Finite Group Databasebeta

← #27182746

Subgroups — #27182746

Order 12928.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic, complemented
2#20265846C211yesmaximal, characteristic, complemented
3D4×C22⋊D101C211yesmaximal, characteristic, complemented
4#20265552C211yesmaximal, characteristic, complemented
5#20265430C211yesmaximal, characteristic, complemented
6#20265427C211yesmaximal, characteristic, complemented
7#20265375C211yesmaximal, characteristic, complemented
8#20265370C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic, complemented
10#20265321C211yesmaximal, characteristic, complemented
11#20265311C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14#20265253C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic, complemented
17C22×C22⋊D101—21no—
18C2×D4×D101C2211yescharacteristic
19#15692282—24no—
20C2×#13108061C2211yescharacteristic, complemented
21C2×D808C2211yescharacteristic, complemented
22C2×#13108059C2211yescharacteristic
23C2×#13108059C2211yescharacteristic, complemented
24C2×#13108059C2211yescharacteristic, complemented
25C2×#13108057C2211yescharacteristic, complemented
26#15692251—21no—
27#15692251C2211yescharacteristic
28#15692251C2211yescharacteristic, complemented
29#15692229—21no—
30#15692229C2211yescharacteristic
31#15692229C2211yescharacteristic, complemented
32#15692227—21no—
33#15692226—21no—
34#15692226C2211yescharacteristic
35#15692226C2211yescharacteristic, complemented
36#15692224—21no—
37#15692223—21no—
38Q8×Dic101C2211yescharacteristic, complemented
39D4×Dic101C2211yescharacteristic
40C4×C22⋊D101—21no—
41C4×C22⋊D101C2211yescharacteristic
42#15692187C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692177C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic, complemented
54C22×D4×C101C2211yescharacteristic
55AGL(1,ℤ/8)×C101—24no—
56C2×SD16×C101C2211yescharacteristic
57C2×SD16×C101C2211yescharacteristic, complemented
58C2×D8×C101C2211yescharacteristic
59C2×D8×C101C2211yescharacteristic, complemented
60C2×M16×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108075—22no—
63#13108075—41no—
64C2×D404—21no—
65C2×D404C2311yescharacteristic
66C2×D404C2311yescharacteristic
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—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101—44no—
77C2×C22⋊D101D411yescharacteristic
78D4×D101—22no—
79D4×D101—41no—
80C22×Dic101—21no—
81C2×C4×D101—21no—
82C2×C4×D101C2311yescharacteristic
83#13108061—22no—
84D808—22no—
85#13108059—22no—
86#13108059—41no—
87#13108059—41no—
88#13108057—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C4⋊Dic101—21no—
92C4⋊Dic101C2311yescharacteristic
93C4⋊Dic101C2311yescharacteristic
94#13108053D411yescharacteristic
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101—21no—
98C22⋊Dic101D411yescharacteristic
99#13108051—21no—
100#13108051—21no—
101#13108051D411yescharacteristic
102C4×Dic101—21no—
103C4×Dic101C2311yescharacteristic
104C24×C101—21no—
105#41×C101—22no—
106#41×C101—41no—
107C2×Q8×C101C2311yescharacteristic
108C2×D4×C101—22no—
109C2×D4×C101—21no—
110C2×D4×C101—21no—
111C2×D4×C101—41no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C22×C4×C101—21no—
116C22×C4×C101C2311yescharacteristic
117SD16×C101—22no—
118SD16×C101D414yescomplemented
119D8×C101—22no—
120D8×C101—22no—
121M16×C101—22no—
122C2×C8×C101C2311yescharacteristic
123C2×C8×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—42no—
128C23×C101—42no—
129C23×C101—81no—
130C23×C101C2×D411yescharacteristic
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—42no—
134C22×D101—42no—
135C22×D101—81no—
136C22×D101C2×D411yescharacteristic
137Q8×C101—21no—
138Q8×C101C2×D412yescomplemented
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—42no—
145D4×C101—42no—
146D4×C101C2×D412yes—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101—42no—
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2411yescharacteristic
153C22⋊D101—22no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—42no—
158C22⋊D101—42no—
159C22⋊D101—81no—
160C22⋊D101—81no—
161C22⋊D101—81no—
162C22⋊D101—81no—
163C2×Dic101—21no—
164C2×Dic101—21no—
165C2×Dic101—42no—
166C2×Dic101C2×D411yescharacteristic
167C2×Dic101C2×D411yescharacteristic
168C2×Dic101C2×D411yescharacteristic
169D404—21no—
170D404—21no—
171D404—21no—
172D404—21no—
173D404—42no—
174C4×D101—21no—
175C4×D101—21no—
176C4×D101—42no—
177C8×C101—21no—
178C8×C101C2×D412yescomplemented
179#11717579—21no—
180#11717579—21no—
181C22×C101—22no—
182C22×C101—21no—
183C22×C101—21no—
184C22×C101—21no—
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—84no—
194C22×C101C22×D411yescharacteristic
195C2×D101—21no—
196C2×D101—21no—
197C2×D101—21no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—41no—
204C2×D101—84no—
205C4×C101—22no—
206C4×C101—21no—
207C4×C101C22×D411yescharacteristic
208C4×C101H(5,𝔽2)11yescharacteristic
209Dic101—21no—
210Dic101—21no—
211Dic101—41no—
212Dic101—41no—
213Dic101AGL(1,ℤ/8)12yescomplemented
214C2×C101—42no—
215C2×C101—41no—
216C2×C101—81no—
217C2×C101—81no—
218C2×C101D4211yescharacteristic
219C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C101#57611yescharacteristic
221D101—41no—
222D101—81no—
223D101—81no—
224#3274—1011nomaximal, Sylow
225C101#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1011no—
227C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
228#550—1011no—
229D42—1011no—
230#493—1011no—
231#469—1011no—
232#469—1011no—
233#460—1011no—
234#459—1011no—
235#449—1011no—
236#449—1011no—
237#447—1011no—
238#447—1011no—
239C4×SD16—1011no—
240#436—1011no—
241C2×#41—1011no—
242C2×#41C2×D10111yescharacteristic, complemented
243C22×D4—1011no—
244C22×D4—2021no—
245C22×D4C2×D10111yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16C2×D10111yescharacteristic
250C2×SD16C2×D10111yescharacteristic, complemented
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8C2×D10111yescharacteristic
256C2×D8C2×D10111yescharacteristic, complemented
257C2×M16—1011no—
258C2×M16C2×D10111yescharacteristic, complemented
259#127—1011no—
260#127—1011no—
261#127—2021no—
262#121—1011no—
263#121—1011no—
264#121—1011no—
265#121—1011no—
266#121—2021no—
267#121—2021no—
268#121—2021no—
269#120—2021no—
270#120—2021no—
271C4×Q8—1011no—
272C4×D4—1011no—
273C4×D4—1011no—
274C4×D4—2021no—
275#106—1011no—
276C4⋊C8—1011no—
277#103—1011no—
278#102—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102—1011no—
283C22⋊C8—1011no—
284C22⋊C8—1011no—
285C4×C8—1011no—
286C24—21no—
287C24—2021no—
288#41—22no—
289#41—41no—
290#41—4041no—
291#41—2022no—
292C2×Q8C22×D10111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1011no—
298C2×D4—1011no—
299C2×D4—1011no—
300C2×D4—4044no—
301C2×D4—4041no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4—2021no—
312C2×D4—2022no—
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C2×D4C22×D10111yescharacteristic
316C22×C4—21no—
317C22×C4—1011no—
318C22×C4—2021no—
319C22×C4—2021no—
320C22×C4C22×D10111yescharacteristic
321SD16—22no—
322SD16—2022no—
323SD16C22⋊D10114yescomplemented
324D8—22no—
325D8—22no—
326D8—4041no—
327D8—4041no—
328D8—2022no—
329D8—2022no—
330M16—22no—
331M16—2022no—
332C2×C8—1011no—
333C2×C8—1011no—
334C2×C8C22×D10111yescharacteristic
335C2×C8C22×D10111yescharacteristic
336C4⋊C4—1011no—
337C4⋊C4—1011no—
338C4⋊C4—1011no—
339C4⋊C4—2021no—
340C22⋊C4—1011no—
341C22⋊C4—1011no—
342C22⋊C4—2021no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C42—1011no—
348C42—2021no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—8081no—
356C23—1011no—
357C23—4042no—
358C23—4042no—
359C23—2021no—
360C23—2021no—
361C23D4×D10111yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D10112yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—8081no—
372D4—8081no—
373D4—8081no—
374D4—8081no—
375D4—4042no—
376D4—4042no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—2022no—
382D4—2021no—
383D4—2021no—
384D4—2021no—
385D4—2021no—
386D4C2×C22⋊D10112yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
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×C4D4×D10111yescharacteristic
401C2×C4C23×D10111yescharacteristic
402C8—21no—
403C8—2021no—
404C8—2021no—
405C8C2×C22⋊D10112yescomplemented
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—8084no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—4041no—
425C22—4041no—
426C22—2021no—
427C22—2021no—
428C22—2021no—
429C22C2×D4×D10111yescharacteristic
430C4—21no—
431C4—22no—
432C4—1012no—
433C4—4041no—
434C4—4041no—
435C4—2021no—
436C4—2021no—
437C4C22×C22⋊D10111yescharacteristic
438C4#1569230011yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—8081no—
444C2—8081no—
445C2—4041no—
446C2D4×C22⋊D10111yescharacteristic
447C2AGL(1,ℤ/8)×D10111yescharacteristic
448C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3274
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265846C2semidirect
3. D4×C22⋊D101C2semidirect
4. #20265552C2semidirect
5. #20265430C2semidirect
6. #20265427C2semidirect
7. #20265375C2semidirect
8. #20265370C2semidirect
9. #20265322C2semidirect
10. #20265321C2semidirect
11. #20265311C2semidirect
12. #20265309C2semidirect
13. SD16×Dic101C2semidirect
14. #20265253C2semidirect
15. C2×AGL(1,ℤ/8)×C101C2semidirect
16. C2×#13108075C22semidirect
20. C2×#13108061C22semidirect
21. C2×D808C22semidirect
23. C2×#13108059C22semidirect
24. C2×#13108059C22semidirect
25. C2×#13108057C22semidirect
28. #15692251C22semidirect
31. #15692229C22semidirect
35. #15692226C22semidirect
38. Q8×Dic101C22semidirect
42. #15692187C22semidirect
43. #15692184C22semidirect
44. #15692179C22semidirect
45. #15692177C22semidirect
46. #15692177C22semidirect
48. #15692176C22semidirect
49. #15692176C22semidirect
50. C22⋊#11717579C22semidirect
51. #15692167C22semidirect
52. C8×Dic101C22semidirect
53. C2×#41×C101C22semidirect
57. C2×SD16×C101C22semidirect
59. C2×D8×C101C22semidirect
60. C2×M16×C101C22semidirect
118. SD16×C101D4semidirect
138. Q8×C101C2×D4semidirect
178. C8×C101C2×D4semidirect
213. Dic101AGL(1,ℤ/8)semidirect
225. C101#3274semidirect
227. C2×AGL(1,ℤ/8)D101semidirect
242. C2×#41C2×D101semidirect
250. C2×SD16C2×D101semidirect
256. C2×D8C2×D101semidirect
258. C2×M16C2×D101semidirect
323. SD16C22⋊D101semidirect
363. Q8C2×C22⋊D101semidirect
405. C8C2×C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202658462
D4×C22⋊D1012
#202655522
#202654302
#202654272
#202653752
#202653702
#202653222
#202653212
#202653112
#202653092
SD16×Dic1012
#202652532
C2×AGL(1,ℤ/8)×C1012
#3274101