Finite Group Databasebeta

← #80357

Subgroups — #80357

Order 384.

430 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 115 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#5718C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C212yesmaximal, complemented
5#5158C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4777C212yesmaximal, complemented
11#4771C212yesmaximal, complemented
12#3332—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×#235—21no—
15C2×C3×D8C2212yes—
16C3×#127C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#940—21no—
19#939—22no—
20#938C2212yes—
21#938C2212yes—
22D4×Dic3—21no—
23C2×#214C2211yescharacteristic
24#924—21no—
25C2×ΓL(1,𝔽25)C2212yes—
26D3×C4⋊C4C2212yes—
27#883—22no—
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic
30C4×Dic6C2211yescharacteristic
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2212yes—
34#832C2211yescharacteristic
35#832C2212yes—
36(C2×D3)⋊C8C2212yes—
37#817C2212yes—
38#816C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#550—31no—
43#547—31no—
44#486—31no—
45#481—32no—
46#475—31no—
47#469—32no—
48#459D311yescharacteristic, complemented
49#449—32no—
50C4×SD16—31no—
51#436—31no—
52#417—32no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2312yes—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60C2×#67—22no—
61C22×Dic3—22no—
62#237—24no—
63#237—42no—
64#235—44no—
65C2×D12D411yescharacteristic
66C2×C4×D3—22no—
67C2×C4×D3C2312yes—
68C2×Dic6—21no—
69C2×Dic6C2311yescharacteristic
70C3×D8—42no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74C3×C42C2311yescharacteristic
75#217—22no—
76#217—22no—
77#214—22no—
78(C2×D3)⋊C4D412yes—
79C4⋊Dic3—21no—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2312yes—
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88ΓL(1,𝔽25)—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16—31no—
92C2×D8C2×D312yes—
93C2×M16—32no—
94C4⋊Q8—31no—
95#127—61no—
96#127C2×D311yescharacteristic
97#123—62no—
98C22⋊Q8—61no—
99#121—32no—
100#121—32no—
101#121—62no—
102C4×Q8—31no—
103C4×D4—31no—
104C4×D4—61no—
105C4×D4C2×D311yescharacteristic
106C2×C4⋊C4—32no—
107#106—32no—
108#106—31no—
109C4⋊C8C2×D311yescharacteristic
110#103—32no—
111#103—31no—
112#102—32no—
113#102—31no—
114#102—32no—
115#102C2×D312yes—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C23×C3—22no—
120C22×D3C2×D412yes—
121C3×D4—21no—
122C3×D4—22no—
123C3×D4—21no—
124C3×D4—22no—
125C3×D4—42no—
126C3×D4—81no—
127C2×C3×C4—21no—
128C2×C3×C4—42no—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—44no—
133#67—44no—
134#67—82no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—44no—
139C2×Dic3C2×D412yes—
140D12—22no—
141C4×D3—22no—
142C4×D3—22no—
143C4×D3—44no—
144Dic6—21no—
145Dic6—42no—
146Dic6#4112yescomplemented
147C3×C8—22no—
148C3⋊C8—21no—
149C3⋊C8—21no—
150#41—64no—
151#41—124no—
152#41—122no—
153C2×Q8—31no—
154C2×Q8—61no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—31no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4C22×D312yes—
162C2×D4C22×D311yescharacteristic
163C22×C4—21no—
164C22×C4—32no—
165C22×C4—62no—
166C22×C4—62no—
167SD16—62no—
168D8—42no—
169M16—122no—
170C2×C8—31no—
171C2×C8—31no—
172C2×C8C22×D312yes—
173C4⋊C4—31no—
174C4⋊C4—32no—
175C4⋊C4—31no—
176C4⋊C4—62no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4C22×D311yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—32no—
182C22⋊C4—62no—
183C22⋊C4—62no—
184C42—61no—
185C42C22×D311yescharacteristic
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—41no—
192C22×C3C22×D411yescharacteristic
193C2×D3—22no—
194C2×D3—22no—
195C2×D3—22no—
196C3×C4—21no—
197C3×C4—41no—
198C3×C4AGL(1,ℤ/8)12yes—
199C3×C4C2×#4111yescharacteristic
200C3×C4H(5,𝔽2)11yescharacteristic
201Dic3—22no—
202Dic3—22no—
203Dic3—42no—
204Dic3—41no—
205C23—21no—
206C23—22no—
207C23—32no—
208Q8—32no—
209Q8—61no—
210Q8—122no—
211D4—21no—
212D4—22no—
213D4—21no—
214D4—22no—
215D4—42no—
216D4—62no—
217D4—81no—
218D4—124no—
219D4—124no—
220D4—242no—
221C2×C4—21no—
222C2×C4—32no—
223C2×C4—42no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4—124no—
230C2×C4—124no—
231C2×C4D3×D411yescharacteristic
232C2×C4D3×D411yescharacteristic
233C2×C4C23×D311yescharacteristic
234C8—22no—
235C8—61no—
236C8—61no—
237C2×C3—82no—
238C2×C3—81no—
239C2×C3#54611yescharacteristic
240C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C3#57711yescharacteristic
242D3—42no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—62no—
250C22—62no—
251C22—62no—
252C22C2×D3×D411yescharacteristic
253C4—21no—
254C4—41no—
255C4—62no—
256C4—62no—
257C4—122no—
258C4—121no—
259C4#91112yes—
260C4#100411yescharacteristic
261C4D3×#4111yescharacteristic
262C3#333211yescharacteristic, complemented, Sylow
263C2—81no—
264C2—82no—
265C2—122no—
266C2#561211yescharacteristic
267C2C2×#91111yescharacteristic
268C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3332
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5551C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5158C2semidirect
6. #5070C2semidirect
7. C4×#214C2semidirect
8. #4851C2semidirect
9. #4838C2semidirect
10. #4777C2semidirect
11. #4771C2semidirect
48. #459D3semidirect
146. Dic6#41semidirect
262. C3#3332semidirect

Maximal subgroups

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

subgroupindex
#57182
#55512
C3×#4592
#51592
#51582
#50702
C4×#2142
#48512
#48382
#47772
#47712
#33323