Finite Group Databasebeta

← #71885

Subgroups — #71885

Order 384.

260 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C3×#538C211yesmaximal, characteristic
2#5050C211yesmaximal, characteristic, complemented
3#5040C211yesmaximal, characteristic, complemented
4#4947C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4537C211yesmaximal, characteristic, complemented
7#4525C211yesmaximal, characteristic, complemented
8#1606—31nomaximal, Sylow
9C22×Dic6C2211yescharacteristic
10C3×#126—21no—
11C3×#124—21no—
12C3×#123—21no—
13C3×C22⋊Q8C2211yescharacteristic
14C3×#121—21no—
15C3×#121C2211yescharacteristic
16C3×#120—21no—
17C3×C4×D4—21no—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217—21no—
20C2×#216—21no—
21C2×#214—21no—
22C3⋊#95—21no—
23C3⋊#95C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26#810—21no—
27#810—21no—
28#538D311yescharacteristic
29#451—31no—
30#448—31no—
31#393—31no—
32#328—31no—
33#327—31no—
34#323—31no—
35C24×C3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4D411yescharacteristic
41C22×C3×C4—21no—
42C22×C3×C4D411yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×Dic3—21no—
45C22×Dic3—21no—
46C2×Dic6—21no—
47C2×Dic6—21no—
48C2×Dic6—42no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4D411yescharacteristic
52C3×C4⋊C4D411yescharacteristic
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C22⋊C4—21no—
60C3×C42—21no—
61#217—42no—
62#216—42no—
63#214—42no—
64C2×C3⋊C8—21no—
65C2×C3⋊C8—21no—
66C22×Q8—31no—
67C2×Q16—61no—
68C2×SD16—61no—
69#126—21no—
70#124—21no—
71#123—21no—
72C22⋊Q8C2×D311yescharacteristic
73#121—21no—
74#121C2×D311yescharacteristic
75#120—21no—
76C4×D4—21no—
77C2×C22⋊C4—61no—
78C2×C22⋊C4C2×D311yescharacteristic
79#103—61no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8—31no—
83#95—31no—
84#95—61no—
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—21no—
88C23×C3—21no—
89C23×C3—41no—
90C23×C3—41no—
91C23×C3C2×D411yescharacteristic
92C3×Q8—41no—
93C3×D4—41no—
94C3×D4—41no—
95C3×D4—41no—
96C3×D4—81no—
97C3×D4—81no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—81no—
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×Dic3—22no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113Dic6—22no—
114Dic6—22no—
115Dic6—41no—
116Dic6—41no—
117C3⋊C8—41no—
118C3⋊C8—41no—
119C24#6711yescharacteristic
120C2×Q8—61no—
121C2×Q8—61no—
122C2×Q8—122no—
123C2×Q8#6711yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4#6711yescharacteristic
128C22×C4—21no—
129C22×C4—61no—
130C22×C4—61no—
131C22×C4#6711yescharacteristic
132C22×C4C22×D311yescharacteristic
133Q16—122no—
134SD16—122no—
135C2×C8—61no—
136C2×C8—61no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4#6711yescharacteristic
140C4⋊C4#6711yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—21no—
148C22⋊C4—122no—
149C42—21no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—42no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—81no—
160C22×C3#12011yescharacteristic
161C22×C3#13511yescharacteristic
162C22×C3#13711yescharacteristic
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—41no—
168C3×C4—41no—
169Dic3—42no—
170Dic3—81no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×#6711yescharacteristic
178Q8—41no—
179Q8—62no—
180Q8—62no—
181Q8—121no—
182Q8—121no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—81no—
187D4—81no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—62no—
195C2×C4—81no—
196C2×C4—121no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4C2×#6711yescharacteristic
202C2×C4C2×#6711yescharacteristic
203C8—121no—
204C8—121no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—81no—
208C2×C3—81no—
209C2×C3#45211yescharacteristic
210C2×C3#45511yescharacteristic
211C2×C3B(4,𝔽2)11yescharacteristic
212C22—21no—
213C22—21no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—81no—
222C22#95011yescharacteristic
223C22#95111yescharacteristic
224C22#95311yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—122no—
232C4—241no—
233C3#160611yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2—81no—
237C2—81no—
238C2#523911yescharacteristic
239C2#524211yescharacteristic
240C2#524411yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#538C2
socleC22×C3#120
hypercenter#538D3
hypocenterC3#1606
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
2. #5050C2semidirect
3. #5040C2semidirect
4. #4947C2semidirect
5. #4538C2semidirect
6. #4537C2semidirect
7. #4525C2semidirect
233. C3#1606semidirect

Maximal subgroups

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

subgroupindex
C3×#5382
#50502
#50402
#49472
#45382
#45372
#45252
#16063