Finite Group Databasebeta

← #80327

Subgroups — #80327

Order 384.

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

normal subgroups only

This group has 418 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#5713C211yesmaximal, characteristic, complemented
2#5553C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5154C212yesmaximal, complemented
5#5150C212yesmaximal, complemented
6#5073C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4787C212yesmaximal, complemented
11#4785C212yesmaximal, complemented
12#3399—31nomaximal, Sylow
13C2×#235—21no—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#940C2212yes—
18#936—21no—
19#936C2212yes—
20#935—22no—
21D4×Dic3—21no—
22D4×Dic3C2212yes—
23C2×#214C2211yescharacteristic
24C2×#213C2212yes—
25C2×#213C2212yes—
26#930—21no—
27C2×#204C2212yes—
28#893C2212yes—
29#884—22no—
30#874C2211yescharacteristic
31C4×Dic6C2211yescharacteristic
32C4×Dic6C2211yescharacteristic
33C3×C4⋊C8C2211yescharacteristic
34C3×#102C2212yes—
35#832C2211yescharacteristic
36#821C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#559—31no—
43#550—31no—
44#496—32no—
45#493—31no—
46#463—32no—
47#460—31no—
48#459—32no—
49#459D311yescharacteristic, complemented
50#446—31no—
51#442—32no—
52C4×SD16—31no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2311yescharacteristic
56C2×C3×D4C2312yes—
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60C22×Dic3—22no—
61#235—44no—
62C2×D12—21no—
63C2×D12—21no—
64C2×D12C2312yes—
65C2×C4×D3—22no—
66C2×Dic6D411yescharacteristic
67C2×Dic6C2311yescharacteristic
68C3×D8—42no—
69C2×C3×C8C2312yes—
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C42C2311yescharacteristic
73#217—22no—
74#217—22no—
75#217—22no—
76#214—22no—
77#213—24no—
78#213—42no—
79(C2×D3)⋊C4—22no—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4D412yes—
84C4×Dic3—21no—
85C4×Dic3C2312yes—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88#204—42no—
89C2×#41—61no—
90C2×SD16—31no—
91C2×SD16—32no—
92C2×D8—32no—
93C2×D8—32no—
94C2×D8C2×D312yes—
95#127—32no—
96#127—31no—
97#127C2×D311yescharacteristic
98#124—32no—
99#124—61no—
100#123—62no—
101#121—62no—
102#121—61no—
103C4×Q8—31no—
104C4×Q8—31no—
105C4×D4—32no—
106C4×D4—32no—
107C4×D4—61no—
108C4×D4C2×D311yescharacteristic
109#106—31no—
110C4⋊C8—32no—
111C4⋊C8C2×D311yescharacteristic
112#103—31no—
113#102—31no—
114#102—32no—
115#102C2×D312yes—
116#97—32no—
117C4×C8—31no—
118C23×C3—22no—
119C23×C3—21no—
120C22×D3—22no—
121C3×D4—22no—
122C3×D4—21no—
123C3×D4—22no—
124C3×D4—21no—
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—82no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—82no—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×D412yes—
139D12—22no—
140D12—22no—
141D12—42no—
142D12—81no—
143C4×D3—44no—
144Dic6—22no—
145Dic6—21no—
146Dic6—21no—
147C3×C8—22no—
148C3⋊C8—21no—
149C3⋊C8#4112yescomplemented
150#41—124no—
151C2×Q8—31no—
152C2×Q8—31no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—32no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—61no—
159C2×D4—61no—
160C2×D4C22×D311yescharacteristic
161C2×D4C22×D312yes—
162C22×C4—21no—
163C22×C4—62no—
164C22×C4—62no—
165SD16—62no—
166SD16—122no—
167D8—42no—
168D8—64no—
169D8—122no—
170C2×C8—31no—
171C2×C8—31no—
172C2×C8C22×D312yes—
173C4⋊C4—31no—
174C4⋊C4—32no—
175C4⋊C4—31no—
176C4⋊C4—61no—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—62no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C42—32no—
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—42no—
194C2×D3—42no—
195C2×D3—42no—
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—41no—
204Dic3—41no—
205C23—21no—
206C23—22no—
207C23—62no—
208Q8—62no—
209Q8—61no—
210Q8—61no—
211D4—21no—
212D4—22no—
213D4—22no—
214D4—21no—
215D4—42no—
216D4—62no—
217D4—62no—
218D4—81no—
219D4—124no—
220D4—122no—
221D4—241no—
222D4—242no—
223C2×C4—21no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—42no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—124no—
230C2×C4—242no—
231C2×C4D3×D411yescharacteristic
232C2×C4D3×D411yescharacteristic
233C2×C4C23×D311yescharacteristic
234C8—22no—
235C8—32no—
236C8—61no—
237C2×C3—81no—
238C2×C3—82no—
239C2×C3#55011yescharacteristic
240C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C3#57611yescharacteristic
242D3—82no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—122no—
250C22—122no—
251C22—122no—
252C22C2×D3×D411yescharacteristic
253C4—21no—
254C4—41no—
255C4—62no—
256C4—62no—
257C4—121no—
258C4—121no—
259C4#91112yes—
260C4#100411yescharacteristic
261C4D3×#4111yescharacteristic
262C3#339911yescharacteristic, complemented, Sylow
263C2—81no—
264C2—82no—
265C2—242no—
266C2#559911yescharacteristic
267C2C2×#91111yescharacteristic
268C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3399
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. #5713C2semidirect
2. #5553C2semidirect
3. C3×#459C2semidirect
4. #5154C2semidirect
5. #5150C2semidirect
6. #5073C2semidirect
7. C4×#214C2semidirect
8. #4848C2semidirect
9. #4831C2semidirect
10. #4787C2semidirect
11. #4785C2semidirect
49. #459D3semidirect
149. C3⋊C8#41semidirect
262. C3#3399semidirect

Maximal subgroups

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

subgroupindex
#57132
#55532
C3×#4592
#51542
#51502
#50732
C4×#2142
#48482
#48312
#47872
#47852
#33993