Finite Group Databasebeta

← #76249

Subgroups — #76249

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#895C211yesmaximal, characteristic
2C2×#872C211yesmaximal, characteristic, complemented
3C3×#389C211yesmaximal, characteristic, complemented
4#4993C212yesmaximal, complemented
5#4991C212yesmaximal, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4972C211yesmaximal, characteristic, complemented
9#4939C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4672C212yesmaximal, complemented
12#2624—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×D12C2212yes—
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C42C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33#895—24no—
34#872—24no—
35C3×#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#121—31no—
39C2×#117—31no—
40#399—32no—
41#397—32no—
42#394—32no—
43#394—32no—
44#390—31no—
45#390—31no—
46#389—31no—
47#389D311yescharacteristic, complemented
48#380—31no—
49C23×D3C2312yes—
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×D12—24no—
61C2×D12—22no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12—42no—
65C2×D12—42no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3D414yes—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—42no—
72C3×C42—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81C4⋊Dic3—22no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—42no—
85C4×Dic3—22no—
86C22×D4—31no—
87C22×D4—32no—
88C23×C4—31no—
89#121—64no—
90#117—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C4⋊C4C2×D311yescharacteristic
97C2×C4⋊C4C2×D312yes—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C42—31no—
107C2×C42C2×D311yescharacteristic
108#95—32no—
109#95—32no—
110#95C2×D312yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3C2×D412yes—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4C2×D412yes—
139C2×C3×C4#4112yes—
140C2×C3×C4#4112yes—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3C2×D412yes—
153D12—48no—
154D12—44no—
155C4×D3—28no—
156C24—31no—
157C24—32no—
158C2×D4—64no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—122no—
163C2×D4—122no—
164C22×C4—31no—
165C22×C4—34no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4C22×D312yes—
172C22×C4C22×D312yes—
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—62no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—122no—
183C22⋊C4—64no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C42—22no—
192C42—62no—
193C22×C3C22×D411yescharacteristic
194C22×C3C2×#4111yescharacteristic
195C22×C3C2×#4111yescharacteristic
196C22×C3H(5,𝔽2)11yescharacteristic
197C22×C3H(5,𝔽2)11yescharacteristic
198C22×C3H(5,𝔽2)11yescharacteristic
199C22×C32−1+411yescharacteristic
200C2×D3—22no—
201C2×D3—22no—
202C2×D3—22no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C3×C4—42no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—42no—
218C3×C4#12314yes—
219Dic3—22no—
220Dic3—41no—
221Dic3—41no—
222Dic3—42no—
223C23—32no—
224C23—61no—
225C23—61no—
226C23—61no—
227C23—61no—
228C23—61no—
229C23—61no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23C23×D311yescharacteristic
238D4—128no—
239D4—124no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—68no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4#23512yes—
266C2×C4D3×D412yes—
267C2×C4#23912yes—
268C2×C3C2×#12311yescharacteristic
269C2×C3#53511yescharacteristic
270C2×C3#53711yescharacteristic
271C2×C3#54011yescharacteristic
272C2×C3#54011yescharacteristic
273C2×C3#54211yescharacteristic
274C2×C3#54211yescharacteristic
275D3—42no—
276D3—82no—
277C22—62no—
278C22—62no—
279C22—62no—
280C22—121no—
281C22—121no—
282C22—121no—
283C22—121no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—122no—
291C22C2×#23511yescharacteristic
292C22C2×D3×D411yescharacteristic
293C22#100411yescharacteristic
294C22C2×#23911yescharacteristic
295C22#100711yescharacteristic
296C22#100911yescharacteristic
297C22#100911yescharacteristic
298C4—42no—
299C4—42no—
300C4—41no—
301C4—41no—
302C4—62no—
303C4—122no—
304C4—121no—
305C4—121no—
306C4#89414yes—
307C3#262411yescharacteristic, complemented, Sylow
308C2—122no—
309C2—242no—
310C2C2×#89411yescharacteristic
311C2#556211yescharacteristic
312C2#558211yescharacteristic
313C2#560711yescharacteristic
314C2#562911yescharacteristic
315C2#569711yescharacteristic
316C2#573311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC2×#895C2
hypercenter#389D3
hypocenterC3#2624
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#872C2semidirect
3. C3×#389C2semidirect
4. #4993C2semidirect
5. #4991C2semidirect
6. #4988C2semidirect
7. #4987C2semidirect
8. #4972C2semidirect
9. #4939C2semidirect
10. #4675C2semidirect
11. #4672C2semidirect
47. #389D3semidirect
307. C3#2624semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8722
C3×#3892
#49932
#49912
#49882
#49872
#49722
#49392
#46752
#46722
#26243