Finite Group Databasebeta

← #80084

Subgroups — #80084

Order 384.

806 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#41C211yesmaximal, characteristic, complemented
2C2×#914C211yesmaximal, characteristic, complemented
3C2×#911C211yesmaximal, characteristic, complemented
4D3×#121C211yesmaximal, characteristic, complemented
5C3×#449C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5170C211yesmaximal, characteristic, complemented
8#5156C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4806C211yesmaximal, characteristic, complemented
11#4803C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4771C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4726C211yesmaximal, characteristic, complemented
16#3006—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18D3×#41—24no—
19D3×#41—24no—
20C2×#239C2211yescharacteristic
21C2×D3×Q8C2211yescharacteristic, complemented
22C2×#237—21no—
23C2×#237C2211yescharacteristic
24C2×D3×D4—21no—
25C2×D3×D4—21no—
26C2×D3×D4C2211yescharacteristic, complemented
27C2×D3×D4C2211yescharacteristic, complemented
28C2×#235—21no—
29C2×#235C2211yescharacteristic, complemented
30C22×C4×D3—21no—
31C22×C4×D3C2211yescharacteristic
32C2×C3×SD16C2211yescharacteristic, complemented
33C2×C3×D8C2211yescharacteristic, complemented
34C3×#121C2211yescharacteristic, complemented
35C2×#215C2211yescharacteristic, complemented
36#939—21no—
37#938C2211yescharacteristic
38C2×#214C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40C2×#213C2211yescharacteristic, complemented
41#930C2211yescharacteristic, complemented
42#914—24no—
43#911—24no—
44C2×D24C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
48#895C2211yescharacteristic
49D3×C4⋊C4C2211yescharacteristic, complemented
50#884—21no—
51D3×C22⋊C4—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59(C2×D3)⋊C8C2211yescharacteristic, complemented
60(C2×D3)⋊C8C2211yescharacteristic, complemented
61#816C2211yescharacteristic, complemented
62#810C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C22×#41—31no—
65C2×AGL(1,ℤ/8)—31no—
66C2×AGL(1,ℤ/8)—31no—
67C2×#121—31no—
68#449—31no—
69#449—31no—
70#449—31no—
71#449D311yescharacteristic, complemented
72#448—31no—
73#448—31no—
74#447—31no—
75#447—31no—
76#417—31no—
77#417—31no—
78#407—31no—
79C23×D3—21no—
80C23×D3—21no—
81C23×D3D411yescharacteristic
82C3×#41—22no—
83C3×#41—22no—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4C2311yescharacteristic
91C2×#67—21no—
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—21no—
97C22×Dic3—21no—
98C22×Dic3D411yescharacteristic
99#239—22no—
100#239D414yescomplemented
101D3×Q8—22no—
102D3×Q8—22no—
103#237—28no—
104#237—22no—
105#237D414yescomplemented
106D3×D4—28no—
107D3×D4—22no—
108D3×D4—22no—
109D3×D4—22no—
110D3×D4—22no—
111D3×D4—44no—
112#235—28no—
113#235—22no—
114#235—22no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12C2311yescharacteristic
118C2×D12C2311yescharacteristic
119C2×C4×D3—28no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—22no—
124C2×C4×D3—22no—
125C2×C4×D3—21no—
126C2×C4×D3D411yescharacteristic
127C2×C4×D3D411yescharacteristic
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2311yescharacteristic
130C2×Dic6—21no—
131C2×Dic6C2311yescharacteristic
132C3×SD16—22no—
133C3×D8—22no—
134C2×C3×C8C2311yescharacteristic
135C2×C3×C8C2311yescharacteristic
136C3×C4⋊C4C2311yescharacteristic
137C3×C22⋊C4—21no—
138#217—21no—
139#215—22no—
140#214—22no—
141#213—22no—
142#213—22no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145C4⋊Dic3C2311yescharacteristic
146Dic3⋊C4—21no—
147C2×C3⋊C8C2311yescharacteristic
148C2×C3⋊C8C2311yescharacteristic
149D24—22no—
150#204—22no—
151ΓL(1,𝔽25)—22no—
152ΓL(1,𝔽25)—22no—
153C2×#41—31no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—61no—
157C2×#41—61no—
158C2×#41—64no—
159C2×#41—64no—
160C2×#41C2×D311yescharacteristic, complemented
161C22×Q8—31no—
162C22×D4—31no—
163C22×D4—31no—
164C22×D4—61no—
165C22×D4—61no—
166C23×C4—31no—
167C23×C4—61no—
168AGL(1,ℤ/8)—64no—
169AGL(1,ℤ/8)—64no—
170C2×SD16—31no—
171C2×SD16—31no—
172C2×SD16—31no—
173C2×SD16C2×D311yescharacteristic, complemented
174C2×D8—31no—
175C2×D8—31no—
176C2×D8—31no—
177C2×D8C2×D311yescharacteristic, complemented
178C2×M16—31no—
179C2×M16—31no—
180#121—31no—
181#121—31no—
182#121—31no—
183#121—61no—
184#121—61no—
185#121C2×D311yescharacteristic, complemented
186C2×C4⋊C4—31no—
187C2×C22⋊C4—61no—
188#103—31no—
189#103—31no—
190#103—31no—
191#103C2×D311yescharacteristic, complemented
192#102—31no—
193#102—31no—
194#102—31no—
195#102C2×D311yescharacteristic, complemented
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8—31no—
199C22⋊C8C2×D311yescharacteristic, complemented
200C23×C3—21no—
201C23×C3—21no—
202C23×C3C2×D411yescharacteristic
203C22×D3—24no—
204C22×D3—21no—
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—21no—
208C22×D3—41no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3C2×D411yescharacteristic
215C22×D3C2×D411yescharacteristic
216C3×Q8—21no—
217C3×Q8C2×D412yescomplemented
218C3×D4—24no—
219C3×D4—21no—
220C3×D4—21no—
221C3×D4—21no—
222C3×D4—42no—
223C3×D4C2×D412yescomplemented
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4C2×D411yescharacteristic
230C2×C3×C4C2411yescharacteristic
231#67—24no—
232#67—24no—
233#67—24no—
234#67—42no—
235#67—42no—
236#67—42no—
237C2×Dic3—21no—
238C2×Dic3—22no—
239C2×Dic3—24no—
240C2×Dic3—21no—
241C2×Dic3C2×D411yescharacteristic
242C2×Dic3C2×D411yescharacteristic
243D12—24no—
244D12—21no—
245D12—21no—
246D12—21no—
247D12—42no—
248D12C2×D412yescomplemented
249C4×D3—24no—
250C4×D3—24no—
251C4×D3—22no—
252C4×D3—21no—
253C4×D3—21no—
254C4×D3—22no—
255C4×D3—24no—
256C4×D3—42no—
257C4×D3C2×D412yes—
258C4×D3C2×D412yes—
259Dic6—24no—
260Dic6—21no—
261Dic6C2×D412yescomplemented
262C3×C8—21no—
263C3×C8—21no—
264C3⋊C8—21no—
265C3⋊C8—21no—
266C24—31no—
267C24—61no—
268C24—61no—
269#41—22no—
270#41—22no—
271#41—34no—
272#41—34no—
273#41—62no—
274#41—62no—
275#41—68no—
276#41—62no—
277#41—62no—
278#41—68no—
279C2×Q8—31no—
280C2×Q8—61no—
281C2×Q8—62no—
282C2×Q8—62no—
283C2×Q8C22×D311yescharacteristic
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—62no—
295C2×D4—68no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—62no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—124no—
302C2×D4C22×D311yescharacteristic
303C2×D4C22×D311yescharacteristic
304C22×C4—21no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—68no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4C22×D311yescharacteristic
319SD16—22no—
320SD16—62no—
321SD16—62no—
322SD16—62no—
323D8—22no—
324D8—62no—
325D8—62no—
326D8—62no—
327M16—62no—
328M16—62no—
329C2×C8—31no—
330C2×C8—31no—
331C2×C8C22×D311yescharacteristic
332C2×C8C22×D311yescharacteristic
333C4⋊C4—31no—
334C4⋊C4—61no—
335C4⋊C4C22×D311yescharacteristic
336C22⋊C4—21no—
337C22⋊C4—61no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22×C3—21no—
341C22×C3—22no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3C22×D411yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—22no—
355C2×D3—44no—
356C2×D3—48no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—84no—
363C2×D3#12012yes—
364C2×D3AGL(1,ℤ/8)12yescomplemented
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—41no—
368C3×C4C22×D411yescharacteristic
369C3×C4C22×D411yescharacteristic
370Dic3—21no—
371Dic3—22no—
372Dic3—41no—
373Dic3#12012yes—
374C23—21no—
375C23—21no—
376C23—31no—
377C23—31no—
378C23—62no—
379C23—62no—
380C23—64no—
381C23—61no—
382C23—61no—
383C23—121no—
384C23—122no—
385C23—122no—
386C23—122no—
387C23—122no—
388C23—122no—
389C23D3×D411yescharacteristic
390Q8—21no—
391Q8—32no—
392Q8—64no—
393Q8—61no—
394Q8D3×D412yescomplemented
395D4—24no—
396D4—21no—
397D4—21no—
398D4—21no—
399D4—32no—
400D4—42no—
401D4—64no—
402D4—64no—
403D4—64no—
404D4—64no—
405D4—61no—
406D4—61no—
407D4—61no—
408D4—122no—
409D4—122no—
410D4—122no—
411D4—122no—
412D4D3×D412yescomplemented
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—31no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—31no—
422C2×C4—62no—
423C2×C4—64no—
424C2×C4—62no—
425C2×C4—64no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—64no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—122no—
434C2×C4D3×D411yescharacteristic
435C2×C4C23×D311yescharacteristic
436C8—21no—
437C8—21no—
438C8—61no—
439C8—61no—
440C2×C3—41no—
441C2×C3—42no—
442C2×C3—81no—
443C2×C3C2×#12011yescharacteristic
444C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C3#57511yescharacteristic
446D3—22no—
447D3—22no—
448D3—42no—
449D3—81no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—32no—
455C22—32no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—61no—
462C22—61no—
463C22—61no—
464C22—61no—
465C22—62no—
466C22—124no—
467C22—128no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22—121no—
472C22—121no—
473C22—244no—
474C22C2×D3×D411yescharacteristic
475C4—21no—
476C4—22no—
477C4—32no—
478C4—41no—
479C4—61no—
480C4—62no—
481C4—121no—
482C4C2×D3×D411yescharacteristic
483C4C2×D3×D411yescharacteristic
484C3#300611yescharacteristic, complemented, Sylow
485C2—41no—
486C2—42no—
487C2—62no—
488C2—62no—
489C2—81no—
490C2—122no—
491C2—241no—
492C2D3×#12011yescharacteristic
493C2#576811yescharacteristic
494C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3006
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×D3×#41C2semidirect
2. C2×#914C2semidirect
3. C2×#911C2semidirect
4. D3×#121C2semidirect
5. C3×#449C2semidirect
6. #5238C2semidirect
7. #5170C2semidirect
8. #5156C2semidirect
9. #5037C2semidirect
10. #4806C2semidirect
11. #4803C2semidirect
12. #4773C2semidirect
13. #4771C2semidirect
14. #4734C2semidirect
15. #4726C2semidirect
17. C2×C3×#41C22semidirect
21. C2×D3×Q8C22semidirect
26. C2×D3×D4C22semidirect
27. C2×D3×D4C22semidirect
29. C2×#235C22semidirect
32. C2×C3×SD16C22semidirect
33. C2×C3×D8C22semidirect
34. C3×#121C22semidirect
35. C2×#215C22semidirect
38. C2×#214C22semidirect
39. C2×#213C22semidirect
40. C2×#213C22semidirect
41. #930C22semidirect
44. C2×D24C22semidirect
45. C2×#204C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. C2×ΓL(1,𝔽25)C22semidirect
49. D3×C4⋊C4C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. (C2×D3)⋊C8C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. #816C22semidirect
62. #810C22semidirect
63. #809C22semidirect
71. #449D3semidirect
100. #239D4semidirect
105. #237D4semidirect
160. C2×#41C2×D3semidirect
173. C2×SD16C2×D3semidirect
177. C2×D8C2×D3semidirect
185. #121C2×D3semidirect
191. #103C2×D3semidirect
195. #102C2×D3semidirect
199. C22⋊C8C2×D3semidirect
217. C3×Q8C2×D4semidirect
223. C3×D4C2×D4semidirect
248. D12C2×D4semidirect
261. Dic6C2×D4semidirect
364. C2×D3AGL(1,ℤ/8)semidirect
394. Q8D3×D4semidirect
412. D4D3×D4semidirect
484. C3#3006semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#412
C2×#9142
C2×#9112
D3×#1212
C3×#4492
#52382
#51702
#51562
#50372
#48062
#48032
#47732
#47712
#47342
#47262
#30063