Finite Group Databasebeta

← #80231

Subgroups — #80231

Order 384.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic, complemented
2C2×#919C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5629C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5169C211yesmaximal, characteristic, complemented
9#5046C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4803C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13D3×#102C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3012—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18#1009—24no—
19#1009—24no—
20C2×#239C2211yescharacteristic
21C2×#239C2211yescharacteristic
22C2×#239C2211yescharacteristic, complemented
23C2×D3×D4—21no—
24C2×D3×D4—21no—
25C2×D3×D4C2211yescharacteristic
26C2×D3×D4C2211yescharacteristic
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×D12—21no—
30C22×D12C2211yescharacteristic
31C2×C3×Q16C2211yescharacteristic, complemented
32C2×C3×SD16C2211yescharacteristic, complemented
33C3×C22⋊Q8C2211yescharacteristic, complemented
34#946C2211yescharacteristic, complemented
35C2×#215C2211yescharacteristic, complemented
36C2×#215C2211yescharacteristic, complemented
37C2×#215C2211yescharacteristic, complemented
38C2×#213C2211yescharacteristic, complemented
39#930C2211yescharacteristic, complemented
40#919—24no—
41#914—24no—
42C2×D24C2211yescharacteristic, complemented
43C2×D24C2211yescharacteristic, complemented
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45C2×D3×C8C2211yescharacteristic, complemented
46#896—21no—
47#895—21no—
48#895C2211yescharacteristic, complemented
49D3×C4⋊C4C2211yescharacteristic, complemented
50#884—21no—
51#884—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59#821C2211yescharacteristic, complemented
60(C2×D3)⋊C8C2211yescharacteristic, complemented
61(C2×D3)⋊C8C2211yescharacteristic, complemented
62#809C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C2×H(5,𝔽2)—31no—
65C2×AGL(1,ℤ/8)—31no—
66C2×#135—31no—
67#537—31no—
68#452—31no—
69#452D311yescharacteristic, complemented
70#450—31no—
71#450—31no—
72#449—31no—
73#449—31no—
74#447—31no—
75#447—31no—
76#417—31no—
77C2×#102—31no—
78#408—31no—
79C23×D3—21no—
80C23×D3—21no—
81C23×D3—21no—
82C3×#41—22no—
83C3×#41—22no—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4—21no—
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67D411yescharacteristic
94C2×#67D411yescharacteristic
95#239—22no—
96#239—22no—
97#239—22no—
98#239—22no—
99#239—22no—
100#239D414yescomplemented
101D3×D4—28no—
102D3×D4—28no—
103D3×D4—22no—
104D3×D4—22no—
105D3×D4—22no—
106D3×D4D414yescomplemented
107#235—28no—
108#235—22no—
109#235—22no—
110C2×D12—28no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—22no—
115C2×D12—22no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12D411yescharacteristic
119C2×D12C2311yescharacteristic
120C2×D12C2311yescharacteristic
121C2×D12C2311yescharacteristic
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2311yescharacteristic
128C2×Dic6D411yescharacteristic
129C3×Q16—22no—
130C3×SD16—22no—
131C2×C3×C8C2311yescharacteristic
132C2×C3×C8C2311yescharacteristic
133C3×C4⋊C4—21no—
134C3×C4⋊C4C2311yescharacteristic
135C3×C22⋊C4—21no—
136#215—22no—
137#215—22no—
138#215—22no—
139#213—22no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3C2311yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148D24—22no—
149D24—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—
159C2×#41C2×D311yescharacteristic
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—31no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166AGL(1,ℤ/8)—64no—
167#135—64no—
168C2×Q16C2×D311yescharacteristic, complemented
169C2×SD16—31no—
170C2×SD16—31no—
171C2×SD16—31no—
172C2×SD16C2×D311yescharacteristic, complemented
173C2×D8—31no—
174C2×D8—31no—
175C2×D8—31no—
176C2×M16—31no—
177C22×C8—31no—
178C22⋊Q8—31no—
179C22⋊Q8—61no—
180C22⋊Q8C2×D311yescharacteristic, complemented
181#121—31no—
182#121—31no—
183#121—61no—
184#121—61no—
185#121—61no—
186C2×C4⋊C4—31no—
187#103—31no—
188#103C2×D311yescharacteristic, complemented
189#102—31no—
190#102—31no—
191#102—31no—
192#102—31no—
193#102—31no—
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—21no—
200C23×C3C2×D411yescharacteristic
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—24no—
204C22×D3—24no—
205C22×D3—24no—
206C22×D3—21no—
207C22×D3—81no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—81no—
213C22×D3C2×D411yescharacteristic
214C22×D3C2×D411yescharacteristic
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8C2×D412yes—
219C3×D4—24no—
220C3×D4—21no—
221C3×D4C2×D412yes—
222C2×C3×C4—24no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4C2×D411yescharacteristic
229C2×C3×C4C2411yescharacteristic
230#67—24no—
231#67—22no—
232#67—22no—
233#67—24no—
234#67—81no—
235C2×Dic3—21no—
236C2×Dic3C2×D411yescharacteristic
237C2×Dic3C2×D411yescharacteristic
238D12—24no—
239D12—24no—
240D12—22no—
241D12—24no—
242D12—21no—
243D12—21no—
244D12—21no—
245D12—21no—
246D12—42no—
247D12—81no—
248D12C2×D412yes—
249D12C2×D412yes—
250C4×D3—24no—
251C4×D3—21no—
252C4×D3—21no—
253C4×D3—24no—
254C4×D3—42no—
255C4×D3—81no—
256C4×D3C2×D412yes—
257C4×D3C2×D412yes—
258Dic6—22no—
259C3×C8—21no—
260C3×C8—21no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C24—61no—
264C24—61no—
265C24—61no—
266#41—22no—
267#41—22no—
268#41—34no—
269#41—62no—
270#41—62no—
271#41—68no—
272#41—62no—
273#41—62no—
274#41—62no—
275#41—62no—
276#41—62no—
277C2×Q8—31no—
278C2×Q8C22×D311yescharacteristic
279C2×Q8C22×D311yescharacteristic
280C2×D4—21no—
281C2×D4—34no—
282C2×D4—31no—
283C2×D4—31no—
284C2×D4—31no—
285C2×D4—31no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—68no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—62no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—62no—
299C2×D4—68no—
300C2×D4—68no—
301C2×D4—62no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4C22×D311yescharacteristic
305C22×C4—21no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4C22×D311yescharacteristic
313Q16—22no—
314SD16—22no—
315SD16—62no—
316SD16—62no—
317SD16—62no—
318D8—62no—
319D8—62no—
320D8—62no—
321M16—62no—
322C2×C8—31no—
323C2×C8—31no—
324C2×C8—62no—
325C2×C8C22×D311yescharacteristic
326C2×C8C22×D311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—31no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4C22×D311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3C22×D411yescharacteristic
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—22no—
347C2×D3—22no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—84no—
358C2×D3—84no—
359C2×D3—84no—
360C2×D3#12012yes—
361C3×C4—21no—
362C3×C4—22no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4C22×D411yescharacteristic
366C3×C4C22×D411yescharacteristic
367Dic3—21no—
368Dic3—41no—
369Dic3#12012yes—
370C23—21no—
371C23—31no—
372C23—31no—
373C23—64no—
374C23—64no—
375C23—64no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—241no—
380C23—241no—
381C23—241no—
382C23—241no—
383C23—241no—
384C23—241no—
385C23D3×D411yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—62no—
390Q8D3×D412yes—
391D4—24no—
392D4—21no—
393D4—32no—
394D4—32no—
395D4—62no—
396D4—64no—
397D4—64no—
398D4—64no—
399D4—62no—
400D4—64no—
401D4—62no—
402D4—64no—
403D4—61no—
404D4—61no—
405D4—61no—
406D4—61no—
407D4—122no—
408D4—241no—
409D4—241no—
410D4D3×D412yes—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—31no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—31no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—122no—
427C2×C4—241no—
428C2×C4D3×D411yescharacteristic
429C2×C4C23×D311yescharacteristic
430C8—21no—
431C8—21no—
432C8—61no—
433C8—61no—
434C2×C3—41no—
435C2×C3—42no—
436C2×C3C2×#12011yescharacteristic
437C2×C3#57611yescharacteristic
438C2×C3#57711yescharacteristic
439D3—22no—
440D3—41no—
441D3—42no—
442D3—42no—
443D3—81no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—32no—
449C22—41no—
450C22—41no—
451C22—61no—
452C22—61no—
453C22—61no—
454C22—61no—
455C22—62no—
456C22—62no—
457C22—61no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22—244no—
466C22—244no—
467C22—244no—
468C22C2×D3×D411yescharacteristic
469C4—21no—
470C4—22no—
471C4—32no—
472C4—41no—
473C4—41no—
474C4—61no—
475C4—121no—
476C4C2×D3×D411yescharacteristic
477C4C2×D3×D411yescharacteristic
478C3#301211yescharacteristic, complemented, Sylow
479C2—41no—
480C2—42no—
481C2—62no—
482C2—121no—
483C2—122no—
484C2—122no—
485C2—241no—
486C2D3×#12011yescharacteristic
487C2#576911yescharacteristic
488C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3012
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C2×#919C2semidirect
3. C2×#914C2semidirect
4. #5629C2semidirect
5. C3×#452C2semidirect
6. #5238C2semidirect
7. #5187C2semidirect
8. #5169C2semidirect
9. #5046C2semidirect
10. #4810C2semidirect
11. #4803C2semidirect
12. #4773C2semidirect
13. D3×#102C2semidirect
14. #4732C2semidirect
15. #4727C2semidirect
22. C2×#239C22semidirect
31. C2×C3×Q16C22semidirect
32. C2×C3×SD16C22semidirect
33. C3×C22⋊Q8C22semidirect
34. #946C22semidirect
35. C2×#215C22semidirect
36. C2×#215C22semidirect
37. C2×#215C22semidirect
38. C2×#213C22semidirect
39. #930C22semidirect
42. C2×D24C22semidirect
43. C2×D24C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
45. C2×D3×C8C22semidirect
48. #895C22semidirect
49. D3×C4⋊C4C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. #821C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. (C2×D3)⋊C8C22semidirect
62. #809C22semidirect
63. #809C22semidirect
69. #452D3semidirect
100. #239D4semidirect
106. D3×D4D4semidirect
168. C2×Q16C2×D3semidirect
172. C2×SD16C2×D3semidirect
180. C22⋊Q8C2×D3semidirect
188. #103C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
478. C3#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C2×#9192
C2×#9142
#56292
C3×#4522
#52382
#51872
#51692
#50462
#48102
#48032
#47732
D3×#1022
#47322
#47272
#30123