Finite Group Databasebeta

← #79397

Subgroups — #79397

Order 384.

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

normal subgroups only

This group has 388 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#5535C211yesmaximal, characteristic, complemented
2#5526C211yesmaximal, characteristic, complemented
3C3×#433C211yesmaximal, characteristic, complemented
4#4844C212yesmaximal, complemented
5#4841C212yesmaximal, complemented
6#4738C212yesmaximal, complemented
7#4734C212yesmaximal, complemented
8#4712C211yesmaximal, characteristic, complemented
9#4703C211yesmaximal, characteristic, complemented
10#4697C211yesmaximal, characteristic, complemented
11#4693C211yesmaximal, characteristic, complemented
12#3221—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C3×#117C2211yescharacteristic
16#930C2212yes—
17C2×D24C2212yes—
18C2×#204C2212yes—
19C2×#204C2212yes—
20C2×#204C2212yes—
21#897—22no—
22#894—22no—
23#886—22no—
24#884—22no—
25#875C2211yescharacteristic
26#875C2211yescharacteristic
27#874C2211yescharacteristic
28C4×D12C2212yes—
29#870C2211yescharacteristic
30#869C2211yescharacteristic
31#869C2211yescharacteristic
32C3×C4⋊C8C2212yes—
33C3×C22⋊C8C2212yes—
34C3×#97C2211yescharacteristic
35C3×C4×C8C2211yescharacteristic
36#821C2212yes—
37#816C2212yes—
38#816C2212yes—
39#816C2212yes—
40#540—31no—
41#536—31no—
42#496—31no—
43#492—31no—
44#491—31no—
45#488—31no—
46#463—32no—
47#461—32no—
48#449—32no—
49#448—32no—
50#433D311yescharacteristic, complemented
51C22×C3×C4C2311yescharacteristic
52C2×#67—21no—
53C2×#67—22no—
54C22×Dic3—21no—
55#235—42no—
56C2×D12—22no—
57C2×D12C2311yescharacteristic
58C2×D12C2312yes—
59C2×C4×D3—21no—
60C2×C4×D3—22no—
61C2×Dic6—21no—
62C2×Dic6—42no—
63C2×Dic6C2311yescharacteristic
64C2×Dic6C2311yescharacteristic
65C2×Dic6C2311yescharacteristic
66C2×C3×C8C2312yes—
67C2×C3×C8C2312yes—
68C3×C4⋊C4D412yes—
69C3×C22⋊C4D412yes—
70C3×C42C2311yescharacteristic
71C3×C42C2311yescharacteristic
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3C2312yes—
79Dic3⋊C4—22no—
80D24—42no—
81#204—42no—
82#204—42no—
83#204—42no—
84C2×#41—31no—
85C22×Q8—31no—
86C2×SD16—32no—
87C2×SD16—32no—
88C2×SD16—32no—
89C2×D8—32no—
90C4⋊Q8—31no—
91C4⋊Q8—31no—
92#127—31no—
93#125—31no—
94#124—31no—
95#124—31no—
96#123—62no—
97#123—62no—
98C22⋊Q8—62no—
99#121—32no—
100#121—62no—
101C4×D4—32no—
102#117C2×D311yescharacteristic
103C4⋊C8C2×D312yes—
104#103—32no—
105#103—32no—
106#103—32no—
107#102—32no—
108C22⋊C8C2×D312yes—
109#97C2×D311yescharacteristic
110C4×C8C2×D311yescharacteristic
111C23×C3C2×D411yescharacteristic
112C22×D3—21no—
113C22×D3—22no—
114C2×C3×C4—22no—
115C2×C3×C4C2×D412yes—
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2411yescharacteristic
119#67—42no—
120#67—82no—
121C2×Dic3—22no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—42no—
126D12—22no—
127D12—22no—
128D12—22no—
129D12—82no—
130C4×D3—42no—
131C4×D3—82no—
132Dic6—22no—
133Dic6—22no—
134Dic6—22no—
135Dic6—42no—
136C3×C8—22no—
137C3×C8—22no—
138#41—122no—
139C2×Q8—31no—
140C2×Q8—31no—
141C2×Q8—31no—
142C2×Q8—61no—
143C2×Q8—122no—
144C2×D4—31no—
145C2×D4—32no—
146C2×D4—61no—
147C2×D4—62no—
148C2×D4—62no—
149C22×C4—61no—
150C22×C4—61no—
151C22×C4—62no—
152C22×C4C22×D311yescharacteristic
153SD16—122no—
154SD16—122no—
155SD16—122no—
156D8—122no—
157C2×C8C22×D312yes—
158C2×C8C22×D312yes—
159C4⋊C4—32no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4D1212yes—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C22⋊C4D1212yes—
169C42C22×D311yescharacteristic
170C42C22×D311yescharacteristic
171C22×C3—21no—
172C22×C3—22no—
173C22×C3C22×D411yescharacteristic
174C2×D3—41no—
175C2×D3—42no—
176C2×D3—42no—
177C2×D3—42no—
178C2×D3—42no—
179C3×C4—21no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4H(5,𝔽2)12yes—
183Dic3—41no—
184Dic3—41no—
185Dic3—41no—
186Dic3—42no—
187C23—61no—
188C23—62no—
189C23C2×D1211yescharacteristic
190Q8—62no—
191Q8—62no—
192Q8—62no—
193Q8—122no—
194D4—62no—
195D4—62no—
196D4—62no—
197D4—122no—
198D4—242no—
199D4—242no—
200C2×C4—22no—
201C2×C4—62no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C4—242no—
208C2×C4C2×D1212yes—
209C2×C4C2×D1212yes—
210C2×C4C2×D1211yescharacteristic
211C2×C4C23×D311yescharacteristic
212C8—22no—
213C8—22no—
214C2×C3—41no—
215C2×C3#53511yescharacteristic
216C2×C3#57712yes—
217D3—81no—
218D3—82no—
219C22—22no—
220C22—21no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22—121no—
225C22—122no—
226C22C22×D1211yescharacteristic
227C4—22no—
228C4—22no—
229C4—21no—
230C4—122no—
231C4—121no—
232C4—121no—
233C4—121no—
234C4#100912yes—
235C3#322111yescharacteristic, complemented, Sylow
236C2—41no—
237C2—242no—
238C2—241no—
239C2#552511yescharacteristic
240C2#575112yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#433C2
socleC22×C3C22×D4
hypercenter#433D3
hypocenterC3#3221
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #5535C2semidirect
2. #5526C2semidirect
3. C3×#433C2semidirect
4. #4844C2semidirect
5. #4841C2semidirect
6. #4738C2semidirect
7. #4734C2semidirect
8. #4712C2semidirect
9. #4703C2semidirect
10. #4697C2semidirect
11. #4693C2semidirect
50. #433D3semidirect
235. C3#3221semidirect

Maximal subgroups

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

subgroupindex
#55352
#55262
C3×#4332
#48442
#48412
#47382
#47342
#47122
#47032
#46972
#46932
#32213