Finite Group Databasebeta

← #36365049

Subgroups — #36365049

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
4D4×D692C211yesmaximal, characteristic, complemented
5#26011881C211yesmaximal, characteristic, complemented
6#26011743C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011687C211yesmaximal, characteristic, complemented
9#26011640C211yesmaximal, characteristic, complemented
10#26011639C211yesmaximal, characteristic, complemented
11#26011626C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011601C211yesmaximal, characteristic, complemented
14#26011561C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C22×D692—21no—
18C2×D4×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic, complemented
20C2×D4×D173C2211yescharacteristic, complemented
21#19149706—24no—
22C2×#14583813C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic, complemented
24C2×D1384C2211yescharacteristic, complemented
25C2×D1384C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic, complemented
29D8×D173—24no—
30C2×#14583809C2211yescharacteristic, complemented
31C2×C8×D173C2211yescharacteristic, complemented
32#19149676C2211yescharacteristic, complemented
33#19149661—21no—
34#19149655C2211yescharacteristic, complemented
35#19149654C2211yescharacteristic
36#19149654C2211yescharacteristic, complemented
37#19149653—21no—
38#19149652—21no—
39#19149650C2211yescharacteristic, complemented
40#19149649—21no—
41#19149649—21no—
42#19149649—21no—
43C4×D692C2211yescharacteristic
44C4×D692C2211yescharacteristic, complemented
45C22⋊C4×D173—21no—
46#19149610C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149590C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63C23×D173—21no—
64#14583827—22no—
65#14583827—41no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692—21no—
71C2×D692—44no—
72C2×D692D411yescharacteristic
73C2×D692C2311yescharacteristic
74C2×D692C2311yescharacteristic
75C2×D692C2311yescharacteristic
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80D4×D173—22no—
81D4×D173—22no—
82D4×D173—22no—
83D4×D173—41no—
84D4×D173—41no—
85D4×D173—41no—
86C2×C4×D173—21no—
87C2×C4×D173—21no—
88C2×C4×D173C2311yescharacteristic
89C2×C4×D173C2311yescharacteristic
90#14583813—22no—
91#14583813—41no—
92D1384—22no—
93D1384—22no—
94#14583811—22no—
95#14583811—41no—
96#14583811D414yescomplemented
97#14583809—22no—
98C2×#13063436C2311yescharacteristic
99C2×#13063436C2311yescharacteristic
100C8×D173—22no—
101C4⋊Dic173D411yescharacteristic
102C4⋊Dic173C2311yescharacteristic
103#14583805—21no—
104C22⋊Dic173—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803D411yescharacteristic
110#14583803D411yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115SD16×C173—22no—
116D8×C173—22no—
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C4⋊C4×C173C2311yescharacteristic
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—21no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—42no—
131C22×D173—42no—
132C22×D173—42no—
133C22×D173—81no—
134C22×D173—81no—
135C22×D173—81no—
136C22×D173C2×D411yescharacteristic
137C22×D173C2×D411yescharacteristic
138D692—22no—
139D692—21no—
140D692—21no—
141D692—21no—
142D692—21no—
143D692—21no—
144D692—42no—
145D692—42no—
146D692—42no—
147D692—42no—
148D692—81no—
149D692C2×D412yescomplemented
150C22⋊D173—42no—
151C22⋊D173—42no—
152C22⋊D173—42no—
153C22⋊D173—81no—
154C2×Dic173—21no—
155C2×Dic173C2×D411yescharacteristic
156C2×Dic173C2×D411yescharacteristic
157C4×D173—21no—
158C4×D173—21no—
159C4×D173—21no—
160C4×D173—21no—
161C4×D173—42no—
162C4×D173—81no—
163#13063436—21no—
164#13063436C2×D412yescomplemented
165C23×C173—21no—
166C23×C173—21no—
167Q8×C173—21no—
168Q8×C173—21no—
169D4×C173—21no—
170D4×C173—21no—
171D4×C173—21no—
172D4×C173C2×D412yes—
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—42no—
176C2×C4×C173C2×D411yescharacteristic
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173—21no—
181C22×C173—22no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173C22×D411yescharacteristic
188C2×D173—21no—
189C2×D173—21no—
190C2×D173—21no—
191C2×D173—22no—
192C2×D173—21no—
193C2×D173—21no—
194C2×D173—41no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—41no—
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×C173—41no—
210C4×C173C22×D411yescharacteristic
211C4×C173H(5,𝔽2)11yescharacteristic
212Dic173—21no—
213Dic173—21no—
214Dic173—41no—
215C2×C173—42no—
216C2×C173—81no—
217C2×C173D4211yescharacteristic
218C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C173#57611yescharacteristic
220D173—22no—
221D173—41no—
222D173—42no—
223D173—81no—
224D173—81no—
225C173#326411yescharacteristic, complemented, Sylow
226#3264—1731nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1731no—
228C22×D8—1731no—
229#546—1731no—
230D42—1731no—
231#496—1731no—
232#469—1731no—
233#465—1731no—
234#463D17311yescharacteristic, complemented
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×SD16—1731no—
250C2×SD16C2×D17311yescharacteristic, complemented
251C2×D8—3464no—
252C2×D8—1731no—
253C2×D8—1731no—
254C2×D8—1731no—
255C2×D8—1731no—
256C2×D8—1731no—
257C2×D8C2×D17311yescharacteristic
258C2×M16—1731no—
259C22×C8—1731no—
260#127—1731no—
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×D4C2×D17311yescharacteristic, complemented
275C2×C22⋊C4—3461no—
276#106—1731no—
277C4⋊C8C2×D17311yescharacteristic, complemented
278#103C2×D17311yescharacteristic, complemented
279#102—1731no—
280#102—1731no—
281#102—1731no—
282#102—1731no—
283#102C2×D17311yescharacteristic, complemented
284C22⋊C8—1731no—
285C22⋊C8—1731no—
286#97—1731no—
287C24—3461no—
288C24—3461no—
289C24—3461no—
290#41—3462no—
291#41—6921no—
292C2×Q8C22×D17311yescharacteristic
293C2×D4—3461no—
294C2×D4—3461no—
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—1731no—
309C2×D4—6924no—
310C2×D4—6921no—
311C2×D4—6921no—
312C2×D4—6921no—
313C2×D4C22×D17311yescharacteristic
314C2×D4C22×D17311yescharacteristic
315C22×C4—21no—
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4—1731no—
319C22×C4—1731no—
320SD16—22no—
321SD16—3462no—
322SD16—6921no—
323D8—22no—
324D8—3462no—
325D8—3462no—
326D8—3462no—
327D8—1734no—
328D8—6921no—
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⋊C4C22×D17311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
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—3461no—
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—3462no—
372D4—3461no—
373D4—3461no—
374D4—3461no—
375D4—3461no—
376D4—3461no—
377D4—13841no—
378D4—13841no—
379D4—1732no—
380D4—6922no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4—6922no—
387D4D4×D17312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—13841no—
397C2×C4—1731no—
398C2×C4—1731no—
399C2×C4—6922no—
400C2×C4D4×D17311yescharacteristic
401C2×C4D4×D17311yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—21no—
404C8—21no—
405C8—3461no—
406C8—1732no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3461no—
414C22—3461no—
415C22—3461no—
416C22—3461no—
417C22—3461no—
418C22—3462no—
419C22—13844no—
420C22—13844no—
421C22—13844no—
422C22—1732no—
423C22—6921no—
424C22—6921no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22—6921no—
431C22C2×D4×D17311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—3461no—
437C4—3461no—
438C4—6921no—
439C4C2×D4×D17311yescharacteristic
440C4#1914972511yescharacteristic
441C2—42no—
442C2—81no—
443C2—3462no—
444C2—13841no—
445C2—13841no—
446C2—6921no—
447C2—6922no—
448C2#2601212511yescharacteristic
449C2AGL(1,ℤ/8)×D17311yescharacteristic
450C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
hypercenter#463D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. C2×D8×D173C2semidirect
3. #26012134C2semidirect
4. D4×D692C2semidirect
5. #26011881C2semidirect
6. #26011743C2semidirect
7. #26011723C2semidirect
8. #26011687C2semidirect
9. #26011640C2semidirect
10. #26011639C2semidirect
11. #26011626C2semidirect
12. #26011623C2semidirect
13. #26011601C2semidirect
14. #26011561C2semidirect
15. #463×C173C2semidirect
16. C2×#14583827C22semidirect
19. C2×D4×D173C22semidirect
20. C2×D4×D173C22semidirect
22. C2×#14583813C22semidirect
23. C2×#14583813C22semidirect
24. C2×D1384C22semidirect
25. C2×D1384C22semidirect
27. C2×#14583811C22semidirect
28. C2×#14583811C22semidirect
30. C2×#14583809C22semidirect
31. C2×C8×D173C22semidirect
32. #19149676C22semidirect
34. #19149655C22semidirect
36. #19149654C22semidirect
39. #19149650C22semidirect
44. C4×D692C22semidirect
46. #19149610C22semidirect
47. #19149601C22semidirect
48. #19149601C22semidirect
49. #19149600C22semidirect
50. #19149599C22semidirect
51. #19149591C22semidirect
52. #19149591C22semidirect
53. #19149590C22semidirect
54. C2×SD16×C173C22semidirect
56. #124×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
96. #14583811D4semidirect
149. D692C2×D4semidirect
164. #13063436C2×D4semidirect
205. C2×D173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
234. #463D173semidirect
250. C2×SD16C2×D173semidirect
261. #124C2×D173semidirect
274. C4×D4C2×D173semidirect
277. C4⋊C8C2×D173semidirect
278. #103C2×D173semidirect
283. #102C2×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
D4×D6922
#260118812
#260117432
#260117232
#260116872
#260116402
#260116392
#260116262
#260116232
#260116012
#260115612
#463×C1732
#3264173