Finite Group Databasebeta

← #83000

Subgroups — #83000

Order 384.

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

normal subgroups only

This group has 576 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×#124C211yesmaximal, characteristic, complemented
2C2×#872C211yesmaximal, characteristic, complemented
3#5231C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic
6#5222C211yesmaximal, characteristic, complemented
7#4957C212yesmaximal, complemented
8#4956C212yesmaximal, complemented
9#4945C212yesmaximal, complemented
10#4944C212yesmaximal, complemented
11#4932C211yesmaximal, characteristic, complemented
12#2610—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C22×C4×D3C2211yescharacteristic
17C3×#124—24no—
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C42C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#872—24no—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#124D311yescharacteristic, complemented
39C2×#117—31no—
40#395—31no—
41#394—31no—
42#393—31no—
43#392—31no—
44#390—32no—
45#388—32no—
46#388—32no—
47#386—32no—
48#376—31no—
49C24×C3C2312yes—
50C23×D3C2311yescharacteristic
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—42no—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—22no—
70C3×C22⋊C4—24no—
71C3×C22⋊C4—24no—
72C3×C42D414yescomplemented
73#217—24no—
74#217—24no—
75#217—42no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—42no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4—42no—
82C4×Dic3—22no—
83C4×Dic3—24no—
84C22×Q8C2×D311yescharacteristic
85C22×D4—32no—
86C22×D4C2×D311yescharacteristic
87C23×C4—31no—
88#124—24no—
89#117—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D312yes—
100C2×C22⋊C4C2×D312yes—
101C2×C42—31no—
102C2×C42—32no—
103C2×C42C2×D311yescharacteristic
104#95—32no—
105#95—31no—
106#95—31no—
107#95—32no—
108#95—32no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3C2411yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3#4112yes—
122C3×Q8—44no—
123C3×D4—44no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134#67—48no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3#4114yes—
147C2×Dic3#4112yes—
148C4×D3—28no—
149C24—31no—
150C24C22×D312yes—
151C2×Q8—22no—
152C2×Q8—42no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—64no—
156C2×D4—64no—
157C2×D4—122no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—32no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4C22×D312yes—
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C42—64no—
182C42—62no—
183C42#6714yescomplemented
184C22×C3—42no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3C22×D411yescharacteristic
192C22×C3C2×#4111yescharacteristic
193C22×C3C2×#4111yescharacteristic
194C22×C3C2×#4112yes—
195C22×C3H(5,𝔽2)12yes—
196C2×D3—22no—
197C2×D3—24no—
198C2×D3—41no—
199C2×D3—41no—
200C2×D3—42no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—42no—
205Dic3—22no—
206Dic3—24no—
207Dic3—41no—
208Dic3—41no—
209Dic3—42no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—32no—
218C23—62no—
219C23—61no—
220C23—62no—
221C23—61no—
222C23C23×D311yescharacteristic
223Q8—44no—
224D4—44no—
225D4—128no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—34no—
234C2×C4—32no—
235C2×C4—68no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—62no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4C2×#6712yes—
248C2×C4C2×#6712yes—
249C2×C4C2×#6712yes—
250C2×C3—82no—
251C2×C3#52511yescharacteristic
252C2×C3#53211yescharacteristic
253C2×C3#53511yescharacteristic
254C2×C3#55112yes—
255C2×C3#55512yes—
256D3—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—64no—
265C22—62no—
266C22—122no—
267C22—121no—
268C22—121no—
269C22#100412yes—
270C22D3×#4111yescharacteristic
271C22D3×#4112yes—
272C22D3×#4111yescharacteristic
273C22C22×#6711yescharacteristic
274C4—22no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—64no—
279C4—62no—
280C4—122no—
281C4—121no—
282C4—121no—
283C3#261011yescharacteristic, complemented, Sylow
284C2—82no—
285C2—122no—
286C2#567012yes—
287C2#567912yes—
288C2#580511yescharacteristic
289C2#582811yescharacteristic
290C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radical#5223C2
hypercenterC2×#124D3
hypocenterC3#2610
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
2. C2×#872C2semidirect
3. #5231C2semidirect
4. #5230C2semidirect
6. #5222C2semidirect
7. #4957C2semidirect
8. #4956C2semidirect
9. #4945C2semidirect
10. #4944C2semidirect
11. #4932C2semidirect
38. C2×#124D3semidirect
72. C3×C42D4semidirect
183. C42#67semidirect
283. C3#2610semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#8722
#52312
#52302
#52232
#52222
#49572
#49562
#49452
#49442
#49322
#26103