Finite Group Databasebeta

← #82504

Subgroups — #82504

Order 384.

450 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 141 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×Q8C211yesmaximal, characteristic, complemented
2C2×#876C211yesmaximal, characteristic
3#5231C211yesmaximal, characteristic
4#5230C211yesmaximal, characteristic, complemented
5#4994C212yesmaximal
6#4992C211yesmaximal, characteristic
7#4983C211yesmaximal, characteristic, complemented
8#4982C212yesmaximal, complemented
9#4940C212yesmaximal
10#4936C211yesmaximal, characteristic, complemented
11#4933C212yesmaximal, complemented
12#2637—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C3×C4×Q8—24no—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2212yes—
18C2×C3×C42C2211yescharacteristic
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30#876—24no—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×C4×Q8D311yescharacteristic, complemented
40#398—32no—
41#398—31no—
42#397—31no—
43#396—32no—
44#395—31no—
45#393—31no—
46#388—32no—
47#387—32no—
48#382—31no—
49C23×D3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—24no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C3×C4⋊C4—24no—
63C3×C4⋊C4—22no—
64C3×C42—24no—
65C3×C42D414yes—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—24no—
70(C2×D3)⋊C4—42no—
71C4⋊Dic3—42no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4—22no—
75Dic3⋊C4—24no—
76Dic3⋊C4—42no—
77C22×Q8C2×D311yescharacteristic
78#126—64no—
79C4×Q8—24no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D312yes—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C42C2×D312yes—
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×C3C2411yescharacteristic
103C22×D3—21no—
104C22×D3—22no—
105C22×D3—21no—
106C22×D3—22no—
107C22×D3—21no—
108C3×Q8—28no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4#4114yes—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—21no—
135C24—31no—
136C2×Q8—24no—
137C2×Q8—22no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4C22×D312yes—
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D311yescharacteristic
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—64no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—122no—
155C4⋊C4—122no—
156C22⋊C4—64no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C22⋊C4—122no—
161C42—24no—
162C42#6714yes—
163C22×C3C22×D411yescharacteristic
164C22×C3C2×#4112yes—
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C32−1+412yes—
167C22×C32−1+411yescharacteristic
168C2×D3—41no—
169C2×D3—41no—
170C2×D3—41no—
171C2×D3—42no—
172C2×D3—42no—
173C3×C4—22no—
174C3×C4—24no—
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—42no—
178Dic3—41no—
179Dic3—41no—
180Dic3—41no—
181Dic3—42no—
182Dic3—42no—
183C23—62no—
184C23—62no—
185C23—61no—
186C23—61no—
187C23—61no—
188C23C23×D311yescharacteristic
189Q8—28no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—61no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4#23514yes—
213C2×C4C2×#6712yes—
214C2×C4C2×#6712yes—
215C2×C4C2×#6712yes—
216C2×C3#52511yescharacteristic
217C2×C3#53611yescharacteristic
218C2×C3#53711yescharacteristic
219C2×C3#54112yes—
220C2×C3#54212yes—
221D3—81no—
222C22—121no—
223C22—121no—
224C22—121no—
225C22—122no—
226C22—122no—
227C22C2×#23512yes—
228C22#100712yes—
229C22#100911yescharacteristic
230C22#101011yescharacteristic
231C22C22×#6711yescharacteristic
232C4—24no—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—121no—
238C4—121no—
239C4—121no—
240C4—122no—
241C4—122no—
242C3#263711yescharacteristic, complemented, Sylow
243C2—241no—
244C2#558212yes—
245C2#558312yes—
246C2#579111yescharacteristic
247C2#581611yescharacteristic
248C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×Q8C2
prime radicalC23×D3C23
hypercenterC2×C4×Q8D3
hypocenterC3#2637
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q8C2semidirect
4. #5230C2semidirect
7. #4983C2semidirect
8. #4982C2semidirect
10. #4936C2semidirect
11. #4933C2semidirect
39. C2×C4×Q8D3semidirect
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×C4×Q82
C2×#8762
#52312
#52302
#49942
#49922
#49832
#49822
#49402
#49362
#49332
#26373