Finite Group Databasebeta

← #69697

Subgroups — #69697

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#5494C211yesmaximal, characteristic, complemented
2C3×#392C211yesmaximal, characteristic, complemented
3#5218C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#4525C211yesmaximal, characteristic, complemented
6#4452C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1600—31nomaximal, Sylow
9C22×C3×D4—21no—
10C22×C3×D4C2211yescharacteristic
11C2×C3×C22⋊C4—21no—
12C2×C3×C22⋊C4C2211yescharacteristic
13#953—21no—
14C2×#214—21no—
15C2×#213—21no—
16#930C2211yescharacteristic
17C4×#67—21no—
18#924C2211yescharacteristic
19#885—21no—
20#884—21no—
21#883—21no—
22#879—21no—
23C3×#95C2211yescharacteristic
24#832—21no—
25#832—21no—
26#830C2211yescharacteristic
27#830C2211yescharacteristic
28#538—31no—
29#450—31no—
30#447—31no—
31#392D311yescharacteristic, complemented
32#328—31no—
33#327—31no—
34#323—31no—
35C24×C3—21no—
36C24×C3D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—42no—
40C2×C3×D4—42no—
41C2×C3×D4—42no—
42C2×C3×D4—42no—
43C22×C3×C4—21no—
44C22×C3×C4D411yescharacteristic
45C22×C3×C4C2311yescharacteristic
46C2×#67—21no—
47C2×#67—21no—
48C2×#67—21no—
49C2×D12D411yescharacteristic, complemented
50C2×C4×D3—21no—
51C2×Dic6D411yescharacteristic, complemented
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—42no—
55#217—21no—
56#217—21no—
57#217—21no—
58#214—42no—
59#213—42no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62C4⋊Dic3D411yescharacteristic, complemented
63C4⋊Dic3D411yescharacteristic, complemented
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66C4×Dic3—21no—
67C2×C3⋊C8—21no—
68C2×C3⋊C8—21no—
69C22×D4—21no—
70C22×D4C2×D311yescharacteristic
71C2×SD16—61no—
72C2×D8—61no—
73#126—61no—
74#124—61no—
75#123—61no—
76C22⋊Q8—31no—
77#121—31no—
78#121—61no—
79#120—61no—
80C4×D4—61no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4C2×D311yescharacteristic
83#102—61no—
84#102—61no—
85C22⋊C8—31no—
86C22⋊C8—31no—
87#95C2×D311yescharacteristic
88C23×C3—22no—
89C23×C3—21no—
90C23×C3—21no—
91C23×C3—21no—
92C23×C3—41no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3—41no—
97C23×C3—41no—
98C23×C3—41no—
99C23×C3—41no—
100C23×C3C2×D411yescharacteristic
101C22×D3—21no—
102C3×D4—22no—
103C3×D4—22no—
104C3×D4—41no—
105C3×D4—41no—
106C3×D4—48no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—41no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×C3×C4—41no—
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D411yescharacteristic
115#67—41no—
116#67—41no—
117#67—81no—
118#67—81no—
119C2×Dic3—21no—
120C2×Dic3—21no—
121C2×Dic3—21no—
122D12—41no—
123C4×D3—81no—
124Dic6—41no—
125C3⋊C8—41no—
126C3⋊C8—41no—
127C24—21no—
128C24#6711yescharacteristic
129C2×Q8—31no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—31no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—61no—
138C2×D4—61no—
139C2×D4—61no—
140C22×C4—21no—
141C22×C4—61no—
142C22×C4#6711yescharacteristic
143C22×C4C22×D311yescharacteristic
144SD16—122no—
145D8—122no—
146C2×C8—61no—
147C2×C8—61no—
148C4⋊C4—31no—
149C4⋊C4—31no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—42no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—61no—
160C42—61no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—42no—
170C22×C3—81no—
171C22×C3—81no—
172C22×C3—81no—
173C22×C3—81no—
174C22×C3—81no—
175C22×C3#12011yescharacteristic
176C22×C3#13511yescharacteristic
177C22×C3AGL(1,ℤ/8)11yescharacteristic
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—41no—
181C3×C4—21no—
182C3×C4—21no—
183C3×C4—42no—
184C3×C4—41no—
185Dic3—41no—
186Dic3—41no—
187Dic3—41no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—61no—
201C23C2×#6711yescharacteristic
202Q8—121no—
203D4—22no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—48no—
208D4—121no—
209D4—121no—
210D4—121no—
211D4—241no—
212D4—241no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—241no—
223C2×C4D3×D411yescharacteristic
224C2×C4D3×D411yescharacteristic
225C8—121no—
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—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—121no—
250C22—121no—
251C22—121no—
252C22#93211yescharacteristic
253C22#93711yescharacteristic
254C22#95011yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—121no—
260C4—121no—
261C4—121no—
262C3#160011yescharacteristic, complemented, Sylow
263C2—21no—
264C2—21no—
265C2—81no—
266C2—82no—
267C2—241no—
268C2#503211yescharacteristic
269C2#503711yescharacteristic
270C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#392C2
socleC22×C3#120
hypercenter#392D3
hypocenterC3#1600
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #5494C2semidirect
2. C3×#392C2semidirect
3. #5218C2semidirect
4. #5217C2semidirect
5. #4525C2semidirect
6. #4452C2semidirect
7. #4451C2semidirect
31. #392D3semidirect
49. C2×D12D4semidirect
51. C2×Dic6D4semidirect
62. C4⋊Dic3D4semidirect
63. C4⋊Dic3D4semidirect
262. C3#1600semidirect

Maximal subgroups

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

subgroupindex
#54942
C3×#3922
#52182
#52172
#45252
#44522
#44512
#16003