C3C2311yescharacteristic
43C2×C3×D4—41no—
44C2×C3×D4—41no—
45C2×C3×D4Q814yescomplemented
46C22×C3×C4—41no—
47C22×C3×C4—42no—
48C22×C3×C4C2312yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3—41no—
52C22×Dic3—42no—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C2×Dic6—24no—
58C2×Dic6—42no—
59C2×Dic6—42no—
60C3×C22⋊C4—41no—
61C3×C22⋊C4—42no—
62#217—24no—
63#217—24no—
64#217—41no—
65#217—42no—
66C4⋊Dic3—41no—
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69C4×Dic3—41no—
70C22×Q8—32no—
71C22×D4C2×D311yescharacteristic
72C23×C4—31no—
73C23×C4C2×D311yescharacteristic
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—31no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4C2×D312yes—
82C2×C22⋊C4C2×D311yescharacteristic
83C2×C42—31no—
84#95—32no—
85#95—32no—
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—21no—
95C23×C3—22no—
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—
126Dic6—48no—
127C24C22×D311yescharacteristic
128C24C22×D311yescharacteristic
129C2×Q8—64no—
130C2×Q8—122no—
131C2×Q8—122no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4Dic614yescomplemented
135C22×C4—32no—
136C22×C4—32no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—122no—
142C22×C4—121no—
143C22×C4C22×D312yes—
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C4⋊C4—122no—
147C4⋊C4—122no—
148C4⋊C4—121no—
149C22⋊C4—42no—
150C22⋊C4—41no—
151C22⋊C4—64no—
152C22⋊C4—64no—
153C22⋊C4—122no—
154C22⋊C4—121no—
155C42—121no—
156C22×C3—22no—
157C22×C3—22no—
158C22×C3—24no—
159C22×C3—24no—
160C22×C3—42no—
161C22×C3—42no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3C22×Q811yescharacteristic
167C22×C3C2×#4112yes—
168C22×C3H(5,𝔽2)11yescharacteristic
169C22×C3H(5,𝔽2)12yes—
170C22×C3H(5,𝔽2)11yescharacteristic
171C3×C4—22no—
172C3×C4—42no—
173C3×C4—42no—
174Dic3—24no—
175Dic3—42no—
176Dic3—42no—
177Dic3—42no—
178C23—22no—
179C23—21no—
180C23—22no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—22no—
185C23—22no—
186C23C2×Dic612yes—
187C23C2×Dic612yes—
188C23C23×D311yescharacteristic
189Q8—128no—
190D4—44no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—34no—
200C2×C4—44no—
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—124no—
213C2×C4C2×Dic612yes—
214C2×C3—42no—
215C2×C3—42no—
216C2×C3#53812yes—
217C2×C3#54311yescharacteristic
218C2×C3#55412yes—
219C2×C3#55911yescharacteristic
220C2×C3H(3,𝔽4)11yescharacteristic
221C22—22no—
222C22—22no—
223C22—24no—
224C22—24no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—42no—
231C22C22×Dic611yescharacteristic
232C22#100411yescharacteristic
233C22#100411yescharacteristic
234C22#100412yes—
235C22D3×#4112yes—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—64no—
240C4—122no—
241C4—122no—
242C4—122no—
243C3#280411yescharacteristic, complemented, Sylow
244C2—42no—
245C2—42no—
246C2#548311yescharacteristic
247C2#553912yes—
248C2#559411yescharacteristic
249C2#561312yes—
250C2#566411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC22×C3×D4C22
hypercenter#386D3
hypocenterC3#2804
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #5210C2semidirect
5. #4947C2semidirect
6. #4947C2semidirect
7. #4946C2semidirect
8. #4660C2semidirect
9. #4646C2semidirect
45. C2×C3×D4Q8semidirect
134. C2×D4Dic6semidirect
243. C3#2804semidirect

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
#49472
#49462
#46602
#46462
#28043