Finite Group Databasebeta

← #84123

Subgroups — #84123

Order 384.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C3×#585C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic, complemented
3C2×#951C212yesmaximal, complemented
4#5246C214yesmaximal, complemented
5#5242C214yesmaximal, complemented
6#5237C211yesmaximal, characteristic, complemented
7#5236C212yesmaximal, complemented
8#3022—31nomaximal, Sylow
9C3×2−1+4—22no—
10C3×2−1+4C2212yes—
11C3×H(5,𝔽2)—22no—
12C3×H(5,𝔽2)—22no—
13C3×H(5,𝔽2)C2212yes—
14C2×C3×#41—24no—
15C2×C3×#41—22no—
16C2×C3×#41C2212yes—
17C2×C3×#41C2211yescharacteristic
18C22×Dic6C2211yescharacteristic, complemented
19#951—24no—
20#951C2218yescomplemented
21#944C2212yescomplemented
22C2×#216—22no—
23#936C2212yescomplemented
24#935—22no—
25C2×#214—22no—
26#926C2211yescharacteristic, complemented
27#924—22no—
28C2×#208C2212yescomplemented
29#837C2218yescomplemented
30#836C2212yescomplemented
31#833C2212yescomplemented
32#585D311yescharacteristic
33C2×#137—32no—
34#536—31no—
35#456—34no—
36#455—34no—
37#421—32no—
38#413—31no—
39C3×#41—24no—
40C3×#41—28no—
41C3×#41—24no—
42C3×#41—24no—
43C3×#41—24no—
44C3×#41—22no—
45C3×#41—22no—
46C3×#41—24no—
47C3×#41—22no—
48C3×#41—22no—
49C3×#41D414yes—
50C3×#41C2314yes—
51C2×C3×Q8—22no—
52C2×C3×Q8—24no—
53C2×C3×Q8D412yes—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×D4—28no—
56C2×C3×D4—22no—
57C2×C3×D4—24no—
58C2×C3×D4—22no—
59C2×C3×D4—24no—
60C2×C3×D4D412yes—
61C2×C3×D4D412yes—
62C2×C3×D4C2311yescharacteristic
63C22×C3×C4—24no—
64C22×C3×C4—22no—
65C22×C3×C4D412yes—
66C22×C3×C4C2311yescharacteristic
67C22×Dic3—21no—
68C2×Dic6—22no—
69C2×Dic6—22no—
70C2×Dic6—21no—
71C2×Dic6C2312yes—
72#217—24no—
73#217—21no—
74#216—28no—
75#214—28no—
76C4⋊Dic3—21no—
77Dic3⋊C4—24no—
78C4×Dic3C2312yes—
79#208—22no—
80#208C2314yes—
81C2×C3⋊C8—22no—
822−1+4—22no—
832−1+4C2×D312yes—
84H(5,𝔽2)—22no—
85H(5,𝔽2)—22no—
86H(5,𝔽2)C2×D312yes—
87C2×#41—24no—
88C2×#41—22no—
89C2×#41C2×D312yes—
90C2×#41C2×D311yescharacteristic
91C22×Q8—31no—
92#137—38no—
93#137—64no—
94C2×Q16—62no—
95C2×SD16—62no—
96C2×M16—32no—
97C4⋊Q8—32no—
98#124—32no—
99#123—62no—
100C22⋊Q8—62no—
101#117—31no—
102#104—38no—
103#101—32no—
104#100—32no—
105C23×C3—24no—
106C23×C3—22no—
107C23×C3C2×D412yes—
108C23×C3C2×D411yescharacteristic
109C3×Q8—24no—
110C3×Q8—22no—
111C3×Q8—22no—
112C3×Q8C2×D414yes—
113C3×D4—28no—
114C3×D4—24no—
115C3×D4—24no—
116C3×D4—24no—
117C3×D4—24no—
118C3×D4—22no—
119C3×D4—22no—
120C3×D4C2×D414yes—
121C2×C3×C4—24no—
122C2×C3×C4—28no—
123C2×C3×C4—22no—
124C2×C3×C4—24no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—24no—
128C2×C3×C4C2×D412yes—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2411yescharacteristic
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—42no—
136Dic6—24no—
137Dic6—41no—
138Dic6—41no—
139C3⋊C8—24no—
140#41—24no—
141#41—28no—
142#41—24no—
143#41—24no—
144#41—24no—
145#41—22no—
146#41—22no—
147#41—24no—
148#41—22no—
149#41—22no—
150#41#6714yes—
151#41C22×D314yes—
152C2×Q8—22no—
153C2×Q8—24no—
154C2×Q8—32no—
155C2×Q8—61no—
156C2×Q8—62no—
157C2×Q8—62no—
158C2×Q8#6712yes—
159C2×Q8C22×D311yescharacteristic
160C2×D4—28no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—22no—
164C2×D4—24no—
165C2×D4#6712yes—
166C2×D4#6712yes—
167C2×D4C22×D311yescharacteristic
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—61no—
171C22×C4#6712yes—
172C22×C4C22×D311yescharacteristic
173Q16—68no—
174SD16—68no—
175M16—34no—
176M16—62no—
177C2×C8—62no—
178C4⋊C4—61no—
179C4⋊C4—64no—
180C22⋊C4—61no—
181C22⋊C4—64no—
182C42—32no—
183C22×C3—22no—
184C22×C3—24no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—82no—
188C22×C3—84no—
189C22×C3#12012yes—
190C22×C3C22×D411yescharacteristic
191C3×C4—24no—
192C3×C4—22no—
193C3×C4#12011yescharacteristic
194C3×C4#12011yescharacteristic
195C3×C4C22×D412yes—
196Dic3—42no—
197Dic3—42no—
198C23—24no—
199C23—22no—
200C23C2×#6712yes—
201C23C2×#6711yescharacteristic
202Q8—24no—
203Q8—22no—
204Q8—22no—
205Q8—64no—
206Q8—121no—
207Q8—121no—
208Q8C2×#6714yes—
209D4—28no—
210D4—24no—
211D4—24no—
212D4—24no—
213D4—24no—
214D4—22no—
215D4—22no—
216D4C2×#6714yes—
217C2×C4—24no—
218C2×C4—28no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—122no—
227C2×C4C2×#6712yes—
228C2×C4C2×#6711yescharacteristic
229C2×C4C2×#6712yes—
230C2×C4C2×#6712yes—
231C2×C4C23×D311yescharacteristic
232C8—64no—
233C2×C3—21no—
234C2×C3—22no—
235C2×C3—42no—
236C2×C3—44no—
237C2×C3C2×#12011yescharacteristic
238C22—22no—
239C22—24no—
240C22—41no—
241C22—42no—
242C22—82no—
243C22—84no—
244C22#95312yes—
245C22C22×#6711yescharacteristic
246C4—24no—
247C4—22no—
248C4—122no—
249C4—122no—
250C4#95311yescharacteristic
251C4#95311yescharacteristic
252C4C22×#6712yes—
253C3#302211yescharacteristic, complemented, Sylow
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#95311yescharacteristic
subgroupG/H
centerC2C2×#953
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#585C2
socleC2×C3C2×#120
hypercenter#585D3
hypocenterC3#3022
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. C2×#951C2semidirect
4. #5246C2semidirect
5. #5242C2semidirect
6. #5237C2semidirect
7. #5236C2semidirect
18. C22×Dic6C22semidirect
20. #951C22semidirect
21. #944C22semidirect
23. #936C22semidirect
26. #926C22semidirect
28. C2×#208C22semidirect
29. #837C22semidirect
30. #836C22semidirect
31. #833C22semidirect
253. C3#3022semidirect

Maximal subgroups

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

subgroupindex
C3×#5852
#58252
C2×#9512
#52462
#52422
#52372
#52362
#30223