Finite Group Databasebeta

← #80806

Subgroups — #80806

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#915C211yesmaximal, characteristic, complemented
2C2×D3×M16C211yesmaximal, characteristic, complemented
3D3×C22⋊Q8C211yesmaximal, characteristic, complemented
4#5606C211yesmaximal, characteristic
5C3×#468C211yesmaximal, characteristic, complemented
6#5171C211yesmaximal, characteristic, complemented
7#5137C211yesmaximal, characteristic, complemented
8#5052C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#4887C211yesmaximal, characteristic, complemented
11#4881C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4802C211yesmaximal, characteristic, complemented
14#4778C211yesmaximal, characteristic, complemented
15#4771C211yesmaximal, characteristic, complemented
16#3056—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic, complemented
26#938C2211yescharacteristic
27#937—21no—
28D4×Dic3—21no—
29C2×#214C2211yescharacteristic, complemented
30#924C2211yescharacteristic
31#924C2211yescharacteristic
32C2×#208C2211yescharacteristic
33#915—24no—
34D3×M16—24no—
35C2×Dic12C2211yescharacteristic, complemented
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×ΓL(1,𝔽25)C2211yescharacteristic
38C2×D3×C8C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic
40#897C2211yescharacteristic
41#897C2211yescharacteristic
42#896—21no—
43#892C2211yescharacteristic
44D3×C4⋊C4—21no—
45D3×C4⋊C4C2211yescharacteristic
46#883—21no—
47#67⋊C4—21no—
48D3×C22⋊C4—21no—
49#878—21no—
50C3×#107C2211yescharacteristic, complemented
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53#835C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#808C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×#137—31no—
63C22×M16—31no—
64#525—31no—
65C2×C22⋊Q8—31no—
66#470—31no—
67#470—31no—
68#468—31no—
69#468D311yescharacteristic, complemented
70#467—31no—
71#467—31no—
72#465—31no—
73#465—31no—
74#428—31no—
75#419—31no—
76#417—31no—
77C23×D3D411yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C22×Dic3—21no—
85C22×Dic3D411yescharacteristic
86D3×Q8—22no—
87D3×Q8—22no—
88#237—22no—
89#237—22no—
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3D411yescharacteristic
95C2×C4×D3D411yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—21no—
99C2×Dic6C2311yescharacteristic
100C2×Dic6C2311yescharacteristic
101C3×SD16—22no—
102C3×M16—22no—
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4—21no—
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110#217—21no—
111#217—21no—
112#217—21no—
113#216—22no—
114#214—22no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127C4×Dic3—21no—
128#208—22no—
129C2×C3⋊C8C2311yescharacteristic
130C2×C3⋊C8C2311yescharacteristic
131Dic12—22no—
132ΓL(1,𝔽25)—22no—
133ΓL(1,𝔽25)D414yescomplemented
134D3×C8—22no—
135D3×C8—22no—
136C2×#41—31no—
137C22×Q8—31no—
138C23×C4—31no—
139#137—64no—
140C2×Q16—31no—
141C2×Q16—31no—
142C2×SD16—31no—
143C2×SD16C2×D311yescharacteristic, complemented
144C2×M16—31no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16—64no—
148C2×M16C2×D311yescharacteristic
149C22×C8—31no—
150C22×C8—31no—
151#123—61no—
152C22⋊Q8—31no—
153C22⋊Q8—31no—
154C22⋊Q8—31no—
155C22⋊Q8—31no—
156C22⋊Q8—31no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C22⋊Q8C2×D311yescharacteristic
160#121—31no—
161#121C2×D311yescharacteristic
162#120—61no—
163C4×D4—61no—
164C4×D4—61no—
165#117—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—61no—
168C2×C22⋊C4—61no—
169#107—31no—
170#107C2×D311yescharacteristic, complemented
171#106—31no—
172#106—31no—
173#103—31no—
174#103—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103C2×D311yescharacteristic, complemented
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181C23×C3—21no—
182C23×C3C2×D411yescharacteristic
183C22×D3—22no—
184C22×D3—22no—
185C22×D3—41no—
186C22×D3C2×D411yescharacteristic
187C22×D3C2×D411yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—42no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2411yescharacteristic
200#67—42no—
201#67—42no—
202C2×Dic3—22no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210C4×D3—22no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—22no—
214C4×D3—42no—
215C4×D3—42no—
216C4×D3C2×D412yes—
217C4×D3#4112yes—
218Dic6—21no—
219Dic6—21no—
220Dic6—21no—
221Dic6—21no—
222Dic6—42no—
223C3×C8—21no—
224C3×C8C2×D412yes—
225C3⋊C8—21no—
226C3⋊C8C2×D412yes—
227C24—31no—
228#41—62no—
229#41—62no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8C22×D311yescharacteristic
236C2×D4—21no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4C22×D311yescharacteristic
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C22×D311yescharacteristic
251Q16—62no—
252Q16—62no—
253SD16—22no—
254SD16—62no—
255M16—22no—
256M16—34no—
257M16—62no—
258M16—62no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8C22×D311yescharacteristic
264C2×C8C22×D311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4C22×D311yescharacteristic
275C4⋊C4C22×D311yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C42—61no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3C22×D411yescharacteristic
293C2×D3—21no—
294C2×D3—21no—
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—44no—
298C2×D3#12112yes—
299C2×D3#13712yescomplemented
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4C22×D411yescharacteristic
305C3×C4C2×#4111yescharacteristic
306Dic3—21no—
307Dic3—41no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3#12112yes—
312C23—21no—
313C23—31no—
314C23—31no—
315C23—62no—
316C23—62no—
317C23—121no—
318C23D3×D411yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—61no—
322Q8—61no—
323Q8—61no—
324Q8—61no—
325Q8—122no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—122no—
330D4—122no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—31no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4D3×D411yescharacteristic
353C2×C4C23×D311yescharacteristic
354C8—21no—
355C8—32no—
356C8—61no—
357C8D3×D412yes—
358C2×C3—41no—
359C2×C3—81no—
360C2×C3C2×#12111yescharacteristic
361C2×C3C2×#13711yescharacteristic
362C2×C3#57511yescharacteristic
363D3—22no—
364D3—22no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—32no—
369C22—32no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—124no—
378C22C2×D3×D411yescharacteristic
379C4—21no—
380C4—32no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—61no—
385C4—121no—
386C4—121no—
387C4—121no—
388C4—121no—
389C4C2×D3×D411yescharacteristic
390C4D3×#4111yescharacteristic
391C3#305611yescharacteristic, complemented, Sylow
392C2—41no—
393C2—62no—
394C2—62no—
395C2—81no—
396C2D3×#12111yescharacteristic
397C2#577311yescharacteristic
398C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radical#5606C2
hypercenter#468D3
hypocenterC3#3056
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
2. C2×D3×M16C2semidirect
3. D3×C22⋊Q8C2semidirect
5. C3×#468C2semidirect
6. #5171C2semidirect
7. #5137C2semidirect
8. #5052C2semidirect
9. #5042C2semidirect
10. #4887C2semidirect
11. #4881C2semidirect
12. #4813C2semidirect
13. #4802C2semidirect
14. #4778C2semidirect
15. #4771C2semidirect
20. C2×C3×SD16C22semidirect
25. C2×#216C22semidirect
29. C2×#214C22semidirect
35. C2×Dic12C22semidirect
50. C3×#107C22semidirect
51. C3×#103C22semidirect
52. C3×#102C22semidirect
53. #835C22semidirect
54. #832C22semidirect
55. #817C22semidirect
56. #816C22semidirect
57. #816C22semidirect
58. #810C22semidirect
59. #810C22semidirect
60. #808C22semidirect
61. #807C22semidirect
69. #468D3semidirect
133. ΓL(1,𝔽25)D4semidirect
143. C2×SD16C2×D3semidirect
170. #107C2×D3semidirect
178. #103C2×D3semidirect
180. #102C2×D3semidirect
299. C2×D3#137semidirect
391. C3#3056semidirect

Maximal subgroups

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

subgroupindex
C2×#9152
C2×D3×M162
D3×C22⋊Q82
#56062
C3×#4682
#51712
#51372
#50522
#50422
#48872
#48812
#48132
#48022
#47782
#47712
#30563