Finite Group Databasebeta

← #70969

Subgroups — #70969

Order 384.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#5617C211yesmaximal, characteristic, complemented
2C3×#447C211yesmaximal, characteristic, complemented
3#5223C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4467C211yesmaximal, characteristic, complemented
8#1600—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C22×#67—21no—
11C2×C3×D8—21no—
12C3×#121C2211yescharacteristic
13C2×#217—21no—
14#939—21no—
15#937—21no—
16D4×Dic3—21no—
17C2×#214—21no—
18C2×(C2×D3)⋊C4C2211yescharacteristic
19#924C2211yescharacteristic
20#898—21no—
21#886—21no—
22#885—21no—
23C3×#102—21no—
24C3×C22⋊C8C2211yescharacteristic
25#832—21no—
26C3⋊#95C2211yescharacteristic
27#830C2211yescharacteristic
28#538—31no—
29#450—31no—
30#447D311yescharacteristic, complemented
31#392—31no—
32#328—31no—
33#327—31no—
34#323—31no—
35C24×C3—21no—
36C23×D3D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4—42no—
41C2×C3×D4D411yescharacteristic, complemented
42C22×C3×C4C2311yescharacteristic
43C2×#67—21no—
44C2×#67—21no—
45C2×#67—42no—
46C2×#67—42no—
47C2×#67—42no—
48C22×Dic3—21no—
49C22×Dic3—21no—
50C22×Dic3D411yescharacteristic
51C2×Dic6D411yescharacteristic, complemented
52C3×D8—42no—
53C2×C3×C8—21no—
54C3×C4⋊C4D411yescharacteristic, complemented
55C3×C22⋊C4—21no—
56#217—21no—
57#217—21no—
58#217—42no—
59#214—42no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64C4⋊Dic3—21no—
65C4⋊Dic3D411yescharacteristic, complemented
66Dic3⋊C4—21no—
67C4×Dic3—21no—
68C2×C3⋊C8—21no—
69C22×D4—61no—
70C22×D4C2×D311yescharacteristic
71C2×SD16—61no—
72C2×D8—21no—
73#126—61no—
74#124—61no—
75#123—61no—
76C22⋊Q8—31no—
77#121—61no—
78#121C2×D311yescharacteristic
79#120—61no—
80C4×D4—61no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—61no—
83#102—21no—
84#102—61no—
85C22⋊C8—31no—
86C22⋊C8C2×D311yescharacteristic
87#95—31no—
88C23×C3—22no—
89C23×C3—21no—
90C23×C3—41no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3C2×D411yescharacteristic
97C22×D3—21no—
98C22×D3—21no—
99C22×D3—21no—
100C22×D3—41no—
101C22×D3—41no—
102C3×D4—22no—
103C3×D4—22no—
104C3×D4—41no—
105C3×D4—41no—
106C3×D4—41no—
107C3×D4—41no—
108C3×D4—41no—
109C2×C3×C4—21no—
110C2×C3×C4—41no—
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113#67—48no—
114#67—81no—
115#67—81no—
116C2×Dic3—22no—
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—21no—
120C2×Dic3—41no—
121C2×Dic3—41no—
122C2×Dic3—41no—
123C2×Dic3—81no—
124Dic6—41no—
125C3×C8—41no—
126C3⋊C8—41no—
127C24—21no—
128C24—31no—
129C2×Q8—31no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—122no—
137C2×D4—122no—
138C2×D4—122no—
139C2×D4#6711yescharacteristic, complemented
140C22×C4—31no—
141C22×C4—61no—
142C22×C4—61no—
143C22×C4C22×D311yescharacteristic
144SD16—122no—
145D8—42no—
146C2×C8—21no—
147C2×C8—61no—
148C4⋊C4—31no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4#6711yescharacteristic, complemented
152C22⋊C4—21no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—122no—
160C42—61no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—42no—
169C22×C3—81no—
170C22×C3—81no—
171C22×C3—81no—
172C22×C3—81no—
173C22×C3#12011yescharacteristic
174C22×C3#13511yescharacteristic
175C22×C3AGL(1,ℤ/8)11yescharacteristic
176C2×D3—42no—
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—81no—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—41no—
184Dic3—42no—
185Dic3—41no—
186Dic3—41no—
187Dic3—41no—
188C23—22no—
189C23—21no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—61no—
197C23—61no—
198C23—61no—
199C23—121no—
200C23—121no—
201C23D3×D411yescharacteristic
202Q8—121no—
203D4—22no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—128no—
211D4—241no—
212D4—241no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—62no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—241no—
223C2×C4D3×D411yescharacteristic
224C2×C4C2×#6711yescharacteristic
225C8—41no—
226C8—121no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—81no—
230C2×C3—82no—
231C2×C3#44911yescharacteristic
232C2×C3#45311yescharacteristic
233C2×C3B(4,𝔽2)11yescharacteristic
234D3—81no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—42no—
242C22—41no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—122no—
248C22—121no—
249C22—121no—
250C22—121no—
251C22—241no—
252C22#91111yescharacteristic
253C22#91211yescharacteristic
254C22#93711yescharacteristic
255C4—21no—
256C4—21no—
257C4—41no—
258C4—122no—
259C4—121no—
260C4—121no—
261C4—121no—
262C3#160011yescharacteristic, complemented, Sylow
263C2—21no—
264C2—21no—
265C2—82no—
266C2—81no—
267C2—241no—
268C2#503211yescharacteristic
269C2#515811yescharacteristic
270C2#519811yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#447C2
socleC22×C3#120
hypercenter#447D3
hypocenterC3#1600
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #5617C2semidirect
2. C3×#447C2semidirect
3. #5223C2semidirect
4. #5218C2semidirect
5. #4537C2semidirect
6. #4468C2semidirect
7. #4467C2semidirect
30. #447D3semidirect
41. C2×C3×D4D4semidirect
51. C2×Dic6D4semidirect
54. C3×C4⋊C4D4semidirect
65. C4⋊Dic3D4semidirect
139. C2×D4#67semidirect
151. C4⋊C4#67semidirect
262. C3#1600semidirect

Maximal subgroups

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

subgroupindex
#56172
C3×#4472
#52232
#52182
#45372
#44682
#44672
#16003