Finite Group Databasebeta

← #83089

Subgroups — #83089

Order 384.

450 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 137 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×#126C211yesmaximal, characteristic
2C2×C4×Dic6C211yesmaximal, characteristic, complemented
3#4982C212yesmaximal, complemented
4#4981C211yesmaximal, characteristic, complemented
5#4979C211yesmaximal, characteristic, complemented
6#4978C212yesmaximal, complemented
7#4953C211yesmaximal, characteristic
8#4948C212yesmaximal
9#4947C211yesmaximal, characteristic
10#4944C212yesmaximal
11#4936C211yesmaximal, characteristic, complemented
12#2637—31nomaximal, Sylow
13C22×Dic6C2211yescharacteristic
14C3×#126—24no—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C4×Dic6—24no—
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126D311yescharacteristic
39C2×C4×Q8—31no—
40#398—31no—
41#398—32no—
42#397—31no—
43#396—32no—
44#395—31no—
45#393—31no—
46#388—32no—
47#387—32no—
48#382—31no—
49C24×C3C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C2×Dic6—22no—
61C2×Dic6—24no—
62C3×C4⋊C4—22no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—22no—
65C3×C22⋊C4—22no—
66C3×C22⋊C4—22no—
67C3×C22⋊C4—22no—
68C3×C42D414yes—
69#217—24no—
70#217—42no—
71C4⋊Dic3—22no—
72C4⋊Dic3—42no—
73Dic3⋊C4—24no—
74Dic3⋊C4—24no—
75Dic3⋊C4—42no—
76C4×Dic3—24no—
77C22×Q8—31no—
78#126—24no—
79C4×Q8—64no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4C2×D311yescharacteristic
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4C2×D311yescharacteristic
91C2×C22⋊C4C2×D311yescharacteristic
92C2×C22⋊C4C2×D311yescharacteristic
93C2×C42—32no—
94C2×C42C2×D311yescharacteristic
95#95—32no—
96#95—31no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95—32no—
102C23×C3—21no—
103C23×C3—22no—
104C23×C3—21no—
105C23×C3—22no—
106C23×C3—21no—
107C23×C3C2411yescharacteristic
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D412yes—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3#4114yes—
134Dic6—28no—
135C24C22×D311yescharacteristic
136C2×Q8—64no—
137C2×Q8—62no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4C22×D311yescharacteristic
143C22×C4C22×D311yescharacteristic
144C22×C4C22×D311yescharacteristic
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D311yescharacteristic
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—64no—
152C4⋊C4—64no—
153C4⋊C4—62no—
154C4⋊C4—122no—
155C4⋊C4—122no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C22⋊C4—64no—
160C22⋊C4—122no—
161C42—64no—
162C42#6714yes—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3C22×D411yescharacteristic
169C22×C3C2×#4112yes—
170C22×C3H(5,𝔽2)11yescharacteristic
171C22×C32−1+412yes—
172C22×C32−1+411yescharacteristic
173C3×C4—22no—
174C3×C4—22no—
175C3×C4—22no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—41no—
179Dic3—24no—
180Dic3—42no—
181Dic3—42no—
182Dic3—42no—
183C23—22no—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—21no—
188C23C23×D311yescharacteristic
189Q8—68no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—34no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4C2×#6712yes—
214C2×C4C2×#6712yes—
215C2×C4C2×#6712yes—
216C2×C3—81no—
217C2×C3#52511yescharacteristic
218C2×C3#53611yescharacteristic
219C2×C3#53711yescharacteristic
220C2×C3#54112yes—
221C2×C3#54212yes—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22D3×#4112yes—
228C22#100911yescharacteristic
229C22#101011yescharacteristic
230C22#101012yes—
231C22C22×#6711yescharacteristic
232C4—22no—
233C4—22no—
234C4—22no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—64no—
239C4—122no—
240C4—122no—
241C4—122no—
242C3#263711yescharacteristic, complemented, Sylow
243C2—81no—
244C2#570112yes—
245C2#571012yes—
246C2#582511yescharacteristic
247C2#582811yescharacteristic
248C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#126C2
prime radicalC24×C3C23
hypercenterC2×#126D3
hypocenterC3#2637
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic6C2semidirect
3. #4982C2semidirect
4. #4981C2semidirect
5. #4979C2semidirect
6. #4978C2semidirect
11. #4936C2semidirect
242. C3#2637semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1262
C2×C4×Dic62
#49822
#49812
#49792
#49782
#49532
#49482
#49472
#49442
#49362
#26373