Finite Group Databasebeta

← #36365038

Subgroups — #36365038

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×#19149703C211yesmaximal, characteristic, complemented
2C2×D8×D173C211yesmaximal, characteristic, complemented
3#26012136C211yesmaximal, characteristic, complemented
4#26012126C211yesmaximal, characteristic, complemented
5#26011878C211yesmaximal, characteristic, complemented
6#26011743C211yesmaximal, characteristic, complemented
7#26011719C211yesmaximal, characteristic, complemented
8#26011707C211yesmaximal, characteristic, complemented
9#26011643C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011626C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011601C211yesmaximal, characteristic, complemented
14#26011561C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic, complemented
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic, complemented
20C2×D4×D173C2211yescharacteristic, complemented
21#19149703—24no—
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583814C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic, complemented
25C2×D1384C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic, complemented
29D8×D173—24no—
30C2×#14583809C2211yescharacteristic, complemented
31C2×C8×D173C2211yescharacteristic, complemented
32#19149667C2211yescharacteristic, complemented
33#19149663—21no—
34#19149659C2211yescharacteristic, complemented
35#19149654C2211yescharacteristic, complemented
36#19149653—21no—
37#19149650—21no—
38#19149650C2211yescharacteristic
39#19149650C2211yescharacteristic, complemented
40#19149649—21no—
41#19149647—21no—
42#19149647—21no—
43C4×D692C2211yescharacteristic
44C4×D692C2211yescharacteristic, complemented
45C2×#14583803—21no—
46#19149610C2211yescharacteristic, complemented
47#19149604C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149590C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic, complemented
55C2×D8×C173C2211yescharacteristic, complemented
56#127×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic, complemented
58C4⋊C8×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63C23×D173—21no—
64#14583825—22no—
65#14583825—41no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692D411yescharacteristic
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77D4×D173—22no—
78D4×D173—22no—
79D4×D173—22no—
80D4×D173—41no—
81D4×D173—41no—
82D4×D173—41no—
83D4×D173—44no—
84C22×Dic173—21no—
85C2×C4×D173—21no—
86C2×C4×D173C2311yescharacteristic
87C2×C4×D173C2311yescharacteristic
88#14583815—22no—
89#14583814—22no—
90#14583814—41no—
91D1384—22no—
92#14583811—22no—
93#14583811—41no—
94#14583811D414yescomplemented
95#14583809—22no—
96C2×#13063436C2311yescharacteristic
97C2×#13063436C2311yescharacteristic
98C8×D173—22no—
99C4⋊Dic173—21no—
100C4⋊Dic173D411yescharacteristic
101C4⋊Dic173C2311yescharacteristic
102C22⋊Dic173—21no—
103C22⋊Dic173—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803D411yescharacteristic
110#14583803D411yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173—21no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C2×D4×C173C2311yescharacteristic
116C22×C4×C173—21no—
117D8×C173—22no—
118D8×C173—22no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C4⋊C4×C173C2311yescharacteristic
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
137Dic346—21no—
138Dic346—21no—
139D692—22no—
140D692—21no—
141D692—21no—
142D692—21no—
143D692—81no—
144D692C2×D412yes—
145C22⋊D173—42no—
146C22⋊D173—42no—
147C22⋊D173—42no—
148C22⋊D173—42no—
149C22⋊D173—42no—
150C22⋊D173—42no—
151C2×Dic173—21no—
152C2×Dic173—21no—
153C2×Dic173—42no—
154C2×Dic173C2×D411yescharacteristic
155C2×Dic173C2×D411yescharacteristic
156C4×D173—21no—
157C4×D173—21no—
158C4×D173—21no—
159C4×D173—21no—
160C4×D173—42no—
161#13063436—21no—
162#13063436C2×D412yescomplemented
163C23×C173—21no—
164C23×C173—21no—
165C23×C173—21no—
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—21no—
171D4×C173—42no—
172D4×C173—81no—
173D4×C173C2×D412yescomplemented
174C2×C4×C173—21no—
175C2×C4×C173—81no—
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×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—21no—
196C2×D173—21no—
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×C173C22×D411yescharacteristic
210C4×C173H(5,𝔽2)11yescharacteristic
211Dic173—21no—
212Dic173—21no—
213Dic173—41no—
214Dic173—41no—
215C2×C173—42no—
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—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#463—1731no—
235#459D17311yescharacteristic, complemented
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×SD16—1731no—
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#124—1731no—
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#103—1731no—
279#102—1731no—
280#102—1731no—
281#102—1731no—
282#102C2×D17311yescharacteristic, complemented
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×Q8—1731no—
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—21no—
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4—1731no—
319C22×C4—1731no—
320SD16—3462no—
321SD16—3462no—
322SD16—6921no—
323D8—22no—
324D8—22no—
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—3461no—
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—3461no—
367Q8—3461no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—3462no—
376D4—3461no—
377D4—3461no—
378D4—3461no—
379D4—13841no—
380D4—1732no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4—6922no—
387D4D4×D17312yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—1731no—
397C2×C4—1731no—
398C2×C4—6922no—
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—41no—
414C22—41no—
415C22—41no—
416C22—3461no—
417C22—3461no—
418C22—3461no—
419C22—3461no—
420C22—3461no—
421C22—3462no—
422C22—13844no—
423C22—13844no—
424C22—13844no—
425C22—1732no—
426C22—6921no—
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#1914972511yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—3462no—
445C2—13841no—
446C2—6921no—
447C2—6922no—
448C2#2601212511yescharacteristic
449C2AGL(1,ℤ/8)×D17311yescharacteristic
450C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×D8×D173C2semidirect
3. #26012136C2semidirect
4. #26012126C2semidirect
5. #26011878C2semidirect
6. #26011743C2semidirect
7. #26011719C2semidirect
8. #26011707C2semidirect
9. #26011643C2semidirect
10. #26011633C2semidirect
11. #26011626C2semidirect
12. #26011623C2semidirect
13. #26011601C2semidirect
14. #26011561C2semidirect
15. #459×C173C2semidirect
16. C2×#14583825C22semidirect
19. C2×D4×D173C22semidirect
20. C2×D4×D173C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583814C22semidirect
27. C2×#14583811C22semidirect
28. C2×#14583811C22semidirect
30. C2×#14583809C22semidirect
31. C2×C8×D173C22semidirect
32. #19149667C22semidirect
34. #19149659C22semidirect
35. #19149654C22semidirect
39. #19149650C22semidirect
44. C4×D692C22semidirect
46. #19149610C22semidirect
47. #19149604C22semidirect
48. #19149601C22semidirect
49. #19149600C22semidirect
50. #19149599C22semidirect
51. #19149591C22semidirect
52. #19149591C22semidirect
53. #19149590C22semidirect
54. C2×D8×C173C22semidirect
55. C2×D8×C173C22semidirect
56. #127×C173C22semidirect
57. C4×D4×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #102×C173C22semidirect
60. #102×C173C22semidirect
94. #14583811D4semidirect
162. #13063436C2×D4semidirect
173. D4×C173C2×D4semidirect
205. C2×D173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
235. #459D173semidirect
256. C2×D8C2×D173semidirect
257. C2×D8C2×D173semidirect
260. #127C2×D173semidirect
274. C4×D4C2×D173semidirect
277. C4⋊C8C2×D173semidirect
282. #102C2×D173semidirect
283. #102C2×D173semidirect
387. D4D4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×D8×D1732
#260121362
#260121262
#260118782
#260117432
#260117192
#260117072
#260116432
#260116332
#260116262
#260116232
#260116012
#260115612
#459×C1732
#3264173