Finite Group Databasebeta

← #27182614

Subgroups — #27182614

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
2C2×D8×D101C211yesmaximal, characteristic, complemented
3#20265822C211yesmaximal, characteristic, complemented
4D4×D404C211yesmaximal, characteristic, complemented
5#20265566C211yesmaximal, characteristic, complemented
6#20265431C211yesmaximal, characteristic, complemented
7#20265407C211yesmaximal, characteristic, complemented
8#20265392C211yesmaximal, characteristic, complemented
9#20265372C211yesmaximal, characteristic, complemented
10#20265329C211yesmaximal, characteristic, complemented
11#20265323C211yesmaximal, characteristic, complemented
12#20265312C211yesmaximal, characteristic, complemented
13#20265312C211yesmaximal, characteristic, complemented
14#20265250C211yesmaximal, characteristic, complemented
15#442×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic, complemented
17C22×D404—21no—
18C2×D4×D101C2211yescharacteristic
19C2×D4×D101C2211yescharacteristic, complemented
20C2×D4×D101C2211yescharacteristic, complemented
21#15692279—24no—
22C2×#13108063C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×D808C2211yescharacteristic, complemented
26C2×D808C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108059C2211yescharacteristic, complemented
30D8×D101—24no—
31C2×#13108057C2211yescharacteristic, complemented
32C2×C8×D101C2211yescharacteristic, complemented
33#15692252C2211yescharacteristic, complemented
34#15692243C2211yescharacteristic, complemented
35#15692239—21no—
36#15692235C2211yescharacteristic, complemented
37#15692230C2211yescharacteristic
38#15692230C2211yescharacteristic, complemented
39#15692230C2211yescharacteristic, complemented
40#15692229—21no—
41#15692229—21no—
42#15692225—21no—
43#15692225—21no—
44#15692225—21no—
45C4×D404C2211yescharacteristic
46C2×#13108051—21no—
47C4⋊#11717579C2211yescharacteristic, complemented
48#15692178C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692175C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692167C2211yescharacteristic, complemented
54C2×D8×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic, complemented
56C4×D4×C101C2211yescharacteristic, complemented
57#106×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63C23×D101—21no—
64#13108073—22no—
65#13108073—41no—
66C2×Dic202C2311yescharacteristic
67C2×D404—21no—
68C2×D404—21no—
69C2×D404—21no—
70C2×D404—21no—
71C2×D404—21no—
72C2×D404—44no—
73C2×D404D411yescharacteristic
74C2×D404C2311yescharacteristic
75C2×D404C2311yescharacteristic
76C2×D404C2311yescharacteristic
77C2×C22⋊D101—21no—
78C2×C22⋊D101—21no—
79C2×C22⋊D101—21no—
80C2×C22⋊D101—21no—
81D4×D101—22no—
82D4×D101—22no—
83D4×D101—22no—
84D4×D101—41no—
85D4×D101—41no—
86D4×D101—41no—
87C22×Dic101—21no—
88C2×C4×D101C2311yescharacteristic
89C2×C4×D101C2311yescharacteristic
90#13108063—22no—
91#13108063—41no—
92#13108062—22no—
93D808—22no—
94D808—41no—
95#13108059—22no—
96#13108059—22no—
97#13108059—22no—
98#13108057—22no—
99C2×#11717579C2311yescharacteristic
100C2×#11717579C2311yescharacteristic
101C8×D101—22no—
102C4⋊Dic101—21no—
103C4⋊Dic101D411yescharacteristic
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109#13108051—21no—
110#13108051D411yescharacteristic
111#13108051D411yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101—21no—
116D8×C101D414yescomplemented
117C2×C8×C101C2311yescharacteristic
118C2×C8×C101C2311yescharacteristic
119C4⋊C4×C101C2311yescharacteristic
120C4⋊C4×C101C2311yescharacteristic
121C22⋊C4×C101—21no—
122C22⋊C4×C101—21no—
123C42×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—42no—
135C22×D101—81no—
136C22×D101—81no—
137C22×D101—81no—
138C22×D101C2×D411yescharacteristic
139C22×D101C2×D411yescharacteristic
140D4×C101—21no—
141D4×C101—21no—
142D4×C101C2×D412yes—
143D4×C101C2×D412yescomplemented
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—42no—
147C2×C4×C101—81no—
148C2×C4×C101C2×D411yescharacteristic
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2411yescharacteristic
151C22⋊D101—42no—
152C22⋊D101—42no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C2×Dic101—21no—
156C2×Dic101—42no—
157C2×Dic101C2×D411yescharacteristic
158C2×Dic101C2×D411yescharacteristic
159D404—22no—
160D404—21no—
161D404—21no—
162D404—21no—
163D404—21no—
164D404—21no—
165D404—21no—
166D404—42no—
167D404—42no—
168D404—42no—
169D404—81no—
170D404—81no—
171C4×D101—21no—
172C4×D101—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—22no—
182C22×C101—22no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101C22×D411yescharacteristic
188C2×D101—21no—
189C2×D101—21no—
190C2×D101—21no—
191C2×D101—21no—
192C2×D101—21no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—84no—
203C2×D101—84no—
204C2×D101—84no—
205C2×D101AGL(1,ℤ/8)12yescomplemented
206C4×C101—21no—
207C4×C101—21no—
208C4×C101—41no—
209C4×C101—41no—
210C4×C101C22×D411yescharacteristic
211C4×C101H(5,𝔽2)11yescharacteristic
212Dic101—21no—
213Dic101—21no—
214Dic101—41no—
215C2×C101—42no—
216C2×C101—42no—
217C2×C101D4211yescharacteristic
218C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C101#57611yescharacteristic
220D101—22no—
221D101—41no—
222D101—81no—
223D101—81no—
224D101—81no—
225#3264—1011nomaximal, Sylow
226C101#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1011no—
228C22×D8—1011no—
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#442D10111yescharacteristic, complemented
241#435—1011no—
242C2×#41—1011no—
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4—1011no—
246C22×D4—2021no—
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16—1011no—
250C2×SD16—1011no—
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8—1011no—
256C2×D8—2024no—
257C2×D8C2×D10111yescharacteristic
258C2×M16—1011no—
259C22×C8—1011no—
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×D4C2×D10111yescharacteristic, complemented
274C4×D4C2×D10111yescharacteristic, complemented
275C2×C22⋊C4—2021no—
276#106C2×D10111yescharacteristic, complemented
277C4⋊C8—1011no—
278#103—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102C2×D10111yescharacteristic, complemented
283#102C2×D10111yescharacteristic, complemented
284C22⋊C8—1011no—
285C22⋊C8—1011no—
286#97C2×D10111yescharacteristic, complemented
287C24—2021no—
288C24—2021no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8—1011no—
293C2×D4—1011no—
294C2×D4—1011no—
295C2×D4—1011no—
296C2×D4—1011no—
297C2×D4—4044no—
298C2×D4—4041no—
299C2×D4—4041no—
300C2×D4—4041no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2022no—
309C2×D4—2022no—
310C2×D4—2022no—
311C2×D4—2021no—
312C2×D4—2021no—
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—1011no—
318C22×C4—1011no—
319C22×C4—2021no—
320SD16—4041no—
321SD16—2022no—
322SD16—2022no—
323D8—4041no—
324D8—2022no—
325D8—2022no—
326D8—2022no—
327D8—2022no—
328D8D40414yescomplemented
329M16—2022no—
330C2×C8—1011no—
331C2×C8—1011no—
332C2×C8—2022no—
333C2×C8C22×D10111yescharacteristic
334C2×C8C22×D10111yescharacteristic
335C4⋊C4—1011no—
336C4⋊C4—2021no—
337C4⋊C4C22×D10111yescharacteristic
338C4⋊C4C22×D10111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1011no—
342C22⋊C4—1011no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4—2021no—
348C22⋊C4—2021no—
349C42C22×D10111yescharacteristic
350C23—21no—
351C23—21no—
352C23—8081no—
353C23—8081no—
354C23—8081no—
355C23—1011no—
356C23—1011no—
357C23—4042no—
358C23—4042no—
359C23—4042no—
360C23—4042no—
361C23—4042no—
362C23—4042no—
363C23—2021no—
364C23—2021no—
365C23—2021no—
366Q8—2021no—
367Q8—2021no—
368D4—21no—
369D4—21no—
370D4—8081no—
371D4—8081no—
372D4—4042no—
373D4—4042no—
374D4—4042no—
375D4—4042no—
376D4—4042no—
377D4—4042no—
378D4—4042no—
379D4—2022no—
380D4—2021no—
381D4—2021no—
382D4—2021no—
383D4—2021no—
384D4—2021no—
385D4—2021no—
386D4C2×D40412yes—
387D4C2×D40412yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—1011no—
393C2×C4—1011no—
394C2×C4—4042no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4D4×D10111yescharacteristic
401C2×C4D4×D10111yescharacteristic
402C2×C4C23×D10111yescharacteristic
403C8—21no—
404C8—2021no—
405C8—2021no—
406C8C2×D40412yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—8084no—
414C22—8084no—
415C22—8084no—
416C22—1012no—
417C22—4041no—
418C22—4041no—
419C22—4041no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—4041no—
425C22—4041no—
426C22—2021no—
427C22—2021no—
428C22—2021no—
429C22—2021no—
430C22—2021no—
431C22C2×D4×D10111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—4041no—
437C4—2021no—
438C4—2021no—
439C4C22×D40411yescharacteristic
440C4#1569230111yescharacteristic
441C2—42no—
442C2—42no—
443C2—8081no—
444C2—8081no—
445C2—8081no—
446C2—4041no—
447C2—2022no—
448C2D4×D40411yescharacteristic
449C2AGL(1,ℤ/8)×D10111yescharacteristic
450C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#442×C101C2
socleC22×C101C22×D4
hypercenter#442D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×D8×D101C2semidirect
3. #20265822C2semidirect
4. D4×D404C2semidirect
5. #20265566C2semidirect
6. #20265431C2semidirect
7. #20265407C2semidirect
8. #20265392C2semidirect
9. #20265372C2semidirect
10. #20265329C2semidirect
11. #20265323C2semidirect
12. #20265312C2semidirect
13. #20265312C2semidirect
14. #20265250C2semidirect
15. #442×C101C2semidirect
16. C2×#13108073C22semidirect
19. C2×D4×D101C22semidirect
20. C2×D4×D101C22semidirect
22. C2×#13108063C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108062C22semidirect
25. C2×D808C22semidirect
26. C2×D808C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108059C22semidirect
31. C2×#13108057C22semidirect
32. C2×C8×D101C22semidirect
33. #15692252C22semidirect
34. #15692243C22semidirect
36. #15692235C22semidirect
38. #15692230C22semidirect
39. #15692230C22semidirect
47. C4⋊#11717579C22semidirect
48. #15692178C22semidirect
49. #15692175C22semidirect
50. #15692175C22semidirect
51. #15692175C22semidirect
52. #15692167C22semidirect
53. #15692167C22semidirect
55. C4×D4×C101C22semidirect
56. C4×D4×C101C22semidirect
57. #106×C101C22semidirect
58. #102×C101C22semidirect
59. #102×C101C22semidirect
60. #97×C101C22semidirect
116. D8×C101D4semidirect
143. D4×C101C2×D4semidirect
178. C8×C101C2×D4semidirect
205. C2×D101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
240. #442D101semidirect
273. C4×D4C2×D101semidirect
274. C4×D4C2×D101semidirect
276. #106C2×D101semidirect
282. #102C2×D101semidirect
283. #102C2×D101semidirect
286. #97C2×D101semidirect
328. D8D404semidirect
387. D4C2×D404semidirect
406. C8C2×D404semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
C2×D8×D1012
#202658222
D4×D4042
#202655662
#202654312
#202654072
#202653922
#202653722
#202653292
#202653232
#202653122
#202652502
#442×C1012
#3264101