Finite Group Databasebeta

← #76210

Subgroups — #76210

Order 384.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#897C211yesmaximal, characteristic, complemented
2C2×#897C211yesmaximal, characteristic, complemented
3C2×#892C211yesmaximal, characteristic, complemented
4C2×C4×D12C211yesmaximal, characteristic
5C3×#389C211yesmaximal, characteristic, complemented
6#4991C212yesmaximal
7#4987C211yesmaximal, characteristic, complemented
8#4984C212yesmaximal, complemented
9#4929C211yesmaximal, characteristic, complemented
10#4674C212yesmaximal, complemented
11#4670C212yesmaximal, complemented
12#2415—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32#897—24no—
33#897—24no—
34#892—24no—
35C4×D12—24no—
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8—31no—
42C2×C4×D4—31no—
43C2×#117—31no—
44#397—32no—
45#394—32no—
46#391—31no—
47#389—32no—
48#389D311yescharacteristic, complemented
49#388—32no—
50#380—31no—
51C23×D3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C2×D12—22no—
63C2×D12—22no—
64C2×D12—22no—
65C2×C4×D3—22no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—22no—
70C2×C4×D3D414yes—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6—22no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—24no—
76C3×C4⋊C4D414yescomplemented
77C3×C42—22no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—22no—
84C4⋊Dic3—22no—
85C4⋊Dic3—24no—
86C4⋊Dic3—22no—
87C4⋊Dic3—24no—
88C4×Dic3—24no—
89C22×Q8—31no—
90C22×D4—31no—
91C23×C4—31no—
92C23×C4—31no—
93C22⋊Q8—64no—
94C22⋊Q8—64no—
95C4×D4—64no—
96#117—64no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4C2×D311yescharacteristic
102C2×C4⋊C4C2×D312yes—
103C2×C4⋊C4C2×D311yescharacteristic
104C2×C22⋊C4—32no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C42—32no—
111C2×C42C2×D311yescharacteristic
112#95—32no—
113#95—32no—
114#95—32no—
115#95C2×D312yes—
116C23×C3C2411yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—21no—
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3C2×D412yes—
130C22×D3#4112yes—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4C2×D414yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yes—
145C2×C3×C4#4112yes—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3C2×D412yes—
158D12—28no—
159C4×D3—28no—
160C4×D3—44no—
161Dic6—44no—
162C24—31no—
163C24—31no—
164C2×Q8—62no—
165C2×Q8—62no—
166C2×Q8—62no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C22×C4—31no—
171C22×C4—34no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—31no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4—62no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D312yes—
182C22×C4C22×D312yes—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—62no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C4⋊C4—64no—
191C4⋊C4D1214yescomplemented
192C22⋊C4—62no—
193C22⋊C4—62no—
194C22⋊C4—64no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C42—22no—
199C42—64no—
200C22×C3C22×D411yescharacteristic
201C22×C3C22×D411yescharacteristic
202C22×C3C2×#4111yescharacteristic
203C22×C3C2×#4111yescharacteristic
204C22×C3C2×#4111yescharacteristic
205C22×C32−1+411yescharacteristic
206C22×C32−1+411yescharacteristic
207C2×D3—22no—
208C2×D3—22no—
209C2×D3—22no—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C3×C4—22no—
222C3×C4—24no—
223C3×C4—42no—
224C3×C4—41no—
225C3×C4#12114yes—
226Dic3—22no—
227Dic3—42no—
228Dic3—41no—
229Dic3—42no—
230C23—32no—
231C23—32no—
232C23—61no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23—61no—
238C23—61no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23C23×D311yescharacteristic
245Q8—124no—
246D4—68no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—32no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—68no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—124no—
271C2×C4C2×D1214yes—
272C2×C4C2×D1212yes—
273C2×C4D3×D412yes—
274C2×C4#23712yes—
275C2×C4#23912yes—
276C2×C3C2×#12111yescharacteristic
277C2×C3#52511yescharacteristic
278C2×C3#53611yescharacteristic
279C2×C3#54711yescharacteristic
280C2×C3#54811yescharacteristic
281C2×C3#55211yescharacteristic
282C2×C3#55611yescharacteristic
283D3—42no—
284D3—42no—
285C22—62no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22—62no—
290C22—62no—
291C22—121no—
292C22—121no—
293C22—121no—
294C22—121no—
295C22—121no—
296C22—121no—
297C22—121no—
298C22—121no—
299C22C22×D1211yescharacteristic
300C22C2×D3×D411yescharacteristic
301C22C2×#23711yescharacteristic
302C22C2×#23911yescharacteristic
303C22#100711yescharacteristic
304C22D3×#4111yescharacteristic
305C22#101011yescharacteristic
306C4—24no—
307C4—22no—
308C4—42no—
309C4—41no—
310C4—62no—
311C4—121no—
312C4—122no—
313C4—122no—
314C4#89514yes—
315C3#241511yescharacteristic, complemented, Sylow
316C2—122no—
317C2—122no—
318C2C2×#89511yescharacteristic
319C2#555511yescharacteristic
320C2#557611yescharacteristic
321C2#560611yescharacteristic
322C2#562811yescharacteristic
323C2#569411yescharacteristic
324C2#573211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2415
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
2. C2×#897C2semidirect
3. C2×#892C2semidirect
5. C3×#389C2semidirect
7. #4987C2semidirect
8. #4984C2semidirect
9. #4929C2semidirect
10. #4674C2semidirect
11. #4670C2semidirect
48. #389D3semidirect
76. C3×C4⋊C4D4semidirect
191. C4⋊C4D12semidirect
315. C3#2415semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8922
C2×C4×D122
C3×#3892
#49912
#49872
#49842
#49292
#46742
#46702
#24153