Finite Group Databasebeta

← #76808

Subgroups — #76808

Order 384.

576 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 135 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×#947C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic, complemented
3C3×#393C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#4960C211yesmaximal, characteristic
6#4958C212yesmaximal, complemented
7#4957C212yesmaximal, complemented
8#4672C211yesmaximal, characteristic, complemented
9#4669C212yesmaximal, complemented
10#4669C212yesmaximal, complemented
11#4648C211yesmaximal, characteristic, complemented
12#2610—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×#67C2212yes—
15C23×Dic3C2211yescharacteristic
16C22×D12C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2211yescharacteristic
20#947—24no—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32#877—24no—
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×#117—31no—
40#395—31no—
41#394—31no—
42#393D311yescharacteristic, complemented
43#392—31no—
44#390—32no—
45#388—32no—
46#388—32no—
47#386—32no—
48#376—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2312yes—
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—24no—
58C2×#67—24no—
59C2×#67—42no—
60C22×Dic3—22no—
61C22×Dic3—22no—
62C22×Dic3—22no—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C2×D12—22no—
69C2×D12—42no—
70C3×C22⋊C4—22no—
71C3×C22⋊C4—42no—
72#217—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—42no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4—42no—
81C4×Dic3—22no—
82C4×Dic3—24no—
83C4×Dic3D414yescomplemented
84C22×Q8C2×D311yescharacteristic
85C22×D4—32no—
86C22×D4—31no—
87C23×C4—31no—
88#124—64no—
89#117—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D312yes—
101C2×C42—31no—
102C2×C42—31no—
103C2×C42—32no—
104#95—32no—
105#95—31no—
106#95—32no—
107#95C2×D311yescharacteristic
108#95C2×D312yes—
109C23×C3—21no—
110C23×C3—21no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3#4112yes—
114C23×C3C2411yescharacteristic
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C3×Q8—44no—
123C2×C3×C4—22no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D412yes—
133#67—48no—
134C2×Dic3—28no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3C2×D412yes—
145C2×Dic3C2×D412yes—
146C2×Dic3#4114yes—
147C2×Dic3#4112yes—
148D12—44no—
149C24—32no—
150C24C22×D311yescharacteristic
151C2×Q8—22no—
152C2×Q8—42no—
153C2×D4—64no—
154C2×D4—64no—
155C2×D4—62no—
156C2×D4—122no—
157C2×D4—122no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4C22×D312yes—
167C22×C4C22×D312yes—
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C22⋊C4—22no—
174C22⋊C4—42no—
175C22⋊C4—64no—
176C22⋊C4—64no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—62no—
180C22⋊C4—122no—
181C42—34no—
182C42—62no—
183C42—64no—
184C22×C3—22no—
185C22×C3—24no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C22×C3C2×#4111yescharacteristic
191C22×C3C2×#4111yescharacteristic
192C22×C3C2×#4112yes—
193C22×C3H(5,𝔽2)12yes—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—42no—
197C2×D3—42no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—42no—
201C3×C4—22no—
202C3×C4—41no—
203C3×C4—42no—
204C3×C4—42no—
205Dic3—22no—
206Dic3—22no—
207Dic3—22no—
208Dic3—24no—
209Dic3—41no—
210C23—21no—
211C23—22no—
212C23—21no—
213C23—22no—
214C23—62no—
215C23—62no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23—62no—
221C23#23912yes—
222C23C23×D311yescharacteristic
223Q8—44no—
224D4—124no—
225D4—128no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—34no—
238C2×C4—32no—
239C2×C4—68no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—62no—
249C2×C4D3×D412yes—
250C2×C3—42no—
251C2×C3#52511yescharacteristic
252C2×C3#53211yescharacteristic
253C2×C3#53511yescharacteristic
254C2×C3#55112yes—
255C2×C3#55512yes—
256D3—82no—
257C22—24no—
258C22—22no—
259C22—41no—
260C22—42no—
261C22—41no—
262C22—122no—
263C22—122no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22—122no—
268C22—122no—
269C22C2×D3×D411yescharacteristic
270C22#100412yes—
271C22C2×#23911yescharacteristic
272C22D3×#4112yes—
273C22D3×#4111yescharacteristic
274C4—22no—
275C4—41no—
276C4—42no—
277C4—42no—
278C4—64no—
279C4—62no—
280C4—62no—
281C4—62no—
282C4—121no—
283C3#261011yescharacteristic, complemented, Sylow
284C2—42no—
285C2—242no—
286C2#559211yescharacteristic
287C2#562711yescharacteristic
288C2#563712yes—
289C2#566811yescharacteristic
290C2#567012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radical#4960C2
hypercenter#393D3
hypocenterC3#2610
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#947C2semidirect
2. C2×#877C2semidirect
3. C3×#393C2semidirect
4. #5230C2semidirect
6. #4958C2semidirect
7. #4957C2semidirect
8. #4672C2semidirect
9. #4669C2semidirect
10. #4669C2semidirect
11. #4648C2semidirect
42. #393D3semidirect
83. C4×Dic3D4semidirect
283. C3#2610semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#8772
C3×#3932
#52302
#49602
#49582
#49572
#46722
#46692
#46482
#26103