Finite Group Databasebeta

← #36365045

Subgroups — #36365045

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
4D4×D692C211yesmaximal, characteristic, complemented
5#26011880C211yesmaximal, characteristic, complemented
6#26011745C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011706C211yesmaximal, characteristic, complemented
9#26011686C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011637C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011626C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#442×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic, complemented
17C22×D692—21no—
18C2×D4×D173C2211yescharacteristic
19C2×D4×D173C2211yescharacteristic, complemented
20C2×D4×D173C2211yescharacteristic, complemented
21#19149703—24no—
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic, complemented
25C2×D1384C2211yescharacteristic, complemented
26C2×D1384C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583811C2211yescharacteristic, complemented
30D8×D173—24no—
31C2×#14583809C2211yescharacteristic, complemented
32C2×C8×D173C2211yescharacteristic, complemented
33#19149676C2211yescharacteristic, complemented
34#19149667C2211yescharacteristic, complemented
35#19149663—21no—
36#19149659C2211yescharacteristic, complemented
37#19149654C2211yescharacteristic
38#19149654C2211yescharacteristic, complemented
39#19149654C2211yescharacteristic, complemented
40#19149653—21no—
41#19149653—21no—
42#19149649—21no—
43#19149649—21no—
44#19149649—21no—
45C4×D692C2211yescharacteristic
46C2×#14583803—21no—
47C4⋊#13063436C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic, complemented
50#19149599C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149591C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55C4×D4×C173C2211yescharacteristic, complemented
56C4×D4×C173C2211yescharacteristic, complemented
57#106×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic, complemented
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63C23×D173—21no—
64#14583825—22no—
65#14583825—41no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—21no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692—44no—
73C2×D692D411yescharacteristic
74C2×D692C2311yescharacteristic
75C2×D692C2311yescharacteristic
76C2×D692C2311yescharacteristic
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173—21no—
81D4×D173—22no—
82D4×D173—22no—
83D4×D173—22no—
84D4×D173—41no—
85D4×D173—41no—
86D4×D173—41no—
87C22×Dic173—21no—
88C2×C4×D173C2311yescharacteristic
89C2×C4×D173C2311yescharacteristic
90#14583815—22no—
91#14583815—41no—
92#14583814—22no—
93D1384—22no—
94D1384—41no—
95#14583811—22no—
96#14583811—22no—
97#14583811—22no—
98#14583809—22no—
99C2×#13063436C2311yescharacteristic
100C2×#13063436C2311yescharacteristic
101C8×D173—22no—
102C4⋊Dic173—21no—
103C4⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803—21no—
110#14583803D411yescharacteristic
111#14583803D411yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173—21no—
116D8×C173D414yescomplemented
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C4⋊C4×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
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
138Dic346—21no—
139Dic346—21no—
140D692—22no—
141D692—21no—
142D692—21no—
143D692—21no—
144D692—21no—
145D692—21no—
146D692—21no—
147D692—42no—
148D692—42no—
149D692—42no—
150D692—81no—
151D692—81no—
152C22⋊D173—42no—
153C22⋊D173—42no—
154C22⋊D173—42no—
155C22⋊D173—42no—
156C2×Dic173—21no—
157C2×Dic173—42no—
158C2×Dic173C2×D411yescharacteristic
159C2×Dic173C2×D411yescharacteristic
160C4×D173—21no—
161C4×D173—21no—
162C4×D173—21no—
163C4×D173—21no—
164#13063436—21no—
165#13063436—21no—
166C23×C173—21no—
167C23×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173C2×D412yes—
171D4×C173C2×D412yescomplemented
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×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173C2×D412yescomplemented
181C22×C173—22no—
182C22×C173—22no—
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—21no—
192C2×D173—21no—
193C2×D173—41no—
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—42no—
217C2×C173D4211yescharacteristic
218C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C173#57611yescharacteristic
220D173—22no—
221D173—41no—
222D173—81no—
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#463—1731no—
235#459—1731no—
236#450—1731no—
237#449—1731no—
238#447—1731no—
239#447—1731no—
240#442D17311yescharacteristic, complemented
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×D8—1731no—
257C2×D8C2×D17311yescharacteristic
258C2×M16—1731no—
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×D4C2×D17311yescharacteristic, complemented
274C4×D4C2×D17311yescharacteristic, complemented
275C2×C22⋊C4—3461no—
276#106C2×D17311yescharacteristic, complemented
277C4⋊C8—1731no—
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#97C2×D17311yescharacteristic, complemented
287C24—3461no—
288C24—3461no—
289C24—3461no—
290#41—3462no—
291#41—6921no—
292C2×Q8—1731no—
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—21no—
317C22×C4—3461no—
318C22×C4—1731no—
319C22×C4—1731no—
320SD16—3462no—
321SD16—3462no—
322SD16—6921no—
323D8—3462no—
324D8—3462no—
325D8—3462no—
326D8—3462no—
327D8—6921no—
328D8D69214yescomplemented
329M16—3462no—
330C2×C8—3462no—
331C2×C8—1731no—
332C2×C8—1731no—
333C2×C8C22×D17311yescharacteristic
334C2×C8C22×D17311yescharacteristic
335C4⋊C4—3461no—
336C4⋊C4—1731no—
337C4⋊C4C22×D17311yescharacteristic
338C4⋊C4C22×D17311yescharacteristic
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—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—3461no—
367Q8—3461no—
368D4—21no—
369D4—21no—
370D4—3462no—
371D4—3461no—
372D4—3461no—
373D4—3461no—
374D4—3461no—
375D4—3461no—
376D4—3461no—
377D4—13841no—
378D4—13841no—
379D4—6922no—
380D4—6922no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4C2×D69212yes—
387D4C2×D69212yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—3461no—
397C2×C4—1731no—
398C2×C4—1731no—
399C2×C4—6922no—
400C2×C4D4×D17311yescharacteristic
401C2×C4D4×D17311yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—21no—
404C8—3461no—
405C8—3461no—
406C8C2×D69212yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3461no—
414C22—3461no—
415C22—3461no—
416C22—3461no—
417C22—3461no—
418C22—13844no—
419C22—13844no—
420C22—13844no—
421C22—1732no—
422C22—6921no—
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—
439C4C22×D69211yescharacteristic
440C4#1914972511yescharacteristic
441C2—42no—
442C2—42no—
443C2—3462no—
444C2—13841no—
445C2—13841no—
446C2—13841no—
447C2—6921no—
448C2D4×D69211yescharacteristic
449C2AGL(1,ℤ/8)×D17311yescharacteristic
450C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#442×C173C2
socleC22×C173C22×D4
hypercenter#442D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×D8×D173C2semidirect
3. #26012136C2semidirect
4. D4×D692C2semidirect
5. #26011880C2semidirect
6. #26011745C2semidirect
7. #26011721C2semidirect
8. #26011706C2semidirect
9. #26011686C2semidirect
10. #26011643C2semidirect
11. #26011637C2semidirect
12. #26011626C2semidirect
13. #26011626C2semidirect
14. #26011564C2semidirect
15. #442×C173C2semidirect
16. C2×#14583825C22semidirect
19. C2×D4×D173C22semidirect
20. C2×D4×D173C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583814C22semidirect
25. C2×D1384C22semidirect
26. C2×D1384C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583811C22semidirect
31. C2×#14583809C22semidirect
32. C2×C8×D173C22semidirect
33. #19149676C22semidirect
34. #19149667C22semidirect
36. #19149659C22semidirect
38. #19149654C22semidirect
39. #19149654C22semidirect
47. C4⋊#13063436C22semidirect
48. #19149602C22semidirect
49. #19149599C22semidirect
50. #19149599C22semidirect
51. #19149599C22semidirect
52. #19149591C22semidirect
53. #19149591C22semidirect
55. C4×D4×C173C22semidirect
56. C4×D4×C173C22semidirect
57. #106×C173C22semidirect
58. #102×C173C22semidirect
59. #102×C173C22semidirect
60. #97×C173C22semidirect
116. D8×C173D4semidirect
171. D4×C173C2×D4semidirect
180. C8×C173C2×D4semidirect
205. C2×D173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
240. #442D173semidirect
273. C4×D4C2×D173semidirect
274. C4×D4C2×D173semidirect
276. #106C2×D173semidirect
282. #102C2×D173semidirect
283. #102C2×D173semidirect
286. #97C2×D173semidirect
328. D8D692semidirect
387. D4C2×D692semidirect
406. C8C2×D692semidirect

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
D4×D6922
#260118802
#260117452
#260117212
#260117062
#260116862
#260116432
#260116372
#260116262
#260115642
#442×C1732
#3264173