Finite Group Databasebeta

← #78964

Subgroups — #78964

Order 384.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
2C22×#216C211yesmaximal, characteristic, complemented
3C2×#907C211yesmaximal, characteristic
4#5526C211yesmaximal, characteristic
5C3×#419C211yesmaximal, characteristic, complemented
6#5120C211yesmaximal, characteristic, complemented
7#4998C211yesmaximal, characteristic, complemented
8#4816C214yesmaximal, complemented
9#4813C214yesmaximal, complemented
10#3059—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C2×#235C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×M16C2211yescharacteristic
15C3×#117C2211yescharacteristic
16#947—21no—
17#946C2214yes—
18#944—21no—
19C2×#216—24no—
20C2×#216C2212yes—
21C2×#216C2212yes—
22#929—22no—
23#926C2211yescharacteristic
24C22×C3⋊C8C2211yescharacteristic
25#907—24no—
26C2×ΓL(1,𝔽25)C2212yes—
27C2×D3×C8C2212yes—
28#897C2214yes—
29#886—22no—
30#875—21no—
31#869—21no—
32C3×#103C2214yes—
33#816C2214yes—
34#808C2212yes—
35#807C2212yes—
36C22×Q16—31no—
37C2×#131—31no—
38#536—31no—
39#536—31no—
40#470—34no—
41#467—34no—
42#427—31no—
43#419—31no—
44#419D311yescharacteristic, complemented
45C2×C3×Q8—21no—
46C2×C3×Q8—22no—
47C2×C3×Q8—22no—
48C2×C3×Q8C2312yes—
49C22×C3×C4—21no—
50C22×C3×C4C2311yescharacteristic
51C2×#67D412yes—
52C22×Dic3—21no—
53#235—22no—
54#235—22no—
55C2×D12D411yescharacteristic
56C2×C4×D3C2312yes—
57C2×Dic6—22no—
58C2×Dic6—22no—
59C2×Dic6—21no—
60C2×Dic6D411yescharacteristic
61C2×Dic6C2312yes—
62C3×M16—22no—
63C2×C3×C8C2312yes—
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—21no—
66C3×C42—21no—
67#217—21no—
68#216—24no—
69#216—24no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—24no—
72C4⋊Dic3—24no—
73C4⋊Dic3C2312yes—
74Dic3⋊C4—24no—
75C4×Dic3—21no—
76C2×C3⋊C8D414yes—
77C2×C3⋊C8C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79ΓL(1,𝔽25)—24no—
80D3×C8—24no—
81C2×#41—31no—
82C22×Q8—31no—
83C22×Q8C2×D311yescharacteristic
84C2×Q16—32no—
85C2×Q16—32no—
86C2×Q16—64no—
87#131—64no—
88C2×M16—32no—
89C2×M16C2×D311yescharacteristic
90C22×C8—31no—
91C22×C8—32no—
92C4⋊Q8—61no—
93C4⋊Q8—61no—
94#124—61no—
95#124—61no—
96#123—62no—
97#123—62no—
98C22⋊Q8—34no—
99C22⋊Q8—34no—
100#117—31no—
101#117C2×D311yescharacteristic
102#107—32no—
103#106—32no—
104#103—34no—
105#103C2×D314yes—
106C23×C3C2×D411yescharacteristic
107C22×D3C2×D412yes—
108C3×Q8—22no—
109C3×Q8—22no—
110C3×Q8—42no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—42no—
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2×D412yes—
116C2×C3×C4#4112yes—
117C2×C3×C4C2411yescharacteristic
118#67—24no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—42no—
122C2×Dic3C2×D412yes—
123D12—22no—
124C4×D3—22no—
125C4×D3—22no—
126Dic6—22no—
127Dic6—22no—
128Dic6—22no—
129Dic6—42no—
130C3×C8—22no—
131C3⋊C8C2×D412yes—
132C3⋊C8C2×D412yes—
133#41—62no—
134#41—62no—
135C2×Q8—21no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—31no—
139C2×Q8—32no—
140C2×Q8—61no—
141C2×Q8—62no—
142C2×Q8—62no—
143C2×Q8C22×D312yes—
144C2×D4—32no—
145C2×D4—31no—
146C22×C4—21no—
147C22×C4—32no—
148C22×C4—61no—
149C22×C4C22×D311yescharacteristic
150Q16—64no—
151Q16—64no—
152M16—22no—
153M16—64no—
154C2×C8—34no—
155C2×C8—31no—
156C2×C8—31no—
157C2×C8—64no—
158C2×C8C22×D312yes—
159C4⋊C4—32no—
160C4⋊C4—64no—
161C4⋊C4—64no—
162C4⋊C4C22×D312yes—
163C22⋊C4—21no—
164C22⋊C4—64no—
165C22⋊C4—64no—
166C22⋊C4—61no—
167C42—21no—
168C42—61no—
169C22×C3—22no—
170C22×C3#12112yes—
171C22×C3C22×D411yescharacteristic
172C2×D3—22no—
173C2×D3—24no—
174C3×C4—42no—
175C3×C4—42no—
176C3×C4#12112yes—
177C3×C4C22×D411yescharacteristic
178C3×C4C2×#4111yescharacteristic
179Dic3—22no—
180Dic3—42no—
181Dic3—42no—
182C23—32no—
183C23D3×D411yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—42no—
187Q8—62no—
188Q8—62no—
189Q8—62no—
190Q8—122no—
191D4—64no—
192D4—62no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—32no—
196C2×C4—42no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—122no—
202C2×C4#23512yes—
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D412yes—
205C2×C4C23×D311yescharacteristic
206C8—22no—
207C8—32no—
208C8—32no—
209C2×C3—22no—
210C2×C3C2×#12111yescharacteristic
211C2×C3#57812yes—
212D3—42no—
213C22—22no—
214C22—62no—
215C22—64no—
216C22#88412yes—
217C22C2×D3×D411yescharacteristic
218C4—42no—
219C4—42no—
220C4—62no—
221C4—122no—
222C4—122no—
223C4#88412yes—
224C4C2×#23511yescharacteristic
225C4C2×D3×D411yescharacteristic
226C3#305911yescharacteristic, complemented, Sylow
227C2—22no—
228C2—122no—
229C2C2×#88411yescharacteristic
230C2#577912yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#419C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenter#419D3
hypocenterC3#3059
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#216C2semidirect
5. C3×#419C2semidirect
6. #5120C2semidirect
7. #4998C2semidirect
8. #4816C2semidirect
9. #4813C2semidirect
44. #419D3semidirect
226. C3#3059semidirect

Maximal subgroups

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

subgroupindex
#58162
C22×#2162
C2×#9072
#55262
C3×#4192
#51202
#49982
#48162
#48132
#30593