Finite Group Databasebeta

← #86610

Subgroups — #86610

Order 384.

1142 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D42C211yesmaximal, characteristic, complemented
2#5806C212yesmaximal, complemented
3#5806C212yesmaximal, complemented
4#5798C212yesmaximal, complemented
5C2×#936C212yesmaximal, complemented
6#5716C211yesmaximal, characteristic, complemented
7#5713C211yesmaximal, characteristic, complemented
8#5616C212yesmaximal, complemented
9#5603C212yesmaximal, complemented
10#5600C212yesmaximal, complemented
11#5598C212yesmaximal, complemented
12#5593C214yesmaximal, complemented
13#5587C214yesmaximal, complemented
14#5557C211yesmaximal, characteristic, complemented
15#5542C211yesmaximal, characteristic, complemented
16D4×Dic6C211yesmaximal, characteristic, complemented
17C4×#237C211yesmaximal, characteristic, complemented
18#3514—31nomaximal, Sylow
19C22×C3×D4C2212yes—
20C22×C3×D4C2212yes—
21C2×#237C2211yescharacteristic
22C2×#237C2212yescomplemented
23C22×Dic6C2212yescomplemented
24C3×#127C2211yescharacteristic
25C3×#121C2212yes—
26C3×#121C2212yes—
27C3×#120C2214yes—
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic
30#953C2214yescomplemented
31C2×#217C2214yescomplemented
32C2×#217C2214yescomplemented
33C2×#217C2214yescomplemented
34#939C2214yescomplemented
35#939C2214yescomplemented
36#938C2211yescharacteristic
37#938C2212yescomplemented
38#938C2212yescomplemented
39#936C2211yescharacteristic
40#936C2212yes—
41#936C2212yescomplemented
42#936C2212yescomplemented
43#936C2212yescomplemented
44#936C2214yescomplemented
45#936C2214yescomplemented
46#935C2214yescomplemented
47#935C2214yescomplemented
48#935C2214yescomplemented
49D4×Dic3C2211yescharacteristic
50D4×Dic3C2212yes—
51D4×Dic3C2212yescomplemented
52D4×Dic3C2212yescomplemented
53D4×Dic3C2212yescomplemented
54D4×Dic3C2212yescomplemented
55D4×Dic3C2214yescomplemented
56D4×Dic3C2214yescomplemented
57C4×#67C2212yescomplemented
58C4×#67C2212yescomplemented
59#926C2212yescomplemented
60#926C2212yescomplemented
61#924C2212yescomplemented
62#924C2212yescomplemented
63C2×C4×Dic3C2212yescomplemented
64#897C2211yescharacteristic, complemented
65#892C2211yescharacteristic, complemented
66#890C2211yescharacteristic, complemented
67#888C2212yescomplemented
68#887C2211yescharacteristic, complemented
69#887C2212yescomplemented
70#885C2214yescomplemented
71#883C2212yescomplemented
72#67⋊C4C2212yescomplemented
73#879C2214yescomplemented
74#879C2212yescomplemented
75#879C2214yescomplemented
76#878C2214yescomplemented
77#878C2214yescomplemented
78#877C2212yescomplemented
79#877C2214yescomplemented
80C42×D3C2211yescharacteristic, complemented
81#869C2211yescharacteristic, complemented
82C4×Dic6C2211yescharacteristic, complemented
83C4×Dic6C2211yescharacteristic, complemented
84#559—31no—
85#556—31no—
86#555—32no—
87#551—34no—
88D4×Q8—31no—
89#548—32no—
90#546—32no—
91#546—32no—
92D42D311yescharacteristic, complemented
93#542—32no—
94#538—34no—
95#532—31no—
96#529—32no—
97#529—31no—
98C2×#124—32no—
99#518—32no—
100C4×#41—31no—
101C24×C3C2314yes—
102C2×C3×D4—28no—
103C2×C3×D4—44no—
104C2×C3×D4C2314yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2312yes—
107C2×C3×D4C2311yescharacteristic
108C2×C3×D4C2312yes—
109C2×C3×D4C2314yes—
110C2×C3×D4C2311yescharacteristic
111C22×C3×C4C2312yes—
112C22×C3×C4C2312yes—
113C2×#67C2312yes—
114C2×#67C2312yes—
115C2×#67C2314yes—
116C22×Dic3C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2314yes—
119C22×Dic3C2314yes—
120C22×Dic3C2312yes—
121#237—24no—
122#237—44no—
123C2×C4×D3C2311yescharacteristic
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×Dic6—44no—
127C2×Dic6C2311yescharacteristic
128C2×Dic6C2311yescharacteristic
129C2×Dic6C2312yes—
130C2×Dic6C2314yes—
131C3×C4⋊C4C2311yescharacteristic
132C3×C4⋊C4C2311yescharacteristic
133C3×C22⋊C4C2312yes—
134C3×C22⋊C4C2314yes—
135C3×C22⋊C4C2312yes—
136C3×C42C2311yescharacteristic
137#217C2312yes—
138#217C2314yes—
139#217C2314yes—
140#217C2314yes—
141#217C2314yes—
142#217C2312yes—
143#217C2314yes—
144#217C2312yes—
145#217C2314yes—
146#217C2314yes—
147#217C2312yes—
148#217C2314yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151C4⋊Dic3C2311yescharacteristic
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163C4×Dic3C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2312yes—
169C4×Dic3C2314yes—
170C4×Dic3C2312yes—
171C2×#41—31no—
172C2×#41—32no—
173C22×Q8—32no—
174C22×D4C2×D312yes—
175C22×D4C2×D312yes—
176C4⋊Q8—32no—
177C4⋊Q8—31no—
178#127C2×D311yescharacteristic
179#126—34no—
180#126—32no—
181#126—34no—
182#125—31no—
183#124—31no—
184#124—32no—
185#124—34no—
186#124—32no—
187#124—32no—
188#124—34no—
189#124—32no—
190#124—34no—
191#123—34no—
192#123—34no—
193#123—34no—
194#123—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—31no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200#121—31no—
201#121—32no—
202#121—32no—
203#121—34no—
204#121—34no—
205#121C2×D312yes—
206#121C2×D312yes—
207#120—34no—
208#120C2×D314yes—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—31no—
213C4×D4—31no—
214C4×D4—34no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—32no—
224C4×D4C2×D311yescharacteristic
225C4×D4C2×D311yescharacteristic
226#117—32no—
227#117—32no—
228#117—31no—
229#117—32no—
230#117—34no—
231C2×C22⋊C4—34no—
232C2×C22⋊C4—34no—
233C2×C22⋊C4—34no—
234C2×C42—32no—
235C2×C42—31no—
236C23×C3—28no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3C2412yes—
240C23×C3C2414yes—
241C23×C3C2412yes—
242C22×D3C2411yescharacteristic
243C3×D4—28no—
244C3×D4—24no—
245C3×D4—24no—
246C3×D4—22no—
247C3×D4—44no—
248C3×D4—42no—
249C3×D4#4114yescomplemented
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4C2412yes—
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256C2×C3×C4C2411yescharacteristic
257#67—24no—
258#67—44no—
259#67—42no—
260C2×Dic3—24no—
261C2×Dic3—44no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3—44no—
265C2×Dic3C2411yescharacteristic
266C2×Dic3C2412yes—
267C2×Dic3C2411yescharacteristic
268C2×Dic3C2412yes—
269C2×Dic3C2411yescharacteristic
270C2×Dic3C2412yes—
271C2×Dic3C2414yes—
272C2×Dic3C2412yes—
273C4×D3—22no—
274C4×D3—22no—
275C4×D3—41no—
276Dic6—22no—
277Dic6—22no—
278Dic6—24no—
279Dic6—44no—
280C24C22×D314yes—
281#41—64no—
282#41—124no—
283C2×Q8—31no—
284C2×Q8—31no—
285C2×Q8—32no—
286C2×Q8—34no—
287C2×Q8—124no—
288C2×D4—28no—
289C2×D4—32no—
290C2×D4—32no—
291C2×D4—34no—
292C2×D4—44no—
293C2×D4C22×D314yes—
294C2×D4C22×D312yes—
295C2×D4C22×D312yes—
296C2×D4C22×D311yescharacteristic
297C2×D4C22×D312yes—
298C2×D4C22×D314yes—
299C2×D4C22×D311yescharacteristic
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—34no—
303C22×C4—34no—
304C22×C4—34no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4C22×D312yes—
309C22×C4C22×D312yes—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—32no—
316C4⋊C4—32no—
317C4⋊C4—34no—
318C4⋊C4—31no—
319C4⋊C4—32no—
320C4⋊C4—34no—
321C4⋊C4—32no—
322C4⋊C4C22×D311yescharacteristic
323C4⋊C4C22×D311yescharacteristic
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—34no—
331C22⋊C4—34no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4—32no—
337C22⋊C4—32no—
338C22⋊C4C22×D312yes—
339C22⋊C4C22×D314yes—
340C22⋊C4C22×D312yes—
341C42—31no—
342C42—32no—
343C42—31no—
344C42—31no—
345C42—32no—
346C42—32no—
347C42—34no—
348C42—32no—
349C42C22×D311yescharacteristic
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—24no—
357C22×C3—22no—
358C22×C3—88no—
359C22×C3C2×#4114yes—
360C22×C3C2511yescharacteristic
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4C2×#4112yes—
368C3×C4H(5,𝔽2)12yes—
369Dic3—22no—
370Dic3—21no—
371Dic3—22no—
372Dic3—24no—
373Dic3—21no—
374Dic3—21no—
375Dic3—22no—
376Dic3—22no—
377C23—28no—
378C23—31no—
379C23—44no—
380C23—44no—
381C23C23×D312yes—
382C23C23×D314yes—
383C23C23×D312yes—
384Q8—62no—
385Q8—62no—
386Q8—64no—
387Q8—124no—
388D4—28no—
389D4—24no—
390D4—22no—
391D4—24no—
392D4—44no—
393D4—42no—
394D4—64no—
395D4—124no—
396D4—122no—
397D4#23714yescomplemented
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—31no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—34no—
408C2×C4—64no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—121no—
412C2×C4—124no—
413C2×C4—124no—
414C2×C4—122no—
415C2×C4—124no—
416C2×C4C23×D312yes—
417C2×C4C23×D312yes—
418C2×C4C23×D311yescharacteristic
419C2×C4C23×D311yescharacteristic
420C2×C4C23×D311yescharacteristic
421C2×C3—24no—
422C2×C3—44no—
423C2×C3—42no—
424C2×C3C22×#4111yescharacteristic
425C2×C3C2×H(5,𝔽2)11yescharacteristic
426C2×C3#58511yescharacteristic
427D3—41no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—24no—
432C22—24no—
433C22—22no—
434C22—24no—
435C22—22no—
436C22—61no—
437C22—61no—
438C22—61no—
439C22—88no—
440C22C2×#23714yes—
441C22C24×D311yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—61no—
446C4—62no—
447C4—61no—
448C4—64no—
449C4—62no—
450C4—61no—
451C4—62no—
452C4—62no—
453C4C2×#23712yes—
454C4#100412yes—
455C3#351411yescharacteristic, complemented, Sylow
456C2—24no—
457C2—42no—
458C2—44no—
459C2—121no—
460C2C22×#23711yescharacteristic
461C2C2×#100411yescharacteristic
462C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D42C2
hypercenterD42D3
hypocenterC3#3514
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C3×D42C2semidirect
2. #5806C2semidirect
3. #5806C2semidirect
4. #5798C2semidirect
5. C2×#936C2semidirect
6. #5716C2semidirect
7. #5713C2semidirect
8. #5616C2semidirect
9. #5603C2semidirect
10. #5600C2semidirect
11. #5598C2semidirect
12. #5593C2semidirect
13. #5587C2semidirect
14. #5557C2semidirect
15. #5542C2semidirect
16. D4×Dic6C2semidirect
17. C4×#237C2semidirect
22. C2×#237C22semidirect
23. C22×Dic6C22semidirect
30. #953C22semidirect
31. C2×#217C22semidirect
32. C2×#217C22semidirect
33. C2×#217C22semidirect
34. #939C22semidirect
35. #939C22semidirect
37. #938C22semidirect
38. #938C22semidirect
41. #936C22semidirect
42. #936C22semidirect
43. #936C22semidirect
44. #936C22semidirect
45. #936C22semidirect
46. #935C22semidirect
47. #935C22semidirect
48. #935C22semidirect
51. D4×Dic3C22semidirect
52. D4×Dic3C22semidirect
53. D4×Dic3C22semidirect
54. D4×Dic3C22semidirect
55. D4×Dic3C22semidirect
56. D4×Dic3C22semidirect
57. C4×#67C22semidirect
58. C4×#67C22semidirect
59. #926C22semidirect
60. #926C22semidirect
61. #924C22semidirect
62. #924C22semidirect
63. C2×C4×Dic3C22semidirect
64. #897C22semidirect
65. #892C22semidirect
66. #890C22semidirect
67. #888C22semidirect
68. #887C22semidirect
69. #887C22semidirect
70. #885C22semidirect
71. #883C22semidirect
72. #67⋊C4C22semidirect
73. #879C22semidirect
74. #879C22semidirect
75. #879C22semidirect
76. #878C22semidirect
77. #878C22semidirect
78. #877C22semidirect
79. #877C22semidirect
80. C42×D3C22semidirect
81. #869C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
92. D42D3semidirect
249. C3×D4#41semidirect
397. D4#237semidirect
455. C3#3514semidirect

Maximal subgroups

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

subgroupindex
C3×D422
#58062
#57982
C2×#9362
#57162
#57132
#56162
#56032
#56002
#55982
#55932
#55872
#55572
#55422
D4×Dic62
C4×#2372
#35143