Finite Group Databasebeta

← #73554

Subgroups — #73554

Order 384.

428 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 145 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×C4×C22⋊C4C211yesmaximal, characteristic
2#5211C211yesmaximal, characteristic
3C4×#217C211yesmaximal, characteristic
4#4948C212yesmaximal
5#4948C212yesmaximal
6#4933C212yesmaximal, complemented
7#4928C212yesmaximal, complemented
8#4655C212yesmaximal, complemented
9#4647C212yesmaximal, complemented
10#2608—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C2×C3×C22⋊C4C2212yes—
13C2×C3×C42C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15C2×#217C2212yes—
16C2×#217C2211yescharacteristic
17C2×#217C2211yescharacteristic
18C2×C4⋊Dic3C2212yes—
19C2×C4⋊Dic3C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2211yescharacteristic
25C2×C4×Dic3C2211yescharacteristic
26C3×#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32#396—32no—
33#396—32no—
34#389—32no—
35#389—32no—
36#387—32no—
37#387—32no—
38#381—31no—
39C4×C22⋊C4—31no—
40C4×C22⋊C4D311yescharacteristic
41C24×C3C2311yescharacteristic
42C22×C3×C4—41no—
43C22×C3×C4—41no—
44C22×C3×C4Q814yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C3×C22⋊C4—24no—
54C3×C42—41no—
55C3×C42—41no—
56#217—24no—
57#217—41no—
58#217—41no—
59C4⋊Dic3—24no—
60C4⋊Dic3—42no—
61Dic3⋊C4—24no—
62Dic3⋊C4—24no—
63Dic3⋊C4—24no—
64Dic3⋊C4—42no—
65C4×Dic3—41no—
66C4×Dic3—41no—
67C23×C4C2×D311yescharacteristic
68C2×C4⋊C4—32no—
69C2×C4⋊C4—32no—
70C2×C4⋊C4—32no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C22⋊C4—32no—
75C2×C22⋊C4—31no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4C2×D312yes—
78C2×C42—31no—
79C2×C42—31no—
80C2×C42C2×D311yescharacteristic
81C2×C42C2×D311yescharacteristic
82#95—32no—
83#95—32no—
84#95—32no—
85#95—32no—
86#95—32no—
87#95C2×D312yes—
88C23×C3—22no—
89C23×C3—22no—
90C23×C3—21no—
91C23×C3—21no—
92C23×C3C2×Q812yes—
93C23×C3C2411yescharacteristic
94C2×C3×C4—22no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—22no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—44no—
104C2×C3×C4C2×Q812yes—
105C2×C3×C4C2×Q812yes—
106C2×C3×C4#4114yes—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3#4114yes—
119C24C22×D311yescharacteristic
120C22×C4—32no—
121C22×C4—32no—
122C22×C4—32no—
123C22×C4—32no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4Dic614yes—
127C22×C4C22×D312yes—
128C22×C4C22×D311yescharacteristic
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D312yes—
131C4⋊C4—64no—
132C4⋊C4—64no—
133C4⋊C4—64no—
134C4⋊C4—64no—
135C4⋊C4—122no—
136C4⋊C4—122no—
137C22⋊C4—24no—
138C22⋊C4—64no—
139C22⋊C4—121no—
140C22⋊C4—121no—
141C42—41no—
142C42—41no—
143C42—121no—
144C42—121no—
145C22×C3—22no—
146C22×C3—22no—
147C22×C3—22no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3C22×Q811yescharacteristic
151C22×C3C2×#4112yes—
152C22×C3C2×#4112yes—
153C22×C3H(5,𝔽2)11yescharacteristic
154C22×C3H(5,𝔽2)11yescharacteristic
155C3×C4—22no—
156C3×C4—22no—
157C3×C4—24no—
158C3×C4—42no—
159Dic3—24no—
160Dic3—42no—
161Dic3—42no—
162Dic3—42no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23C2×Dic612yes—
168C23C23×D311yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—34no—
179C2×C4—44no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4C2×Dic612yes—
192C2×C4C2×Dic612yes—
193C2×C4#23514yes—
194C2×C3—42no—
195C2×C3#53311yescharacteristic
196C2×C3#53311yescharacteristic
197C2×C3#54311yescharacteristic
198C2×C3#55312yes—
199C2×C3#55312yes—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—42no—
204C22—42no—
205C22C22×Dic611yescharacteristic
206C22C2×#23512yes—
207C22#100411yescharacteristic
208C22#100411yescharacteristic
209C22D3×#4112yes—
210C4—22no—
211C4—22no—
212C4—24no—
213C4—42no—
214C4—64no—
215C4—122no—
216C4—122no—
217C4—122no—
218C3#260811yescharacteristic, complemented, Sylow
219C2—42no—
220C2#547011yescharacteristic
221C2#548311yescharacteristic
222C2#551711yescharacteristic
223C2#554012yes—
224C2#555812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
prime radicalC24×C3C23
hypercenterC4×C22⋊C4D3
hypocenterC3#2608
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #4933C2semidirect
7. #4928C2semidirect
8. #4655C2semidirect
9. #4647C2semidirect
218. C3#2608semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊C42
#52112
C4×#2172
#49482
#49332
#49282
#46552
#46472
#26083