Finite Group Databasebeta

← #80721

Subgroups — #80721

Order 384.

424 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#5606C212yesmaximal, complemented
2C3×#466C211yesmaximal, characteristic, complemented
3#5158C212yesmaximal, complemented
4#5116C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6#5041C212yesmaximal, complemented
7#4888C212yesmaximal, complemented
8#4784C212yesmaximal, complemented
9#4783C212yesmaximal, complemented
10#3191—31nomaximal, Sylow
11C2×#237C2212yes—
12C22×C4×D3C2211yescharacteristic
13C2×C3×D8C2211yescharacteristic, complemented
14C22×C3×C8C2211yescharacteristic
15C3×#121C2212yes—
16#938C2212yes—
17#937—22no—
18D4×Dic3—22no—
19C2×#214C2212yescomplemented
20#924C2212yes—
21C22×C3⋊C8C2211yescharacteristic
22C2×Dic12C2211yescharacteristic, complemented
23#897C2212yes—
24#892C2212yes—
25#883—22no—
26#67⋊C4—22no—
27C3×#107C2211yescharacteristic, complemented
28C3×#102C2212yescomplemented
29#832C2212yescomplemented
30(C2×D3)⋊C8C2212yes—
31(C2×D3)⋊C8C2212yes—
32#818C2211yescharacteristic, complemented
33#816C2212yescomplemented
34#810C2212yescomplemented
35#808C2212yescomplemented
36#525—32no—
37#485—32no—
38#482—32no—
39#467—31no—
40#466D311yescharacteristic, complemented
41#465—32no—
42#452—32no—
43#449—32no—
44C2×C22⋊C8—31no—
45C23×D3D411yescharacteristic
46C2×C3×D4—22no—
47C2×C3×D4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C2×#67—22no—
50C2×#67—22no—
51C22×Dic3—22no—
52C22×Dic3D411yescharacteristic
53#237—42no—
54#237—42no—
55C2×C4×D3—41no—
56C2×C4×D3—41no—
57C2×C4×D3D412yes—
58C2×C4×D3C2312yes—
59C2×Dic6C2312yes—
60C3×D8—41no—
61C2×C3×C8—41no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—22no—
66#217—22no—
67#217—22no—
68#214—42no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71C4⋊Dic3—22no—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75C2×C3⋊C8—41no—
76C2×C3⋊C8C2312yes—
77Dic12—41no—
78C2×#41—32no—
79C23×C4—31no—
80C2×Q16—31no—
81C2×SD16—32no—
82C2×D8C2×D311yescharacteristic, complemented
83C22×C8—31no—
84C22×C8C2×D311yescharacteristic
85#123—62no—
86C22⋊Q8—32no—
87C22⋊Q8—32no—
88#121—32no—
89#121C2×D312yes—
90#120—62no—
91C4×D4—62no—
92C4×D4—62no—
93#117—32no—
94#107—31no—
95#107C2×D311yescharacteristic, complemented
96#106—32no—
97#103—32no—
98#103—32no—
99#102—32no—
100#102C2×D312yescomplemented
101C22⋊C8—32no—
102C22⋊C8—32no—
103C23×C3—22no—
104C23×C3C2×D411yescharacteristic
105C22×D3—22no—
106C22×D3—41no—
107C22×D3—41no—
108C22×D3C2×D412yes—
109C3×D4—22no—
110C3×D4—22no—
111C3×D4—44no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2411yescharacteristic
117#67—44no—
118#67—44no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—41no—
122C2×Dic3—44no—
123C2×Dic3C2×D412yes—
124C4×D3—22no—
125C4×D3—22no—
126C4×D3—42no—
127Dic6—22no—
128Dic6—22no—
129C3×C8—21no—
130C3×C8—21no—
131C3⋊C8—22no—
132C24—31no—
133#41—122no—
134#41—122no—
135C2×Q8—32no—
136C2×D4—22no—
137C2×D4—62no—
138C2×D4—62no—
139C2×D4C22×D312yes—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—62no—
144C22×C4—121no—
145C22×C4—121no—
146C22×C4C22×D311yescharacteristic
147Q16—121no—
148SD16—122no—
149D8—41no—
150C2×C8—32no—
151C2×C8—41no—
152C2×C8—121no—
153C2×C8C22×D311yescharacteristic
154C2×C8C22×D311yescharacteristic
155C4⋊C4—32no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4C22×D312yes—
159C22⋊C4—22no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C42—62no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3—42no—
170C22×C3#13512yes—
171C22×C3C22×D411yescharacteristic
172C2×D3—22no—
173C2×D3—22no—
174C2×D3—22no—
175C2×D3—44no—
176C3×C4—21no—
177C3×C4—42no—
178C3×C4H(5,𝔽2)11yescharacteristic
179C3×C4H(5,𝔽2)11yescharacteristic
180Dic3—22no—
181Dic3—42no—
182Dic3—42no—
183C23—22no—
184C23—32no—
185C23—62no—
186C23—121no—
187C23—121no—
188C23D3×D411yescharacteristic
189Q8—62no—
190Q8—62no—
191D4—22no—
192D4—22no—
193D4—44no—
194D4—124no—
195D4—124no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—32no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—122no—
205C2×C4—121no—
206C2×C4—124no—
207C2×C4D3×D411yescharacteristic
208C2×C4C23×D311yescharacteristic
209C8—21no—
210C8—21no—
211C8—62no—
212C2×C3—22no—
213C2×C3—82no—
214C2×C3#53411yescharacteristic
215C2×C3C2×#13511yescharacteristic
216C2×C3#57511yescharacteristic
217D3—42no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—62no—
224C22—62no—
225C22—62no—
226C22—124no—
227C22#91212yes—
228C22C2×D3×D411yescharacteristic
229C4—21no—
230C4—42no—
231C4—62no—
232C4—122no—
233C4—122no—
234C4#100411yescharacteristic
235C4#100911yescharacteristic
236C3#319111yescharacteristic, complemented, Sylow
237C2—22no—
238C2—82no—
239C2—122no—
240C2#560511yescharacteristic
241C2C2×#91211yescharacteristic
242C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3191
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #5606C2semidirect
2. C3×#466C2semidirect
3. #5158C2semidirect
4. #5116C2semidirect
5. C2×(C2×D3)⋊C8C2semidirect
6. #5041C2semidirect
7. #4888C2semidirect
8. #4784C2semidirect
9. #4783C2semidirect
13. C2×C3×D8C22semidirect
19. C2×#214C22semidirect
22. C2×Dic12C22semidirect
27. C3×#107C22semidirect
28. C3×#102C22semidirect
29. #832C22semidirect
32. #818C22semidirect
33. #816C22semidirect
34. #810C22semidirect
35. #808C22semidirect
40. #466D3semidirect
82. C2×D8C2×D3semidirect
95. #107C2×D3semidirect
100. #102C2×D3semidirect
236. C3#3191semidirect

Maximal subgroups

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

subgroupindex
#56062
C3×#4662
#51582
#51162
C2×(C2×D3)⋊C82
#50412
#48882
#47842
#47832
#31913