Finite Group Databasebeta

← #86012

Subgroups — #86012

Order 384.

1560 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1004C211yesmaximal, characteristic, complemented
2C22×#235C211yesmaximal, characteristic, complemented
3C2×C3×#123C211yesmaximal, characteristic, complemented
4#5804C211yesmaximal, characteristic, complemented
5C2×#929C211yesmaximal, characteristic, complemented
6#5656C212yesmaximal, complemented
7#5655C214yesmaximal, complemented
8#5654C214yesmaximal, complemented
9#5653C212yesmaximal, complemented
10D3×#123C212yesmaximal, complemented
11#5648C212yesmaximal, complemented
12#5507C214yesmaximal, complemented
13#5493C211yesmaximal, characteristic, complemented
14#5490C214yesmaximal, complemented
15#5486C211yesmaximal, characteristic, complemented
16#3485—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic, complemented
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic, complemented
22C22×#67C2212yescomplemented
23#1004—28no—
24C2×#237C2212yes—
25C2×#237C2212yescomplemented
26C2×D3×D4C2212yes—
27C2×D3×D4C2212yescomplemented
28C2×#235—28no—
29C2×#235C2214yescomplemented
30C2×#235C2214yescomplemented
31C2×#235C2212yescomplemented
32C22×D12C2211yescharacteristic, complemented
33C22×C4×D3C2211yescharacteristic, complemented
34C22×C4×D3C2211yescharacteristic, complemented
35C22×Dic6C2211yescharacteristic, complemented
36C3×#123C2212yescomplemented
37C3×#123C2212yescomplemented
38C3×#123C2214yescomplemented
39C2×C3×C4⋊C4C2212yescomplemented
40C2×C3×C22⋊C4C2211yescharacteristic, complemented
41C2×C3×C22⋊C4C2212yescomplemented
42#953C2212yescomplemented
43C2×#217C2211yescharacteristic, complemented
44#939C2214yescomplemented
45#937C2212yescomplemented
46#935C2212yescomplemented
47#930C2214yescomplemented
48#929C2212yescomplemented
49#929C2212yescomplemented
50#929C2214yescomplemented
51#929C2212yescomplemented
52C4×#67C2214yescomplemented
53C2×(C2×D3)⋊C4C2212yescomplemented
54C2×Dic3⋊C4C2212yescomplemented
55#896C2214yescomplemented
56#896C2214yescomplemented
57#895C2214yescomplemented
58#894C2214yescomplemented
59#894C2214yescomplemented
60#893C2214yescomplemented
61D3×C4⋊C4C2214yescomplemented
62#888C2214yescomplemented
63#886C2212yescomplemented
64#886C2212yescomplemented
65#885C2214yescomplemented
66#884C2214yescomplemented
67#884C2214yescomplemented
68#883C2214yescomplemented
69#882C2214yescomplemented
70#882C2212yescomplemented
71#882C2212yescomplemented
72#67⋊C4C2214yescomplemented
73#67⋊C4C2214yescomplemented
74D3×C22⋊C4C2212yescomplemented
75D3×C22⋊C4C2214yescomplemented
76#878C2214yescomplemented
77#877C2212yescomplemented
78C2×H(5,𝔽2)—31no—
79C22×#41—31no—
80#547—34no—
81#547—34no—
82#546—34no—
83#546—34no—
84#536—32no—
85#534—31no—
86#534—32no—
87#534—31no—
88#525—32no—
89C2×#123—31no—
90C2×#123—32no—
91C2×#123D311yescharacteristic, complemented
92#518—31no—
93C24×C3C2311yescharacteristic
94C24×C3C2311yescharacteristic
95C23×D3C2312yes—
96C23×D3C2311yescharacteristic
97C23×D3C2311yescharacteristic
98C23×D3C2311yescharacteristic
99C2×C3×D4—24no—
100C2×C3×D4C2312yes—
101C2×C3×D4C2312yes—
102C22×C3×C4—24no—
103C22×C3×C4C2312yes—
104C22×C3×C4C2312yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2312yes—
109C2×#67—24no—
110C2×#67—24no—
111C2×#67—28no—
112C2×#67D418yescomplemented
113C2×#67C2312yes—
114C2×#67C2312yes—
115C2×#67C2312yes—
116C2×#67C2312yes—
117C2×#67C2312yes—
118C2×#67C2312yes—
119C2×#67C2314yes—
120C2×#67C2312yes—
121C2×#67C2312yes—
122C22×Dic3C2312yes—
123C22×Dic3C2311yescharacteristic
124C22×Dic3C2311yescharacteristic
125C22×Dic3C2312yes—
126C22×Dic3C2311yescharacteristic
127#237—28no—
128#237—28no—
129D3×D4—28no—
130D3×D4—28no—
131#235—216no—
132#235—216no—
133#235—28no—
134C2×D12—24no—
135C2×D12C2314yes—
136C2×D12C2312yes—
137C2×D12C2314yes—
138C2×C4×D3—24no—
139C2×C4×D3—24no—
140C2×C4×D3C2314yes—
141C2×C4×D3C2314yes—
142C2×C4×D3C2312yes—
143C2×C4×D3C2312yes—
144C2×C4×D3C2314yes—
145C2×Dic6—24no—
146C2×Dic6C2314yes—
147C2×Dic6C2312yes—
148C3×C4⋊C4C2314yes—
149C3×C4⋊C4C2314yes—
150C3×C22⋊C4C2312yes—
151C3×C22⋊C4C2314yes—
152C3×C22⋊C4C2314yes—
153C3×C22⋊C4C2312yes—
154#217C2312yes—
155#217C2314yes—
156#217C2312yes—
157(C2×D3)⋊C4C2314yes—
158(C2×D3)⋊C4C2314yes—
159(C2×D3)⋊C4C2314yes—
160(C2×D3)⋊C4C2314yes—
161(C2×D3)⋊C4C2314yes—
162(C2×D3)⋊C4C2314yes—
163C4⋊Dic3C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166Dic3⋊C4C2314yes—
167Dic3⋊C4C2314yes—
168C4×Dic3C2314yes—
169H(5,𝔽2)—68no—
170C2×#41—32no—
171C2×#41—32no—
172C2×#41—34no—
173C2×#41—34no—
174C2×#41—32no—
175C2×#41—68no—
176C22×Q8—31no—
177C22×D4—31no—
178C22×D4—31no—
179C22×D4—31no—
180C22×D4—32no—
181C22×D4—32no—
182C22×D4—32no—
183C22×D4—31no—
184C22×D4C2×D311yescharacteristic
185C23×C4—31no—
186C23×C4—31no—
187C23×C4C2×D311yescharacteristic, complemented
188C4⋊Q8—34no—
189#124—34no—
190#123—32no—
191#123—32no—
192#123—32no—
193#123—34no—
194#123—32no—
195#123—34no—
196#123—34no—
197#123—32no—
198#123—34no—
199#123—32no—
200#123C2×D312yescomplemented
201#123C2×D312yescomplemented
202#123C2×D314yescomplemented
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206#121—34no—
207#121—34no—
208#121—34no—
209#121—34no—
210#121—34no—
211#120—32no—
212#120—32no—
213#120—32no—
214#120—34no—
215#120—32no—
216C4×D4—34no—
217C4×D4—34no—
218C4×D4—34no—
219C4×D4—34no—
220#117—32no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—34no—
223C2×C4⋊C4C2×D312yescomplemented
224C2×C22⋊C4—31no—
225C2×C22⋊C4—32no—
226C2×C22⋊C4—32no—
227C2×C22⋊C4—34no—
228C2×C22⋊C4C2×D311yescharacteristic, complemented
229C2×C22⋊C4C2×D312yescomplemented
230C23×C3—22no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3C2×D414yes—
234C23×C3C2412yes—
235C23×C3C2411yescharacteristic
236C23×C3C2411yescharacteristic
237C23×C3C2411yescharacteristic
238C22×D3—22no—
239C22×D3—22no—
240C22×D3—24no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—44no—
244C22×D3—42no—
245C22×D3C2×D414yes—
246C22×D3C2412yes—
247C22×D3C2412yes—
248C22×D3C2412yes—
249C3×D4—24no—
250C3×D4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—42no—
257C2×C3×C4C2414yes—
258C2×C3×C4C2412yes—
259C2×C3×C4C2412yes—
260C2×C3×C4C2412yes—
261#67—24no—
262#67—24no—
263#67—24no—
264#67—24no—
265#67—24no—
266#67—24no—
267#67—44no—
268#67C2×D418yes—
269#67C2×D418yes—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—24no—
273C2×Dic3—42no—
274C2×Dic3C2×D414yes—
275C2×Dic3C2414yes—
276C2×Dic3C2412yes—
277C2×Dic3C2412yes—
278C2×Dic3C2412yes—
279D12—28no—
280D12—24no—
281D12—44no—
282C4×D3—28no—
283C4×D3—24no—
284C4×D3—28no—
285C4×D3—24no—
286C4×D3—44no—
287Dic6—28no—
288Dic6—24no—
289C24—31no—
290C24—32no—
291C24—31no—
292C24—31no—
293C24C22×D311yescharacteristic
294C24C22×D311yescharacteristic
295#41—616no—
296#41—616no—
297#41—68no—
298#41—68no—
299#41—68no—
300C2×Q8—34no—
301C2×Q8—32no—
302C2×Q8—64no—
303C2×D4—24no—
304C2×D4—38no—
305C2×D4—32no—
306C2×D4—32no—
307C2×D4—32no—
308C2×D4—32no—
309C2×D4—32no—
310C2×D4—32no—
311C2×D4—32no—
312C2×D4—32no—
313C2×D4—34no—
314C2×D4—34no—
315C2×D4—32no—
316C2×D4—34no—
317C2×D4—64no—
318C2×D4—68no—
319C2×D4—68no—
320C2×D4—68no—
321C2×D4—64no—
322C2×D4—64no—
323C2×D4C22×D312yes—
324C2×D4C22×D312yes—
325C22×C4—24no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—34no—
332C22×C4—32no—
333C22×C4—34no—
334C22×C4—32no—
335C22×C4—34no—
336C22×C4—64no—
337C22×C4—64no—
338C22×C4C22×D312yes—
339C22×C4C22×D312yes—
340C22×C4C22×D311yescharacteristic
341C22×C4C22×D311yescharacteristic
342C22×C4C22×D311yescharacteristic
343C22×C4C22×D312yes—
344C4⋊C4—34no—
345C4⋊C4—34no—
346C4⋊C4—34no—
347C4⋊C4—34no—
348C4⋊C4—34no—
349C4⋊C4C22×D314yes—
350C4⋊C4C22×D314yes—
351C22⋊C4—32no—
352C22⋊C4—32no—
353C22⋊C4—34no—
354C22⋊C4—34no—
355C22⋊C4—34no—
356C22⋊C4—34no—
357C22⋊C4—34no—
358C22⋊C4—34no—
359C22⋊C4—34no—
360C22⋊C4C22×D312yes—
361C22⋊C4C22×D314yes—
362C22⋊C4C22×D312yes—
363C22⋊C4C22×D314yes—
364C42—34no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—24no—
370C22×C3—44no—
371C22×C3—82no—
372C22×C3C22×D412yes—
373C22×C3C22×D412yes—
374C22×C3C22×D412yes—
375C22×C3C2511yescharacteristic
376C2×D3—24no—
377C2×D3—22no—
378C2×D3—22no—
379C2×D3—24no—
380C2×D3—24no—
381C2×D3—44no—
382C2×D3—44no—
383C2×D3—84no—
384C2×D3—82no—
385C2×D3C22×D414yes—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—24no—
390Dic3—22no—
391Dic3—22no—
392Dic3—24no—
393Dic3C22×D414yes—
394C23—22no—
395C23—32no—
396C23—34no—
397C23—32no—
398C23—32no—
399C23—42no—
400C23—42no—
401C23—64no—
402C23—62no—
403C23—62no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23—124no—
408C23D3×D414yes—
409C23C23×D312yes—
410C23C23×D311yescharacteristic
411C23C23×D311yescharacteristic
412C23C23×D311yescharacteristic
413Q8—68no—
414Q8—64no—
415D4—24no—
416D4—24no—
417D4—38no—
418D4—38no—
419D4—64no—
420D4—68no—
421D4—64no—
422D4—64no—
423D4—64no—
424D4—64no—
425D4—64no—
426D4—64no—
427D4—124no—
428D4—124no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—34no—
435C2×C4—32no—
436C2×C4—34no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—42no—
440C2×C4—68no—
441C2×C4—64no—
442C2×C4—68no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—124no—
448C2×C4—122no—
449C2×C4C23×D314yes—
450C2×C4C23×D312yes—
451C2×C4C23×D312yes—
452C2×C4C23×D312yes—
453C2×C3—22no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—42no—
457C2×C3C23×D411yescharacteristic
458C2×C3#58512yes—
459D3—24no—
460D3—42no—
461D3—42no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—24no—
467C22—34no—
468C22—44no—
469C22—64no—
470C22—62no—
471C22—62no—
472C22—64no—
473C22—64no—
474C22—82no—
475C22—124no—
476C22—124no—
477C22—242no—
478C22—244no—
479C22C2×D3×D412yes—
480C22C2×D3×D412yes—
481C22C2×D3×D412yes—
482C22C24×D311yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—24no—
487C4—34no—
488C4—62no—
489C4—62no—
490C4—64no—
491C3#348511yescharacteristic, complemented, Sylow
492C2—22no—
493C2—22no—
494C2—22no—
495C2—42no—
496C2—64no—
497C2—122no—
498C2—122no—
499C2C22×D3×D411yescharacteristic
500C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×#123C2
hypercenterC2×#123D3
hypocenterC3#3485
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C22×#235C2semidirect
3. C2×C3×#123C2semidirect
4. #5804C2semidirect
5. C2×#929C2semidirect
6. #5656C2semidirect
7. #5655C2semidirect
8. #5654C2semidirect
9. #5653C2semidirect
10. D3×#123C2semidirect
11. #5648C2semidirect
12. #5507C2semidirect
13. #5493C2semidirect
14. #5490C2semidirect
15. #5486C2semidirect
18. C23×C3×C4C22semidirect
21. C22×#67C22semidirect
22. C22×#67C22semidirect
25. C2×#237C22semidirect
27. C2×D3×D4C22semidirect
29. C2×#235C22semidirect
30. C2×#235C22semidirect
31. C2×#235C22semidirect
32. C22×D12C22semidirect
33. C22×C4×D3C22semidirect
34. C22×C4×D3C22semidirect
35. C22×Dic6C22semidirect
36. C3×#123C22semidirect
37. C3×#123C22semidirect
38. C3×#123C22semidirect
39. C2×C3×C4⋊C4C22semidirect
40. C2×C3×C22⋊C4C22semidirect
41. C2×C3×C22⋊C4C22semidirect
42. #953C22semidirect
43. C2×#217C22semidirect
44. #939C22semidirect
45. #937C22semidirect
46. #935C22semidirect
47. #930C22semidirect
48. #929C22semidirect
49. #929C22semidirect
50. #929C22semidirect
51. #929C22semidirect
52. C4×#67C22semidirect
53. C2×(C2×D3)⋊C4C22semidirect
54. C2×Dic3⋊C4C22semidirect
55. #896C22semidirect
56. #896C22semidirect
57. #895C22semidirect
58. #894C22semidirect
59. #894C22semidirect
60. #893C22semidirect
61. D3×C4⋊C4C22semidirect
62. #888C22semidirect
63. #886C22semidirect
64. #886C22semidirect
65. #885C22semidirect
66. #884C22semidirect
67. #884C22semidirect
68. #883C22semidirect
69. #882C22semidirect
70. #882C22semidirect
71. #882C22semidirect
72. #67⋊C4C22semidirect
73. #67⋊C4C22semidirect
74. D3×C22⋊C4C22semidirect
75. D3×C22⋊C4C22semidirect
76. #878C22semidirect
77. #877C22semidirect
91. C2×#123D3semidirect
112. C2×#67D4semidirect
187. C23×C4C2×D3semidirect
200. #123C2×D3semidirect
201. #123C2×D3semidirect
202. #123C2×D3semidirect
223. C2×C4⋊C4C2×D3semidirect
228. C2×C22⋊C4C2×D3semidirect
229. C2×C22⋊C4C2×D3semidirect
491. C3#3485semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C22×#2352
C2×C3×#1232
#58042
C2×#9292
#56562
#56552
#56542
#56532
D3×#1232
#56482
#55072
#54932
#54902
#54862
#34853