Finite Group Databasebeta

← #80004

Subgroups — #80004

Order 384.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1004C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3C2×D3×D8C211yesmaximal, characteristic, complemented
4#5607C211yesmaximal, characteristic, complemented
5C3×#447C211yesmaximal, characteristic, complemented
6#5217C211yesmaximal, characteristic, complemented
7#5158C211yesmaximal, characteristic, complemented
8#5156C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4781C211yesmaximal, characteristic, complemented
11#4773C211yesmaximal, characteristic, complemented
12#4772C211yesmaximal, characteristic, complemented
13#4770C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3008—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic, complemented
18C22×#67—21no—
19C22×#67—21no—
20#1004—24no—
21#1004—24no—
22C2×#237—21no—
23C2×#237C2211yescharacteristic
24C2×#237C2211yescharacteristic, complemented
25C2×D3×D4—21no—
26C2×D3×D4C2211yescharacteristic
27C2×D3×D4C2211yescharacteristic, complemented
28C2×D3×D4C2211yescharacteristic, complemented
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic, complemented
31C2×C3×D8C2211yescharacteristic, complemented
32C2×C3×D8C2211yescharacteristic, complemented
33C3×#121C2211yescharacteristic, complemented
34#940—21no—
35#938C2211yescharacteristic, complemented
36#937—21no—
37C2×#214C2211yescharacteristic, complemented
38C2×#213C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40C2×#213C2211yescharacteristic, complemented
41#930C2211yescharacteristic, complemented
42#911—24no—
43D3×D8—24no—
44C2×D24C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47C2×D3×C8C2211yescharacteristic, complemented
48#895C2211yescharacteristic, complemented
49#892C2211yescharacteristic, complemented
50#885—21no—
51#882—21no—
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#832C2211yescharacteristic, 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
64C2×H(5,𝔽2)—31no—
65C2×AGL(1,ℤ/8)—31no—
66C22×D8—31no—
67#535—31no—
68#449—31no—
69#449—31no—
70#449—31no—
71#449—31no—
72#447—31no—
73#447—31no—
74#447—31no—
75#447D311yescharacteristic, complemented
76#418—31no—
77#416—31no—
78#408—31no—
79C24×C3—21no—
80C23×D3—21no—
81C23×D3—21no—
82C23×D3—21no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—28no—
92C2×#67—28no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—21no—
97C2×#67—21no—
98C2×#67—21no—
99C2×#67—21no—
100C2×#67—21no—
101C2×#67D411yescharacteristic
102C2×#67D411yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105#237—28no—
106#237—22no—
107#237—22no—
108#237—22no—
109#237D414yescomplemented
110D3×D4—28no—
111D3×D4—22no—
112D3×D4—22no—
113D3×D4—22no—
114D3×D4—22no—
115D3×D4—22no—
116D3×D4D414yescomplemented
117#235—22no—
118#235—22no—
119#235—22no—
120#235—22no—
121C2×D12—21no—
122C2×D12D411yescharacteristic
123C2×D12C2311yescharacteristic
124C2×D12C2311yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2311yescharacteristic
128C2×Dic6D411yescharacteristic
129C2×Dic6C2311yescharacteristic
130C3×D8—22no—
131C3×D8—22no—
132C2×C3×C8C2311yescharacteristic
133C2×C3×C8C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C22⋊C4—21no—
136#217—21no—
137#214—22no—
138#213—22no—
139#213—22no—
140#213—22no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144C4⋊Dic3C2311yescharacteristic
145C4×Dic3—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148D24—22no—
149#204—22no—
150ΓL(1,𝔽25)—22no—
151D3×C8—22no—
152H(5,𝔽2)—64no—
153H(5,𝔽2)—64no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—31no—
157C2×#41—31no—
158C2×#41—61no—
159C22×D4—31no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—61no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4C2×D311yescharacteristic, complemented
166AGL(1,ℤ/8)—64no—
167C2×SD16—31no—
168C2×SD16—31no—
169C2×D8—31no—
170C2×D8—31no—
171C2×D8—31no—
172C2×D8—31no—
173C2×D8—64no—
174C2×D8C2×D311yescharacteristic, complemented
175C2×D8C2×D311yescharacteristic, complemented
176C2×M16—31no—
177C22×C8—31no—
178#127—61no—
179#124—61no—
180#121—31no—
181#121—31no—
182#121—31no—
183#121C2×D311yescharacteristic, complemented
184#120—61no—
185#120—61no—
186#117—31no—
187#103—31no—
188#103—31no—
189#102—31no—
190#102—31no—
191#102—31no—
192#102—31no—
193#102C2×D311yescharacteristic, complemented
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—24no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3C2×D411yescharacteristic
206C22×D3—21no—
207C22×D3—24no—
208C22×D3—24no—
209C22×D3—21no—
210C22×D3—42no—
211C22×D3—81no—
212C22×D3—81no—
213C22×D3—81no—
214C22×D3—81no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3C2×D411yescharacteristic
218C22×D3C2×D411yescharacteristic
219C3×D4—24no—
220C3×D4—21no—
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—81no—
225C3×D4C2×D412yes—
226C3×D4C2×D412yes—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4C2×D411yescharacteristic
231C2×C3×C4C2411yescharacteristic
232#67—24no—
233#67—24no—
234#67—24no—
235#67—22no—
236#67—24no—
237#67—22no—
238#67—24no—
239#67—42no—
240#67—81no—
241#67—81no—
242C2×Dic3—24no—
243C2×Dic3—21no—
244C2×Dic3—24no—
245C2×Dic3—21no—
246C2×Dic3C2×D411yescharacteristic
247C2×Dic3C2×D411yescharacteristic
248D12—22no—
249D12—21no—
250D12—21no—
251D12—21no—
252D12—81no—
253D12C2×D412yes—
254C4×D3—24no—
255C4×D3—21no—
256C4×D3—21no—
257C4×D3C2×D412yes—
258C4×D3C2×D412yes—
259Dic6—21no—
260Dic6—22no—
261Dic6C2×D412yes—
262C3×C8—21no—
263C3×C8—21no—
264C3⋊C8—21no—
265C3⋊C8—21no—
266C24—21no—
267C24—61no—
268C24—61no—
269C24—61no—
270#41—34no—
271#41—62no—
272#41—62no—
273#41—62no—
274#41—62no—
275#41—62no—
276#41—62no—
277#41—68no—
278#41—62no—
279C2×Q8—31no—
280C2×Q8—31no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—34no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—31no—
289C2×D4—31no—
290C2×D4—31no—
291C2×D4—61no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—62no—
299C2×D4—68no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—68no—
303C2×D4—68no—
304C2×D4—61no—
305C2×D4—62no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4C22×D311yescharacteristic
309C2×D4C22×D311yescharacteristic
310C2×D4C22×D311yescharacteristic
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4C22×D311yescharacteristic
317SD16—62no—
318SD16—62no—
319D8—22no—
320D8—22no—
321D8—62no—
322D8—62no—
323D8—62no—
324D8—62no—
325M16—62no—
326C2×C8—31no—
327C2×C8—31no—
328C2×C8—62no—
329C2×C8C22×D311yescharacteristic
330C2×C8C22×D311yescharacteristic
331C4⋊C4—31no—
332C4⋊C4C22×D311yescharacteristic
333C22⋊C4—21no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C42—61no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—84no—
352C22×C3C22×D411yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—22no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—84no—
365C2×D3—84no—
366C2×D3—84no—
367C2×D3#12012yes—
368C3×C4—21no—
369C3×C4—41no—
370C3×C4C22×D411yescharacteristic
371C3×C4C22×D411yescharacteristic
372Dic3—21no—
373Dic3—22no—
374Dic3—41no—
375Dic3#12012yes—
376C23—24no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—31no—
381C23—31no—
382C23—64no—
383C23—64no—
384C23—61no—
385C23—61no—
386C23—81no—
387C23—81no—
388C23—122no—
389C23—241no—
390C23—241no—
391C23—241no—
392C23—241no—
393C23—241no—
394C23—241no—
395C23D3×D411yescharacteristic
396Q8—32no—
397Q8—62no—
398Q8—61no—
399D4—24no—
400D4—21no—
401D4—21no—
402D4—21no—
403D4—21no—
404D4—32no—
405D4—62no—
406D4—62no—
407D4—64no—
408D4—62no—
409D4—64no—
410D4—64no—
411D4—64no—
412D4—64no—
413D4—61no—
414D4—61no—
415D4—61no—
416D4—81no—
417D4—122no—
418D4—241no—
419D4—241no—
420D4—241no—
421D4D3×D412yes—
422D4D3×D412yes—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—31no—
427C2×C4—32no—
428C2×C4—32no—
429C2×C4—31no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4D3×D411yescharacteristic
438C2×C4C23×D311yescharacteristic
439C8—21no—
440C8—21no—
441C8—61no—
442C8—61no—
443C2×C3—42no—
444C2×C3—42no—
445C2×C3—41no—
446C2×C3—81no—
447C2×C3C2×#12011yescharacteristic
448C2×C3#57611yescharacteristic
449C2×C3#57611yescharacteristic
450D3—22no—
451D3—41no—
452D3—42no—
453D3—81no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—32no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—62no—
472C22—61no—
473C22—84no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22—244no—
480C22—244no—
481C22—244no—
482C22C2×D3×D411yescharacteristic
483C4—21no—
484C4—32no—
485C4—41no—
486C4—61no—
487C4—62no—
488C4—121no—
489C4C2×D3×D411yescharacteristic
490C4C2×D3×D411yescharacteristic
491C3#300811yescharacteristic, complemented, Sylow
492C2—41no—
493C2—42no—
494C2—42no—
495C2—62no—
496C2—81no—
497C2—121no—
498C2—122no—
499C2—241no—
500C2D3×#12011yescharacteristic
501C2#575711yescharacteristic
502C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3008
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C2×#911C2semidirect
3. C2×D3×D8C2semidirect
4. #5607C2semidirect
5. C3×#447C2semidirect
6. #5217C2semidirect
7. #5158C2semidirect
8. #5156C2semidirect
9. #5037C2semidirect
10. #4781C2semidirect
11. #4773C2semidirect
12. #4772C2semidirect
13. #4770C2semidirect
14. #4734C2semidirect
15. #4727C2semidirect
17. C22×C3×D4C22semidirect
24. C2×#237C22semidirect
27. C2×D3×D4C22semidirect
28. C2×D3×D4C22semidirect
30. C2×#235C22semidirect
31. C2×C3×D8C22semidirect
32. C2×C3×D8C22semidirect
33. C3×#121C22semidirect
35. #938C22semidirect
37. C2×#214C22semidirect
38. C2×#213C22semidirect
39. C2×#213C22semidirect
40. C2×#213C22semidirect
41. #930C22semidirect
44. C2×D24C22semidirect
45. C2×#204C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. C2×D3×C8C22semidirect
48. #895C22semidirect
49. #892C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #832C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. (C2×D3)⋊C8C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. #816C22semidirect
62. #810C22semidirect
63. #809C22semidirect
75. #447D3semidirect
109. #237D4semidirect
116. D3×D4D4semidirect
165. C22×D4C2×D3semidirect
174. C2×D8C2×D3semidirect
175. C2×D8C2×D3semidirect
183. #121C2×D3semidirect
193. #102C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
491. C3#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C2×#9112
C2×D3×D82
#56072
C3×#4472
#52172
#51582
#51562
#50372
#47812
#47732
#47722
#47702
#47342
#47272
#30083