Finite Group Databasebeta

← #68290

Subgroups — #68290

Order 384.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#5605C211yesmaximal, characteristic
2#5034C211yesmaximal, characteristic, complemented
3#4735C211yesmaximal, characteristic, complemented
4#4677C211yesmaximal, characteristic, complemented
5C3×#327C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1613—31nomaximal, Sylow
9C2×D3×D4—21no—
10C3×#121C2211yescharacteristic
11#939—21no—
12#937—21no—
13#930C2211yescharacteristic
14C2×(C2×D3)⋊C4—21no—
15C2×(C2×D3)⋊C4C2211yescharacteristic
16C2×C4⋊Dic3—21no—
17C2×C4⋊Dic3C2211yescharacteristic, complemented
18#894—21no—
19#883—21no—
20#882—21no—
21C3×C22⋊C8C2211yescharacteristic, complemented
22C3×#95C2211yescharacteristic, complemented
23#832—21no—
24#830C2211yescharacteristic, complemented
25#821—21no—
26#817—21no—
27#808—21no—
28#534—31no—
29#481—31no—
30#481—31no—
31#399—31no—
32#327—31no—
33#327D311yescharacteristic, complemented
34#323—31no—
35C23×D3—21no—
36C23×D3D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4D411yescharacteristic
39C22×C3×C4—21no—
40C22×C3×C4C2311yescharacteristic
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—21no—
44C22×Dic3—21no—
45C22×Dic3D411yescharacteristic
46D3×D4—42no—
47D3×D4—42no—
48C2×D12—21no—
49C2×D12D411yescharacteristic
50C2×C4×D3—21no—
51C2×C3×C8—21no—
52C3×C4⋊C4D411yescharacteristic
53C3×C22⋊C4—21no—
54#217—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—42no—
60C4⋊Dic3—21no—
61C4⋊Dic3—21no—
62C4⋊Dic3—42no—
63C4⋊Dic3D411yescharacteristic
64Dic3⋊C4—21no—
65C2×C3⋊C8—21no—
66C22×D4—61no—
67#123—61no—
68#123—61no—
69#121—31no—
70#121—61no—
71#121C2×D311yescharacteristic
72#120—61no—
73#120—61no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4—61no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—61no—
78#106—61no—
79#106—61no—
80#102—61no—
81#102—61no—
82C22⋊C8—31no—
83C22⋊C8C2×D311yescharacteristic, complemented
84#95C2×D311yescharacteristic, complemented
85C23×C3—21no—
86C23×C3C2×D411yescharacteristic
87C22×D3—21no—
88C22×D3—21no—
89C22×D3—21no—
90C22×D3—42no—
91C22×D3—41no—
92C22×D3—41no—
93C22×D3—81no—
94C22×D3—81no—
95C22×D3SD1612yes—
96C3×D4—41no—
97C3×D4—41no—
98C3×D4—41no—
99C2×C3×C4—22no—
100C2×C3×C4—21no—
101C2×C3×C4—41no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106#67—42no—
107#67—41no—
108#67—41no—
109#67—81no—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C2×Dic3—41no—
115C2×Dic3—41no—
116D12—41no—
117D12—42no—
118C4×D3—42no—
119C3×C8—41no—
120C3⋊C8—41no—
121C24—31no—
122C24—61no—
123C2×D4—21no—
124C2×D4—31no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4—61no—
128C2×D4—61no—
129C2×D4—122no—
130C2×D4—122no—
131C2×D4D1211yescharacteristic
132C22×C4—21no—
133C22×C4—31no—
134C22×C4—61no—
135C22×C4—61no—
136C22×C4C22×D311yescharacteristic
137C2×C8—21no—
138C2×C8—61no—
139C4⋊C4—31no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4—122no—
144C4⋊C4D1211yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—122no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3#12011yescharacteristic
158C22×C3C2×SD1611yescharacteristic
159C22×C3AGL(1,ℤ/8)11yescharacteristic
160C2×D3—22no—
161C2×D3—41no—
162C2×D3—41no—
163C2×D3—42no—
164C2×D3—41no—
165C2×D3—41no—
166C2×D3—41no—
167C2×D3—81no—
168C2×D3—84no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—42no—
172C3×C4—41no—
173Dic3—41no—
174Dic3—41no—
175Dic3—81no—
176C23—21no—
177C23—32no—
178C23—61no—
179C23—61no—
180C23—61no—
181C23—121no—
182C23—121no—
183C23—122no—
184C23—241no—
185C23—241no—
186C23D3×D411yescharacteristic
187D4—41no—
188D4—41no—
189D4—41no—
190D4—121no—
191D4—122no—
192D4—122no—
193D4—121no—
194D4—121no—
195D4—241no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—122no—
204C2×C4—121no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4—121no—
208C2×C4C2×D1211yescharacteristic
209C2×C4D3×D411yescharacteristic
210C8—41no—
211C8—121no—
212C2×C3—21no—
213C2×C3—21no—
214C2×C3—81no—
215C2×C3#45011yescharacteristic
216C2×C3#45311yescharacteristic
217C2×C3#45811yescharacteristic
218D3—42no—
219D3—81no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—62no—
226C22—122no—
227C22—121no—
228C22—121no—
229C22—121no—
230C22—121no—
231C22—121no—
232C22—241no—
233C22—244no—
234C22#88211yescharacteristic
235C22#91111yescharacteristic
236C22D3×SD1611yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—41no—
241C4—121no—
242C4—121no—
243C4—241no—
244C3#161311yescharacteristic, complemented, Sylow
245C2—21no—
246C2—21no—
247C2—81no—
248C2—122no—
249C2—241no—
250C2#474211yescharacteristic
251C2#477611yescharacteristic
252C2#482211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3#120
prime radical#5605C2
hypercenter#327D3
hypocenterC3#1613
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #5034C2semidirect
3. #4735C2semidirect
4. #4677C2semidirect
5. C3×#327C2semidirect
6. #4468C2semidirect
7. #4451C2semidirect
17. C2×C4⋊Dic3C22semidirect
21. C3×C22⋊C8C22semidirect
22. C3×#95C22semidirect
24. #830C22semidirect
33. #327D3semidirect
83. C22⋊C8C2×D3semidirect
84. #95C2×D3semidirect
244. C3#1613semidirect

Maximal subgroups

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

subgroupindex
#56052
#50342
#47352
#46772
C3×#3272
#44682
#44512
#16133