Finite Group Databasebeta

← #80987

Subgroups — #80987

Order 384.

352 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 115 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#5732C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic
3C3×#474C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5C4×#214C211yesmaximal, characteristic, complemented
6#4888C212yesmaximal, complemented
7#4880C212yesmaximal, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4784C212yesmaximal, complemented
11#4778C212yesmaximal, complemented
12#3340—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×D4C2211yescharacteristic
15#947—21no—
16Q8×Dic3—21no—
17#938C2212yes—
18C2×#214C2211yescharacteristic
19#930—21no—
20#926—21no—
21C2×D3×C8C2212yes—
22#898—22no—
23#897C2212yes—
24#892C2212yes—
25#892C2212yes—
26#885—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#107C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34(C2×D3)⋊C8C2212yes—
35#818C2212yes—
36#816C2212yes—
37#810C2211yescharacteristic
38#808C2211yescharacteristic
39#808C2212yes—
40#808C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#555—31no—
44#498—31no—
45#485—32no—
46#482—32no—
47#477—31no—
48#474D311yescharacteristic, complemented
49#465—32no—
50C4×SD16—31no—
51C8×D4—31no—
52#427—32no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57C2×D12D411yescharacteristic
58C2×C4×D3C2312yes—
59C2×Dic6—21no—
60C2×Dic6C2311yescharacteristic
61C2×C3×C8C2312yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4C2312yes—
64C3×C4⋊C4C2311yescharacteristic
65C3×C22⋊C4—21no—
66C3×C42C2311yescharacteristic
67#217—22no—
68#214—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4D412yes—
73C4⋊Dic3—21no—
74C4⋊Dic3—22no—
75C4⋊Dic3D411yescharacteristic
76C4⋊Dic3C2312yes—
77C4⋊Dic3C2311yescharacteristic
78Dic3⋊C4—21no—
79C4×Dic3—21no—
80C4×Dic3—22no—
81C4×Dic3—22no—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84D3×C8—24no—
85C2×SD16—31no—
86C22×C8—32no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D311yescharacteristic
89#126—62no—
90#124—62no—
91#124—61no—
92C22⋊Q8—32no—
93#121—32no—
94#121—61no—
95C4×Q8—31no—
96C4×Q8—61no—
97C4×D4—31no—
98C4×D4C2×D311yescharacteristic
99#117—32no—
100#117—32no—
101#117—61no—
102#107—32no—
103#107C2×D312yes—
104#106—31no—
105#106—32no—
106#106—32no—
107C4⋊C8C2×D311yescharacteristic
108#103—32no—
109#103—31no—
110#102—31no—
111#102C2×D312yes—
112C22⋊C8—32no—
113C4×C8—31no—
114C23×C3—21no—
115C22×D3C2×D412yes—
116C3×Q8—42no—
117C3×D4—21no—
118C3×D4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—42no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2411yescharacteristic
125#67—82no—
126C2×Dic3—21no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3C2×D412yes—
130D12—22no—
131C4×D3—22no—
132C4×D3—22no—
133Dic6—21no—
134Dic6—21no—
135Dic6—81no—
136C3×C8—22no—
137C3⋊C8—21no—
138C3⋊C8#4112yes—
139C2×Q8—21no—
140C2×Q8—31no—
141C2×Q8—61no—
142C2×D4—31no—
143C2×D4—62no—
144C2×D4C22×D311yescharacteristic
145C22×C4—21no—
146C22×C4—32no—
147SD16—62no—
148C2×C8—31no—
149C2×C8—31no—
150C2×C8—64no—
151C2×C8C22×D312yes—
152C4⋊C4—21no—
153C4⋊C4—31no—
154C4⋊C4—31no—
155C4⋊C4—32no—
156C4⋊C4—62no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4C22×D312yes—
160C4⋊C4C22×D311yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—32no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C42—62no—
168C42—62no—
169C42—61no—
170C42C22×D311yescharacteristic
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3C22×D411yescharacteristic
175C2×D3—22no—
176C2×D3—22no—
177C2×D3—22no—
178C3×C4—21no—
179C3×C4—41no—
180C3×C4—42no—
181C3×C4#13512yes—
182C3×C4C2×#4111yescharacteristic
183C3×C42−1+411yescharacteristic
184Dic3—22no—
185Dic3—41no—
186Dic3—42no—
187Dic3—41no—
188C23—21no—
189C23—32no—
190Q8—42no—
191Q8—61no—
192Q8—61no—
193Q8—241no—
194D4—21no—
195D4—21no—
196D4—62no—
197D4—242no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—32no—
201C2×C4—42no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—61no—
205C2×C4—62no—
206C2×C4—61no—
207C2×C4D3×D411yescharacteristic
208C2×C4D3×D411yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—22no—
211C8—32no—
212C8—61no—
213C2×C3—81no—
214C2×C3#54711yescharacteristic
215C2×C3C2×#13511yescharacteristic
216C2×C3#57711yescharacteristic
217D3—42no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—62no—
222C22—62no—
223C22—62no—
224C22C2×D3×D411yescharacteristic
225C4—21no—
226C4—41no—
227C4—42no—
228C4—62no—
229C4—122no—
230C4—121no—
231C4—121no—
232C4#91212yes—
233C4#100711yescharacteristic
234C4D3×#4111yescharacteristic
235C3#334011yescharacteristic, complemented, Sylow
236C2—81no—
237C2—122no—
238C2#563111yescharacteristic
239C2C2×#91211yescharacteristic
240C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radical#5563C2
hypercenter#474D3
hypocenterC3#3340
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
3. C3×#474C2semidirect
4. #5080C2semidirect
5. C4×#214C2semidirect
6. #4888C2semidirect
7. #4880C2semidirect
8. #4850C2semidirect
9. #4834C2semidirect
10. #4784C2semidirect
11. #4778C2semidirect
48. #474D3semidirect
235. C3#3340semidirect

Maximal subgroups

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

subgroupindex
#57322
#55632
C3×#4742
#50802
C4×#2142
#48882
#48802
#48502
#48342
#47842
#47782
#33403