Finite Group Databasebeta

← #80323

Subgroups — #80323

Order 384.

384 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 117 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
2D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic
4#5153C212yesmaximal, complemented
5#5152C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4765C212yesmaximal, complemented
11#4761C212yesmaximal, complemented
12#3373—31nomaximal, Sylow
13C22×Dic6—21no—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#936—21no—
18#936C2212yes—
19#935—22no—
20D4×Dic3—21no—
21D4×Dic3C2212yes—
22D4×Dic3C2212yes—
23C2×#214C2211yescharacteristic
24#924—21no—
25#888C2212yes—
26#878—22no—
27#869C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#102C2212yes—
32#832C2211yescharacteristic
33#832C2212yes—
34#832C2212yes—
35#817C2212yes—
36#816C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#559—31no—
43D4×Q8—31no—
44#489—32no—
45#486—31no—
46#476—32no—
47#475—31no—
48#463—32no—
49#459D311yescharacteristic
50#446—31no—
51#442—32no—
52C4×SD16—31no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2312yes—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C22×Dic3—22no—
59C22×Dic3—22no—
60C2×Dic6—21no—
61C2×Dic6—22no—
62C2×Dic6—44no—
63C2×Dic6D411yescharacteristic
64C2×Dic6C2311yescharacteristic
65C3×D8—24no—
66C2×C3×C8C2312yes—
67C3×C4⋊C4C2311yescharacteristic
68C3×C22⋊C4—21no—
69C3×C42C2311yescharacteristic
70#217—22no—
71#217—22no—
72#217—22no—
73#217—22no—
74#214—22no—
75C4⋊Dic3—21no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3C2311yescharacteristic
78C4⋊Dic3C2312yes—
79Dic3⋊C4—21no—
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3—21no—
83C4×Dic3C2312yes—
84C2×C3⋊C8C2311yescharacteristic
85C2×C3⋊C8C2311yescharacteristic
86C22×Q8—61no—
87C2×SD16—31no—
88C2×D8C2×D312yes—
89C4⋊Q8—32no—
90C4⋊Q8—31no—
91#127C2×D311yescharacteristic
92#124—32no—
93#124—61no—
94#123—62no—
95C22⋊Q8—62no—
96C22⋊Q8—61no—
97C4×Q8—31no—
98C4×Q8—31no—
99C4×D4—32no—
100C4×D4—32no—
101C4×D4—61no—
102C4×D4C2×D311yescharacteristic
103#106—32no—
104#106—31no—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—32no—
108#103—31no—
109#102—31no—
110#102—32no—
111#102—32no—
112#102C2×D312yes—
113#97—32no—
114C4×C8—31no—
115C23×C3—21no—
116C23×C3—22no—
117C3×D4—22no—
118C3×D4—21no—
119C3×D4—22no—
120C3×D4—42no—
121C3×D4—81no—
122C3×D4#4112yes—
123C2×C3×C4—21no—
124C2×C3×C4—42no—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—44no—
132C2×Dic3—82no—
133C2×Dic3C2×D412yes—
134C2×Dic3C2×D412yes—
135Dic6—22no—
136Dic6—21no—
137Dic6—21no—
138Dic6—44no—
139Dic6—42no—
140C3×C8—22no—
141C3⋊C8—21no—
142C3⋊C8—21no—
143C2×Q8—31no—
144C2×Q8—31no—
145C2×Q8—62no—
146C2×Q8—61no—
147C2×Q8—124no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C22×D312yes—
151C2×D4C22×D311yescharacteristic
152C22×C4—21no—
153C22×C4—62no—
154C22×C4—62no—
155SD16—62no—
156D8—24no—
157C2×C8—31no—
158C2×C8—31no—
159C2×C8C22×D312yes—
160C4⋊C4—31no—
161C4⋊C4—32no—
162C4⋊C4—32no—
163C4⋊C4—31no—
164C4⋊C4—62no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C4⋊C4C22×D311yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C22⋊C4—62no—
173C42—32no—
174C42—61no—
175C42C22×D311yescharacteristic
176C22×C3—22no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3—42no—
182C22×C3C22×D411yescharacteristic
183C3×C4—21no—
184C3×C4—41no—
185C3×C4AGL(1,ℤ/8)12yes—
186C3×C4C2×#4111yescharacteristic
187C3×C42−1+411yescharacteristic
188Dic3—22no—
189Dic3—22no—
190Dic3—41no—
191Dic3—42no—
192Dic3—41no—
193C23—21no—
194C23—22no—
195Q8—62no—
196Q8—61no—
197Q8—61no—
198Q8—124no—
199Q8—122no—
200D4—22no—
201D4—21no—
202D4—22no—
203D4—42no—
204D4—81no—
205D4#23712yes—
206C2×C4—21no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—42no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—62no—
213C2×C4—124no—
214C2×C4—242no—
215C2×C4D3×D411yescharacteristic
216C2×C4D3×D411yescharacteristic
217C2×C4C23×D311yescharacteristic
218C8—22no—
219C8—61no—
220C8—61no—
221C2×C3—42no—
222C2×C3—82no—
223C2×C3#54811yescharacteristic
224C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
225C2×C3#57711yescharacteristic
226C22—22no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C2×D3×D411yescharacteristic
233C4—21no—
234C4—41no—
235C4—62no—
236C4—62no—
237C4—122no—
238C4—121no—
239C4—121no—
240C4#91112yes—
241C4C2×#23711yescharacteristic
242C4#101011yescharacteristic
243C3#337311yescharacteristic, complemented, Sylow
244C2—42no—
245C2—82no—
246C2#559811yescharacteristic
247C2C2×#91111yescharacteristic
248C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3373
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. D4×Dic6C2semidirect
4. #5153C2semidirect
5. #5152C2semidirect
6. #5070C2semidirect
7. C4×#214C2semidirect
8. #4851C2semidirect
9. #4831C2semidirect
10. #4765C2semidirect
11. #4761C2semidirect
243. C3#3373semidirect

Maximal subgroups

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

subgroupindex
#57132
D4×Dic62
C3×#4592
#51532
#51522
#50702
C4×#2142
#48512
#48312
#47652
#47612
#33733