Finite Group Databasebeta

← #36365048

Subgroups — #36365048

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×#19149707C211yesmaximal, characteristic, complemented
2C22×D1384C211yesmaximal, characteristic, complemented
3#26012137C211yesmaximal, characteristic, complemented
4D4×D692C211yesmaximal, characteristic, complemented
5#26011880C211yesmaximal, characteristic, complemented
6#26011744C211yesmaximal, characteristic, complemented
7#26011720C211yesmaximal, characteristic, complemented
8#26011707C211yesmaximal, characteristic, complemented
9#26011687C211yesmaximal, characteristic, complemented
10#26011644C211yesmaximal, characteristic, complemented
11#26011638C211yesmaximal, characteristic, complemented
12#26011627C211yesmaximal, characteristic, complemented
13#26011627C211yesmaximal, characteristic, complemented
14#26011604C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C22×D692C2211yescharacteristic
18C22×D692C2211yescharacteristic, complemented
19C22×D692C2211yescharacteristic, complemented
20C2×D4×D173—21no—
21#19149707—24no—
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic, complemented
25C2×D1384—24no—
26C2×D1384C2211yescharacteristic
27C2×D1384C2211yescharacteristic
28C2×D1384C2211yescharacteristic, complemented
29C2×D1384C2211yescharacteristic, complemented
30C2×D1384C2211yescharacteristic, complemented
31C2×D1384C2211yescharacteristic, complemented
32#19149676C2211yescharacteristic, complemented
33#19149667C2211yescharacteristic, complemented
34#19149664—21no—
35#19149658C2211yescharacteristic, complemented
36#19149654—21no—
37#19149653C2211yescharacteristic
38#19149653C2211yescharacteristic, complemented
39#19149653C2211yescharacteristic, complemented
40#19149652—21no—
41#19149649—21no—
42#19149649—21no—
43#19149647—21no—
44C4×D692C2211yescharacteristic, complemented
45C4×D692C2211yescharacteristic, complemented
46C22⋊C4×D173—21no—
47#19149611C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#19149601C2211yescharacteristic, complemented
53#19149601C2211yescharacteristic, complemented
54C2×M16×C173C2211yescharacteristic, complemented
55C22×C8×C173C2211yescharacteristic, complemented
56C4×D4×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C22⋊C8×C173C2211yescharacteristic, complemented
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63C23×D173—21no—
64#14583826—22no—
65#14583826—41no—
66C2×Dic346C2311yescharacteristic
67C2×D692—22no—
68C2×D692—22no—
69C2×D692—22no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692—21no—
73C2×D692—21no—
74C2×D692—21no—
75C2×D692—41no—
76C2×D692—41no—
77C2×D692—41no—
78C2×D692C2311yescharacteristic
79C2×D692C2311yescharacteristic
80C2×D692C2311yescharacteristic
81C2×D692C2311yescharacteristic
82C2×D692C2311yescharacteristic
83C2×C22⋊D173—21no—
84C2×C22⋊D173—21no—
85C2×C22⋊D173—21no—
86C2×C22⋊D173—21no—
87D4×D173—44no—
88C2×C4×D173—21no—
89C2×C4×D173—21no—
90C2×C4×D173—21no—
91#14583815—22no—
92#14583815—22no—
93#14583815—41no—
94D1384—22no—
95D1384—22no—
96D1384—22no—
97D1384—22no—
98D1384—41no—
99D1384D414yescomplemented
100C4⋊Dic173C2311yescharacteristic
101C4⋊Dic173C2311yescharacteristic
102#14583805—21no—
103C22⋊Dic173—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109#14583803—21no—
110#14583803—21no—
111C2×D4×C173D411yescharacteristic
112C22×C4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114M16×C173—22no—
115C2×C8×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C22⋊C4×C173D411yescharacteristic
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—21no—
127C22×D173—21no—
128C22×D173—21no—
129C22×D173—42no—
130C22×D173—42no—
131C22×D173—42no—
132C22×D173—42no—
133C22×D173—42no—
134C22×D173—42no—
135C22×D173—81no—
136C22×D173—81no—
137C22×D173—81no—
138Dic346—21no—
139Dic346—21no—
140D692—21no—
141D692—21no—
142D692—21no—
143D692—21no—
144D692—21no—
145D692—21no—
146D692—21no—
147D692—21no—
148D692—42no—
149D692—42no—
150D692—42no—
151D692—42no—
152D692—81no—
153D692C2×D412yes—
154D692C2×D412yescomplemented
155C22⋊D173—42no—
156C22⋊D173—42no—
157C22⋊D173—42no—
158C22⋊D173—81no—
159C2×Dic173—21no—
160C2×Dic173—21no—
161C2×Dic173—21no—
162C4×D173—42no—
163C4×D173—42no—
164C4×D173—81no—
165C23×C173C2×D411yescharacteristic
166C23×C173C2×D411yescharacteristic
167D4×C173—22no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173C2×D411yescharacteristic
173C2×C4×C173C2×D411yescharacteristic
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2×D411yescharacteristic
176C2×C4×C173C2411yescharacteristic
177C8×C173—21no—
178C8×C173—21no—
179C8×C173—21no—
180C8×C173C2×D412yescomplemented
181C22×C173—21no—
182C22×C173—21no—
183C22×C173—21no—
184C22×C173—21no—
185C22×C173—21no—
186C22×C173AGL(1,ℤ/8)12yescomplemented
187C22×C173C22×D411yescharacteristic
188C2×D173—22no—
189C2×D173—22no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
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—41no—
203C2×D173—84no—
204C2×D173—84no—
205C2×D173—84no—
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—21no—
209C4×C173—21no—
210C4×C173C22×D411yescharacteristic
211C4×C173H(5,𝔽2)11yescharacteristic
212Dic173—41no—
213Dic173—41no—
214Dic173—41no—
215C2×C173—22no—
216C2×C173—41no—
217C2×C173D4211yescharacteristic
218C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C173#57611yescharacteristic
220D173—42no—
221D173—42no—
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#442—1731no—
241#435D17311yescharacteristic, complemented
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×D8—1731no—
258C2×M16C2×D17311yescharacteristic, complemented
259C22×C8C2×D17311yescharacteristic, complemented
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×D4C2×D17311yescharacteristic
275C2×C22⋊C4—3461no—
276#106—1731no—
277C4⋊C8C2×D17311yescharacteristic, complemented
278#103—1731no—
279#102—1731no—
280#102—1731no—
281#102—1731no—
282#102—1731no—
283#102—1731no—
284C22⋊C8C2×D17311yescharacteristic, complemented
285C22⋊C8C2×D17311yescharacteristic, complemented
286#97C2×D17311yescharacteristic, complemented
287C24—3461no—
288C24—3461no—
289C24—3461no—
290#41—3462no—
291#41—6921no—
292C2×Q8—1731no—
293C2×D4—3462no—
294C2×D4—3462no—
295C2×D4—3462no—
296C2×D4—3461no—
297C2×D4—3461no—
298C2×D4—3461no—
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—1731no—
310C2×D4—6921no—
311C2×D4—6921no—
312C2×D4—6921no—
313C2×D4—6924no—
314C2×D4D69211yescharacteristic
315C22×C4—3461no—
316C22×C4—3461no—
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—22no—
330C2×C8—22no—
331C2×C8C22×D17311yescharacteristic
332C2×C8C22×D17311yescharacteristic
333C2×C8C22×D17311yescharacteristic
334C2×C8C22×D17311yescharacteristic
335C4⋊C4—3461no—
336C4⋊C4—1731no—
337C4⋊C4—1731no—
338C4⋊C4D69211yescharacteristic
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⋊C4D69211yescharacteristic
348C22⋊C4D69211yescharacteristic
349C42C22×D17311yescharacteristic
350C23—3461no—
351C23—3461no—
352C23—3461no—
353C23—3461no—
354C23—3461no—
355C23—13841no—
356C23—13841no—
357C23—13841no—
358C23—6922no—
359C23—6922no—
360C23—6922no—
361C23—6922no—
362C23—6922no—
363C23—6922no—
364C23C2×D69211yescharacteristic
365C23C2×D69211yescharacteristic
366Q8—3461no—
367Q8—3461no—
368D4—22no—
369D4—3461no—
370D4—3461no—
371D4—3461no—
372D4—3461no—
373D4—3461no—
374D4—3461no—
375D4—3461no—
376D4—3461no—
377D4—13841no—
378D4—13841no—
379D4—1732no—
380D4—1732no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4—6922no—
387D4—6922no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—13841no—
396C2×C4—6922no—
397C2×C4—6922no—
398C2×C4C2×D69211yescharacteristic
399C2×C4C2×D69211yescharacteristic
400C2×C4C2×D69211yescharacteristic
401C2×C4C2×D69211yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D17312yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—3462no—
413C22—3462no—
414C22—13844no—
415C22—13844no—
416C22—13844no—
417C22—6921no—
418C22—6921no—
419C22—6921no—
420C22—6921no—
421C22—6921no—
422C22—6921no—
423C22—6921no—
424C22—6921no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22#1914970712yescomplemented
431C22C22×D69211yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—6921no—
437C4—6921no—
438C4—6921no—
439C4C2×D4×D17311yescharacteristic
440C4#1914972511yescharacteristic
441C2—22no—
442C2—41no—
443C2—13841no—
444C2—13841no—
445C2—13841no—
446C2—6922no—
447C2—6922no—
448C2D4×D69211yescharacteristic
449C2C2×#1914970711yescharacteristic
450C2#2601229611yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
hypercenter#435D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19149707C2semidirect
2. C22×D1384C2semidirect
3. #26012137C2semidirect
4. D4×D692C2semidirect
5. #26011880C2semidirect
6. #26011744C2semidirect
7. #26011720C2semidirect
8. #26011707C2semidirect
9. #26011687C2semidirect
10. #26011644C2semidirect
11. #26011638C2semidirect
12. #26011627C2semidirect
13. #26011627C2semidirect
14. #26011604C2semidirect
15. #435×C173C2semidirect
16. C2×#14583826C22semidirect
18. C22×D692C22semidirect
19. C22×D692C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583815C22semidirect
24. C2×#14583815C22semidirect
28. C2×D1384C22semidirect
29. C2×D1384C22semidirect
30. C2×D1384C22semidirect
31. C2×D1384C22semidirect
32. #19149676C22semidirect
33. #19149667C22semidirect
35. #19149658C22semidirect
38. #19149653C22semidirect
39. #19149653C22semidirect
44. C4×D692C22semidirect
45. C4×D692C22semidirect
47. #19149611C22semidirect
48. #19149604C22semidirect
49. #19149601C22semidirect
50. #19149601C22semidirect
51. #19149601C22semidirect
52. #19149601C22semidirect
53. #19149601C22semidirect
54. C2×M16×C173C22semidirect
55. C22×C8×C173C22semidirect
57. C4⋊C8×C173C22semidirect
58. C22⋊C8×C173C22semidirect
59. C22⋊C8×C173C22semidirect
60. #97×C173C22semidirect
99. D1384D4semidirect
154. D692C2×D4semidirect
180. C8×C173C2×D4semidirect
186. C22×C173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
241. #435D173semidirect
258. C2×M16C2×D173semidirect
259. C22×C8C2×D173semidirect
277. C4⋊C8C2×D173semidirect
284. C22⋊C8C2×D173semidirect
285. C22⋊C8C2×D173semidirect
286. #97C2×D173semidirect
406. C8D4×D173semidirect
430. C22#19149707semidirect

Maximal subgroups

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

subgroupindex
C2×#191497072
C22×D13842
#260121372
D4×D6922
#260118802
#260117442
#260117202
#260117072
#260116872
#260116442
#260116382
#260116272
#260116042
#435×C1732
#3264173