Finite Group Databasebeta

← #82925

Subgroups — #82925

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#525C211yesmaximal, characteristic
2C2×#951C211yesmaximal, characteristic, complemented
3C2×#924C211yesmaximal, characteristic, complemented
4C22×#208C211yesmaximal, characteristic, complemented
5#5232C211yesmaximal, characteristic, complemented
6#5052C212yesmaximal, complemented
7#5051C212yesmaximal, complemented
8#5042C212yesmaximal, complemented
9#5041C212yesmaximal, complemented
10#5004C211yesmaximal, characteristic, complemented
11#4995C211yesmaximal, characteristic, complemented
12#3056—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C3×#123—21no—
17C3×C22⋊Q8C2211yescharacteristic
18C3×C22⋊Q8C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C3×#120—21no—
22C3×C4×D4—21no—
23C3×C4×D4—21no—
24C3×#117C2211yescharacteristic
25C2×#217—21no—
26#951—24no—
27C2×#216C2212yescomplemented
28C2×#214C2212yescomplemented
29C2×C4⋊Dic3C2211yescharacteristic
30#924—21no—
31#924—21no—
32#924C2212yes—
33#924C2212yes—
34C2×Dic3⋊C4—21no—
35C2×#208—24no—
36C2×#208C2211yescharacteristic
37C2×#208C2212yes—
38C2×#208C2211yescharacteristic
39C22×C3⋊C8C2212yes—
40#835C2212yescomplemented
41#832C2212yescomplemented
42#810C2212yescomplemented
43#810C2212yescomplemented
44#808C2212yescomplemented
45#807C2212yescomplemented
46C2×#137—31no—
47C22×M16—31no—
48#525D311yescharacteristic
49C2×C22⋊Q8—31no—
50#470—32no—
51#468—32no—
52#467—32no—
53#465—32no—
54#428—31no—
55#419—31no—
56#417—31no—
57C24×C3D411yescharacteristic
58C3×#41—22no—
59C3×#41—22no—
60C2×C3×Q8C2311yescharacteristic
61C2×C3×D4—21no—
62C2×C3×D4—21no—
63C2×C3×D4—21no—
64C2×C3×D4C2311yescharacteristic
65C22×C3×C4—22no—
66C22×C3×C4—22no—
67C22×C3×C4—21no—
68C22×C3×C4D411yescharacteristic
69C22×C3×C4D411yescharacteristic
70C22×C3×C4D411yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×Dic3—21no—
75C22×Dic3—21no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—21no—
79C2×Dic6C2312yes—
80C3×C4⋊C4—21no—
81C3×C4⋊C4—21no—
82C3×C4⋊C4C2311yescharacteristic
83C3×C4⋊C4C2311yescharacteristic
84C3×C22⋊C4—21no—
85C3×C22⋊C4—21no—
86C3×C22⋊C4—21no—
87C3×C22⋊C4—21no—
88C3×C22⋊C4—21no—
89C3×C42—21no—
90#217—22no—
91#217—22no—
92#216—24no—
93#214—24no—
94C4⋊Dic3—21no—
95C4⋊Dic3C2312yes—
96Dic3⋊C4—22no—
97Dic3⋊C4—22no—
98#208—22no—
99#208—24no—
100#208D414yescomplemented
101C2×C3⋊C8—24no—
102C2×C3⋊C8C2312yes—
103C2×C3⋊C8C2312yes—
104C2×#41C2×D311yescharacteristic
105C22×Q8—31no—
106C23×C4C2×D311yescharacteristic
107#137—64no—
108C2×Q16—32no—
109C2×SD16—32no—
110C2×M16—31no—
111C2×M16—31no—
112C2×M16—32no—
113C2×M16—64no—
114C22×C8—32no—
115#123—21no—
116C22⋊Q8—32no—
117C22⋊Q8—32no—
118C22⋊Q8—61no—
119C22⋊Q8—61no—
120C22⋊Q8C2×D311yescharacteristic
121C22⋊Q8C2×D311yescharacteristic
122#121C2×D311yescharacteristic
123#121C2×D311yescharacteristic
124#120—21no—
125C4×D4—21no—
126C4×D4—21no—
127#117C2×D311yescharacteristic
128C2×C4⋊C4—31no—
129C2×C4⋊C4—61no—
130C2×C22⋊C4—61no—
131#107—32no—
132#106—32no—
133#103—32no—
134#103—32no—
135#103—32no—
136#102—32no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—21no—
140C23×C3—41no—
141C23×C3C2×D411yescharacteristic
142C23×C3C2×D411yescharacteristic
143C23×C3C2×D411yescharacteristic
144C3×Q8—21no—
145C3×Q8—21no—
146C3×D4—21no—
147C3×D4—21no—
148C3×D4—42no—
149C3×D4—42no—
150C3×D4—42no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—42no—
162C2×C3×C4C2×D412yes—
163C2×C3×C4C2×D411yescharacteristic
164C2×C3×C4C2×D411yescharacteristic
165C2×C3×C4C2×D411yescharacteristic
166C2×C3×C4#4112yes—
167C2×C3×C4C2411yescharacteristic
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172Dic6—22no—
173Dic6—22no—
174Dic6—42no—
175C3⋊C8—22no—
176C3⋊C8C2×D414yes—
177C24#6711yescharacteristic
178#41—22no—
179#41—22no—
180C2×Q8—32no—
181C2×Q8—61no—
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8C22×D311yescharacteristic
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4C22×D311yescharacteristic
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—61no—
193C22×C4—61no—
194C22×C4#6711yescharacteristic
195C22×C4#6711yescharacteristic
196C22×C4#6711yescharacteristic
197C22×C4C22×D311yescharacteristic
198C22×C4C22×D311yescharacteristic
199C22×C4C22×D311yescharacteristic
200Q16—64no—
201SD16—64no—
202M16—34no—
203M16—64no—
204M16—62no—
205C2×C8—32no—
206C2×C8—32no—
207C2×C8—64no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—32no—
211C4⋊C4—61no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4C22×D311yescharacteristic
215C4⋊C4C22×D311yescharacteristic
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C22⋊C4—21no—
221C22⋊C4—62no—
222C22⋊C4—62no—
223C42—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—44no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3#12112yes—
236C22×C3#13712yescomplemented
237C22×C3C22×D411yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4#12112yes—
244C3×C4C22×D411yescharacteristic
245C3×C4C2×#4111yescharacteristic
246Dic3—42no—
247Dic3—42no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—41no—
252C23C2×#6711yescharacteristic
253C23C2×#6711yescharacteristic
254C23C2×#6711yescharacteristic
255Q8—21no—
256Q8—21no—
257Q8—62no—
258Q8—62no—
259Q8—122no—
260D4—21no—
261D4—21no—
262D4—42no—
263D4—42no—
264D4—42no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4D3×D412yes—
281C2×C4#23712yes—
282C2×C4C2×#6711yescharacteristic
283C2×C4C2×#6711yescharacteristic
284C2×C4C2×#6711yescharacteristic
285C2×C4C23×D311yescharacteristic
286C8—34no—
287C8—62no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—41no—
291C2×C3—81no—
292C2×C3C2×#12111yescharacteristic
293C2×C3C2×#13711yescharacteristic
294C2×C3#57511yescharacteristic
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—44no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22#93912yes—
307C22#95112yescomplemented
308C22C22×#6711yescharacteristic
309C4—21no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—122no—
315C4—122no—
316C4#93912yes—
317C4C2×D3×D411yescharacteristic
318C4C2×#23711yescharacteristic
319C3#305611yescharacteristic, complemented, Sylow
320C2—22no—
321C2—22no—
322C2—41no—
323C2—81no—
324C2C2×#93911yescharacteristic
325C2#582211yescharacteristic
326C2C2×#95111yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#525C2
socleC22×C3C22×D4
hypercenter#525D3
hypocenterC3#3056
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
2. C2×#951C2semidirect
3. C2×#924C2semidirect
4. C22×#208C2semidirect
5. #5232C2semidirect
6. #5052C2semidirect
7. #5051C2semidirect
8. #5042C2semidirect
9. #5041C2semidirect
10. #5004C2semidirect
11. #4995C2semidirect
27. C2×#216C22semidirect
28. C2×#214C22semidirect
40. #835C22semidirect
41. #832C22semidirect
42. #810C22semidirect
43. #810C22semidirect
44. #808C22semidirect
45. #807C22semidirect
100. #208D4semidirect
236. C22×C3#137semidirect
307. C22#951semidirect
319. C3#3056semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
C2×#9512
C2×#9242
C22×#2082
#52322
#50522
#50512
#50422
#50412
#50042
#49952
#30563