Finite Group Databasebeta

← #78007

Subgroups — #78007

Order 384.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#896C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic
3C3×#398C211yesmaximal, characteristic, complemented
4#4993C212yesmaximal, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4991C212yesmaximal, complemented
7#4983C211yesmaximal, characteristic, complemented
8#4673C212yesmaximal, complemented
9#4672C211yesmaximal, characteristic, complemented
10#4672C212yesmaximal, complemented
11#4664C211yesmaximal, characteristic, complemented
12#2816—31nomaximal, Sylow
13C22×D12C2212yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31#896—24no—
32#895—24no—
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×C22⋊Q8—31no—
39C2×#121—31no—
40#399—32no—
41#398—31no—
42#398D311yescharacteristic, complemented
43#394—31no—
44#394—32no—
45#394—31no—
46#394—32no—
47#393—32no—
48#381—31no—
49C23×D3C2312yes—
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×D12—24no—
61C2×D12—22no—
62C2×D12—42no—
63C2×D12—42no—
64C2×D12—42no—
65C2×C4×D3—42no—
66C2×C4×D3D414yes—
67C2×Dic6—22no—
68C2×Dic6—42no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—42no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4—42no—
80Dic3⋊C4—22no—
81Dic3⋊C4—22no—
82Dic3⋊C4—42no—
83C22×Q8—31no—
84C22×D4—32no—
85C22×D4—31no—
86C23×C4—31no—
87C22⋊Q8—64no—
88#121—64no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4C2×D311yescharacteristic
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D312yes—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103#95—32no—
104#95—31no—
105#95—32no—
106#95C2×D311yescharacteristic
107#95C2×D312yes—
108C23×C3C2411yescharacteristic
109C22×D3—22no—
110C22×D3—21no—
111C22×D3—21no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3C2×D412yes—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3C2×D412yes—
144D12—44no—
145D12—48no—
146C4×D3—44no—
147Dic6—44no—
148C24—31no—
149C24—32no—
150C2×Q8—62no—
151C2×Q8—122no—
152C2×D4—64no—
153C2×D4—62no—
154C2×D4—122no—
155C2×D4—122no—
156C2×D4—122no—
157C22×C4—31no—
158C22×C4—34no—
159C22×C4—32no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—122no—
163C22×C4C22×D312yes—
164C22×C4C22×D312yes—
165C22×C4C22×D311yescharacteristic
166C22×C4C22×D311yescharacteristic
167C22×C4C22×D311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—122no—
174C22⋊C4—64no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22×C3C22×D411yescharacteristic
183C22×C3H(5,𝔽2)11yescharacteristic
184C22×C3H(5,𝔽2)12yes—
185C22×C3H(5,𝔽2)11yescharacteristic
186C22×C32−1+412yes—
187C2×D3—22no—
188C2×D3—22no—
189C2×D3—22no—
190C2×D3—42no—
191C2×D3—42no—
192C2×D3—42no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—42no—
197C2×D3—42no—
198C2×D3—42no—
199C3×C4—22no—
200C3×C4—41no—
201C3×C4—42no—
202C3×C4—41no—
203C3×C4—42no—
204Dic3—22no—
205Dic3—41no—
206Dic3—41no—
207Dic3—42no—
208C23—32no—
209C23—61no—
210C23—62no—
211C23—62no—
212C23—61no—
213C23—62no—
214C23—62no—
215C23—62no—
216C23—62no—
217C23—62no—
218C23—62no—
219C23—62no—
220C23C23×D311yescharacteristic
221Q8—124no—
222D4—124no—
223D4—128no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—32no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—124no—
247C2×C4D3×D412yes—
248C2×C3#53411yescharacteristic
249C2×C3#53511yescharacteristic
250C2×C3#53611yescharacteristic
251C2×C3#56212yes—
252C2×C3#56212yes—
253D3—42no—
254D3—82no—
255C22—62no—
256C22—62no—
257C22—62no—
258C22—122no—
259C22—122no—
260C22—122no—
261C22—122no—
262C22—122no—
263C22—122no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22C2×D3×D411yescharacteristic
268C22#100411yescharacteristic
269C22#100712yes—
270C22#100912yes—
271C22#100911yescharacteristic
272C4—22no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—62no—
278C4—121no—
279C4—122no—
280C4—121no—
281C3#281611yescharacteristic, complemented, Sylow
282C2—122no—
283C2—242no—
284C2#560711yescharacteristic
285C2#564712yes—
286C2#565311yescharacteristic
287C2#570012yes—
288C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC2×#895C2
hypercenter#398D3
hypocenterC3#2816
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
3. C3×#398C2semidirect
4. #4993C2semidirect
5. #4991C2semidirect
6. #4991C2semidirect
7. #4983C2semidirect
8. #4673C2semidirect
9. #4672C2semidirect
10. #4672C2semidirect
11. #4664C2semidirect
42. #398D3semidirect
281. C3#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8952
C3×#3982
#49932
#49912
#49832
#46732
#46722
#46642
#28163