Finite Group Databasebeta

← #37545329

Subgroups — #37545329

Order 23168.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153844C211yesmaximal, characteristic, complemented
2C2×#19153843C211yesmaximal, characteristic, complemented
3#26041926C211yesmaximal, characteristic, complemented
4#26041920C211yesmaximal, characteristic, complemented
5#26041660C211yesmaximal, characteristic, complemented
6#26041526C211yesmaximal, characteristic, complemented
7#26041497C211yesmaximal, characteristic, complemented
8#26041493C211yesmaximal, characteristic, complemented
9#26041444C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041414C211yesmaximal, characteristic, complemented
12#26041412C211yesmaximal, characteristic, complemented
13#26041361C211yesmaximal, characteristic, complemented
14#26041355C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668449C2211yescharacteristic
18C22×D724C2211yescharacteristic, complemented
19C2×D4×D181C2211yescharacteristic, complemented
20#19153844—24no—
21#19153843—24no—
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668438C2211yescharacteristic, complemented
24C2×#15668438C2211yescharacteristic, complemented
25C2×#15668436C2211yescharacteristic, complemented
26C2×D1448C2211yescharacteristic
27C2×D1448C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C2×#15668434C2211yescharacteristic
30C2×#15668432C2211yescharacteristic, complemented
31#19153805C2211yescharacteristic, complemented
32#19153803C2211yescharacteristic, complemented
33#19153801—21no—
34#19153798—21no—
35#19153796C2211yescharacteristic, complemented
36#19153791—21no—
37#19153791C2211yescharacteristic
38#19153787—21no—
39#19153786—21no—
40#19153784—21no—
41#19153778C2211yescharacteristic, complemented
42#19153778C2211yescharacteristic, complemented
43C4×C22⋊D181C2211yescharacteristic
44C22⋊C4×D181—21no—
45C22⋊C4×D181—21no—
46#19153745C2211yescharacteristic, complemented
47#19153739C2211yescharacteristic, complemented
48#19153736C2211yescharacteristic, complemented
49#19153736C2211yescharacteristic, complemented
50#19153736C2211yescharacteristic, complemented
51C22⋊#13089678C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C2×SD16×C181C2211yescharacteristic, complemented
55C2×M16×C181C2211yescharacteristic, complemented
56C22⋊Q8×C181C2211yescharacteristic, complemented
57#121×C181C2211yescharacteristic
58#107×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62C23×D181—21no—
63#15668450—22no—
64#15668450—41no—
65#15668449—22no—
66#15668449—41no—
67C2×Dic362C2311yescharacteristic
68C2×D724—22no—
69C2×D724—21no—
70C2×D724—21no—
71C2×D724—21no—
72C2×D724—41no—
73C2×D724C2311yescharacteristic
74C2×D724C2311yescharacteristic
75C2×D724C2311yescharacteristic
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78C2×C22⋊D181D411yescharacteristic
79D4×D181—22no—
80D4×D181—41no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181—21no—
85C2×C4×D181C2311yescharacteristic
86#15668439—41no—
87#15668438—22no—
88#15668438—22no—
89#15668436—22no—
90D1448—22no—
91D1448D414yescomplemented
92#15668434—22no—
93#15668434—41no—
94#15668432—22no—
95C2×#13089678C2311yescharacteristic
96C2×#13089678C2311yescharacteristic
97C4⋊Dic181—21no—
98#15668428D411yescharacteristic
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181D411yescharacteristic
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426D411yescharacteristic
109C4×Dic181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181C2311yescharacteristic
114SD16×C181—22no—
115M16×C181—22no—
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C4⋊C4×C181—21no—
119C4⋊C4×C181C2311yescharacteristic
120C4⋊C4×C181C2311yescharacteristic
121C22⋊C4×C181—21no—
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—21no—
126C22×D181—42no—
127C22×D181—42no—
128C22×D181—42no—
129C22×D181—42no—
130C22×D181—81no—
131C22×D181—81no—
132C22×D181C2×D411yescharacteristic
133Dic362—21no—
134Dic362—21no—
135D724—21no—
136D724—21no—
137D724—21no—
138D724—21no—
139D724—42no—
140D724—42no—
141D724—81no—
142D724C2×D412yes—
143D724C2×D412yes—
144C22⋊D181—41no—
145C22⋊D181—42no—
146C22⋊D181—42no—
147C2×Dic181—21no—
148C2×Dic181C2×D411yescharacteristic
149C2×Dic181C2×D411yescharacteristic
150C2×Dic181C2×D411yescharacteristic
151C4×D181—21no—
152C4×D181—21no—
153C4×D181—42no—
154C4×D181—42no—
155C4×D181—81no—
156C4×D181—81no—
157#13089678—21no—
158#13089678—21no—
159C23×C181—21no—
160C23×C181C2×D411yescharacteristic
161Q8×C181—21no—
162Q8×C181—21no—
163D4×C181—21no—
164D4×C181—21no—
165D4×C181—81no—
166C2×C4×C181—21no—
167C2×C4×C181—21no—
168C2×C4×C181—21no—
169C2×C4×C181—21no—
170C2×C4×C181—21no—
171C2×C4×C181C2×D411yescharacteristic
172C2×C4×C181C2411yescharacteristic
173C8×C181—21no—
174C8×C181C2×D412yes—
175C22×C181—21no—
176C22×C181—21no—
177C22×C181—21no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181C22×D411yescharacteristic
182C2×D181—21no—
183C2×D181—22no—
184C2×D181—22no—
185C2×D181—21no—
186C2×D181—21no—
187C2×D181—41no—
188C2×D181—41no—
189C2×D181—41no—
190C2×D181—41no—
191C2×D181—41no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—84no—
195C2×D181—84no—
196C4×C181—21no—
197C4×C181—41no—
198C4×C181—41no—
199C4×C181—41no—
200C4×C181C22×D411yescharacteristic
201C4×C181H(5,𝔽2)11yescharacteristic
202Dic181—21no—
203Dic181—21no—
204Dic181—21no—
205Dic181—41no—
206C2×C181—41no—
207C2×C181—81no—
208C2×C181D4211yescharacteristic
209C2×C181#57511yescharacteristic
210C2×C181#57611yescharacteristic
211D181—41no—
212D181—42no—
213D181—42no—
214D181—81no—
215C181#327311yescharacteristic, complemented, Sylow
216#3273—1811nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1811no—
218C2×AGL(1,ℤ/8)—1811no—
219#546—1811no—
220#546—1811no—
221#495—1811no—
222#468—1811no—
223#468D18111yescharacteristic, complemented
224#463—1811no—
225#463—1811no—
226#452—1811no—
227#452—1811no—
228#447—1811no—
229#447—1811no—
230C4×D8—1811no—
231#436—1811no—
232C2×#41—1811no—
233C2×#41—1811no—
234C22×D4—1811no—
235C22×D4—1811no—
236AGL(1,ℤ/8)—3624no—
237AGL(1,ℤ/8)—3624no—
238C2×Q16—1811no—
239C2×SD16—1811no—
240C2×SD16—1811no—
241C2×SD16—1811no—
242C2×SD16C2×D18111yescharacteristic, complemented
243C2×D8—1811no—
244C2×D8—1811no—
245C2×D8—1811no—
246C2×D8—1811no—
247C2×M16—1811no—
248C2×M16C2×D18111yescharacteristic, complemented
249#124—1811no—
250#124—1811no—
251#123—3621no—
252#123—3621no—
253C22⋊Q8—1811no—
254C22⋊Q8C2×D18111yescharacteristic, complemented
255#121—3621no—
256#121—3621no—
257#121—1811no—
258#121C2×D18111yescharacteristic
259#120—3621no—
260#120—3621no—
261C4×D4—1811no—
262C4×D4—1811no—
263C4×D4—1811no—
264C2×C22⋊C4—3621no—
265C2×C22⋊C4—3621no—
266#107C2×D18111yescharacteristic, complemented
267C4⋊C8—1811no—
268#103—1811no—
269#103C2×D18111yescharacteristic, complemented
270#102—1811no—
271#102—1811no—
272#102—1811no—
273#102C2×D18111yescharacteristic, complemented
274C22⋊C8—1811no—
275C22⋊C8—1811no—
276C4×C8—1811no—
277C24—3621no—
278C24—3621no—
279#41—3622no—
280#41—3622no—
281#41—7241no—
282#41—7241no—
283C2×Q8—1811no—
284C2×Q8C22×D18111yescharacteristic
285C2×D4—21no—
286C2×D4—3622no—
287C2×D4—3621no—
288C2×D4—3621no—
289C2×D4—3622no—
290C2×D4—3621no—
291C2×D4—3621no—
292C2×D4—3621no—
293C2×D4—1811no—
294C2×D4—1811no—
295C2×D4—1811no—
296C2×D4—1811no—
297C2×D4—7241no—
298C2×D4—7241no—
299C2×D4C22×D18111yescharacteristic
300C22×C4—3621no—
301C22×C4—3621no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4—1811no—
305C22×C4C22×D18111yescharacteristic
306Q16—7241no—
307SD16—22no—
308SD16—3622no—
309SD16—3622no—
310SD16—3622no—
311D8—3622no—
312D8—3622no—
313D8—1814no—
314D8—7241no—
315M16—22no—
316M16—3622no—
317C2×C8—1811no—
318C2×C8—1811no—
319C2×C8C22×D18111yescharacteristic
320C2×C8C22×D18111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3621no—
323C4⋊C4—1811no—
324C4⋊C4C22×D18111yescharacteristic
325C4⋊C4C22×D18111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—3621no—
329C22⋊C4—3621no—
330C22⋊C4—3621no—
331C22⋊C4—3621no—
332C22⋊C4—3621no—
333C22⋊C4—3621no—
334C22⋊C4—3621no—
335C22⋊C4—3621no—
336C22⋊C4—1811no—
337C22⋊C4—1811no—
338C42—1811no—
339C23—21no—
340C23—3621no—
341C23—3621no—
342C23—3621no—
343C23—14481no—
344C23—14481no—
345C23—1811no—
346C23—7242no—
347C23—7242no—
348C23—7242no—
349C23—7242no—
350C23D4×D18111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3621no—
354Q8—3621no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—3621no—
359D4—3621no—
360D4—3621no—
361D4—3621no—
362D4—14481no—
363D4—1812no—
364D4—1812no—
365D4—7242no—
366D4—7242no—
367D4—7241no—
368D4—7242no—
369D4—7242no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3621no—
376C2×C4—3621no—
377C2×C4—3621no—
378C2×C4—14481no—
379C2×C4—14481no—
380C2×C4—1811no—
381C2×C4—1811no—
382C2×C4—1811no—
383C2×C4—7242no—
384C2×C4—7242no—
385C2×C4D4×D18111yescharacteristic
386C2×C4C23×D18111yescharacteristic
387C8—21no—
388C8—3621no—
389C8—3621no—
390C8D4×D18112yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3621no—
398C22—3622no—
399C22—3622no—
400C22—3621no—
401C22—3621no—
402C22—14484no—
403C22—14484no—
404C22—7241no—
405C22—7241no—
406C22—7241no—
407C22—7241no—
408C22—7241no—
409C22—7241no—
410C22—7241no—
411C22C2×D4×D18111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3621no—
417C4—3621no—
418C4—3621no—
419C4—7241no—
420C4C2×D4×D18111yescharacteristic
421C4#1915386211yescharacteristic
422C2—41no—
423C2—81no—
424C2—14481no—
425C2—7241no—
426C2—7242no—
427C2—7242no—
428C2#2604191311yescharacteristic
429C2#2604207811yescharacteristic
430C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
hypercenter#468D181
hypocenterC181#3273
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. C2×#19153843C2semidirect
3. #26041926C2semidirect
4. #26041920C2semidirect
5. #26041660C2semidirect
6. #26041526C2semidirect
7. #26041497C2semidirect
8. #26041493C2semidirect
9. #26041444C2semidirect
10. #26041439C2semidirect
11. #26041414C2semidirect
12. #26041412C2semidirect
13. #26041361C2semidirect
14. #26041355C2semidirect
15. #468×C181C2semidirect
18. C22×D724C22semidirect
19. C2×D4×D181C22semidirect
22. C2×#15668439C22semidirect
23. C2×#15668438C22semidirect
24. C2×#15668438C22semidirect
25. C2×#15668436C22semidirect
30. C2×#15668432C22semidirect
31. #19153805C22semidirect
32. #19153803C22semidirect
35. #19153796C22semidirect
41. #19153778C22semidirect
42. #19153778C22semidirect
46. #19153745C22semidirect
47. #19153739C22semidirect
48. #19153736C22semidirect
49. #19153736C22semidirect
50. #19153736C22semidirect
51. C22⋊#13089678C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
54. C2×SD16×C181C22semidirect
55. C2×M16×C181C22semidirect
56. C22⋊Q8×C181C22semidirect
58. #107×C181C22semidirect
59. #103×C181C22semidirect
60. #102×C181C22semidirect
91. D1448D4semidirect
215. C181#3273semidirect
223. #468D181semidirect
242. C2×SD16C2×D181semidirect
248. C2×M16C2×D181semidirect
254. C22⋊Q8C2×D181semidirect
266. #107C2×D181semidirect
269. #103C2×D181semidirect
273. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×#191538432
#260419262
#260419202
#260416602
#260415262
#260414972
#260414932
#260414442
#260414392
#260414142
#260414122
#260413612
#260413552
#468×C1812
#3273181