Finite Group Databasebeta

← #84009

Subgroups — #84009

Order 384.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#576C211yesmaximal, characteristic, complemented
2#5836C211yesmaximal, characteristic, complemented
3#5835C211yesmaximal, characteristic, complemented
4#5770C211yesmaximal, characteristic, complemented
5#5242C211yesmaximal, characteristic, complemented
6#5241C211yesmaximal, characteristic, complemented
7#5201C211yesmaximal, characteristic, complemented
8#5200C211yesmaximal, characteristic, complemented
9#5199C211yesmaximal, characteristic, complemented
10#5198C211yesmaximal, characteristic, complemented
11#5197C211yesmaximal, characteristic, complemented
12#5160C211yesmaximal, characteristic, complemented
13#5154C211yesmaximal, characteristic, complemented
14#5145C211yesmaximal, characteristic, complemented
15#5141C211yesmaximal, characteristic, complemented
16#3288—31nomaximal, Sylow
17C3×H(5,𝔽2)C2211yescharacteristic
18C3×H(5,𝔽2)C2211yescharacteristic, complemented
19#1010C2211yescharacteristic, complemented
20#1009C2211yescharacteristic, complemented
21C3×AGL(1,ℤ/8)—21no—
22C3×AGL(1,ℤ/8)—21no—
23C3×AGL(1,ℤ/8)C2211yescharacteristic
24C3×AGL(1,ℤ/8)C2211yescharacteristic, complemented
25C3×#135—21no—
26C3×#135C2211yescharacteristic
27C2×C3×D8—21no—
28C2×C3×D8C2211yescharacteristic, complemented
29C3×#131C2211yescharacteristic, complemented
30#951—21no—
31#951C2211yescharacteristic, complemented
32#951C2211yescharacteristic, complemented
33#950—21no—
34#950—21no—
35#950C2211yescharacteristic
36#949—21no—
37#949C2211yescharacteristic
38#949C2211yescharacteristic, complemented
39#948C2211yescharacteristic, complemented
40#948C2211yescharacteristic, complemented
41#940C2211yescharacteristic, complemented
42#936C2211yescharacteristic, complemented
43C2×#214—21no—
44C2×#214C2211yescharacteristic, complemented
45#932—21no—
46#932—21no—
47#932C2211yescharacteristic, complemented
48#932C2211yescharacteristic, complemented
49C2×#213—21no—
50C2×#213C2211yescharacteristic, complemented
51#918—21no—
52#916—21no—
53D3×SD16—21no—
54#911—21no—
55C2×#204C2211yescharacteristic, complemented
56#901C2211yescharacteristic, complemented
57#837C2211yescharacteristic, complemented
58#837C2211yescharacteristic, complemented
59#837C2211yescharacteristic, complemented
60#833C2211yescharacteristic, complemented
61#833C2211yescharacteristic, complemented
62#825C2211yescharacteristic, complemented
63#824C2211yescharacteristic, complemented
64#823C2211yescharacteristic, complemented
65#822C2211yescharacteristic, complemented
66#819C2211yescharacteristic, complemented
67#815C2211yescharacteristic, complemented
68#577—31no—
69#577—31no—
70#576—31no—
71#576D311yescharacteristic, complemented
72#496—31no—
73#472—31no—
74#472—31no—
75#471—31no—
76#471—31no—
77#455—31no—
78#454—31no—
79#453—31no—
80#453—31no—
81#444—31no—
82#444—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41—21no—
86C3×#41—21no—
87C3×#41C2311yescharacteristic
88C3×#41C2311yescharacteristic
89C3×#41C2311yescharacteristic
90C2×C3×D4—21no—
91C2×C3×D4—21no—
92C2×C3×D4—21no—
93C2×C3×D4—21no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4C2311yescharacteristic
97C2×C3×D4C2311yescharacteristic
98C2×#67—41no—
99#239—21no—
100D3×Q8—21no—
101#237—21no—
102#237—41no—
103D3×D4—21no—
104D3×D4—41no—
105#235—21no—
106#235—21no—
107#235C2311yescharacteristic
108C2×D12—21no—
109C2×D12C2311yescharacteristic
110C2×Dic6—21no—
111C2×Dic6C2311yescharacteristic
112C3×Q16D411yescharacteristic, complemented
113C3×SD16—21no—
114C3×SD16—21no—
115C3×SD16D411yescharacteristic, complemented
116C3×SD16D411yescharacteristic, complemented
117C3×D8—21no—
118C3×D8—21no—
119C3×D8—21no—
120C3×D8—21no—
121C3×D8D411yescharacteristic, complemented
122C3×M16—21no—
123C3×M16C2311yescharacteristic
124C2×C3×C8—21no—
125C2×C3×C8C2311yescharacteristic
126#217—41no—
127#216—21no—
128#216—21no—
129#216D411yescharacteristic, complemented
130#215—21no—
131#215—21no—
132#215D411yescharacteristic, complemented
133#214—21no—
134#214—21no—
135#214—21no—
136#214—21no—
137#214—21no—
138#214—21no—
139#214D411yescharacteristic, complemented
140#213—21no—
141#213—21no—
142#213—21no—
143#213—21no—
144#213—21no—
145#213—21no—
146#213D411yescharacteristic, complemented
147C4×Dic3C2311yescharacteristic
148#208—21no—
149#208—21no—
150#208C2311yescharacteristic
151#208C2311yescharacteristic
152#208C2311yescharacteristic
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8C2311yescharacteristic
156#204—21no—
157#204—21no—
158ΓL(1,𝔽25)—21no—
159D3×C8—21no—
1602−1+4—31no—
161H(5,𝔽2)—31no—
162H(5,𝔽2)C2×D311yescharacteristic
163H(5,𝔽2)C2×D311yescharacteristic, complemented
164#137—31no—
165#137—31no—
166#137—61no—
167#137—61no—
168AGL(1,ℤ/8)—21no—
169AGL(1,ℤ/8)—21no—
170AGL(1,ℤ/8)—31no—
171AGL(1,ℤ/8)—31no—
172AGL(1,ℤ/8)—31no—
173AGL(1,ℤ/8)—31no—
174AGL(1,ℤ/8)—61no—
175AGL(1,ℤ/8)—61no—
176AGL(1,ℤ/8)—61no—
177AGL(1,ℤ/8)—61no—
178AGL(1,ℤ/8)C2×D311yescharacteristic
179AGL(1,ℤ/8)C2×D311yescharacteristic, complemented
180#135—21no—
181#135—31no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135C2×D311yescharacteristic
186C2×SD16—31no—
187C2×SD16—31no—
188C2×SD16—61no—
189C2×SD16—61no—
190C2×D8—21no—
191C2×D8—31no—
192C2×D8—61no—
193C2×D8C2×D311yescharacteristic, complemented
194#131—31no—
195#131—31no—
196#131—31no—
197#131C2×D311yescharacteristic, complemented
198#127—31no—
199#124—31no—
200#108—31no—
201#108—31no—
202#104—31no—
203#104—31no—
204#104—31no—
205#104—31no—
206#101—31no—
207#100—31no—
208#100—31no—
209#100—31no—
210#97—31no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C22×D3—21no—
216C22×D3—41no—
217C3×Q8C2×D411yescharacteristic
218C3×Q8C2×D411yescharacteristic
219C3×Q8C2×D411yescharacteristic
220C3×D4—21no—
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—21no—
226C3×D4—21no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4—41no—
230C3×D4—41no—
231C3×D4C2×D411yescharacteristic
232C3×D4C2×D411yescharacteristic
233C3×D4C2×D411yescharacteristic
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4C2411yescharacteristic
240#67—21no—
241#67—41no—
242#67—41no—
243#67—82no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—41no—
247D12—21no—
248D12—21no—
249D12—41no—
250D12C2×D411yescharacteristic
251C4×D3—21no—
252C4×D3—41no—
253C4×D3—41no—
254Dic6—21no—
255Dic6—21no—
256Dic6—41no—
257Dic6C2×D411yescharacteristic
258C3×C8—21no—
259C3×C8C2×D411yescharacteristic
260C3×C8C2×D411yescharacteristic
261C3⋊C8—21no—
262C3⋊C8—21no—
263C3⋊C8—21no—
264C3⋊C8C2×D411yescharacteristic
265C3⋊C8C2×D411yescharacteristic
266#41—21no—
267#41—21no—
268#41—21no—
269#41—21no—
270#41—31no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—121no—
276#41C22×D311yescharacteristic
277#41C22×D311yescharacteristic
278#41C22×D311yescharacteristic
279C2×Q8—31no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—31no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4C22×D311yescharacteristic
294C2×D4C22×D311yescharacteristic
295Q16—31no—
296Q16—61no—
297Q16—61no—
298Q16#6711yescharacteristic, complemented
299SD16—21no—
300SD16—21no—
301SD16—31no—
302SD16—31no—
303SD16—61no—
304SD16—61no—
305SD16—61no—
306SD16—61no—
307SD16—61no—
308SD16—61no—
309SD16—61no—
310SD16—61no—
311SD16—61no—
312SD16—61no—
313SD16#6711yescharacteristic, complemented
314SD16#6711yescharacteristic, complemented
315D8—21no—
316D8—21no—
317D8—21no—
318D8—21no—
319D8—31no—
320D8—61no—
321D8—61no—
322D8—61no—
323D8—61no—
324D8—61no—
325D8—61no—
326D8#6711yescharacteristic, complemented
327M16—21no—
328M16—31no—
329M16—31no—
330M16—31no—
331M16—61no—
332M16—61no—
333M16—61no—
334M16C22×D311yescharacteristic
335C2×C8—21no—
336C2×C8—31no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8C22×D311yescharacteristic
341C22⋊C4—121no—
342C42—31no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—161no—
351C22×C3—161no—
352C22×C3H(5,𝔽2)11yescharacteristic
353C2×D3—21no—
354C2×D3—41no—
355C2×D3—81no—
356C2×D3—161no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4C22×D411yescharacteristic
361C3×C4C22×D411yescharacteristic
362Dic3—21no—
363Dic3—41no—
364Dic3—41no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—61no—
370C23—121no—
371Q8—31no—
372Q8—61no—
373Q8—61no—
374Q8—121no—
375Q8D3×D411yescharacteristic
376Q8C2×#6711yescharacteristic
377Q8C2×#6711yescharacteristic
378D4—21no—
379D4—21no—
380D4—21no—
381D4—21no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—31no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—61no—
391D4—61no—
392D4—61no—
393D4—121no—
394D4—121no—
395D4—121no—
396D4—242no—
397D4D3×D411yescharacteristic
398D4C2×#6711yescharacteristic
399D4C2×#6711yescharacteristic
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4C23×D311yescharacteristic
412C8—21no—
413C8—31no—
414C8—31no—
415C8—61no—
416C8—61no—
417C8—61no—
418C8C2×#6711yescharacteristic
419C8C2×#6711yescharacteristic
420C2×C3—21no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—81no—
425C2×C3—81no—
426C2×C3D4211yescharacteristic
427D3—41no—
428D3—81no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—61no—
435C22—81no—
436C22—81no—
437C22—121no—
438C22—161no—
439C22—161no—
440C22—241no—
441C22—481no—
442C22#100411yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—61no—
447C4—121no—
448C4—121no—
449C4C2×D3×D411yescharacteristic
450C4C22×#6711yescharacteristic
451C3#328811yescharacteristic, complemented, Sylow
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—81no—
458C2—121no—
459C2—241no—
460C2D4×#6711yescharacteristic
subgroupG/H
centerC2D4×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#576C2
socleC2×C3D42
hypercenter#576D3
hypocenterC3#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. C3×#576C2semidirect
2. #5836C2semidirect
3. #5835C2semidirect
4. #5770C2semidirect
5. #5242C2semidirect
6. #5241C2semidirect
7. #5201C2semidirect
8. #5200C2semidirect
9. #5199C2semidirect
10. #5198C2semidirect
11. #5197C2semidirect
12. #5160C2semidirect
13. #5154C2semidirect
14. #5145C2semidirect
15. #5141C2semidirect
18. C3×H(5,𝔽2)C22semidirect
19. #1010C22semidirect
20. #1009C22semidirect
24. C3×AGL(1,ℤ/8)C22semidirect
28. C2×C3×D8C22semidirect
29. C3×#131C22semidirect
31. #951C22semidirect
32. #951C22semidirect
38. #949C22semidirect
39. #948C22semidirect
40. #948C22semidirect
41. #940C22semidirect
42. #936C22semidirect
44. C2×#214C22semidirect
47. #932C22semidirect
48. #932C22semidirect
50. C2×#213C22semidirect
55. C2×#204C22semidirect
56. #901C22semidirect
57. #837C22semidirect
58. #837C22semidirect
59. #837C22semidirect
60. #833C22semidirect
61. #833C22semidirect
62. #825C22semidirect
63. #824C22semidirect
64. #823C22semidirect
65. #822C22semidirect
66. #819C22semidirect
67. #815C22semidirect
71. #576D3semidirect
112. C3×Q16D4semidirect
115. C3×SD16D4semidirect
116. C3×SD16D4semidirect
121. C3×D8D4semidirect
129. #216D4semidirect
132. #215D4semidirect
139. #214D4semidirect
146. #213D4semidirect
163. H(5,𝔽2)C2×D3semidirect
179. AGL(1,ℤ/8)C2×D3semidirect
193. C2×D8C2×D3semidirect
197. #131C2×D3semidirect
298. Q16#67semidirect
313. SD16#67semidirect
314. SD16#67semidirect
326. D8#67semidirect
451. C3#3288semidirect

Maximal subgroups

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

subgroupindex
C3×#5762
#58362
#58352
#57702
#52422
#52412
#52012
#52002
#51992
#51982
#51972
#51602
#51542
#51452
#51412
#32883