Finite Group Databasebeta

← #37545391

Subgroups — #37545391

Order 23168.

620 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668434C211yesmaximal, characteristic, complemented
2C2×D8×D181C211yesmaximal, characteristic, complemented
3#26041913C211yesmaximal, characteristic, complemented
4D4×C22⋊D181C211yesmaximal, characteristic, complemented
5#26041654C211yesmaximal, characteristic, complemented
6#26041514C211yesmaximal, characteristic, complemented
7#26041513C211yesmaximal, characteristic, complemented
8#26041473C211yesmaximal, characteristic, complemented
9#26041472C211yesmaximal, characteristic, complemented
10#26041415C211yesmaximal, characteristic, complemented
11#26041414C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13#26041359C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#447×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic, complemented
17C22×C22⋊D181—21no—
18C2×D4×D181—21no—
19C2×D4×D181C2211yescharacteristic, complemented
20C2×D4×D181C2211yescharacteristic, complemented
21C2×D1448C2211yescharacteristic, complemented
22C2×D1448C2211yescharacteristic, complemented
23C2×#15668434—24no—
24C2×#15668434C2211yescharacteristic
25C2×#15668434C2211yescharacteristic, complemented
26C2×#15668434C2211yescharacteristic, complemented
27C2×#15668434C2211yescharacteristic, complemented
28C2×#15668434C2211yescharacteristic, complemented
29D8×D181—24no—
30C22×#13089678C2211yescharacteristic, complemented
31C2×C8×D181C2211yescharacteristic, complemented
32#19153812C2211yescharacteristic, complemented
33#19153812C2211yescharacteristic, complemented
34#19153791C2211yescharacteristic, complemented
35#19153790C2211yescharacteristic, complemented
36#19153789—21no—
37#19153788—21no—
38#19153787C2211yescharacteristic, complemented
39#19153786—21no—
40#19153785—21no—
41#19153784—21no—
42#19153784—21no—
43D4×Dic181C2211yescharacteristic, complemented
44#19153778C2211yescharacteristic, complemented
45C4×C22⋊D181C2211yescharacteristic
46#19153749C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153738C2211yescharacteristic, complemented
49#19153738C2211yescharacteristic, complemented
50#19153737C2211yescharacteristic, complemented
51#19153736C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C22×D4×C181C2211yescharacteristic, complemented
55C2×D8×C181C2211yescharacteristic, complemented
56C2×D8×C181C2211yescharacteristic, complemented
57#121×C181C2211yescharacteristic, complemented
58#102×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic, complemented
60C22⋊C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62C23×D181—21no—
63C23×D181—21no—
64C2×D724—22no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78C2×C22⋊D181—21no—
79C2×C22⋊D181—44no—
80C2×C22⋊D181D411yescharacteristic
81D4×D181—22no—
82D4×D181—22no—
83D4×D181—41no—
84D4×D181—41no—
85D4×D181—44no—
86C22×Dic181—21no—
87C2×C4×D181—21no—
88C2×C4×D181C2311yescharacteristic
89D1448—22no—
90D1448—41no—
91#15668434—22no—
92#15668434—22no—
93#15668434—22no—
94#15668434—22no—
95#15668434D414yescomplemented
96C2×#13089678—22no—
97C2×#13089678C2311yescharacteristic
98C2×#13089678C2311yescharacteristic
99C8×D181—22no—
100C4⋊Dic181C2311yescharacteristic
101#15668428D411yescharacteristic
102C22⋊Dic181—21no—
103C22⋊Dic181—21no—
104C22⋊Dic181D411yescharacteristic
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426D411yescharacteristic
109C4×Dic181C2311yescharacteristic
110C24×C181—21no—
111C2×D4×C181—22no—
112C2×D4×C181—21no—
113C2×D4×C181—21no—
114C2×D4×C181—41no—
115C2×D4×C181C2311yescharacteristic
116C2×D4×C181C2311yescharacteristic
117C2×D4×C181C2311yescharacteristic
118C22×C4×C181C2311yescharacteristic
119D8×C181—22no—
120D8×C181—41no—
121C2×C8×C181C2311yescharacteristic
122C2×C8×C181C2311yescharacteristic
123C4⋊C4×C181C2311yescharacteristic
124C22⋊C4×C181—21no—
125C22×D181—21no—
126C22×D181—21no—
127C22×D181—21no—
128C22×D181—42no—
129C22×D181—42no—
130C22×D181—42no—
131C22×D181—42no—
132C22×D181—42no—
133C22×D181—42no—
134C22×D181—81no—
135C22×D181—81no—
136C22×D181—81no—
137C22×D181C2×D411yescharacteristic
138D724—21no—
139D724—21no—
140D724—21no—
141D724—21no—
142D724—21no—
143D724—42no—
144D724—42no—
145D724C2×D412yescomplemented
146C22⋊D181—42no—
147C22⋊D181—42no—
148C22⋊D181—42no—
149C22⋊D181—42no—
150C22⋊D181—42no—
151C22⋊D181—42no—
152C22⋊D181—81no—
153C22⋊D181—81no—
154C22⋊D181D814yescomplemented
155C2×Dic181—21no—
156C2×Dic181—42no—
157C2×Dic181C2×D411yescharacteristic
158C2×Dic181C2×D411yescharacteristic
159C2×Dic181C2×D411yescharacteristic
160C4×D181—21no—
161C4×D181—21no—
162C4×D181—42no—
163#13089678—21no—
164#13089678C2×D412yescomplemented
165C23×C181—21no—
166C23×C181—21no—
167C23×C181—21no—
168C23×C181—42no—
169C23×C181—42no—
170C23×C181—81no—
171C23×C181C2×D411yescharacteristic
172D4×C181—21no—
173D4×C181—21no—
174D4×C181—21no—
175D4×C181—21no—
176D4×C181—21no—
177D4×C181—42no—
178D4×C181—42no—
179D4×C181C2×D412yescomplemented
180C2×C4×C181—21no—
181C2×C4×C181—21no—
182C2×C4×C181—21no—
183C2×C4×C181C2×D411yescharacteristic
184C2×C4×C181C2411yescharacteristic
185C8×C181—21no—
186C8×C181—21no—
187C22×C181—21no—
188C22×C181—21no—
189C22×C181—22no—
190C22×C181—41no—
191C22×C181—41no—
192C22×C181—41no—
193C22×C181—41no—
194C22×C181—41no—
195C22×C181—41no—
196C22×C181—41no—
197C22×C181—41no—
198C22×C181—84no—
199C22×C181C2×D812yescomplemented
200C22×C181C22×D411yescharacteristic
201C2×D181—21no—
202C2×D181—21no—
203C2×D181—22no—
204C2×D181—41no—
205C2×D181—41no—
206C2×D181—41no—
207C2×D181—41no—
208C2×D181—41no—
209C2×D181—41no—
210C2×D181—41no—
211C2×D181—41no—
212C2×D181—84no—
213C2×D181—84no—
214C2×D181—84no—
215C2×D181C2×D812yescomplemented
216C4×C181—21no—
217C4×C181—41no—
218C4×C181C22×D411yescharacteristic
219C4×C181H(5,𝔽2)11yescharacteristic
220Dic181—21no—
221Dic181—21no—
222Dic181—41no—
223Dic181C2×D812yescomplemented
224C2×C181—22no—
225C2×C181—42no—
226C2×C181—81no—
227C2×C181—81no—
228C2×C181D4211yescharacteristic
229C2×C181C22×D811yescharacteristic
230C2×C181#57611yescharacteristic
231D181—22no—
232D181—42no—
233D181—81no—
234D181—81no—
235C181D4×D811yescharacteristic, complemented, Sylow
236D4×D8—1811nomaximal, Sylow
237C22×D8—1811no—
238C22×D8—1811no—
239D42—1811no—
240D42—1811no—
241#493—1811no—
242#466—1811no—
243#466—1811no—
244#459—1811no—
245#459—1811no—
246#447—1811no—
247#447—1811no—
248#447—1811no—
249#447D18111yescharacteristic, complemented
250C4×D8—1811no—
251C8×D4—1811no—
252C22×D4—3621no—
253C22×D4—3621no—
254C22×D4—1811no—
255C22×D4—1811no—
256C22×D4—1811no—
257C22×D4C2×D18111yescharacteristic, complemented
258C2×D8—3624no—
259C2×D8—3624no—
260C2×D8—1811no—
261C2×D8—1811no—
262C2×D8—1811no—
263C2×D8—1811no—
264C2×D8—1811no—
265C2×D8—1811no—
266C2×D8—1811no—
267C2×D8C2×D18111yescharacteristic, complemented
268C2×D8C2×D18111yescharacteristic, complemented
269C22×C8—1811no—
270C22×C8—1811no—
271#127—1811no—
272#127—1811no—
273#121—3621no—
274#121—3621no—
275#121—1811no—
276#121—1811no—
277#121—1811no—
278#121C2×D18111yescharacteristic, complemented
279#120—3621no—
280#120—3621no—
281#120—3621no—
282#120—3621no—
283C4×D4—1811no—
284C4×D4—1811no—
285C4×D4—1811no—
286#107—1811no—
287C4⋊C8—1811no—
288#102—1811no—
289#102—1811no—
290#102—1811no—
291#102—1811no—
292#102C2×D18111yescharacteristic, complemented
293#102C2×D18111yescharacteristic, complemented
294C22⋊C8—1811no—
295C22⋊C8C2×D18111yescharacteristic, complemented
296C4×C8—1811no—
297C24—21no—
298C24—3621no—
299C24—3621no—
300C24—3621no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—3622no—
306C2×D4—3621no—
307C2×D4—3622no—
308C2×D4—3622no—
309C2×D4—3621no—
310C2×D4—3621no—
311C2×D4—3621no—
312C2×D4—3621no—
313C2×D4—3621no—
314C2×D4—3621no—
315C2×D4—3621no—
316C2×D4—3621no—
317C2×D4—3621no—
318C2×D4—1811no—
319C2×D4—1811no—
320C2×D4—1811no—
321C2×D4—1811no—
322C2×D4—7241no—
323C2×D4—7241no—
324C2×D4—7241no—
325C2×D4—7244no—
326C2×D4—7244no—
327C2×D4C22×D18111yescharacteristic
328C2×D4C22×D18111yescharacteristic
329C2×D4C22×D18111yescharacteristic
330C22×C4—3621no—
331C22×C4—3621no—
332C22×C4—1811no—
333C22×C4C22×D18111yescharacteristic
334D8—22no—
335D8—41no—
336D8—3622no—
337D8—3622no—
338D8—3622no—
339D8—3622no—
340D8—3622no—
341D8—1814no—
342D8—7241no—
343C2×C8—3622no—
344C2×C8—3622no—
345C2×C8—1811no—
346C2×C8—1811no—
347C2×C8C22×D18111yescharacteristic
348C2×C8C22×D18111yescharacteristic
349C4⋊C4—1811no—
350C4⋊C4—1811no—
351C4⋊C4C22×D18111yescharacteristic
352C22⋊C4—21no—
353C22⋊C4—3621no—
354C22⋊C4—3621no—
355C22⋊C4—3621no—
356C22⋊C4—3621no—
357C22⋊C4—3621no—
358C22⋊C4—1811no—
359C22⋊C4—1811no—
360C42—1811no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—3621no—
368C23—3621no—
369C23—3621no—
370C23—14481no—
371C23—14481no—
372C23—14481no—
373C23—1811no—
374C23—7242no—
375C23—7242no—
376C23—7242no—
377C23—7242no—
378C23—7242no—
379C23—7242no—
380C23D4×D18111yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—21no—
386D4—42no—
387D4—42no—
388D4—3621no—
389D4—3621no—
390D4—3621no—
391D4—3621no—
392D4—3621no—
393D4—14481no—
394D4—14481no—
395D4—1814no—
396D4—1812no—
397D4—7242no—
398D4—7242no—
399D4—7242no—
400D4—7242no—
401D4—7242no—
402D4—7242no—
403D4—7242no—
404D4—7242no—
405D4D4×D18112yescomplemented
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—3621no—
410C2×C4—3621no—
411C2×C4—3621no—
412C2×C4—1811no—
413C2×C4—1811no—
414C2×C4—1811no—
415C2×C4—7242no—
416C2×C4—7242no—
417C2×C4D4×D18111yescharacteristic
418C2×C4C23×D18111yescharacteristic
419C8—21no—
420C8—21no—
421C8—3621no—
422C8—1812no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—84no—
435C22—3621no—
436C22—3621no—
437C22—3622no—
438C22—14484no—
439C22—14484no—
440C22—14484no—
441C22—1812no—
442C22—7241no—
443C22—7241no—
444C22—7241no—
445C22—7241no—
446C22—7241no—
447C22—7241no—
448C22—7241no—
449C22—7241no—
450C22D8×D18112yescomplemented
451C22C2×D4×D18111yescharacteristic
452C4—21no—
453C4—41no—
454C4—3621no—
455C4—3621no—
456C4—1812no—
457C4—7241no—
458C4C2×D4×D18111yescharacteristic
459C4#1915386111yescharacteristic
460C2—22no—
461C2—42no—
462C2—81no—
463C2—81no—
464C2—3622no—
465C2—14481no—
466C2—14481no—
467C2—7242no—
468C2#2604191111yescharacteristic
469C2C2×D8×D18111yescharacteristic
470C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181D4×D8
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#15668434C2semidirect
2. C2×D8×D181C2semidirect
3. #26041913C2semidirect
4. D4×C22⋊D181C2semidirect
5. #26041654C2semidirect
6. #26041514C2semidirect
7. #26041513C2semidirect
8. #26041473C2semidirect
9. #26041472C2semidirect
10. #26041415C2semidirect
11. #26041414C2semidirect
12. #26041411C2semidirect
13. #26041359C2semidirect
14. #26041342C2semidirect
15. #447×C181C2semidirect
16. C22×D724C22semidirect
19. C2×D4×D181C22semidirect
20. C2×D4×D181C22semidirect
21. C2×D1448C22semidirect
22. C2×D1448C22semidirect
25. C2×#15668434C22semidirect
26. C2×#15668434C22semidirect
27. C2×#15668434C22semidirect
28. C2×#15668434C22semidirect
30. C22×#13089678C22semidirect
31. C2×C8×D181C22semidirect
32. #19153812C22semidirect
33. #19153812C22semidirect
34. #19153791C22semidirect
35. #19153790C22semidirect
38. #19153787C22semidirect
43. D4×Dic181C22semidirect
44. #19153778C22semidirect
46. #19153749C22semidirect
47. #19153745C22semidirect
48. #19153738C22semidirect
49. #19153738C22semidirect
50. #19153737C22semidirect
51. #19153736C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
54. C22×D4×C181C22semidirect
55. C2×D8×C181C22semidirect
56. C2×D8×C181C22semidirect
57. #121×C181C22semidirect
58. #102×C181C22semidirect
59. #102×C181C22semidirect
60. C22⋊C8×C181C22semidirect
95. #15668434D4semidirect
145. D724C2×D4semidirect
154. C22⋊D181D8semidirect
164. #13089678C2×D4semidirect
179. D4×C181C2×D4semidirect
199. C22×C181C2×D8semidirect
215. C2×D181C2×D8semidirect
223. Dic181C2×D8semidirect
235. C181D4×D8semidirect
249. #447D181semidirect
257. C22×D4C2×D181semidirect
267. C2×D8C2×D181semidirect
268. C2×D8C2×D181semidirect
278. #121C2×D181semidirect
292. #102C2×D181semidirect
293. #102C2×D181semidirect
295. C22⋊C8C2×D181semidirect
405. D4D4×D181semidirect
450. C22D8×D181semidirect

Maximal subgroups

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

subgroupindex
C22×#156684342
C2×D8×D1812
#260419132
D4×C22⋊D1812
#260416542
#260415142
#260415132
#260414732
#260414722
#260414152
#260414142
#260414112
#260413592
#260413422
#447×C1812
D4×D8181