Finite Group Databasebeta

← #83646

Subgroups — #83646

Order 384.

352 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 123 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×#555C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic, complemented
3#5091C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5041C212yesmaximal, complemented
6#5035C212yesmaximal, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8D4×C3⋊C8C211yesmaximal, characteristic, complemented
9#5007C211yesmaximal, characteristic, complemented
10#4998C212yesmaximal, complemented
11#4996C212yesmaximal, complemented
12#3340—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×#124—22no—
15C3×#121—21no—
16C3×#121C2212yes—
17C3×C4×D4C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic
19C3×#117—21no—
20C3×#117C2212yes—
21#936—21no—
22C2×#214C2211yescharacteristic
23#926C2212yes—
24#924C2212yes—
25C22×C3⋊C8C2212yes—
26#887—21no—
27#879—22no—
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30#832C2212yes—
31#832C2211yescharacteristic
32#830C2212yes—
33#810C2211yescharacteristic
34#810C2212yes—
35#808C2212yes—
36#808C2211yescharacteristic
37#808C2212yes—
38#807C2212yes—
39#807C2212yes—
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#555D311yescharacteristic
44#498—31no—
45#485—32no—
46#482—32no—
47#477—31no—
48#474—31no—
49#465—32no—
50C4×SD16—31no—
51C8×D4—31no—
52#427—32no—
53C2×C3×Q8—21no—
54C2×C3×D4—22no—
55C2×C3×D4D411yescharacteristic
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4C2312yes—
59C2×Dic6—21no—
60C2×Dic6C2311yescharacteristic
61C3×C4⋊C4—21no—
62C3×C4⋊C4D411yescharacteristic
63C3×C4⋊C4C2311yescharacteristic
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—22no—
66C3×C22⋊C4—22no—
67C3×C22⋊C4—21no—
68C3×C22⋊C4D412yes—
69C3×C42—22no—
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—22no—
73#214—22no—
74C4⋊Dic3—21no—
75C4⋊Dic3C2311yescharacteristic
76C4⋊Dic3C2312yes—
77Dic3⋊C4—21no—
78Dic3⋊C4—22no—
79C4×Dic3—21no—
80C4×Dic3—22no—
81C2×C3⋊C8—24no—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2312yes—
84C2×C3⋊C8C2311yescharacteristic
85C2×SD16—31no—
86C22×C8—32no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D311yescharacteristic
89#126—62no—
90#124—22no—
91#124—61no—
92C22⋊Q8—32no—
93#121—21no—
94#121C2×D312yes—
95C4×Q8—31no—
96C4×Q8—61no—
97C4×D4C2×D311yescharacteristic
98C4×D4C2×D311yescharacteristic
99#117—21no—
100#117—32no—
101#117C2×D312yes—
102#107—32no—
103#107—32no—
104#106—31no—
105#106—32no—
106#106—32no—
107C4⋊C8—31no—
108#103—31no—
109#103—32no—
110#102—31no—
111#102—32no—
112C22⋊C8—32no—
113C4×C8—31no—
114C23×C3—21no—
115C23×C3C2×D412yes—
116C3×Q8—81no—
117C3×D4—22no—
118C3×D4—21no—
119C3×D4—21no—
120C3×D4—82no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—42no—
126C2×C3×C4C2×D412yes—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2411yescharacteristic
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133Dic6—21no—
134Dic6—21no—
135Dic6—42no—
136C3⋊C8—22no—
137C3⋊C8—21no—
138C3⋊C8#4112yes—
139C2×Q8—21no—
140C2×Q8—31no—
141C2×Q8—61no—
142C2×D4—22no—
143C2×D4#6711yescharacteristic
144C2×D4C22×D311yescharacteristic
145C22×C4—21no—
146C22×C4C22×D312yes—
147SD16—62no—
148C2×C8—31no—
149C2×C8—31no—
150C2×C8—32no—
151C2×C8—64no—
152C4⋊C4—21no—
153C4⋊C4—32no—
154C4⋊C4—31no—
155C4⋊C4—61no—
156C4⋊C4—62no—
157C4⋊C4—61no—
158C4⋊C4#6711yescharacteristic
159C4⋊C4C22×D312yes—
160C4⋊C4C22×D311yescharacteristic
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—21no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4#6712yes—
167C42—22no—
168C42—62no—
169C42—61no—
170C42C22×D311yescharacteristic
171C22×C3—22no—
172C22×C3—22no—
173C22×C3—22no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3C22×D411yescharacteristic
178C3×C4—22no—
179C3×C4—21no—
180C3×C4—41no—
181C3×C4—42no—
182C3×C4#13512yes—
183C3×C4C2×#4111yescharacteristic
184C3×C42−1+411yescharacteristic
185Dic3—41no—
186Dic3—42no—
187Dic3—41no—
188C23—21no—
189C23C2×#6712yes—
190Q8—61no—
191Q8—61no—
192Q8—81no—
193Q8—122no—
194D4—22no—
195D4—21no—
196D4—21no—
197D4—82no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—61no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4C2×#6712yes—
207C2×C4C2×#6711yescharacteristic
208C2×C4C2×#6711yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—32no—
211C8—62no—
212C8—61no—
213C2×C3—42no—
214C2×C3—81no—
215C2×C3#54711yescharacteristic
216C2×C3C2×#13511yescharacteristic
217C2×C3#57711yescharacteristic
218C22—22no—
219C22—22no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22C22×#6711yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—122no—
230C4—121no—
231C4—121no—
232C4#95012yes—
233C4D3×#4111yescharacteristic
234C4#101011yescharacteristic
235C3#334011yescharacteristic, complemented, Sylow
236C2—42no—
237C2—81no—
238C2C2×#95011yescharacteristic
239C2#582411yescharacteristic
240C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#555C2
socleC22×C3C22×D4
hypercenter#555D3
hypocenterC3#3340
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
2. #5542C2semidirect
3. #5091C2semidirect
4. #5080C2semidirect
5. #5041C2semidirect
6. #5035C2semidirect
7. C4×#214C2semidirect
8. D4×C3⋊C8C2semidirect
9. #5007C2semidirect
10. #4998C2semidirect
11. #4996C2semidirect
235. C3#3340semidirect

Maximal subgroups

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

subgroupindex
C3×#5552
#55422
#50912
#50802
#50412
#50352
C4×#2142
D4×C3⋊C82
#50072
#49982
#49962
#33403