Finite Group Databasebeta

← #78804

Subgroups — #78804

Order 384.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#5816C211yesmaximal, characteristic
2#5504C211yesmaximal, characteristic
3C2×C3×#103C211yesmaximal, characteristic, complemented
4#5225C211yesmaximal, characteristic, complemented
5#5110C211yesmaximal, characteristic
6#5106C211yesmaximal, characteristic, complemented
7#4964C211yesmaximal, characteristic, complemented
8#4815C212yesmaximal, complemented
9#4814C212yesmaximal, complemented
10#4811C212yesmaximal, complemented
11#4805C212yesmaximal, complemented
12#3097—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×C3×C8C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17#947—21no—
18#946C2212yes—
19#946C2212yes—
20#944—21no—
21#929—22no—
22C4×#67—22no—
23#926C2211yescharacteristic
24#926C2211yescharacteristic
25C2×#208C2211yescharacteristic
26#893—21no—
27#892C2212yes—
28D3×C4⋊C4C2212yes—
29#887—21no—
30C3×#103C2212yes—
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34(C2×D3)⋊C8C2212yes—
35(C2×D3)⋊C8C2212yes—
36#818C2211yescharacteristic
37#818C2211yescharacteristic
38#817C2211yescharacteristic
39#817C2211yescharacteristic
40#808C2212yes—
41#807C2212yes—
42#536—31no—
43#520—31no—
44#485—32no—
45#485—32no—
46#484—32no—
47#483—32no—
48#428—31no—
49#427—31no—
50#419—31no—
51C2×#103D311yescharacteristic, complemented
52#408—31no—
53C2×C3×Q8—21no—
54C2×C3×Q8—22no—
55C2×C3×Q8—22no—
56C2×C3×Q8C2312yes—
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4C2311yescharacteristic
60C2×#67D412yes—
61#235—22no—
62#235—22no—
63C2×D12D411yescharacteristic
64C2×C4×D3—22no—
65C2×C4×D3C2312yes—
66C2×Dic6D411yescharacteristic
67C2×C3×C8—22no—
68C2×C3×C8C2311yescharacteristic
69C2×C3×C8C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4C2312yes—
72#217—21no—
73#217—21no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77C4⋊Dic3C2311yescharacteristic
78C4⋊Dic3C2311yescharacteristic
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2311yescharacteristic
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84C4×Dic3—21no—
85C4×Dic3—21no—
86C4×Dic3—22no—
87#208—22no—
88C2×C3⋊C8C2312yes—
89C2×#41—31no—
90C22×Q8C2×D311yescharacteristic
91C2×M16—31no—
92C22×C8C2×D311yescharacteristic
93C4⋊Q8—61no—
94#124—61no—
95#123—62no—
96C22⋊Q8—32no—
97C22⋊Q8—32no—
98C4×Q8—61no—
99C4×D4—61no—
100C4×D4—62no—
101#117—31no—
102#117—31no—
103#117—32no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4C2×D311yescharacteristic
106#107—31no—
107#107—31no—
108#107—32no—
109#106—31no—
110#106—31no—
111#106—32no—
112#103—32no—
113#103—32no—
114#103C2×D312yes—
115#103C2×D312yes—
116C22⋊C8—32no—
117C22⋊C8—32no—
118C23×C3C2×D411yescharacteristic
119C22×D3C2×D412yes—
120C3×Q8—22no—
121C3×Q8—22no—
122C3×Q8—42no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—42no—
126C2×C3×C4—81no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4#4112yes—
129C2×C3×C4#4112yes—
130C2×C3×C4C2411yescharacteristic
131#67—24no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3C2×D412yes—
137D12—22no—
138C4×D3—22no—
139C4×D3—22no—
140C4×D3—82no—
141Dic6—22no—
142C3×C8—21no—
143C3×C8—21no—
144C3⋊C8—22no—
145#41—62no—
146#41—62no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—31no—
151C2×Q8C22×D312yes—
152C2×D4—32no—
153C2×D4—31no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—32no—
157C22×C4—62no—
158C22×C4C22×D311yescharacteristic
159M16—62no—
160C2×C8—22no—
161C2×C8—32no—
162C2×C8C22×D311yescharacteristic
163C2×C8C22×D311yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—31no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4C22×D312yes—
173C22⋊C4—62no—
174C22⋊C4—62no—
175C22⋊C4—62no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C42—62no—
179C42—61no—
180C42—61no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3#12312yes—
184C22×C3C22×D411yescharacteristic
185C2×D3—22no—
186C2×D3—22no—
187C2×D3—22no—
188C3×C4—42no—
189C3×C4—42no—
190C3×C4#12312yes—
191C3×C4C2×#4111yescharacteristic
192C3×C4C2×#4111yescharacteristic
193Dic3—22no—
194Dic3—41no—
195Dic3—41no—
196Dic3—41no—
197Dic3—41no—
198C23—32no—
199C23D3×D411yescharacteristic
200Q8—22no—
201Q8—22no—
202Q8—42no—
203Q8—62no—
204D4—64no—
205D4—62no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—32no—
209C2×C4—42no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—81no—
217C2×C4—242no—
218C2×C4#23512yes—
219C2×C4D3×D411yescharacteristic
220C2×C4#23712yes—
221C2×C4C23×D311yescharacteristic
222C8—21no—
223C8—21no—
224C8—62no—
225C2×C3—22no—
226C2×C3C2×#12311yescharacteristic
227C2×C3#57711yescharacteristic
228C2×C3#57811yescharacteristic
229D3—42no—
230C22—21no—
231C22—21no—
232C22—62no—
233C22—62no—
234C22—62no—
235C22#88312yes—
236C22C2×D3×D411yescharacteristic
237C4—42no—
238C4—42no—
239C4—62no—
240C4—121no—
241C4—121no—
242C4—121no—
243C4—121no—
244C4#88312yes—
245C4C2×#23511yescharacteristic
246C4C2×#23711yescharacteristic
247C3#309711yescharacteristic, complemented, Sylow
248C2—22no—
249C2—122no—
250C2C2×#88311yescharacteristic
251C2#576211yescharacteristic
252C2#576611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#103C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenterC2×#103D3
hypocenterC3#3097
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×C3×#103C2semidirect
4. #5225C2semidirect
6. #5106C2semidirect
7. #4964C2semidirect
8. #4815C2semidirect
9. #4814C2semidirect
10. #4811C2semidirect
11. #4805C2semidirect
51. C2×#103D3semidirect
247. C3#3097semidirect

Maximal subgroups

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

subgroupindex
#58162
#55042
C2×C3×#1032
#52252
#51102
#51062
#49642
#48152
#48142
#48112
#48052
#30973