Finite Group Databasebeta

← #36365042

Subgroups — #36365042

Order 22144.

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×#19149706C211yesmaximal, characteristic, complemented
2C2×D8×D173C211yesmaximal, characteristic, complemented
3#26012134C211yesmaximal, characteristic, complemented
4#26012126C211yesmaximal, characteristic, complemented
5#26011741C211yesmaximal, characteristic, complemented
6#26011717C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011683C211yesmaximal, characteristic, complemented
9#26011640C211yesmaximal, characteristic, complemented
10#26011635C211yesmaximal, characteristic, complemented
11#26011623C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011604C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic, complemented
20C2×D4×D173C2211yescharacteristic, complemented
21#19149706—24no—
22C2×#14583813C2211yescharacteristic, complemented
23C2×D1384C2211yescharacteristic
24C2×#14583811C2211yescharacteristic
25C2×#14583811C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic, complemented
27D8×D173—24no—
28C2×#14583809C2211yescharacteristic, complemented
29C2×C8×D173C2211yescharacteristic, complemented
30#19149661—21no—
31#19149655C2211yescharacteristic, complemented
32#19149652—21no—
33#19149650—21no—
34#19149650C2211yescharacteristic
35#19149650C2211yescharacteristic, complemented
36#19149650C2211yescharacteristic, complemented
37#19149649—21no—
38#19149647—21no—
39#19149647—21no—
40C4×D692C2211yescharacteristic
41C4×D692C2211yescharacteristic, complemented
42C4×D692C2211yescharacteristic, complemented
43C22⋊C4×D173—21no—
44#19149611C2211yescharacteristic, complemented
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149591C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic, complemented
56C2×D8×C173C2211yescharacteristic, complemented
57C2×D8×C173C2211yescharacteristic, complemented
58#127×C173C2211yescharacteristic, complemented
59#124×C173C2211yescharacteristic, complemented
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63C23×D173—21no—
64#14583827—22no—
65#14583827—41no—
66C2×D692—21no—
67C2×D692D411yescharacteristic
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—22no—
77D4×D173—22no—
78D4×D173—22no—
79D4×D173—41no—
80D4×D173—41no—
81D4×D173—41no—
82D4×D173—44no—
83C2×C4×D173—21no—
84C2×C4×D173—21no—
85C2×C4×D173—21no—
86C2×C4×D173C2311yescharacteristic
87C2×C4×D173C2311yescharacteristic
88#14583813—22no—
89D1384D414yescomplemented
90#14583811—22no—
91#14583811—22no—
92#14583811—22no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C8×D173—22no—
97C4⋊Dic173D411yescharacteristic
98C4⋊Dic173C2311yescharacteristic
99C4⋊Dic173C2311yescharacteristic
100#14583805—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803D411yescharacteristic
108#14583803D411yescharacteristic
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173—21no—
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115SD16×C173—22no—
116SD16×C173—41no—
117D8×C173—22no—
118D8×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—42no—
131C22×D173—42no—
132C22×D173—81no—
133C22×D173—81no—
134C22×D173—81no—
135C22×D173C2×D411yescharacteristic
136C22×D173C2×D411yescharacteristic
137D692—22no—
138D692—21no—
139D692—21no—
140D692—42no—
141D692C2×D412yes—
142D692C2×D412yescomplemented
143C22⋊D173—42no—
144C22⋊D173—42no—
145C22⋊D173—42no—
146C22⋊D173—42no—
147C22⋊D173—42no—
148C22⋊D173—81no—
149C2×Dic173—21no—
150C2×Dic173—21no—
151C2×Dic173C2×D411yescharacteristic
152C2×Dic173C2×D411yescharacteristic
153C4×D173—21no—
154C4×D173—21no—
155C4×D173—21no—
156C4×D173—21no—
157C4×D173—42no—
158C4×D173—42no—
159C4×D173—81no—
160#13063436—21no—
161#13063436—21no—
162C23×C173—21no—
163C23×C173—21no—
164C23×C173—21no—
165Q8×C173—21no—
166Q8×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—21no—
171D4×C173—21no—
172D4×C173—21no—
173D4×C173—42no—
174D4×C173—81no—
175C2×C4×C173—21no—
176C2×C4×C173C2×D411yescharacteristic
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173C2×D412yescomplemented
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173C22×D411yescharacteristic
191C2×D173—21no—
192C2×D173—21no—
193C2×D173—21no—
194C2×D173—22no—
195C2×D173—22no—
196C2×D173—21no—
197C2×D173—21no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—84no—
203C2×D173—84no—
204C2×D173—84no—
205C2×D173AGL(1,ℤ/8)12yescomplemented
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—41no—
209C4×C173C22×D411yescharacteristic
210C4×C173H(5,𝔽2)11yescharacteristic
211Dic173—21no—
212Dic173—21no—
213Dic173—41no—
214Dic173—41no—
215C2×C173—81no—
216C2×C173—81no—
217C2×C173—81no—
218C2×C173D4211yescharacteristic
219C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C173#57611yescharacteristic
221D173—22no—
222D173—41no—
223D173—42no—
224D173—42no—
225C173#326411yescharacteristic, complemented, Sylow
226#3264—1731nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1731no—
228C22×D8—1731no—
229#546—1731no—
230D42—1731no—
231#496D17311yescharacteristic, complemented
232#469—1731no—
233#465—1731no—
234#463—1731no—
235#459—1731no—
236#450—1731no—
237#449—1731no—
238#447—1731no—
239#447—1731no—
240#442—1731no—
241#435—1731no—
242C2×#41—1731no—
243C22×D4—3461no—
244C22×D4—1731no—
245C22×D4—1731no—
246C22×D4—1731no—
247AGL(1,ℤ/8)—3464no—
248C2×SD16—1731no—
249C2×SD16C2×D17311yescharacteristic, complemented
250C2×SD16C2×D17311yescharacteristic, complemented
251C2×D8—3464no—
252C2×D8—1731no—
253C2×D8—1731no—
254C2×D8—1731no—
255C2×D8—1731no—
256C2×D8C2×D17311yescharacteristic, complemented
257C2×D8C2×D17311yescharacteristic, complemented
258C2×M16—1731no—
259C22×C8—1731no—
260#127C2×D17311yescharacteristic, complemented
261#124C2×D17311yescharacteristic, complemented
262#123—3461no—
263C22⋊Q8—1731no—
264#121—3461no—
265#121—3461no—
266#121—1731no—
267#121—1731no—
268#121—1731no—
269#120—3461no—
270#120—3461no—
271#120—3461no—
272C4×D4—1731no—
273C4×D4—1731no—
274C4×D4—1731no—
275C2×C22⋊C4—3461no—
276#106—1731no—
277C4⋊C8—1731no—
278#103—1731no—
279#102—1731no—
280#102—1731no—
281#102—1731no—
282#102—1731no—
283#102—1731no—
284C22⋊C8—1731no—
285C22⋊C8—1731no—
286#97C2×D17311yescharacteristic, complemented
287C24—3461no—
288C24—3461no—
289C24—3461no—
290#41—3462no—
291#41—6921no—
292C2×Q8C22×D17311yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3461no—
296C2×D4—3462no—
297C2×D4—3462no—
298C2×D4—3462no—
299C2×D4—3461no—
300C2×D4—3461no—
301C2×D4—3461no—
302C2×D4—3461no—
303C2×D4—3461no—
304C2×D4—3461no—
305C2×D4—1731no—
306C2×D4—1731no—
307C2×D4—1731no—
308C2×D4—6921no—
309C2×D4—6921no—
310C2×D4—6921no—
311C2×D4—6924no—
312C2×D4C22×D17311yescharacteristic
313C2×D4C22×D17311yescharacteristic
314C2×D4C22×D17311yescharacteristic
315C22×C4—3461no—
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4—1731no—
319C22×C4—1731no—
320SD16—22no—
321SD16—41no—
322SD16—3462no—
323D8—22no—
324D8—41no—
325D8—3462no—
326D8—3462no—
327D8—3462no—
328D8—1734no—
329M16—3462no—
330C2×C8—3462no—
331C2×C8—1731no—
332C2×C8—1731no—
333C2×C8C22×D17311yescharacteristic
334C2×C8C22×D17311yescharacteristic
335C4⋊C4—3461no—
336C4⋊C4—1731no—
337C4⋊C4—1731no—
338C4⋊C4—1731no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3461no—
342C22⋊C4—3461no—
343C22⋊C4—3461no—
344C22⋊C4—3461no—
345C22⋊C4—3461no—
346C22⋊C4—3461no—
347C22⋊C4—1731no—
348C22⋊C4—1731no—
349C42C22×D17311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—3461no—
354C23—3461no—
355C23—13841no—
356C23—13841no—
357C23—13841no—
358C23—1731no—
359C23—1731no—
360C23—6922no—
361C23—6922no—
362C23—6922no—
363C23—6922no—
364C23—6922no—
365C23—6922no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—3462no—
377D4—3461no—
378D4—3461no—
379D4—13841no—
380D4—1732no—
381D4—1732no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4—6922no—
387D4—6922no—
388C2×C4—21no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—13841no—
396C2×C4—1731no—
397C2×C4—1731no—
398C2×C4—6922no—
399C2×C4—6922no—
400C2×C4D4×D17311yescharacteristic
401C2×C4D4×D17311yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—21no—
404C8—3461no—
405C8—3461no—
406C8D4×D17312yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3461no—
417C22—3461no—
418C22—3461no—
419C22—3461no—
420C22—3461no—
421C22—3462no—
422C22—3462no—
423C22—13844no—
424C22—13844no—
425C22—13844no—
426C22—1732no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22—6921no—
431C22C2×D4×D17311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3461no—
436C4—3461no—
437C4—6921no—
438C4—6921no—
439C4C2×D4×D17311yescharacteristic
440C4#1914972411yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—3462no—
445C2—6921no—
446C2—6922no—
447C2—6922no—
448C2#2601212611yescharacteristic
449C2AGL(1,ℤ/8)×D17311yescharacteristic
450C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×D8×D173C2semidirect
3. #26012134C2semidirect
4. #26012126C2semidirect
5. #26011741C2semidirect
6. #26011717C2semidirect
7. #26011703C2semidirect
8. #26011683C2semidirect
9. #26011640C2semidirect
10. #26011635C2semidirect
11. #26011623C2semidirect
12. #26011623C2semidirect
13. #26011604C2semidirect
14. #26011564C2semidirect
15. #496×C173C2semidirect
16. C2×#14583827C22semidirect
19. C2×D4×D173C22semidirect
20. C2×D4×D173C22semidirect
22. C2×#14583813C22semidirect
25. C2×#14583811C22semidirect
26. C2×#14583811C22semidirect
28. C2×#14583809C22semidirect
29. C2×C8×D173C22semidirect
31. #19149655C22semidirect
35. #19149650C22semidirect
36. #19149650C22semidirect
41. C4×D692C22semidirect
42. C4×D692C22semidirect
44. #19149611C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149603C22semidirect
47. #19149601C22semidirect
48. #19149601C22semidirect
49. #19149600C22semidirect
50. #19149600C22semidirect
51. #19149600C22semidirect
52. #19149591C22semidirect
53. #19149591C22semidirect
54. C2×SD16×C173C22semidirect
55. C2×SD16×C173C22semidirect
56. C2×D8×C173C22semidirect
57. C2×D8×C173C22semidirect
58. #127×C173C22semidirect
59. #124×C173C22semidirect
60. #97×C173C22semidirect
89. D1384D4semidirect
142. D692C2×D4semidirect
180. C8×C173C2×D4semidirect
205. C2×D173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
231. #496D173semidirect
249. C2×SD16C2×D173semidirect
250. C2×SD16C2×D173semidirect
256. C2×D8C2×D173semidirect
257. C2×D8C2×D173semidirect
260. #127C2×D173semidirect
261. #124C2×D173semidirect
286. #97C2×D173semidirect
406. C8D4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C2×D8×D1732
#260121342
#260121262
#260117412
#260117172
#260117032
#260116832
#260116402
#260116352
#260116232
#260116042
#260115642
#496×C1732
#3264173