Finite Group Databasebeta

← #80288

Subgroups — #80288

Order 384.

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

normal subgroups only

This group has 448 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
1#5838C212yesmaximal, complemented
2#5775C212yesmaximal, complemented
3C3×#454C211yesmaximal, characteristic, complemented
4#5200C212yesmaximal, complemented
5#5061C211yesmaximal, characteristic, complemented
6#5060C211yesmaximal, characteristic, complemented
7#4826C212yesmaximal, complemented
8#4823C212yesmaximal, complemented
9#4755C212yesmaximal, complemented
10#3289—31nomaximal, Sylow
11C3×2−1+4C2211yescharacteristic, complemented
12#1010C2212yes—
13#1004C2211yescharacteristic
14C3×AGL(1,ℤ/8)C2212yes—
15C3×#124C2211yescharacteristic
16#951—22no—
17#951C2212yescomplemented
18#950—22no—
19#950C2212yes—
20#948C2212yescomplemented
21C2×#216—22no—
22C2×#216C2212yescomplemented
23#942—22no—
24#942C2212yes—
25C2×#214C2212yes—
26#915—22no—
27D3×SD16—22no—
28#912—22no—
29#911—22no—
30#909C2212yes—
31#907C2212yes—
32#869C2211yescharacteristic, complemented
33C3×#104C2212yescomplemented
34C3×#101C2211yescharacteristic, complemented
35#833C2211yescharacteristic
36#824C2212yescomplemented
37#822C2212yescomplemented
38#805C2212yes—
39#803C2211yescharacteristic, complemented
40#578—32no—
41#577—32no—
42#497—31no—
43#473—32no—
44#471—32no—
45#456—32no—
46#455—31no—
47#454D311yescharacteristic, complemented
48#444—32no—
49C3×#41—22no—
50C3×#41—41no—
51C3×#41C2312yes—
52C2×C3×Q8—22no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C2×#67—41no—
56D3×Q8—22no—
57#237—22no—
58#237—22no—
59#237—42no—
60D3×D4—22no—
61#235—22no—
62#235C2312yes—
63C2×Dic6—22no—
64C2×Dic6C2312yes—
65C3×SD16—22no—
66C3×D8—22no—
67C3×M16C2312yes—
68C3×C22⋊C4—41no—
69C3×C42C2311yescharacteristic
70#216—22no—
71#216—22no—
72#216—22no—
73#216—22no—
74#216D412yes—
75#215—22no—
76#215D412yes—
77#214—22no—
78#214—22no—
79#214—22no—
80#214D412yes—
81#213D412yes—
82C4⋊Dic3—41no—
83#208—22no—
84#208C2312yes—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2312yes—
87Dic12—22no—
88#204—22no—
89ΓL(1,𝔽25)—22no—
90D3×C8—22no—
912−1+4—32no—
922−1+4C2×D311yescharacteristic, complemented
93H(5,𝔽2)—31no—
94#137—32no—
95#137—32no—
96#137—32no—
97#137—62no—
98#137—62no—
99#137—62no—
100AGL(1,ℤ/8)—62no—
101AGL(1,ℤ/8)C2×D312yes—
102#135—32no—
103#135—62no—
104#135—62no—
105C2×Q16—32no—
106C2×Q16—62no—
107C2×SD16—32no—
108C2×SD16—62no—
109#131—32no—
110#131—32no—
111C4⋊Q8—31no—
112#124C2×D311yescharacteristic
113#108—32no—
114#104—32no—
115#104C2×D312yescomplemented
116#101—32no—
117#101C2×D311yescharacteristic, complemented
118#100—31no—
119#97—31no—
120C23×C3—21no—
121C22×D3—41no—
122C3×Q8—22no—
123C3×Q8—41no—
124C3×Q8C2×D412yes—
125C3×D4—22no—
126C3×D4—22no—
127C3×D4—41no—
128C3×D4C2×D412yes—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—42no—
133C2×C3×C4C2411yescharacteristic
134#67—22no—
135#67—42no—
136#67—42no—
137C2×Dic3—22no—
138C2×Dic3—41no—
139C2×Dic3—42no—
140D12C2×D412yes—
141C4×D3—22no—
142C4×D3—42no—
143Dic6—22no—
144Dic6—22no—
145Dic6—42no—
146Dic6C2×D412yes—
147C3×C8—22no—
148C3⋊C8—22no—
149C3⋊C8C2×D412yes—
150C3⋊C8C2×D412yes—
151#41—22no—
152#41—32no—
153#41—41no—
154#41—62no—
155#41—62no—
156#41—62no—
157#41—122no—
158#41C22×D312yes—
159C2×Q8—22no—
160C2×Q8—32no—
161C2×Q8—62no—
162C2×Q8—62no—
163C2×Q8C22×D311yescharacteristic
164C2×D4—62no—
165C2×D4—121no—
166C2×D4C22×D311yescharacteristic
167Q16—32no—
168Q16—62no—
169Q16—62no—
170Q16—62no—
171Q16—62no—
172Q16—62no—
173SD16—22no—
174SD16—32no—
175SD16—32no—
176SD16—62no—
177SD16—62no—
178SD16—62no—
179SD16—62no—
180SD16—62no—
181D8—22no—
182D8—32no—
183M16—32no—
184M16—62no—
185M16—62no—
186M16C22×D312yes—
187C2×C8—32no—
188C2×C8—62no—
189C2×C8—62no—
190C4⋊C4—121no—
191C22⋊C4—41no—
192C42C22×D311yescharacteristic
193C22×C3—22no—
194C22×C3—41no—
195C22×C3—81no—
196C22×C3H(5,𝔽2)11yescharacteristic
197C2×D3—22no—
198C2×D3—161no—
199C3×C4—22no—
200C3×C4—41no—
201C3×C4—41no—
202C3×C4C22×D412yes—
203Dic3—22no—
204Dic3—42no—
205C23—21no—
206C23—121no—
207Q8—22no—
208Q8—32no—
209Q8—41no—
210Q8—62no—
211Q8—62no—
212Q8—122no—
213Q8D3×D412yes—
214D4—22no—
215D4—22no—
216D4—32no—
217D4—41no—
218D4—62no—
219D4—122no—
220D4—122no—
221D4D3×D412yes—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—122no—
229C2×C4—121no—
230C2×C4—122no—
231C2×C4C23×D311yescharacteristic
232C8—22no—
233C8—32no—
234C8—32no—
235C8—62no—
236C2×C3—21no—
237C2×C3—42no—
238C2×C3—81no—
239C2×C3D4211yescharacteristic
240D3—42no—
241C22—22no—
242C22—41no—
243C22—62no—
244C22—81no—
245C22—481no—
246C22#100411yescharacteristic
247C4—22no—
248C4—41no—
249C4—41no—
250C4—62no—
251C4—122no—
252C4C2×D3×D412yes—
253C3#328911yescharacteristic, complemented, Sylow
254C2—21no—
255C2—42no—
256C2—81no—
257C2—122no—
258C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#454C2
socleC2×C3D42
hypercenter#454D3
hypocenterC3#3289
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #5838C2semidirect
2. #5775C2semidirect
3. C3×#454C2semidirect
4. #5200C2semidirect
5. #5061C2semidirect
6. #5060C2semidirect
7. #4826C2semidirect
8. #4823C2semidirect
9. #4755C2semidirect
11. C3×2−1+4C22semidirect
17. #951C22semidirect
20. #948C22semidirect
22. C2×#216C22semidirect
32. #869C22semidirect
33. C3×#104C22semidirect
34. C3×#101C22semidirect
36. #824C22semidirect
37. #822C22semidirect
39. #803C22semidirect
47. #454D3semidirect
92. 2−1+4C2×D3semidirect
115. #104C2×D3semidirect
117. #101C2×D3semidirect
253. C3#3289semidirect

Maximal subgroups

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

subgroupindex
#58382
#57752
C3×#4542
#52002
#50612
#50602
#48262
#48232
#47552
#32893