Finite Group Databasebeta

← #83659

Subgroups — #83659

Order 384.

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

normal subgroups only

This group has 340 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×#556C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic
3#5094C211yesmaximal, characteristic, complemented
4#5078C211yesmaximal, characteristic, complemented
5#5051C212yesmaximal, complemented
6#5045C212yesmaximal, complemented
7C4×#216C211yesmaximal, characteristic, complemented
8#5020C211yesmaximal, characteristic, complemented
9D4×C3⋊C8C211yesmaximal, characteristic
10#4998C212yesmaximal, complemented
11#4996C212yesmaximal, complemented
12#3342—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×#126—22no—
15C3×#124—21no—
16C3×C22⋊Q8C2212yes—
17C3×C4×Q8—21no—
18C3×C4×Q8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20C3×#117C2212yes—
21C2×#216C2211yescharacteristic
22#936—21no—
23#926C2212yes—
24#924C2212yes—
25C22×C3⋊C8C2212yes—
26#887—21no—
27#879—22no—
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30#835C2212yes—
31#835C2211yescharacteristic
32#830C2212yes—
33#810C2211yescharacteristic
34#810C2212yes—
35#808C2212yes—
36#808C2212yes—
37#807C2212yes—
38#807C2211yescharacteristic
39#807C2212yes—
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#556D311yescharacteristic
44#500—31no—
45#485—32no—
46#485—32no—
47#475—31no—
48#475—31no—
49#467—32no—
50C4×Q16—31no—
51C8×D4—31no—
52#427—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×D4D411yescharacteristic
56C22×C3×C4C2312yes—
57C2×Dic6—21no—
58C2×Dic6C2311yescharacteristic
59C3×C4⋊C4—22no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4—21no—
62C3×C4⋊C4D411yescharacteristic
63C3×C4⋊C4C2312yes—
64C3×C4⋊C4C2311yescharacteristic
65C3×C22⋊C4—22no—
66C3×C22⋊C4—22no—
67C3×C22⋊C4D412yes—
68C3×C42—22no—
69C3×C42—21no—
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—22no—
73#216—22no—
74C4⋊Dic3—21no—
75C4⋊Dic3C2312yes—
76C4⋊Dic3C2311yescharacteristic
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×Q16—31no—
86C22×C8—32no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D311yescharacteristic
89#126—22no—
90#126—62no—
91#124—21no—
92#124—61no—
93C22⋊Q8—32no—
94C22⋊Q8C2×D312yes—
95C4×Q8—21no—
96C4×Q8—31no—
97C4×Q8—61no—
98C4×Q8C2×D311yescharacteristic
99C4×D4C2×D311yescharacteristic
100#117—32no—
101#117C2×D312yes—
102#107—31no—
103#107—32no—
104#107—32no—
105#106—32no—
106#106—32no—
107C4⋊C8—31no—
108#103—31no—
109#103—32no—
110#103—31no—
111#103—32no—
112C22⋊C8—32no—
113C4×C8—31no—
114C23×C3C2×D412yes—
115C3×Q8—21no—
116C3×Q8—21no—
117C3×Q8—42no—
118C3×D4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4C2×D412yes—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131Dic6—21no—
132Dic6—21no—
133Dic6—42no—
134C3⋊C8—22no—
135C3⋊C8—21no—
136C3⋊C8#4112yes—
137C2×Q8—21no—
138C2×Q8—31no—
139C2×Q8—61no—
140C2×Q8C22×D311yescharacteristic
141C2×D4#6711yescharacteristic
142C22×C4C22×D312yes—
143Q16—62no—
144C2×C8—31no—
145C2×C8—31no—
146C2×C8—32no—
147C2×C8—64no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—31no—
152C4⋊C4—32no—
153C4⋊C4—61no—
154C4⋊C4—62no—
155C4⋊C4—61no—
156C4⋊C4#6711yescharacteristic
157C4⋊C4C22×D311yescharacteristic
158C4⋊C4C22×D312yes—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4#6712yes—
164C42—22no—
165C42—21no—
166C42—62no—
167C42—61no—
168C42C22×D311yescharacteristic
169C22×C3—22no—
170C22×C3—22no—
171C22×C3—22no—
172C22×C3C22×D411yescharacteristic
173C3×C4—22no—
174C3×C4—21no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4#13512yes—
179C3×C4C2×#4111yescharacteristic
180C3×C42−1+411yescharacteristic
181Dic3—41no—
182Dic3—42no—
183Dic3—41no—
184C23C2×#6712yes—
185Q8—21no—
186Q8—21no—
187Q8—42no—
188Q8—61no—
189Q8—61no—
190Q8—122no—
191D4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—61no—
198C2×C4—62no—
199C2×C4—61no—
200C2×C4C2×#6712yes—
201C2×C4C2×#6711yescharacteristic
202C2×C4C2×#6711yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—32no—
205C8—62no—
206C8—61no—
207C2×C3—42no—
208C2×C3#54711yescharacteristic
209C2×C3C2×#13511yescharacteristic
210C2×C3#57811yescharacteristic
211C22—22no—
212C22—22no—
213C22—22no—
214C22C22×#6711yescharacteristic
215C4—22no—
216C4—21no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—122no—
221C4—121no—
222C4—121no—
223C4#95012yes—
224C4D3×#4111yescharacteristic
225C4#101011yescharacteristic
226C3#334211yescharacteristic, complemented, Sylow
227C2—42no—
228C2C2×#95011yescharacteristic
229C2#582411yescharacteristic
230C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#556C2
socleC22×C3C22×D4
prime radicalC2×C3×D4D4
hypercenter#556D3
hypocenterC3#3342
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
3. #5094C2semidirect
4. #5078C2semidirect
5. #5051C2semidirect
6. #5045C2semidirect
7. C4×#216C2semidirect
8. #5020C2semidirect
10. #4998C2semidirect
11. #4996C2semidirect
226. C3#3342semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#55422
#50942
#50782
#50512
#50452
C4×#2162
#50202
D4×C3⋊C82
#49982
#49962
#33423