Finite Group Databasebeta

← #27182612

Subgroups — #27182612

Order 12928.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692279C211yesmaximal, characteristic, complemented
2#20265821C211yesmaximal, characteristic, complemented
3D4×C22⋊D101C211yesmaximal, characteristic, complemented
4#20265565C211yesmaximal, characteristic, complemented
5#20265427C211yesmaximal, characteristic, complemented
6#20265406C211yesmaximal, characteristic, complemented
7#20265390C211yesmaximal, characteristic, complemented
8#20265370C211yesmaximal, characteristic, complemented
9#20265328C211yesmaximal, characteristic, complemented
10#20265319C211yesmaximal, characteristic, complemented
11#20265311C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265289C211yesmaximal, characteristic, complemented
14#20265248C211yesmaximal, characteristic, complemented
15C22×D8×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic, complemented
17C22×C22⋊D101—21no—
18C2×D4×D101C2211yescharacteristic, complemented
19#15692279—24no—
20C2×#13108063C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108059C2211yescharacteristic, complemented
24C2×#13108059C2211yescharacteristic, complemented
25C2×#13108057C2211yescharacteristic, complemented
26#15692251C2211yescharacteristic, complemented
27#15692242C2211yescharacteristic, complemented
28#15692236—21no—
29#15692234C2211yescharacteristic, complemented
30#15692229C2211yescharacteristic
31#15692227—21no—
32#15692227—21no—
33#15692226C2211yescharacteristic, complemented
34#15692226C2211yescharacteristic, complemented
35#15692224—21no—
36#15692224—21no—
37#15692223—21no—
38D4×Dic101C2211yescharacteristic, complemented
39D4×Dic101C2211yescharacteristic, complemented
40C4×C22⋊D101C2211yescharacteristic
41C2×C22⋊Dic101—21no—
42#15692187C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic, complemented
46#15692176C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52#15692165C2211yescharacteristic, complemented
53C22×D4×C101C2211yescharacteristic
54C22×D4×C101C2211yescharacteristic, complemented
55C2×D8×C101—24no—
56C2×D8×C101C2211yescharacteristic
57C2×D8×C101C2211yescharacteristic
58C2×D8×C101C2211yescharacteristic, complemented
59C2×D8×C101C2211yescharacteristic, complemented
60C22×C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108073—22no—
63#13108073—41no—
64C2×Dic202C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—44no—
74C2×C22⋊D101D411yescharacteristic
75D4×D101—22no—
76D4×D101—41no—
77C22×Dic101—21no—
78C22×Dic101—21no—
79C22×Dic101—21no—
80C2×C4×D101C2311yescharacteristic
81#13108063—22no—
82#13108062—22no—
83#13108062—41no—
84#13108059—22no—
85#13108059—41no—
86#13108057—22no—
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90C4⋊Dic101C2311yescharacteristic
91#13108053—21no—
92#13108053D411yescharacteristic
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101—21no—
98C22⋊Dic101—21no—
99C22⋊Dic101—21no—
100C22⋊Dic101D411yescharacteristic
101#13108051—21no—
102#13108051D411yescharacteristic
103C4×Dic101C2311yescharacteristic
104C24×C101—21no—
105C24×C101—21no—
106C2×D4×C101—22no—
107C2×D4×C101—22no—
108C2×D4×C101—21no—
109C2×D4×C101—21no—
110C2×D4×C101—41no—
111C2×D4×C101—41no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C2×D4×C101C2311yescharacteristic
116C22×C4×C101C2311yescharacteristic
117D8×C101—22no—
118D8×C101—22no—
119D8×C101—22no—
120D8×C101D414yescomplemented
121C2×C8×C101—22no—
122C2×C8×C101C2311yescharacteristic
123C2×C8×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—21no—
128C23×C101—42no—
129C23×C101—42no—
130C23×C101—42no—
131C23×C101—42no—
132C23×C101—81no—
133C23×C101—81no—
134C23×C101C2×D411yescharacteristic
135C22×D101—21no—
136C22×D101—42no—
137C22×D101—42no—
138C22×D101—81no—
139C22×D101C2×D411yescharacteristic
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—21no—
146D4×C101—42no—
147D4×C101—42no—
148D4×C101C2×D412yes—
149D4×C101C2×D412yescomplemented
150C2×C4×C101—21no—
151C2×C4×C101—21no—
152C2×C4×C101C2×D411yescharacteristic
153C2×C4×C101C2411yescharacteristic
154C22⋊D101—22no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—42no—
158C22⋊D101—42no—
159C22⋊D101—42no—
160C22⋊D101—81no—
161C22⋊D101—81no—
162C2×Dic101—21no—
163C2×Dic101—21no—
164C2×Dic101—21no—
165C2×Dic101—42no—
166C2×Dic101—42no—
167C2×Dic101—81no—
168C2×Dic101C2×D411yescharacteristic
169C2×Dic101C2×D411yescharacteristic
170C2×Dic101C2×D411yescharacteristic
171D404—21no—
172D404—21no—
173C4×D101—21no—
174C4×D101—21no—
175Dic202—21no—
176Dic202—21no—
177C8×C101—21no—
178C8×C101C2×D412yescomplemented
179#11717579—21no—
180#11717579—21no—
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—22no—
184C22×C101—22no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101—41no—
191C22×C101—41no—
192C22×C101—41no—
193C22×C101—41no—
194C22×C101—41no—
195C22×C101—84no—
196C22×C101—84no—
197C22×C101AGL(1,ℤ/8)12yescomplemented
198C22×C101C22×D411yescharacteristic
199C2×D101—21no—
200C2×D101—21no—
201C2×D101—21no—
202C2×D101—41no—
203C2×D101—41no—
204C2×D101—41no—
205C2×D101—84no—
206C4×C101—21no—
207C4×C101C22×D411yescharacteristic
208C4×C101H(5,𝔽2)11yescharacteristic
209Dic101—21no—
210Dic101—21no—
211Dic101—21no—
212Dic101—41no—
213Dic101—41no—
214Dic101—41no—
215C2×C101—22no—
216C2×C101—42no—
217C2×C101—42no—
218C2×C101—81no—
219C2×C101—81no—
220C2×C101D4211yescharacteristic
221C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C101#57611yescharacteristic
223D101—41no—
224D101—81no—
225#3264—1011nomaximal, Sylow
226C101#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1011no—
228C22×D8D10111yescharacteristic, complemented
229#546—1011no—
230D42—1011no—
231#496—1011no—
232#469—1011no—
233#465—1011no—
234#463—1011no—
235#459—1011no—
236#450—1011no—
237#449—1011no—
238#447—1011no—
239#447—1011no—
240#442—1011no—
241#435—1011no—
242C2×#41—1011no—
243C22×D4—1011no—
244C22×D4—2021no—
245C22×D4C2×D10111yescharacteristic
246C22×D4C2×D10111yescharacteristic, complemented
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16—1011no—
250C2×SD16—1011no—
251C2×D8—24no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8C2×D10111yescharacteristic
255C2×D8C2×D10111yescharacteristic
256C2×D8C2×D10111yescharacteristic, complemented
257C2×D8C2×D10111yescharacteristic, complemented
258C2×M16—1011no—
259C22×C8C2×D10111yescharacteristic, complemented
260#127—1011no—
261#124—1011no—
262#123—2021no—
263C22⋊Q8—1011no—
264#121—1011no—
265#121—1011no—
266#121—1011no—
267#121—2021no—
268#121—2021no—
269#120—2021no—
270#120—2021no—
271#120—2021no—
272C4×D4—1011no—
273C4×D4—1011no—
274C4×D4—1011no—
275C2×C22⋊C4—2021no—
276#106—1011no—
277C4⋊C8—1011no—
278#103—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102—1011no—
283#102—1011no—
284C22⋊C8—1011no—
285C22⋊C8—1011no—
286#97—1011no—
287C24—21no—
288C24—21no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8—1011no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—1011no—
300C2×D4—1011no—
301C2×D4—4044no—
302C2×D4—4041no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2022no—
311C2×D4C22×D10111yescharacteristic
312C2×D4C22×D10111yescharacteristic
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—1011no—
316C22×C4—2021no—
317C22×C4—2021no—
318C22×C4—2021no—
319C22×C4C22×D10111yescharacteristic
320SD16—4041no—
321SD16—2022no—
322SD16—2022no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—4041no—
327D8—2022no—
328D8C22⋊D10114yescomplemented
329M16—2022no—
330C2×C8—22no—
331C2×C8—1011no—
332C2×C8—1011no—
333C2×C8C22×D10111yescharacteristic
334C2×C8C22×D10111yescharacteristic
335C4⋊C4—1011no—
336C4⋊C4—1011no—
337C4⋊C4—1011no—
338C4⋊C4—2021no—
339C22⋊C4—1011no—
340C22⋊C4—1011no—
341C22⋊C4—2021no—
342C22⋊C4—2021no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4—2021no—
348C22⋊C4—2021no—
349C42—1011no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—8081no—
361C23—1011no—
362C23—4042no—
363C23—4042no—
364C23—2021no—
365C23D4×D10111yescharacteristic
366Q8—2021no—
367Q8—2021no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—8081no—
377D4—8081no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—4042no—
382D4—4042no—
383D4—2022no—
384D4—2021no—
385D4—2021no—
386D4C2×C22⋊D10112yes—
387D4C2×C22⋊D10112yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—8081no—
391C2×C4—1011no—
392C2×C4—1011no—
393C2×C4—1011no—
394C2×C4—4042no—
395C2×C4—4042no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4—2021no—
401C2×C4D4×D10111yescharacteristic
402C2×C4C23×D10111yescharacteristic
403C8—21no—
404C8—2021no—
405C8—2021no—
406C8C2×C22⋊D10112yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—8084no—
424C22—4041no—
425C22—4041no—
426C22—4041no—
427C22—2021no—
428C22—2021no—
429C22—2021no—
430C22#1569227912yescomplemented
431C22C2×D4×D10111yescharacteristic
432C4—21no—
433C4—4041no—
434C4—4041no—
435C4—4041no—
436C4—2021no—
437C4—2021no—
438C4—2021no—
439C4C22×C22⋊D10111yescharacteristic
440C4#1569230011yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—8081no—
447C2—4041no—
448C2D4×C22⋊D10111yescharacteristic
449C2C2×#1569227911yescharacteristic
450C2#2026597811yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×D8×C101C2
socleC22×C101C22×D4
hypercenterC22×D8D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. #20265821C2semidirect
3. D4×C22⋊D101C2semidirect
4. #20265565C2semidirect
5. #20265427C2semidirect
6. #20265406C2semidirect
7. #20265390C2semidirect
8. #20265370C2semidirect
9. #20265328C2semidirect
10. #20265319C2semidirect
11. #20265311C2semidirect
12. #20265309C2semidirect
13. #20265289C2semidirect
14. #20265248C2semidirect
15. C22×D8×C101C2semidirect
16. C2×#13108073C22semidirect
18. C2×D4×D101C22semidirect
20. C2×#13108063C22semidirect
21. C2×#13108062C22semidirect
22. C2×#13108062C22semidirect
23. C2×#13108059C22semidirect
24. C2×#13108059C22semidirect
25. C2×#13108057C22semidirect
26. #15692251C22semidirect
27. #15692242C22semidirect
29. #15692234C22semidirect
33. #15692226C22semidirect
34. #15692226C22semidirect
38. D4×Dic101C22semidirect
39. D4×Dic101C22semidirect
42. #15692187C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692177C22semidirect
46. #15692176C22semidirect
47. #15692176C22semidirect
48. #15692176C22semidirect
49. #15692176C22semidirect
50. C22⋊#11717579C22semidirect
51. #15692167C22semidirect
52. #15692165C22semidirect
54. C22×D4×C101C22semidirect
58. C2×D8×C101C22semidirect
59. C2×D8×C101C22semidirect
60. C22×C8×C101C22semidirect
120. D8×C101D4semidirect
149. D4×C101C2×D4semidirect
178. C8×C101C2×D4semidirect
197. C22×C101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
228. C22×D8D101semidirect
246. C22×D4C2×D101semidirect
256. C2×D8C2×D101semidirect
257. C2×D8C2×D101semidirect
259. C22×C8C2×D101semidirect
328. D8C22⋊D101semidirect
387. D4C2×C22⋊D101semidirect
406. C8C2×C22⋊D101semidirect
430. C22#15692279semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#202658212
D4×C22⋊D1012
#202655652
#202654272
#202654062
#202653902
#202653702
#202653282
#202653192
#202653112
#202653092
#202652892
#202652482
C22×D8×C1012
#3264101