Finite Group Databasebeta

← #74585

Subgroups — #74585

Order 384.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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×#385C211yesmaximal, characteristic
2#5210C212yesmaximal
3#4985C212yesmaximal, complemented
4#4979C212yesmaximal, complemented
5#4953C211yesmaximal, characteristic
6#4952C212yesmaximal
7#4947C211yesmaximal, characteristic
8#4660C212yesmaximal, complemented
9#4658C212yesmaximal, complemented
10#2835—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C22⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2212yes—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×Dic3⋊C4C2212yes—
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C3×#95C2212yes—
23C3⋊#95C2211yescharacteristic
24C3⋊#95C2212yes—
25C3⋊#95C2212yes—
26C3⋊#95C2212yes—
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32#400—32no—
33#400—32no—
34#400—32no—
35#397—32no—
36#397—31no—
37#395—32no—
38#393—31no—
39#385—32no—
40#385D311yescharacteristic
41C24×C3C2311yescharacteristic
42C22×C3×C4—41no—
43C22×C3×C4—42no—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×Dic6—41no—
53C2×Dic6—42no—
54C3×C4⋊C4—42no—
55C3×C22⋊C4—22no—
56C3×C22⋊C4—41no—
57#217—24no—
58#217—42no—
59C4⋊Dic3—41no—
60Dic3⋊C4—42no—
61Dic3⋊C4—42no—
62Dic3⋊C4—42no—
63C22×Q8—31no—
64C23×C4C2×D311yescharacteristic
65C2×C4⋊C4—32no—
66C2×C4⋊C4—32no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4C2×D312yes—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—32no—
72C2×C22⋊C4C2×D311yescharacteristic
73C2×C22⋊C4C2×D311yescharacteristic
74#95—32no—
75#95—32no—
76#95—31no—
77#95—32no—
78#95—32no—
79#95—32no—
80#95—32no—
81#95—32no—
82#95—31no—
83#95C2×D312yes—
84C23×C3—22no—
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—22no—
88C23×C3#4112yes—
89C23×C3C2411yescharacteristic
90C2×C3×C4—22no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—22no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—44no—
101C2×Dic3—22no—
102C2×Dic3—22no—
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113Dic6—44no—
114C24C22×D311yescharacteristic
115C2×Q8—121no—
116C2×Q8—122no—
117C22×C4—32no—
118C22×C4—32no—
119C22×C4—32no—
120C22×C4—32no—
121C22×C4—42no—
122C22×C4—41no—
123C22×C4C22×D311yescharacteristic
124C22×C4C22×D312yes—
125C22×C4C22×D311yescharacteristic
126C22×C4C22×D312yes—
127C4⋊C4—42no—
128C4⋊C4—122no—
129C4⋊C4—122no—
130C4⋊C4—122no—
131C4⋊C4—121no—
132C22⋊C4—22no—
133C22⋊C4—41no—
134C22⋊C4—64no—
135C22⋊C4—122no—
136C22×C3—22no—
137C22×C3—24no—
138C22×C3—41no—
139C22×C3—42no—
140C22×C3—41no—
141C22×C3C2×#4111yescharacteristic
142C22×C3H(5,𝔽2)12yes—
143C22×C3H(5,𝔽2)11yescharacteristic
144C22×C32−1+411yescharacteristic
145C22×C32−1+412yes—
146C3×C4—42no—
147C3×C4—41no—
148C3×C4—42no—
149C3×C4—41no—
150Dic3—42no—
151Dic3—42no—
152Dic3—42no—
153Dic3—42no—
154C23—21no—
155C23—22no—
156C23—21no—
157C23—22no—
158C23#23512yes—
159C23C23×D311yescharacteristic
160Q8—124no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—44no—
172C2×C4—62no—
173C2×C4—62no—
174C2×C4—62no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C3—42no—
185C2×C3#53911yescharacteristic
186C2×C3#53912yes—
187C2×C3H(3,𝔽4)11yescharacteristic
188C2×C3#56312yes—
189C2×C3#56311yescharacteristic
190C22—22no—
191C22—24no—
192C22—42no—
193C22—41no—
194C22—41no—
195C22C2×#23511yescharacteristic
196C22#100412yes—
197C22#100411yescharacteristic
198C22#100711yescharacteristic
199C22#101012yes—
200C4—41no—
201C4—42no—
202C4—42no—
203C4—41no—
204C4—122no—
205C4—122no—
206C4—122no—
207C4—122no—
208C3#283511yescharacteristic, complemented, Sylow
209C2—42no—
210C2#551511yescharacteristic
211C2#556412yes—
212C2#559411yescharacteristic
213C2#564612yes—
214C2#565111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×C3C23
hypercenter#385D3
hypocenterC3#2835
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #4985C2semidirect
4. #4979C2semidirect
8. #4660C2semidirect
9. #4658C2semidirect
208. C3#2835semidirect

Maximal subgroups

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

subgroupindex
C3×#3852
#52102
#49852
#49792
#49532
#49522
#49472
#46602
#46582
#28353