Finite Group Databasebeta

← #79617

Subgroups — #79617

Order 384.

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

normal subgroups only

This group has 386 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#5563C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic, complemented
3C3×C4×D8C211yesmaximal, characteristic, complemented
4#5159C212yesmaximal, complemented
5#5008C211yesmaximal, characteristic, complemented
6#5007C211yesmaximal, characteristic, complemented
7#4881C212yesmaximal, complemented
8#4779C212yesmaximal, complemented
9#4777C212yesmaximal, complemented
10#4688C211yesmaximal, characteristic, complemented
11#4680C211yesmaximal, characteristic, complemented
12#3354—31nomaximal, Sylow
13C2×#235—21no—
14C2×C3×D8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#938C2212yes—
18#938C2212yes—
19#930—21no—
20#926—21no—
21#924—21no—
22C2×ΓL(1,𝔽25)C2212yes—
23#892C2212yes—
24D3×C4⋊C4C2212yes—
25#885—22no—
26#883—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29#869C2211yescharacteristic
30C3×#107C2211yescharacteristic
31C3×#102C2211yescharacteristic
32C3×#102C2211yescharacteristic
33C3×C4×C8C2211yescharacteristic
34#832C2212yes—
35#832C2212yes—
36(C2×D3)⋊C8C2212yes—
37#817C2212yes—
38#817C2212yes—
39#808C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#555—31no—
43#547—31no—
44#498—31no—
45#482—32no—
46#481—32no—
47#476—31no—
48#474—31no—
49#469—32no—
50C4×D8D311yescharacteristic, complemented
51#436—31no—
52#428—32no—
53C2×C3×D4C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C2×#67—22no—
58C2×#67—22no—
59#235—44no—
60C2×D12D411yescharacteristic
61C2×C4×D3—22no—
62C2×C4×D3C2312yes—
63C2×Dic6—21no—
64C2×Dic6—21no—
65C3×D8—22no—
66C2×C3×C8C2311yescharacteristic
67C2×C3×C8C2311yescharacteristic
68C3×C4⋊C4C2311yescharacteristic
69C3×C4⋊C4C2311yescharacteristic
70C3×C22⋊C4—21no—
71C3×C22⋊C4—21no—
72C3×C42C2311yescharacteristic
73#217—22no—
74#217—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4D412yes—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3D411yescharacteristic
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2312yes—
82Dic3⋊C4—22no—
83C4×Dic3—22no—
84C2×C3⋊C8C2312yes—
85ΓL(1,𝔽25)—24no—
86C2×#41—61no—
87C2×D8C2×D311yescharacteristic
88C2×M16—32no—
89C4⋊Q8—31no—
90C4⋊Q8—31no—
91#124—62no—
92#123—62no—
93C22⋊Q8—61no—
94#121—32no—
95#121—32no—
96#121—61no—
97C4×D4—31no—
98C4×D4C2×D311yescharacteristic
99C4×D4C2×D311yescharacteristic
100#117—32no—
101#117—61no—
102C2×C4⋊C4—32no—
103#107—32no—
104#107C2×D311yescharacteristic
105#106—32no—
106#106—32no—
107#106—32no—
108C4⋊C8—31no—
109#102—32no—
110#102—32no—
111#102C2×D311yescharacteristic
112#102C2×D311yescharacteristic
113C22⋊C8—32no—
114C4×C8C2×D311yescharacteristic
115C23×C3—21no—
116C23×C3—21no—
117C22×D3C2×D412yes—
118C3×D4—21no—
119C3×D4—21no—
120C3×D4—21no—
121C3×D4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2411yescharacteristic
129#67—44no—
130#67—82no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3C2×D412yes—
134D12—22no—
135C4×D3—22no—
136C4×D3—22no—
137C4×D3—44no—
138Dic6—42no—
139Dic6—81no—
140C3×C8—21no—
141C3×C8#4112yescomplemented
142C3⋊C8—22no—
143#41—124no—
144C2×Q8—61no—
145C2×Q8—61no—
146C2×D4—31no—
147C2×D4—62no—
148C2×D4—62no—
149C2×D4C22×D311yescharacteristic
150C2×D4C22×D311yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—32no—
154C22×C4—62no—
155D8—22no—
156M16—64no—
157C2×C8—32no—
158C2×C8C22×D311yescharacteristic
159C2×C8C22×D311yescharacteristic
160C4⋊C4—31no—
161C4⋊C4—32no—
162C4⋊C4—32no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4C22×D311yescharacteristic
167C4⋊C4C22×D311yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—32no—
171C22⋊C4—62no—
172C22⋊C4—62no—
173C22⋊C4—62no—
174C42—62no—
175C42C22×D311yescharacteristic
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3C22×D411yescharacteristic
183C2×D3—22no—
184C2×D3—22no—
185C2×D3—22no—
186C3×C4—21no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4AGL(1,ℤ/8)12yes—
190C3×C4C2×#4111yescharacteristic
191C3×C42−1+411yescharacteristic
192Dic3—22no—
193Dic3—42no—
194Dic3—42no—
195C23—21no—
196C23—21no—
197C23—32no—
198Q8—122no—
199Q8—241no—
200D4—21no—
201D4—21no—
202D4—21no—
203D4—21no—
204D4—62no—
205D4—124no—
206D4—242no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—32no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—124no—
217C2×C4D3×D411yescharacteristic
218C2×C4D3×D411yescharacteristic
219C2×C4C23×D311yescharacteristic
220C8—21no—
221C8—62no—
222C8#23512yescomplemented
223C2×C3—81no—
224C2×C3—81no—
225C2×C3#54711yescharacteristic
226C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
227C2×C3#57711yescharacteristic
228D3—42no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—62no—
236C22—62no—
237C22—62no—
238C22C2×D3×D411yescharacteristic
239C4—21no—
240C4—41no—
241C4—41no—
242C4—62no—
243C4—122no—
244C4—122no—
245C4#91112yes—
246C4C2×#23511yescharacteristic
247C4#101011yescharacteristic
248C3#335411yescharacteristic, complemented, Sylow
249C2—81no—
250C2—81no—
251C2—122no—
252C2#555111yescharacteristic
253C2C2×#91111yescharacteristic
254C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3354
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5563C2semidirect
2. #5551C2semidirect
3. C3×C4×D8C2semidirect
4. #5159C2semidirect
5. #5008C2semidirect
6. #5007C2semidirect
7. #4881C2semidirect
8. #4779C2semidirect
9. #4777C2semidirect
10. #4688C2semidirect
11. #4680C2semidirect
50. C4×D8D3semidirect
141. C3×C8#41semidirect
222. C8#235semidirect
248. C3#3354semidirect

Maximal subgroups

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

subgroupindex
#55632
#55512
C3×C4×D82
#51592
#50082
#50072
#48812
#47792
#47772
#46882
#46802
#33543