Finite Group Databasebeta

← #37543222

Subgroups — #37543222

Order 23168.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153843C211yesmaximal, characteristic, complemented
2C2×D8×D181C211yesmaximal, characteristic, complemented
3#26041847C211yesmaximal, characteristic, complemented
4#26041751C211yesmaximal, characteristic, complemented
5#26041497C211yesmaximal, characteristic, complemented
6#26041496C211yesmaximal, characteristic, complemented
7#26041491C211yesmaximal, characteristic, complemented
8#26041475C211yesmaximal, characteristic, complemented
9#26041474C211yesmaximal, characteristic, complemented
10#26041473C211yesmaximal, characteristic, complemented
11#26041472C211yesmaximal, characteristic, complemented
12#26041304C211yesmaximal, characteristic, complemented
13#26041282C211yesmaximal, characteristic, complemented
14#26041274C211yesmaximal, characteristic, complemented
15#463×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×D4×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153843—24no—
20C2×#15668436C2211yescharacteristic, complemented
21C2×D1448C2211yescharacteristic
22C2×D1448C2211yescharacteristic, complemented
23C2×#15668434C2211yescharacteristic
24C2×#15668434C2211yescharacteristic
25C2×#15668434C2211yescharacteristic, complemented
26D8×D181—24no—
27C2×#15668432C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29#19153813C2211yescharacteristic, complemented
30#19153812C2211yescharacteristic, complemented
31#19153805C2211yescharacteristic, complemented
32#19153789—21no—
33#19153789—21no—
34#19153786—21no—
35#19153786—21no—
36C4×D724C2211yescharacteristic, complemented
37D4×Dic181C2211yescharacteristic, complemented
38#19153778C2211yescharacteristic, complemented
39C4×C22⋊D181—21no—
40#19153776—21no—
41#19153774C2211yescharacteristic
42#19153773C2211yescharacteristic
43C2×#15668426—21no—
44C22⋊C4×D181—21no—
45C4⋊#13089678C2211yescharacteristic, complemented
46#19153745C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153740C2211yescharacteristic, complemented
49#19153738C2211yescharacteristic, complemented
50#19153738C2211yescharacteristic, complemented
51#19153737C2211yescharacteristic, complemented
52#19153736C2211yescharacteristic, complemented
53#19153736C2211yescharacteristic, complemented
54C2×SD16×C181C2211yescharacteristic, complemented
55C2×D8×C181C2211yescharacteristic
56#124×C181C2211yescharacteristic, complemented
57C4×D4×C181C2211yescharacteristic, complemented
58C4⋊C8×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#102×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62C23×D181—21no—
63#15668450—22no—
64#15668450—22no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—21no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75D4×D181—22no—
76D4×D181—22no—
77D4×D181—22no—
78D4×D181D414yescomplemented
79C22×Dic181—21no—
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181C2311yescharacteristic
84#15668436—22no—
85D1448—22no—
86D1448—22no—
87#15668434—22no—
88#15668434—22no—
89#15668434—22no—
90#15668432—22no—
91C2×#13089678C2311yescharacteristic
92C2×#13089678C2311yescharacteristic
93C8×D181—22no—
94C4⋊Dic181C2311yescharacteristic
95#15668428D411yescharacteristic
96#15668428D411yescharacteristic
97C22⋊Dic181—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426D411yescharacteristic
104#15668426D411yescharacteristic
105C4×Dic181—21no—
106C4×Dic181C2311yescharacteristic
107C2×Q8×C181C2311yescharacteristic
108C2×D4×C181C2311yescharacteristic
109C2×D4×C181C2311yescharacteristic
110C22×C4×C181—21no—
111SD16×C181—22no—
112D8×C181—22no—
113C2×C8×C181C2311yescharacteristic
114C2×C8×C181C2311yescharacteristic
115C4⋊C4×C181C2311yescharacteristic
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C42×C181C2311yescharacteristic
120C22×D181—21no—
121C22×D181—24no—
122C22×D181—21no—
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—81no—
126C22×D181—81no—
127C22×D181—81no—
128C22×D181—81no—
129C22×D181C2×D411yescharacteristic
130D724—21no—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—21no—
135D724—42no—
136D724—81no—
137D724—81no—
138D724C2×D412yes—
139C22⋊D181—24no—
140C22⋊D181—42no—
141C22⋊D181—81no—
142C22⋊D181—81no—
143C2×Dic181—21no—
144C2×Dic181—81no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C2×Dic181C2×D411yescharacteristic
148C4×D181—42no—
149C4×D181—81no—
150C4×D181—81no—
151C4×D181C2×D412yes—
152C4×D181#4112yes—
153#13089678—21no—
154#13089678—21no—
155C23×C181—21no—
156C23×C181—21no—
157Q8×C181—21no—
158Q8×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181C2×D412yes—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—81no—
166C2×C4×C181C2×D411yescharacteristic
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2411yescharacteristic
169C8×C181—21no—
170C8×C181—21no—
171C22×C181—22no—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181C22×D411yescharacteristic
178C2×D181—21no—
179C2×D181—21no—
180C2×D181—22no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—41no—
189C2×D181—84no—
190C2×D181—84no—
191C2×D181#12112yes—
192C4×C181—21no—
193C4×C181—21no—
194C4×C181—41no—
195C4×C181—41no—
196C4×C181C22×D411yescharacteristic
197C4×C181C2×#4111yescharacteristic
198Dic181—21no—
199Dic181—21no—
200Dic181—41no—
201Dic181#12112yes—
202C2×C181—42no—
203C2×C181—81no—
204C2×C181C2×#12111yescharacteristic
205C2×C181#57611yescharacteristic
206C2×C181#57611yescharacteristic
207D181—22no—
208D181—42no—
209D181—81no—
210D181—81no—
211C181#304111yescharacteristic, complemented, Sylow
212#3041—1811nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1811no—
214C22×D8—1811no—
215#535—1811no—
216#518—1811no—
217#463—1811no—
218#463—1811no—
219#463—1811no—
220#463D18111yescharacteristic, complemented
221#459—1811no—
222#459—1811no—
223#459—1811no—
224#459—1811no—
225#424—1811no—
226#418—1811no—
227#416—1811no—
228C2×#41—1811no—
229C22×D4—1811no—
230C22×D4—1811no—
231AGL(1,ℤ/8)—3624no—
232C2×SD16—1811no—
233C2×SD16C2×D18111yescharacteristic, complemented
234C2×D8—3624no—
235C2×D8—1811no—
236C2×D8—1811no—
237C2×D8—1811no—
238C2×D8—1811no—
239C2×D8—1811no—
240C2×D8C2×D18111yescharacteristic
241C2×M16—1811no—
242C22×C8—1811no—
243#127—1811no—
244#127—1811no—
245#124—1811no—
246#124C2×D18111yescharacteristic, complemented
247#121—3621no—
248#121—3621no—
249#120—3621no—
250#120—3621no—
251C4×D4—3621no—
252C4×D4—3621no—
253C4×D4—1811no—
254C4×D4—1811no—
255C4×D4—1811no—
256C4×D4C2×D18111yescharacteristic, complemented
257#117—1811no—
258#117—1811no—
259C2×C22⋊C4—3621no—
260C2×C22⋊C4—3621no—
261C4⋊C8—1811no—
262C4⋊C8—1811no—
263C4⋊C8—1811no—
264C4⋊C8C2×D18111yescharacteristic, complemented
265#103—1811no—
266#103C2×D18111yescharacteristic, complemented
267#102—1811no—
268#102—1811no—
269#102—1811no—
270#102—1811no—
271#102—1811no—
272#102C2×D18111yescharacteristic, complemented
273C24—3621no—
274C24—3621no—
275#41—3622no—
276#41—3622no—
277C2×Q8C22×D18111yescharacteristic
278C2×D4—3621no—
279C2×D4—3622no—
280C2×D4—3622no—
281C2×D4—3622no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—3621no—
288C2×D4—1814no—
289C2×D4—1811no—
290C2×D4—1811no—
291C2×D4—1811no—
292C2×D4C22×D18111yescharacteristic
293C2×D4C22×D18111yescharacteristic
294C22×C4—21no—
295C22×C4—3621no—
296C22×C4—3621no—
297C22×C4—3621no—
298C22×C4—3621no—
299C22×C4—1811no—
300SD16—22no—
301SD16—3622no—
302D8—22no—
303D8—3622no—
304D8—3622no—
305D8—3622no—
306D8—3622no—
307D8—3622no—
308M16—3622no—
309C2×C8—3622no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8C22×D18111yescharacteristic
313C2×C8C22×D18111yescharacteristic
314C4⋊C4—1811no—
315C4⋊C4—1811no—
316C4⋊C4—1811no—
317C4⋊C4C22×D18111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—1811no—
328C22⋊C4—1811no—
329C42—3621no—
330C42—1811no—
331C42C22×D18111yescharacteristic
332C23—21no—
333C23—21no—
334C23—3624no—
335C23—3621no—
336C23—3621no—
337C23—3621no—
338C23—14481no—
339C23—14481no—
340C23—14481no—
341C23—14481no—
342C23—1811no—
343C23—7242no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—3624no—
350D4—3621no—
351D4—3621no—
352D4—3621no—
353D4—3621no—
354D4—3621no—
355D4—14481no—
356D4—14481no—
357D4—14481no—
358D4—14481no—
359D4—1812no—
360D4—7242no—
361D4—7242no—
362D4D4×D18112yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
366C2×C4—3621no—
367C2×C4—14481no—
368C2×C4—14481no—
369C2×C4—14481no—
370C2×C4—1811no—
371C2×C4—1811no—
372C2×C4—1811no—
373C2×C4—1812no—
374C2×C4—1812no—
375C2×C4—7242no—
376C2×C4D4×D18111yescharacteristic
377C2×C4D4×D18111yescharacteristic
378C2×C4C23×D18111yescharacteristic
379C8—21no—
380C8—21no—
381C8—3621no—
382C8—3621no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3621no—
390C22—3621no—
391C22—3622no—
392C22—14484no—
393C22—14484no—
394C22—1812no—
395C22—7241no—
396C22—7241no—
397C22—7241no—
398C22—7241no—
399C22—7241no—
400C22—7241no—
401C22—7241no—
402C22—7241no—
403C22C2×D4×D18111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—3621no—
409C4—3621no—
410C4—1812no—
411C4—7241no—
412C4C2×D4×D18111yescharacteristic
413C4#41×D18111yescharacteristic
414C2—42no—
415C2—81no—
416C2—3622no—
417C2—14481no—
418C2—14481no—
419C2—7242no—
420C2#121×D18111yescharacteristic
421C2#2604208111yescharacteristic
422C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#463×C181C2
socleC22×C181C22×D4
hypercenter#463D181
hypocenterC181#3041
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19153843C2semidirect
2. C2×D8×D181C2semidirect
3. #26041847C2semidirect
4. #26041751C2semidirect
5. #26041497C2semidirect
6. #26041496C2semidirect
7. #26041491C2semidirect
8. #26041475C2semidirect
9. #26041474C2semidirect
10. #26041473C2semidirect
11. #26041472C2semidirect
12. #26041304C2semidirect
13. #26041282C2semidirect
14. #26041274C2semidirect
15. #463×C181C2semidirect
20. C2×#15668436C22semidirect
22. C2×D1448C22semidirect
25. C2×#15668434C22semidirect
29. #19153813C22semidirect
30. #19153812C22semidirect
31. #19153805C22semidirect
36. C4×D724C22semidirect
37. D4×Dic181C22semidirect
38. #19153778C22semidirect
45. C4⋊#13089678C22semidirect
46. #19153745C22semidirect
47. #19153745C22semidirect
48. #19153740C22semidirect
49. #19153738C22semidirect
50. #19153738C22semidirect
51. #19153737C22semidirect
52. #19153736C22semidirect
53. #19153736C22semidirect
54. C2×SD16×C181C22semidirect
56. #124×C181C22semidirect
57. C4×D4×C181C22semidirect
58. C4⋊C8×C181C22semidirect
59. #103×C181C22semidirect
60. #102×C181C22semidirect
78. D4×D181D4semidirect
211. C181#3041semidirect
220. #463D181semidirect
233. C2×SD16C2×D181semidirect
246. #124C2×D181semidirect
256. C4×D4C2×D181semidirect
264. C4⋊C8C2×D181semidirect
266. #103C2×D181semidirect
272. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
C2×D8×D1812
#260418472
#260417512
#260414972
#260414962
#260414912
#260414752
#260414742
#260414732
#260414722
#260413042
#260412822
#260412742
#463×C1812
#3041181