Finite Group Databasebeta

← #82374

Subgroups — #82374

Order 384.

584 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×C3×C4×D4C211yesmaximal, characteristic
2C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#5220C212yesmaximal
4#5212C212yesmaximal, complemented
5#5212C211yesmaximal, characteristic, complemented
6#4986C212yesmaximal, complemented
7#4949C212yesmaximal, complemented
8#4946C212yesmaximal, complemented
9#4928C212yesmaximal, complemented
10#2487—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×C3×C4C2211yescharacteristic
14C23×Dic3C2212yes—
15C3×C4×D4—24no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2214yes—
22C2×C4⋊Dic3—24no—
23C2×C4⋊Dic3C2212yes—
24C2×C4⋊Dic3C2212yes—
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2214yes—
27C3⋊#95C2214yes—
28C3⋊#95C2214yes—
29C3⋊#95C2212yes—
30C2×C4×D4D311yescharacteristic
31C22×C4⋊C4—31no—
32#400—32no—
33#399—32no—
34#389—32no—
35#386—32no—
36#385—32no—
37#380—32no—
38#380—31no—
39C24×C3C2311yescharacteristic
40C24×C3C2311yescharacteristic
41C2×C3×D4—22no—
42C2×C3×D4—22no—
43C2×C3×D4Q814yes—
44C22×C3×C4—24no—
45C22×C3×C4—22no—
46C22×C3×C4—24no—
47C22×C3×C4D414yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×Dic3—24no—
54C22×Dic3—44no—
55C22×Dic3C2314yes—
56C22×Dic3C2314yes—
57C3×C4⋊C4—22no—
58C3×C22⋊C4—22no—
59C3×C22⋊C4—22no—
60C3×C42—22no—
61#217—24no—
62#217—44no—
63C4⋊Dic3—24no—
64C4⋊Dic3—24no—
65C4⋊Dic3—24no—
66Dic3⋊C4—44no—
67C22×D4C2×D311yescharacteristic
68C23×C4—32no—
69C23×C4C2×D311yescharacteristic
70C23×C4C2×D311yescharacteristic
71C4×D4—24no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—34no—
76C2×C4⋊C4—64no—
77C2×C4⋊C4C2×D311yescharacteristic
78C2×C22⋊C4—32no—
79C2×C22⋊C4—34no—
80C2×C22⋊C4C2×D311yescharacteristic
81C2×C22⋊C4C2×D311yescharacteristic
82C2×C42C2×D311yescharacteristic
83#95—34no—
84#95—32no—
85#95—34no—
86C23×C3—22no—
87C23×C3—22no—
88C23×C3—21no—
89C23×C3—22no—
90C23×C3—21no—
91C23×C3—22no—
92C23×C3C2×D412yes—
93C23×C3C2×Q812yes—
94C23×C3C2×Q812yes—
95C23×C3#4112yes—
96C23×C3C2411yescharacteristic
97C3×D4—28no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—21no—
101C2×C3×C4—28no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—44no—
107C2×C3×C4C2×D412yes—
108C2×C3×C4C2×D412yes—
109C2×C3×C4C2×Q812yes—
110C2×C3×C4#4114yes—
111C2×Dic3—24no—
112C2×Dic3—24no—
113C2×Dic3—24no—
114C2×Dic3—24no—
115C2×Dic3—24no—
116C2×Dic3—24no—
117C2×Dic3—48no—
118C24C22×D311yescharacteristic
119C24C22×D311yescharacteristic
120C2×D4—22no—
121C2×D4—22no—
122C2×D4Dic614yes—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—34no—
127C22×C4—34no—
128C22×C4—64no—
129C22×C4—124no—
130C22×C4#6714yes—
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D311yescharacteristic
133C22×C4C22×D311yescharacteristic
134C22×C4C22×D311yescharacteristic
135C22×C4C22×D311yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—64no—
138C4⋊C4—64no—
139C4⋊C4—64no—
140C4⋊C4—124no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—64no—
144C22⋊C4—124no—
145C42—22no—
146C22×C3—22no—
147C22×C3—22no—
148C22×C3—24no—
149C22×C3—24no—
150C22×C3—22no—
151C22×C3—24no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—42no—
155C22×C3C22⋊Q814yes—
156C22×C3C22×D411yescharacteristic
157C22×C3C22×Q811yescharacteristic
158C22×C3C2×#4111yescharacteristic
159C22×C3C2×#4112yes—
160C22×C3H(5,𝔽2)12yes—
161C3×C4—22no—
162C3×C4—22no—
163C3×C4—41no—
164C3×C4—41no—
165C3×C4#12314yes—
166Dic3—44no—
167Dic3—44no—
168C23—22no—
169C23—22no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23—21no—
174C23C2×Dic612yes—
175C23C2×Dic612yes—
176C23#23512yes—
177C23C2×#6712yes—
178C23C23×D311yescharacteristic
179D4—28no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—28no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—44no—
189C2×C4—64no—
190C2×C4—64no—
191C2×C4—64no—
192C2×C4—64no—
193C2×C4—64no—
194C2×C4—64no—
195C2×C4—128no—
196C2×C4C2×Dic612yes—
197C2×C4#23714yes—
198C2×C4C2×#6712yes—
199C2×C4C2×#6712yes—
200C2×C3—24no—
201C2×C3—42no—
202C2×C3C2×C22⋊Q811yescharacteristic
203C2×C3C2×#12311yescharacteristic
204C2×C3#53511yescharacteristic
205C2×C3#55312yes—
206C2×C3#55412yes—
207C22—22no—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—41no—
214C22—42no—
215C22—41no—
216C22#92414yes—
217C22C22×Dic611yescharacteristic
218C22C2×#23511yescharacteristic
219C22C2×#23712yes—
220C22#100912yes—
221C22C22×#6711yescharacteristic
222C4—22no—
223C4—22no—
224C4—41no—
225C4—41no—
226C4—124no—
227C4—124no—
228C4#93514yes—
229C3#248711yescharacteristic, complemented, Sylow
230C2—24no—
231C2—42no—
232C2#554312yes—
233C2#556512yes—
234C2C2×#92411yescharacteristic
235C2C2×#93511yescharacteristic
236C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic3C2semidirect
4. #5212C2semidirect
5. #5212C2semidirect
6. #4986C2semidirect
7. #4949C2semidirect
8. #4946C2semidirect
9. #4928C2semidirect
229. C3#2487semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C22×C4⋊Dic32
#52202
#52122
#49862
#49492
#49462
#49282
#24873