Finite Group Databasebeta

← #76984

Subgroups — #76984

Order 384.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#940C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4#5222C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4961C212yesmaximal, complemented
7#4959C212yesmaximal, complemented
8#4957C212yesmaximal, complemented
9#4957C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic, complemented
11#4648C211yesmaximal, characteristic, complemented
12#2641—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C22×D12C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22#940—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34#889—24no—
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×#127—31no—
39C2×#125—31no—
40#394—31no—
41#394—31no—
42#394—32no—
43#394—32no—
44#394—31no—
45#394D311yescharacteristic, complemented
46#390—32no—
47#390—32no—
48#376—31no—
49C24×C3C2312yes—
50C23×D3C2312yes—
51C2×C3×D4—22no—
52C2×C3×D4—42no—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—24no—
58C2×#67—24no—
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—42no—
62C2×#67—42no—
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C2×D12—22no—
69C2×D12—42no—
70C3×C4⋊C4—22no—
71C3×C22⋊C4—42no—
72C3×C22⋊C4—42no—
73#217—42no—
74#217—42no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84C4×Dic3—24no—
85C4×Dic3D414yes—
86C22×D4—32no—
87C22×D4—32no—
88C22×D4—31no—
89C22×D4C2×D311yescharacteristic
90#127—64no—
91#125—64no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4C2×D312yes—
105C2×C22⋊C4C2×D312yes—
106C2×C42—31no—
107C2×C42—32no—
108#95—32no—
109#95—31no—
110#95C2×D311yescharacteristic
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3C2411yescharacteristic
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C3×D4—44no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4C2×D412yes—
137#67—48no—
138#67—48no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3C2×D412yes—
150C2×Dic3C2×D412yes—
151C2×Dic3#4114yes—
152D12—44no—
153C24—32no—
154C24C22×D312yes—
155C2×D4—22no—
156C2×D4—42no—
157C2×D4—64no—
158C2×D4—64no—
159C2×D4—64no—
160C2×D4—64no—
161C2×D4—62no—
162C2×D4—122no—
163C2×D4—122no—
164C2×D4—122no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4C22×D312yes—
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—62no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—122no—
183C22⋊C4—122no—
184C22⋊C4—122no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C42—34no—
189C42—64no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3C22×D411yescharacteristic
198C22×C3C2×#4112yes—
199C22×C3H(5,𝔽2)12yes—
200C22×C3H(5,𝔽2)11yescharacteristic
201C22×C3H(5,𝔽2)11yescharacteristic
202C2×D3—42no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—42no—
209C3×C4—22no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—42no—
213Dic3—22no—
214Dic3—41no—
215Dic3—41no—
216Dic3—42no—
217Dic3#12414yes—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—62no—
226C23—62no—
227C23—62no—
228C23—62no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23C23×D311yescharacteristic
233D4—44no—
234D4—124no—
235D4—128no—
236D4—128no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—32no—
247C2×C4—34no—
248C2×C4—32no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4D3×D412yes—
260C2×C3—82no—
261C2×C3C2×#12411yescharacteristic
262C2×C3#53511yescharacteristic
263C2×C3#53511yescharacteristic
264C2×C3#54012yes—
265C2×C3#54012yes—
266D3—82no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—122no—
275C22—122no—
276C22—122no—
277C22—122no—
278C22—122no—
279C22—122no—
280C22—122no—
281C22C2×D3×D411yescharacteristic
282C22#100412yes—
283C22#100411yescharacteristic
284C22D3×#4112yes—
285C22#100911yescharacteristic
286C4—22no—
287C4—34no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—62no—
292C4—122no—
293C4—121no—
294C4—121no—
295C3#264111yescharacteristic, complemented, Sylow
296C2—82no—
297C2—242no—
298C2#559211yescharacteristic
299C2#560711yescharacteristic
300C2#561512yes—
301C2#566312yes—
302C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#940C2
hypercenter#394D3
hypocenterC3#2641
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#889C2semidirect
3. C3×#394C2semidirect
4. #5222C2semidirect
5. #4991C2semidirect
6. #4961C2semidirect
7. #4959C2semidirect
8. #4957C2semidirect
9. #4957C2semidirect
10. #4672C2semidirect
11. #4648C2semidirect
45. #394D3semidirect
295. C3#2641semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
C2×#8892
C3×#3942
#52222
#49912
#49612
#49592
#49572
#46722
#46482
#26413