Finite Group Databasebeta

← #36365037

Subgroups — #36365037

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×D8×D173C211yesmaximal, characteristic, complemented
2#26012129C211yesmaximal, characteristic, complemented
3D4×C22⋊D173C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011744C211yesmaximal, characteristic, complemented
6#26011717C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011640C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011603C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173—21no—
17C2×D4×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic, complemented
19C2×#14583813C2211yescharacteristic, complemented
20C2×D1384C2211yescharacteristic, complemented
21C2×#14583811C2211yescharacteristic
22C2×#14583811C2211yescharacteristic, complemented
23C2×#14583811C2211yescharacteristic, complemented
24D8×D173—24no—
25C2×C8×D173C2211yescharacteristic, complemented
26#19149675C2211yescharacteristic, complemented
27#19149668C2211yescharacteristic, complemented
28#19149662—21no—
29#19149655C2211yescharacteristic, complemented
30#19149653C2211yescharacteristic, complemented
31#19149653C2211yescharacteristic, complemented
32#19149651—21no—
33#19149651—21no—
34#19149650C2211yescharacteristic
35#19149648—21no—
36#19149647—21no—
37#19149647—21no—
38D4×Dic173C2211yescharacteristic, complemented
39D4×Dic173C2211yescharacteristic, complemented
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583803—21no—
42#19149611C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50C22⋊#13063436C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52#19149589C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C173—24no—
56C2×SD16×C173C2211yescharacteristic, complemented
57C2×SD16×C173C2211yescharacteristic, complemented
58C2×D8×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic, complemented
60C2×M16×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—44no—
73C2×C22⋊D173D411yescharacteristic
74D4×D173—22no—
75D4×D173—22no—
76D4×D173—41no—
77D4×D173—41no—
78C22×Dic173—21no—
79C22×Dic173—21no—
80C2×C4×D173C2311yescharacteristic
81#14583813—41no—
82D1384—22no—
83#14583811—22no—
84#14583811—22no—
85#14583811—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C8×D173—22no—
89C4⋊Dic173C2311yescharacteristic
90C4⋊Dic173C2311yescharacteristic
91#14583805—21no—
92#14583805D411yescharacteristic
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173D411yescharacteristic
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803D411yescharacteristic
103C4×Dic173C2311yescharacteristic
104C24×C173—21no—
105#41×C173—22no—
106#41×C173—41no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—22no—
109C2×D4×C173—21no—
110C2×D4×C173—21no—
111C2×D4×C173—41no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118SD16×C173—22no—
119D8×C173—22no—
120D8×C173D414yescomplemented
121M16×C173—22no—
122C2×C8×C173C2311yescharacteristic
123C2×C8×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—81no—
131C22×D173—81no—
132C22×D173C2×D411yescharacteristic
133D692—21no—
134D692—21no—
135D692—21no—
136D692—21no—
137C22⋊D173—22no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C22⋊D173—42no—
143C22⋊D173—81no—
144C22⋊D173—81no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—42no—
148C2×Dic173—81no—
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C2×Dic173C2×D411yescharacteristic
152C4×D173—21no—
153C4×D173—21no—
154#13063436—21no—
155#13063436—21no—
156C23×C173—21no—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—42no—
160C23×C173—42no—
161C23×C173—81no—
162C23×C173C2×D411yescharacteristic
163Q8×C173—21no—
164Q8×C173—21no—
165D4×C173—21no—
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—42no—
170D4×C173—42no—
171D4×C173C2×D412yes—
172D4×C173C2×D412yescomplemented
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173—42no—
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173C2×D412yescomplemented
181C22×C173—21no—
182C22×C173—22no—
183C22×C173—22no—
184C22×C173—21no—
185C22×C173—21no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—41no—
191C22×C173—41no—
192C22×C173—41no—
193C22×C173—84no—
194C22×C173C22×D411yescharacteristic
195C2×D173—21no—
196C2×D173—21no—
197C2×D173—41no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—84no—
204C2×D173—84no—
205C2×D173AGL(1,ℤ/8)12yescomplemented
206C4×C173—21no—
207C4×C173—41no—
208C4×C173C22×D411yescharacteristic
209C4×C173H(5,𝔽2)11yescharacteristic
210Dic173—21no—
211Dic173—21no—
212Dic173—21no—
213Dic173—41no—
214Dic173—41no—
215C2×C173—42no—
216C2×C173—42no—
217C2×C173—41no—
218C2×C173—81no—
219C2×C173D4211yescharacteristic
220C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C173#57611yescharacteristic
222D173—22no—
223D173—81no—
224D173—81no—
225C173#326411yescharacteristic, complemented, Sylow
226#3264—1731nomaximal, Sylow
227C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
228C22×D8—1731no—
229#546—1731no—
230D42—1731no—
231#496—1731no—
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×#41C2×D17311yescharacteristic, complemented
243C22×D4—3461no—
244C22×D4—1731no—
245C22×D4—1731no—
246C22×D4C2×D17311yescharacteristic, complemented
247AGL(1,ℤ/8)—24no—
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
257C2×D8C2×D17311yescharacteristic, complemented
258C2×M16C2×D17311yescharacteristic, complemented
259C22×C8—1731no—
260#127—1731no—
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×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#97—1731no—
287C24—21no—
288C24—3461no—
289C24—3461no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D17311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
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—3461no—
306C2×D4—1731no—
307C2×D4—1731no—
308C2×D4—1731no—
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×C4C22×D17311yescharacteristic
320SD16—22no—
321SD16—22no—
322SD16—6921no—
323D8—22no—
324D8—3462no—
325D8—3462no—
326D8—3462no—
327D8—6921no—
328D8C22⋊D17314yescomplemented
329M16—22no—
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—3461no—
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—
349C42—1731no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—3461no—
357C23—3461no—
358C23—13841no—
359C23—13841no—
360C23—1731no—
361C23—6922no—
362C23—6922no—
363C23—6922no—
364C23—6922no—
365C23D4×D17311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—3462no—
375D4—3461no—
376D4—3461no—
377D4—3461no—
378D4—3461no—
379D4—13841no—
380D4—13841no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4C2×C22⋊D17312yes—
387D4C2×C22⋊D17312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—13841no—
397C2×C4—1731no—
398C2×C4—1731no—
399C2×C4—1731no—
400C2×C4—6922no—
401C2×C4D4×D17311yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—21no—
404C8—3461no—
405C8—3461no—
406C8C2×C22⋊D17312yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—84no—
420C22—3461no—
421C22—3461no—
422C22—13844no—
423C22—13844no—
424C22—1732no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22—6921no—
431C22C2×D4×D17311yescharacteristic
432C4—21no—
433C4—41no—
434C4—3461no—
435C4—3461no—
436C4—3461no—
437C4—6921no—
438C4—6921no—
439C4C22×C22⋊D17311yescharacteristic
440C4#1914972411yescharacteristic
441C2—41no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—3462no—
446C2—13841no—
447C2—13841no—
448C2D4×C22⋊D17311yescharacteristic
449C2AGL(1,ℤ/8)×D17311yescharacteristic
450C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
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×D8×D173C2semidirect
2. #26012129C2semidirect
3. D4×C22⋊D173C2semidirect
4. #26011877C2semidirect
5. #26011744C2semidirect
6. #26011717C2semidirect
7. #26011708C2semidirect
8. #26011684C2semidirect
9. #26011640C2semidirect
10. #26011636C2semidirect
11. #26011625C2semidirect
12. #26011623C2semidirect
13. #26011603C2semidirect
14. #26011560C2semidirect
15. C2×AGL(1,ℤ/8)×C173C2semidirect
18. C2×D4×D173C22semidirect
19. C2×#14583813C22semidirect
20. C2×D1384C22semidirect
22. C2×#14583811C22semidirect
23. C2×#14583811C22semidirect
25. C2×C8×D173C22semidirect
26. #19149675C22semidirect
27. #19149668C22semidirect
29. #19149655C22semidirect
30. #19149653C22semidirect
31. #19149653C22semidirect
38. D4×Dic173C22semidirect
39. D4×Dic173C22semidirect
42. #19149611C22semidirect
43. #19149608C22semidirect
44. #19149603C22semidirect
45. #19149601C22semidirect
46. #19149601C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
49. #19149600C22semidirect
50. C22⋊#13063436C22semidirect
51. #19149591C22semidirect
52. #19149589C22semidirect
53. C2×#41×C173C22semidirect
54. C22×D4×C173C22semidirect
56. C2×SD16×C173C22semidirect
57. C2×SD16×C173C22semidirect
59. C2×D8×C173C22semidirect
60. C2×M16×C173C22semidirect
120. D8×C173D4semidirect
172. D4×C173C2×D4semidirect
180. C8×C173C2×D4semidirect
205. C2×D173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
227. C2×AGL(1,ℤ/8)D173semidirect
242. C2×#41C2×D173semidirect
246. C22×D4C2×D173semidirect
249. C2×SD16C2×D173semidirect
250. C2×SD16C2×D173semidirect
257. C2×D8C2×D173semidirect
258. C2×M16C2×D173semidirect
328. D8C22⋊D173semidirect
387. D4C2×C22⋊D173semidirect
406. C8C2×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1732
#260121292
D4×C22⋊D1732
#260118772
#260117442
#260117172
#260117082
#260116842
#260116402
#260116362
#260116252
#260116232
#260116032
#260115602
C2×AGL(1,ℤ/8)×C1732
#3264173