Finite Group Databasebeta

← #36365044

Subgroups — #36365044

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×#19149704C211yesmaximal, characteristic, complemented
2C22×#14583811C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011881C211yesmaximal, characteristic, complemented
5#26011742C211yesmaximal, characteristic, complemented
6#26011722C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011686C211yesmaximal, characteristic, complemented
9#26011641C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13#26011601C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15#546×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×D692C2211yescharacteristic, complemented
18C2×D4×D173—21no—
19#19149704—24no—
20C2×#14583813C2211yescharacteristic, complemented
21C2×#14583813C2211yescharacteristic, complemented
22C2×#14583813C2211yescharacteristic, complemented
23C2×#14583811—24no—
24C2×#14583811C2211yescharacteristic
25C2×#14583811C2211yescharacteristic
26C2×#14583811C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583811C2211yescharacteristic, complemented
30C2×#14583810C2211yescharacteristic, complemented
31C22×#13063436C2211yescharacteristic, complemented
32#19149676C2211yescharacteristic, complemented
33#19149654—21no—
34#19149653C2211yescharacteristic
35#19149653C2211yescharacteristic, complemented
36#19149649—21no—
37#19149649—21no—
38C4×D692C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149600C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic, complemented
44#19149599C2211yescharacteristic, complemented
45#19149599C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic, complemented
48C22⋊#13063436C2211yescharacteristic, complemented
49#19149590C2211yescharacteristic, complemented
50C2×#41×C173C2211yescharacteristic, complemented
51C22×D4×C173C2211yescharacteristic, complemented
52#124×C173C2211yescharacteristic, complemented
53#123×C173—21no—
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173—21no—
56#121×C173C2211yescharacteristic, complemented
57#120×C173—21no—
58C4×D4×C173C2211yescharacteristic
59C4×D4×C173C2211yescharacteristic, complemented
60C2×C22⋊C4×C173—21no—
61C23×D173—21no—
62C23×D173—21no—
63C2×D692—22no—
64C2×D692—22no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—41no—
70C2×D692—41no—
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—44no—
77C2×C4×D173—21no—
78#14583813—22no—
79#14583813—22no—
80#14583813—41no—
81#14583811—22no—
82#14583811—22no—
83#14583811—22no—
84#14583811—22no—
85#14583811—41no—
86#14583811D414yescomplemented
87#14583810—22no—
88C2×#13063436—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C2×#13063436C2311yescharacteristic
93C4⋊Dic173C2311yescharacteristic
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C24×C173—21no—
98#41×C173—22no—
99#41×C173—41no—
100C2×Q8×C173C2311yescharacteristic
101C2×D4×C173—22no—
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173—21no—
105C2×D4×C173—41no—
106C2×D4×C173D411yescharacteristic
107C2×D4×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109C22×C4×C173—21no—
110C22×C4×C173—21no—
111C22×C4×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113C4⋊C4×C173—21no—
114C4⋊C4×C173D411yescharacteristic
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22⋊C4×C173D411yescharacteristic
122C22⋊C4×C173D411yescharacteristic
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—81no—
132C22×D173—81no—
133D692—21no—
134D692—21no—
135D692—21no—
136D692—21no—
137D692—21no—
138D692—42no—
139D692—42no—
140D692—42no—
141D692—81no—
142D692C2×D412yes—
143C22⋊D173—42no—
144C22⋊D173—42no—
145C2×Dic173—21no—
146C4×D173—42no—
147#13063436—21no—
148#13063436—21no—
149#13063436—21no—
150#13063436C2×D412yescomplemented
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—42no—
154C23×C173—42no—
155C23×C173—81no—
156C23×C173C2×D411yescharacteristic
157C23×C173C2×D411yescharacteristic
158Q8×C173—21no—
159Q8×C173—21no—
160D4×C173—22no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—42no—
165D4×C173—42no—
166D4×C173—81no—
167D4×C173C2×D412yescomplemented
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173—42no—
175C2×C4×C173—81no—
176C2×C4×C173C2×D411yescharacteristic
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2×D411yescharacteristic
179C2×C4×C173C2×D411yescharacteristic
180C2×C4×C173C2411yescharacteristic
181C22×C173—21no—
182C22×C173—21no—
183C22×C173—21no—
184C22×C173—22no—
185C22×C173—21no—
186C22×C173—21no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—41no—
191C22×C173—41no—
192C22×C173—84no—
193C22×C173AGL(1,ℤ/8)12yescomplemented
194C22×C173C22×D411yescharacteristic
195C2×D173—22no—
196C2×D173—41no—
197C2×D173—41no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—41no—
204C2×D173—84no—
205C2×D173—84no—
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—21no—
209C4×C173—21no—
210C4×C173—41no—
211C4×C173—41no—
212C4×C173C22×D411yescharacteristic
213C4×C173H(5,𝔽2)11yescharacteristic
214Dic173—41no—
215C2×C173—22no—
216C2×C173—41no—
217C2×C173—42no—
218C2×C173—81no—
219C2×C173D4211yescharacteristic
220C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C173#57611yescharacteristic
222D173—42no—
223D173—81no—
224D173—81no—
225C173#326411yescharacteristic, complemented, Sylow
226#3264—1731nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1731no—
228C22×D8—1731no—
229#546D17311yescharacteristic, complemented
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)—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×D8—1731no—
257C2×D8—1731no—
258C2×M16—1731no—
259C22×C8—1731no—
260#127—1731no—
261#124C2×D17311yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D17311yescharacteristic, complemented
264#121—21no—
265#121—3461no—
266#121—1731no—
267#121—1731no—
268#121C2×D17311yescharacteristic, complemented
269#120—21no—
270#120—3461no—
271#120—3461no—
272C4×D4—1731no—
273C4×D4C2×D17311yescharacteristic
274C4×D4C2×D17311yescharacteristic, complemented
275C2×C22⋊C4—21no—
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—21no—
297C2×D4—41no—
298C2×D4—3462no—
299C2×D4—3462no—
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—21no—
317C22×C4—3461no—
318C22×C4C22×D17311yescharacteristic
319C22×C4C22×D17311yescharacteristic
320SD16—3462no—
321SD16—3462no—
322SD16—6921no—
323D8—3462no—
324D8—3462no—
325D8—3462no—
326D8—3462no—
327D8—1734no—
328D8—6921no—
329M16—3462no—
330C2×C8—3462no—
331C2×C8—1731no—
332C2×C8—1731no—
333C2×C8—1731no—
334C2×C8—1731no—
335C4⋊C4—21no—
336C4⋊C4—1731no—
337C4⋊C4C22⋊D17311yescharacteristic
338C4⋊C4C22×D17311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—3461no—
345C22⋊C4—3461no—
346C22⋊C4—3461no—
347C22⋊C4C22⋊D17311yescharacteristic
348C22⋊C4C22⋊D17311yescharacteristic
349C42C22×D17311yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—3461no—
356C23—3461no—
357C23—3461no—
358C23—13841no—
359C23—13841no—
360C23—6922no—
361C23—6922no—
362C23—6922no—
363C23—6922no—
364C23C2×C22⋊D17311yescharacteristic
365C23C2×C22⋊D17311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—3461no—
376D4—3461no—
377D4—3461no—
378D4—3461no—
379D4—3461no—
380D4—13841no—
381D4—1732no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4—6922no—
387D4D4×D17312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—3461no—
397C2×C4—6922no—
398C2×C4C2×C22⋊D17311yescharacteristic
399C2×C4C2×C22⋊D17311yescharacteristic
400C2×C4C2×C22⋊D17311yescharacteristic
401C2×C4C2×C22⋊D17311yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—3461no—
404C8—3461no—
405C8—3461no—
406C8—1732no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—3462no—
420C22—13844no—
421C22—13844no—
422C22—6921no—
423C22—6921no—
424C22—6921no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22#1914970412yescomplemented
431C22C22×C22⋊D17311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—6921no—
439C4C2×D4×D17311yescharacteristic
440C4#1914972411yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—13841no—
446C2—13841no—
447C2—6922no—
448C2D4×C22⋊D17311yescharacteristic
449C2C2×#1914970411yescharacteristic
450C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#546×C173C2
socleC22×C173C22×D4
hypercenter#546D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19149704C2semidirect
2. C22×#14583811C2semidirect
3. D4×D692C2semidirect
4. #26011881C2semidirect
5. #26011742C2semidirect
6. #26011722C2semidirect
7. #26011703C2semidirect
8. #26011686C2semidirect
9. #26011641C2semidirect
10. #26011636C2semidirect
11. #26011625C2semidirect
12. #26011624C2semidirect
13. #26011601C2semidirect
14. #26011563C2semidirect
15. #546×C173C2semidirect
17. C22×D692C22semidirect
20. C2×#14583813C22semidirect
21. C2×#14583813C22semidirect
22. C2×#14583813C22semidirect
26. C2×#14583811C22semidirect
27. C2×#14583811C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583811C22semidirect
30. C2×#14583810C22semidirect
31. C22×#13063436C22semidirect
32. #19149676C22semidirect
35. #19149653C22semidirect
38. C4×D692C22semidirect
39. #19149610C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149603C22semidirect
42. #19149600C22semidirect
43. #19149600C22semidirect
44. #19149599C22semidirect
45. #19149599C22semidirect
46. #19149599C22semidirect
47. C22⋊#13063436C22semidirect
48. C22⋊#13063436C22semidirect
49. #19149590C22semidirect
50. C2×#41×C173C22semidirect
51. C22×D4×C173C22semidirect
52. #124×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #121×C173C22semidirect
59. C4×D4×C173C22semidirect
86. #14583811D4semidirect
150. #13063436C2×D4semidirect
167. D4×C173C2×D4semidirect
193. C22×C173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
229. #546D173semidirect
242. C2×#41C2×D173semidirect
246. C22×D4C2×D173semidirect
261. #124C2×D173semidirect
263. C22⋊Q8C2×D173semidirect
268. #121C2×D173semidirect
274. C4×D4C2×D173semidirect
387. D4D4×D173semidirect
430. C22#19149704semidirect

Maximal subgroups

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

subgroupindex
C2×#191497042
C22×#145838112
D4×D6922
#260118812
#260117422
#260117222
#260117032
#260116862
#260116412
#260116362
#260116252
#260116242
#260116012
#260115632
#546×C1732
#3264173