Finite Group Databasebeta

← #80024

Subgroups — #80024

Order 384.

620 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#67C211yesmaximal, characteristic, complemented
2C22×#213C211yesmaximal, characteristic, complemented
3C2×D3×D8C211yesmaximal, characteristic, complemented
4#5609C211yesmaximal, characteristic, complemented
5C3×#447C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5151C211yesmaximal, characteristic, complemented
8#5150C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4786C211yesmaximal, characteristic, complemented
11#4775C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16D4×D8—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic, complemented
18C22×#67—21no—
19C2×D3×D4—21no—
20C2×D3×D4C2211yescharacteristic, complemented
21C2×D3×D4C2211yescharacteristic, complemented
22C22×D12C2211yescharacteristic, complemented
23C2×C3×D8C2211yescharacteristic, complemented
24C2×C3×D8C2211yescharacteristic, complemented
25C3×#121C2211yescharacteristic, complemented
26#953—21no—
27#940C2211yescharacteristic, complemented
28#940C2211yescharacteristic, complemented
29#939—21no—
30#938C2211yescharacteristic, complemented
31#937—21no—
32#937—21no—
33D4×Dic3C2211yescharacteristic, complemented
34C2×#213—24no—
35C2×#213C2211yescharacteristic
36C2×#213C2211yescharacteristic, complemented
37C2×#213C2211yescharacteristic, complemented
38C2×#213C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40#930C2211yescharacteristic, complemented
41C4×#67C2211yescharacteristic
42C22×C3⋊C8C2211yescharacteristic, complemented
43D3×D8—24no—
44C2×D24C2211yescharacteristic, complemented
45C2×D24C2211yescharacteristic, complemented
46C2×D3×C8C2211yescharacteristic, complemented
47#895C2211yescharacteristic, complemented
48#893C2211yescharacteristic, complemented
49#884—21no—
50#882—21no—
51C3×#102C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#821C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C22×D8—31no—
63C22×D8—31no—
64D42—31no—
65D42—31no—
66#493—31no—
67#466—31no—
68#466—31no—
69#459—31no—
70#459—31no—
71#447—31no—
72#447—31no—
73#447—31no—
74#447D311yescharacteristic, complemented
75C4×D8—31no—
76C8×D4—31no—
77C24×C3—21no—
78C23×D3—21no—
79C23×D3—21no—
80C23×D3—21no—
81C2×C3×D4—22no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—21no—
97C2×#67—44no—
98C2×#67D411yescharacteristic
99C22×Dic3—21no—
100D3×D4—22no—
101D3×D4—22no—
102D3×D4—41no—
103D3×D4—41no—
104D3×D4—44no—
105C2×D12—22no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—41no—
109C2×D12C2311yescharacteristic
110C2×D12C2311yescharacteristic
111C2×D12C2311yescharacteristic
112C2×C4×D3—21no—
113C2×C4×D3C2311yescharacteristic
114C3×D8—22no—
115C3×D8—41no—
116C2×C3×C8C2311yescharacteristic
117C2×C3×C8C2311yescharacteristic
118C3×C4⋊C4C2311yescharacteristic
119C3×C22⋊C4—21no—
120#217—21no—
121#217—21no—
122#217D411yescharacteristic
123#213—22no—
124#213—22no—
125#213—22no—
126#213—22no—
127#213D414yescomplemented
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4—21no—
130(C2×D3)⋊C4—21no—
131(C2×D3)⋊C4D411yescharacteristic
132C4⋊Dic3C2311yescharacteristic
133Dic3⋊C4D411yescharacteristic
134C4×Dic3C2311yescharacteristic
135C2×C3⋊C8—22no—
136C2×C3⋊C8C2311yescharacteristic
137C2×C3⋊C8C2311yescharacteristic
138D24—22no—
139D24—41no—
140D3×C8—22no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145C22×D4—61no—
146C22×D4C2×D311yescharacteristic, complemented
147C2×D8—31no—
148C2×D8—31no—
149C2×D8—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×D8—64no—
155C2×D8—64no—
156C2×D8C2×D311yescharacteristic, complemented
157C2×D8C2×D311yescharacteristic, complemented
158C22×C8—31no—
159C22×C8—31no—
160#127—31no—
161#127—31no—
162#121—31no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#121C2×D311yescharacteristic, complemented
168#120—61no—
169#120—61no—
170#120—61no—
171#120—61no—
172C4×D4—31no—
173C4×D4—31no—
174C4×D4—31no—
175#107—31no—
176C4⋊C8—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181#102C2×D311yescharacteristic, complemented
182#102C2×D311yescharacteristic, complemented
183C22⋊C8—31no—
184C22⋊C8C2×D311yescharacteristic, complemented
185C4×C8—31no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—81no—
192C23×C3C2×D411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—42no—
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—42no—
202C22×D3—81no—
203C22×D3—81no—
204C22×D3—81no—
205C22×D3C2×D411yescharacteristic
206C3×D4—21no—
207C3×D4—21no—
208C3×D4—21no—
209C3×D4—21no—
210C3×D4—21no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4C2×D412yescomplemented
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4C2×D411yescharacteristic
218C2×C3×C4C2411yescharacteristic
219#67—42no—
220#67—42no—
221#67—42no—
222#67—42no—
223#67—42no—
224#67—42no—
225#67—81no—
226#67—81no—
227#67D814yescomplemented
228C2×Dic3—21no—
229C2×Dic3—42no—
230C2×Dic3C2×D411yescharacteristic
231C2×Dic3C2×D411yescharacteristic
232C2×Dic3C2×D411yescharacteristic
233D12—21no—
234D12—21no—
235D12—21no—
236D12—21no—
237D12—21no—
238D12—42no—
239D12—42no—
240D12C2×D412yescomplemented
241C4×D3—21no—
242C4×D3—21no—
243C4×D3—42no—
244C3×C8—21no—
245C3×C8—21no—
246C3⋊C8—21no—
247C3⋊C8C2×D412yescomplemented
248C24—21no—
249C24—61no—
250C24—61no—
251C24—61no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—31no—
256C2×D4—31no—
257C2×D4—31no—
258C2×D4—31no—
259C2×D4—41no—
260C2×D4—62no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—124no—
277C2×D4—124no—
278C2×D4C22×D311yescharacteristic
279C2×D4C22×D311yescharacteristic
280C2×D4C22×D311yescharacteristic
281C22×C4—31no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4C22×D311yescharacteristic
285D8—22no—
286D8—34no—
287D8—41no—
288D8—62no—
289D8—62no—
290D8—62no—
291D8—62no—
292D8—62no—
293D8—121no—
294C2×C8—31no—
295C2×C8—31no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8C22×D311yescharacteristic
299C2×C8C22×D311yescharacteristic
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4C22×D311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—31no—
305C22⋊C4—31no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C42—31no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—84no—
324C22×C3C2×D812yescomplemented
325C22×C3C22×D411yescharacteristic
326C2×D3—21no—
327C2×D3—21no—
328C2×D3—22no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—84no—
338C2×D3—84no—
339C2×D3—84no—
340C2×D3C2×D812yescomplemented
341C3×C4—21no—
342C3×C4—41no—
343C3×C4C22×D411yescharacteristic
344C3×C4H(5,𝔽2)11yescharacteristic
345Dic3—21no—
346Dic3—21no—
347Dic3—41no—
348Dic3C2×D812yescomplemented
349C23—21no—
350C23—21no—
351C23—21no—
352C23—31no—
353C23—42no—
354C23—42no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—81no—
359C23—122no—
360C23—122no—
361C23—122no—
362C23—122no—
363C23—122no—
364C23—122no—
365C23—241no—
366C23—241no—
367C23—241no—
368C23D3×D411yescharacteristic
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—34no—
375D4—32no—
376D4—42no—
377D4—42no—
378D4—61no—
379D4—61no—
380D4—61no—
381D4—61no—
382D4—61no—
383D4—122no—
384D4—122no—
385D4—122no—
386D4—122no—
387D4—122no—
388D4—122no—
389D4—122no—
390D4—122no—
391D4—241no—
392D4—241no—
393D4D3×D412yescomplemented
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4D3×D411yescharacteristic
406C2×C4C23×D311yescharacteristic
407C8—21no—
408C8—21no—
409C8—32no—
410C8—61no—
411C2×C3—22no—
412C2×C3—42no—
413C2×C3—81no—
414C2×C3—81no—
415C2×C3D4211yescharacteristic
416C2×C3C22×D811yescharacteristic
417C2×C3#57611yescharacteristic
418D3—22no—
419D3—42no—
420D3—81no—
421D3—81no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—32no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—61no—
435C22—61no—
436C22—62no—
437C22—84no—
438C22—121no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—244no—
447C22—244no—
448C22—244no—
449C22D3×D812yescomplemented
450C22C2×D3×D411yescharacteristic
451C4—21no—
452C4—32no—
453C4—41no—
454C4—61no—
455C4—61no—
456C4—121no—
457C4C2×D3×D411yescharacteristic
458C4#100411yescharacteristic
459C3D4×D811yescharacteristic, complemented, Sylow
460C2—22no—
461C2—42no—
462C2—62no—
463C2—81no—
464C2—81no—
465C2—122no—
466C2—241no—
467C2—241no—
468C2#559011yescharacteristic
469C2C2×D3×D811yescharacteristic
470C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3D4×D8
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. D4×#67C2semidirect
2. C22×#213C2semidirect
3. C2×D3×D8C2semidirect
4. #5609C2semidirect
5. C3×#447C2semidirect
6. #5156C2semidirect
7. #5151C2semidirect
8. #5150C2semidirect
9. #5038C2semidirect
10. #4786C2semidirect
11. #4775C2semidirect
12. #4773C2semidirect
13. #4756C2semidirect
14. #4732C2semidirect
15. #67⋊C8C2semidirect
17. C22×C3×D4C22semidirect
20. C2×D3×D4C22semidirect
21. C2×D3×D4C22semidirect
22. C22×D12C22semidirect
23. C2×C3×D8C22semidirect
24. C2×C3×D8C22semidirect
25. C3×#121C22semidirect
27. #940C22semidirect
28. #940C22semidirect
30. #938C22semidirect
33. D4×Dic3C22semidirect
36. C2×#213C22semidirect
37. C2×#213C22semidirect
38. C2×#213C22semidirect
39. C2×#213C22semidirect
40. #930C22semidirect
42. C22×C3⋊C8C22semidirect
44. C2×D24C22semidirect
45. C2×D24C22semidirect
46. C2×D3×C8C22semidirect
47. #895C22semidirect
48. #893C22semidirect
51. C3×#102C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. #821C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #809C22semidirect
61. #807C22semidirect
74. #447D3semidirect
127. #213D4semidirect
146. C22×D4C2×D3semidirect
156. C2×D8C2×D3semidirect
157. C2×D8C2×D3semidirect
167. #121C2×D3semidirect
181. #102C2×D3semidirect
182. #102C2×D3semidirect
184. C22⋊C8C2×D3semidirect
213. C3×D4C2×D4semidirect
227. #67D8semidirect
240. D12C2×D4semidirect
247. C3⋊C8C2×D4semidirect
324. C22×C3C2×D8semidirect
340. C2×D3C2×D8semidirect
348. Dic3C2×D8semidirect
393. D4D3×D4semidirect
449. C22D3×D8semidirect
459. C3D4×D8semidirect

Maximal subgroups

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

subgroupindex
D4×#672
C22×#2132
C2×D3×D82
#56092
C3×#4472
#51562
#51512
#51502
#50382
#47862
#47752
#47732
#47562
#47322
#67⋊C82
D4×D83