Finite Group Databasebeta

← #70015

Subgroups — #70015

Order 384.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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#5804C211yesmaximal, characteristic
2C2×C3×#99C211yesmaximal, characteristic, complemented
3C2×#833C211yesmaximal, characteristic, complemented
4#4472C212yesmaximal, complemented
5#4471C212yesmaximal, complemented
6#2121—31nomaximal, Sylow
7C22×C3×D4C2211yescharacteristic
8C22×#67C411yescharacteristic, complemented
9C2×C3×C22⋊C4—21no—
10#953—21no—
11C2×#217C411yescharacteristic, complemented
12#939—21no—
13#939—21no—
14#937C411yescharacteristic, complemented
15#937C2211yescharacteristic
16#935C411yescharacteristic, complemented
17#935C2211yescharacteristic
18#929—21no—
19C2×#208—21no—
20C3×#99—21no—
21C3×#99C2212yes—
22#833—21no—
23#833C2212yes—
24#534—31no—
25C2×#100—31no—
26C2×#99D311yescharacteristic, complemented
27#352—32no—
28AΣL(1,𝔽16)—32no—
29C24×C3C2×C411yescharacteristic
30C24×C3D411yescharacteristic
31C23×D3C2×C411yescharacteristic, complemented
32C2×C3×D4—41no—
33C2×C3×D4—41no—
34C2×C3×D4C2×C411yescharacteristic
35C2×C3×D4D411yescharacteristic
36C2×C3×D4D411yescharacteristic
37C2×C3×D4C2311yescharacteristic
38C22×C3×C4—41no—
39C22×C3×C4D411yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—21no—
44C2×#67—41no—
45C2×#67—41no—
46C22×Dic3—21no—
47C22×Dic3C2×C411yescharacteristic, complemented
48C3×C22⋊C4—22no—
49C3×C22⋊C4—41no—
50#217—21no—
51#217—21no—
52#217—21no—
53#217C2×C411yescharacteristic, complemented
54#217C2×C411yescharacteristic, complemented
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59#208—22no—
60#208—41no—
61C2×C3⋊C8—41no—
62C22×D4—31no—
63C22×D4C2×D311yescharacteristic
64C2×M16—61no—
65#123—31no—
66#123—31no—
67#123—61no—
68#121—61no—
69#121—61no—
70#120—31no—
71#120—31no—
72#120—61no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—31no—
75#100—32no—
76#100—61no—
77#99—21no—
78#99C2×D312yes—
79C23×C3—21no—
80C23×C3—21no—
81C23×C3—21no—
82C23×C3—21no—
83C23×C3—81no—
84C23×C3—81no—
85C23×C3C22⋊C411yescharacteristic
86C23×C3C22⋊C411yescharacteristic
87C23×C3C22⋊C411yescharacteristic
88C23×C3C22×C411yescharacteristic
89C23×C3C2×D411yescharacteristic
90C22×D3—21no—
91C22×D3—41no—
92C22×D3—81no—
93C3×D4—41no—
94C3×D4—41no—
95C3×D4—41no—
96C3×D4—41no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—42no—
100C2×C3×C4—82no—
101C2×C3×C4C22⋊C411yescharacteristic
102C2×C3×C4C2×D411yescharacteristic
103#67—42no—
104#67—41no—
105#67—41no—
106#67—81no—
107#67—81no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—41no—
112C2×Dic3—81no—
113C3⋊C8—42no—
114C24—31no—
115C24C4×D311yescharacteristic
116C24#6711yescharacteristic
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—61no—
120C2×D4—61no—
121C2×D4—61no—
122C2×D4—61no—
123C2×D4—121no—
124C2×D4—121no—
125C2×D4C4×D311yescharacteristic
126C2×D4D1211yescharacteristic
127C2×D4#6711yescharacteristic
128C2×D4C22×D311yescharacteristic
129C22×C4—31no—
130C22×C4—41no—
131C22×C4—61no—
132C22×C4D1211yescharacteristic
133M16—62no—
134M16—121no—
135C2×C8—121no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C22⋊C4—22no—
139C22⋊C4—31no—
140C22⋊C4—31no—
141C22⋊C4—41no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—41no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—42no—
157C22×C3—81no—
158C22×C3—81no—
159C22×C3—81no—
160C22×C3#9911yescharacteristic
161C22×C3#9911yescharacteristic
162C22×C3C2×C22⋊C411yescharacteristic
163C2×D3—41no—
164C2×D3—41no—
165C2×D3—41no—
166C2×D3—82no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—82no—
170Dic3—41no—
171Dic3—41no—
172Dic3—41no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—61no—
178C23—81no—
179C23—81no—
180C23—121no—
181C23—241no—
182C23(C2×D3)⋊C411yescharacteristic
183C23(C2×D3)⋊C411yescharacteristic
184C23(C2×D3)⋊C411yescharacteristic
185C23C2×C4×D311yescharacteristic
186C23C2×#6711yescharacteristic
187D4—41no—
188D4—41no—
189D4—41no—
190D4—41no—
191D4—122no—
192D4—121no—
193D4—121no—
194D4—241no—
195D4—241no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—82no—
203C2×C4—121no—
204C2×C4—241no—
205C2×C4(C2×D3)⋊C411yescharacteristic
206C2×C4C2×D1211yescharacteristic
207C8—122no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3C2×#9911yescharacteristic
216D3—81no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—121no—
231C22—121no—
232C22—121no—
233C22—242no—
234C22#80611yescharacteristic
235C22#80611yescharacteristic
236C22C2×(C2×D3)⋊C411yescharacteristic
237C4—21no—
238C4—21no—
239C4—82no—
240C4—121no—
241C4—121no—
242C4—121no—
243C3#212111yescharacteristic, complemented, Sylow
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—241no—
252C2C2×#80611yescharacteristic
subgroupG/H
centerC2C2×#806
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC2×C3×#99C2
socleC2×C3C2×#99
prime radical#5804C2
cosocleC2×C3×D4C23
hypercenterC2×#99D3
hypocenterC3#2121
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×C3×#99C2semidirect
3. C2×#833C2semidirect
4. #4472C2semidirect
5. #4471C2semidirect
8. C22×#67C4semidirect
11. C2×#217C4semidirect
14. #937C4semidirect
16. #935C4semidirect
26. C2×#99D3semidirect
31. C23×D3C2×C4semidirect
47. C22×Dic3C2×C4semidirect
53. #217C2×C4semidirect
54. #217C2×C4semidirect
243. C3#2121semidirect

Maximal subgroups

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

subgroupindex
#58042
C2×C3×#992
C2×#8332
#44722
#44712
#21213