Finite Group Databasebeta

← #81665

Subgroups — #81665

Order 384.

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

normal subgroups only

This group has 364 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#5728C211yesmaximal, characteristic, complemented
2#5680C211yesmaximal, characteristic
3C3×#489C211yesmaximal, characteristic, complemented
4#5081C211yesmaximal, characteristic, complemented
5#5060C211yesmaximal, characteristic, complemented
6#4816C212yesmaximal, complemented
7#4790C212yesmaximal, complemented
8#4780C212yesmaximal
9#4758C212yesmaximal, complemented
10#4707C211yesmaximal, characteristic, complemented
11#4703C211yesmaximal, characteristic, complemented
12#3172—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×#124C2211yescharacteristic
15#946—21no—
16#946C2211yescharacteristic
17Q8×Dic3—21no—
18Q8×Dic3C2211yescharacteristic
19C2×#216C2212yes—
20#938C2211yescharacteristic
21D4×Dic3C2211yescharacteristic
22C2×#214C2212yes—
23C2×ΓL(1,𝔽25)C2212yes—
24#897C2212yes—
25#892—21no—
26#890—21no—
27#887C2212yes—
28#883—21no—
29#67⋊C4—21no—
30#879—21no—
31#877—21no—
32C42×D3C2211yescharacteristic
33#870C2211yescharacteristic
34#869C2211yescharacteristic
35C3×#103C2212yes—
36C3×#102C2212yes—
37C3×#97C2211yescharacteristic
38#816C2212yes—
39#816C2212yes—
40C8×Dic3C2212yes—
41#808C2212yes—
42#807C2212yes—
43#803C2211yescharacteristic
44#533—31no—
45#529—31no—
46#501—31no—
47#497—31no—
48#491—31no—
49#489D311yescharacteristic, complemented
50#470—32no—
51#468—32no—
52C4×Q16—32no—
53C4×SD16—32no—
54C4×M16—31no—
55C2×C3×Q8—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4C2311yescharacteristic
58C2×#67—21no—
59C22×Dic3—21no—
60C2×C4×D3D411yescharacteristic
61C2×C4×D3D411yescharacteristic
62C2×C4×D3C2311yescharacteristic
63C2×Dic6C2312yes—
64C2×C3×C8C2312yes—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2312yes—
67C3×C22⋊C4—21no—
68C3×C22⋊C4—21no—
69C3×C42C2311yescharacteristic
70#217—21no—
71#217—21no—
72#216—24no—
73#214—24no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—22no—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—22no—
86C4×Dic3—21no—
87C4×Dic3—22no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C2×C3⋊C8C2312yes—
92ΓL(1,𝔽25)—24no—
93C2×Q16—32no—
94C2×SD16—32no—
95C2×M16—32no—
96C4⋊Q8—31no—
97C4⋊Q8C2×D311yescharacteristic
98#126—61no—
99#125—31no—
100#125—61no—
101#124C2×D311yescharacteristic
102#123—61no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—61no—
106#121—31no—
107C4×Q8—31no—
108C4×Q8—32no—
109C4×Q8—61no—
110C4×D4—31no—
111C4×D4—61no—
112#117—61no—
113#117—61no—
114C2×C42—31no—
115#107—32no—
116#106—32no—
117#103—32no—
118#103—32no—
119#103C2×D312yes—
120#102C2×D312yes—
121#97—31no—
122#97C2×D311yescharacteristic
123C4×C8—32no—
124C23×C3—21no—
125C22×D3C2×D411yescharacteristic
126C3×Q8—22no—
127C3×Q8—42no—
128C3×D4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2411yescharacteristic
134#67—42no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3—42no—
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141C2×Dic3C2×D411yescharacteristic
142C4×D3—22no—
143C4×D3—22no—
144C4×D3—41no—
145Dic6—22no—
146Dic6—22no—
147C3×C8—22no—
148C3⋊C8#4114yes—
149C2×Q8—21no—
150C2×Q8—32no—
151C2×Q8C22×D311yescharacteristic
152C2×D4—61no—
153C2×D4C22×D311yescharacteristic
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—61no—
158Q16—64no—
159SD16—64no—
160M16—64no—
161C2×C8—32no—
162C2×C8C22×D312yes—
163C4⋊C4—21no—
164C4⋊C4—31no—
165C4⋊C4—31no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—62no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4C22×D312yes—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—62no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C42—31no—
182C42—31no—
183C42—31no—
184C42—61no—
185C42—62no—
186C42C22×D311yescharacteristic
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3C22×D411yescharacteristic
191C2×D3—21no—
192C2×D3—21no—
193C2×D3—21no—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—41no—
197C3×C4—42no—
198C3×C4C2×#4112yes—
199Dic3—21no—
200Dic3—21no—
201Dic3—42no—
202Dic3—41no—
203Dic3—41no—
204Dic3#13712yes—
205C23—21no—
206C23—31no—
207Q8—22no—
208Q8—42no—
209Q8—62no—
210Q8—62no—
211D4—22no—
212D4—122no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—121no—
224C2×C4—122no—
225C2×C4D3×D411yescharacteristic
226C2×C4D3×D411yescharacteristic
227C2×C4C23×D311yescharacteristic
228C8—22no—
229C8—34no—
230C2×C3—81no—
231C2×C3#53211yescharacteristic
232C2×C3C2×#13711yescharacteristic
233C2×C3#57511yescharacteristic
234D3—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—61no—
239C22—61no—
240C22—61no—
241C22C2×D3×D411yescharacteristic
242C4—21no—
243C4—21no—
244C4—32no—
245C4—41no—
246C4—42no—
247C4—61no—
248C4—61no—
249C4—122no—
250C4—121no—
251C4—121no—
252C4D3×#4112yes—
253C3#317211yescharacteristic, complemented, Sylow
254C2—81no—
255C2—121no—
256C2#566811yescharacteristic
257C2#577311yescharacteristic
258C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#489C2
socleC22×C3C22×D4
prime radical#938C22
hypercenter#489D3
hypocenterC3#3172
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#489C2semidirect
4. #5081C2semidirect
5. #5060C2semidirect
6. #4816C2semidirect
7. #4790C2semidirect
9. #4758C2semidirect
10. #4707C2semidirect
11. #4703C2semidirect
49. #489D3semidirect
253. C3#3172semidirect

Maximal subgroups

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

subgroupindex
#57282
#56802
C3×#4892
#50812
#50602
#48162
#47902
#47802
#47582
#47072
#47032
#31723