Finite Group Databasebeta

← #37545327

Subgroups — #37545327

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×#19153843C211yesmaximal, characteristic, complemented
2C2×#19153840C211yesmaximal, characteristic, complemented
3#26041924C211yesmaximal, characteristic, complemented
4#26041922C211yesmaximal, characteristic, complemented
5#26041661C211yesmaximal, characteristic, complemented
6#26041528C211yesmaximal, characteristic, complemented
7#26041524C211yesmaximal, characteristic, complemented
8#26041495C211yesmaximal, characteristic, complemented
9#26041444C211yesmaximal, characteristic, complemented
10#26041438C211yesmaximal, characteristic, complemented
11#26041414C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13C4×#15668434C211yesmaximal, characteristic, complemented
14#26041356C211yesmaximal, characteristic, complemented
15#463×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic, complemented
19C2×D4×D181C2211yescharacteristic, complemented
20#19153843—24no—
21#19153840—24no—
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668438C2211yescharacteristic, complemented
24C2×#15668437C2211yescharacteristic, complemented
25C2×#15668436C2211yescharacteristic, complemented
26C2×D1448C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C2×#15668432C2211yescharacteristic, complemented
30C2×#15668432C2211yescharacteristic, complemented
31#19153804C2211yescharacteristic, complemented
32#19153800—21no—
33#19153798—21no—
34#19153796C2211yescharacteristic, complemented
35#19153792C2211yescharacteristic, complemented
36#19153791C2211yescharacteristic
37#19153790—21no—
38#19153789—21no—
39#19153787C2211yescharacteristic
40#19153786—21no—
41#19153786—21no—
42C4×D724C2211yescharacteristic
43C4×D724C2211yescharacteristic, complemented
44C2×#15668426—21no—
45C22⋊C4×D181—21no—
46#19153749C2211yescharacteristic, complemented
47#19153741C2211yescharacteristic, complemented
48#19153738C2211yescharacteristic, complemented
49#19153737C2211yescharacteristic, complemented
50#19153736C2211yescharacteristic, complemented
51#19153728C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C4×#13089678C2211yescharacteristic, 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—41no—
65#15668448—22no—
66#15668448—41no—
67C2×Dic362C2311yescharacteristic
68C2×D724—21no—
69C2×D724—21no—
70C2×D724D411yescharacteristic
71C2×D724C2311yescharacteristic
72C2×D724C2311yescharacteristic
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77D4×D181—22no—
78D4×D181—22no—
79D4×D181—41no—
80D4×D181—41no—
81C22×Dic181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181C2311yescharacteristic
85C2×C4×D181C2311yescharacteristic
86#15668439—41no—
87#15668438—22no—
88#15668437—22no—
89#15668436—22no—
90D1448—22no—
91#15668434—41no—
92#15668434D414yescomplemented
93#15668432—22no—
94#15668432—22no—
95C2×#13089678C2311yescharacteristic
96C2×#13089678C2311yescharacteristic
97C4⋊Dic181—21no—
98C4⋊Dic181D411yescharacteristic
99C4⋊Dic181C2311yescharacteristic
100#15668428—21no—
101C22⋊Dic181—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426D411yescharacteristic
109#15668426D411yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181—21no—
114SD16×C181—22no—
115D8×C181—22no—
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C4⋊C4×C181C2311yescharacteristic
119C22⋊C4×C181—21no—
120C22⋊C4×C181—21no—
121C22⋊C4×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—42no—
128C22×D181—42no—
129C22×D181—81no—
130C22×D181—81no—
131C22×D181C2×D411yescharacteristic
132C22×D181C2×D411yescharacteristic
133Dic362—21no—
134Dic362—21no—
135D724—21no—
136D724—21no—
137D724—21no—
138D724—41no—
139D724—42no—
140D724—81no—
141D724C2×D412yes—
142C22⋊D181—42no—
143C22⋊D181—42no—
144C22⋊D181—42no—
145C22⋊D181—81no—
146C2×Dic181—21no—
147C2×Dic181—21no—
148C2×Dic181—42no—
149C2×Dic181C2×D411yescharacteristic
150C2×Dic181C2×D411yescharacteristic
151C4×D181—21no—
152C4×D181—21no—
153C4×D181—21no—
154C4×D181—21no—
155C4×D181—42no—
156C4×D181—81no—
157#13089678—21no—
158#13089678C2×D412yes—
159C23×C181—21no—
160C23×C181—21no—
161Q8×C181—21no—
162Q8×C181—21no—
163D4×C181—21no—
164D4×C181—21no—
165D4×C181—21no—
166D4×C181C2×D412yes—
167C2×C4×C181—21no—
168C2×C4×C181—21no—
169C2×C4×C181—81no—
170C2×C4×C181C2×D411yescharacteristic
171C2×C4×C181C2×D411yescharacteristic
172C2×C4×C181C2411yescharacteristic
173C8×C181—21no—
174C8×C181—21no—
175C22×C181—22no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181C22×D411yescharacteristic
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—22no—
185C2×D181—21no—
186C2×D181—21no—
187C2×D181—21no—
188C2×D181—21no—
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—21no—
198C4×C181—41no—
199C4×C181—41no—
200C4×C181C22×D411yescharacteristic
201C4×C181H(5,𝔽2)11yescharacteristic
202Dic181—21no—
203Dic181—21no—
204Dic181—41no—
205Dic181—41no—
206C2×C181—42no—
207C2×C181—81no—
208C2×C181D4211yescharacteristic
209C2×C181#57511yescharacteristic
210C2×C181#57611yescharacteristic
211D181—41no—
212D181—41no—
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#468—1811no—
224#463—1811no—
225#463D18111yescharacteristic, complemented
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×D8C2×D18111yescharacteristic
247C2×M16—1811no—
248C2×M16—1811no—
249#124—1811no—
250#124C2×D18111yescharacteristic, complemented
251#123—3621no—
252#123—3621no—
253C22⋊Q8—1811no—
254C22⋊Q8—1811no—
255#121—3621no—
256#121—3621no—
257#121—1811no—
258#121—1811no—
259#120—3621no—
260#120—3621no—
261C4×D4—1811no—
262C4×D4—1811no—
263C4×D4C2×D18111yescharacteristic, complemented
264C2×C22⋊C4—3621no—
265C2×C22⋊C4—3621no—
266#107—1811no—
267C4⋊C8C2×D18111yescharacteristic, complemented
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—3621no—
286C2×D4—3621no—
287C2×D4—3622no—
288C2×D4—3622no—
289C2×D4—3621no—
290C2×D4—3621no—
291C2×D4—3621no—
292C2×D4—3621no—
293C2×D4—1811no—
294C2×D4—1811no—
295C2×D4—1811no—
296C2×D4—7241no—
297C2×D4—7241no—
298C2×D4C22×D18111yescharacteristic
299C2×D4C22×D18111yescharacteristic
300C22×C4—21no—
301C22×C4—3621no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4—1811no—
305C22×C4—1811no—
306Q16—7241no—
307SD16—22no—
308SD16—3622no—
309SD16—3622no—
310SD16—3622no—
311D8—22no—
312D8—3622no—
313D8—1814no—
314D8—7241no—
315M16—3622no—
316M16—3622no—
317C2×C8—1811no—
318C2×C8—1811no—
319C2×C8C22×D18111yescharacteristic
320C2×C8C22×D18111yescharacteristic
321C4⋊C4—3621no—
322C4⋊C4—3621no—
323C4⋊C4—1811no—
324C4⋊C4—1811no—
325C4⋊C4C22×D18111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
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—
338C42C22×D18111yescharacteristic
339C23—21no—
340C23—21no—
341C23—3621no—
342C23—3621no—
343C23—14481no—
344C23—14481no—
345C23—1811no—
346C23—1811no—
347C23—7242no—
348C23—7242no—
349C23—7242no—
350C23—7242no—
351Q8—21no—
352Q8—21no—
353Q8—3621no—
354Q8—3621no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—3621no—
359D4—3621no—
360D4—3621no—
361D4—14481no—
362D4—14481no—
363D4—1812no—
364D4—7241no—
365D4—7242no—
366D4—7242no—
367D4—7242no—
368D4—7242no—
369D4D4×D18112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—3621no—
374C2×C4—3621no—
375C2×C4—3621no—
376C2×C4—3621no—
377C2×C4—3621no—
378C2×C4—3621no—
379C2×C4—14481no—
380C2×C4—1811no—
381C2×C4—1811no—
382C2×C4—7242no—
383C2×C4—7242no—
384C2×C4D4×D18111yescharacteristic
385C2×C4D4×D18111yescharacteristic
386C2×C4C23×D18111yescharacteristic
387C8—21no—
388C8—21no—
389C8—3621no—
390C8—1812no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3621no—
398C22—3621no—
399C22—3622no—
400C22—3621no—
401C22—3621no—
402C22—3621no—
403C22—3621no—
404C22—14484no—
405C22—14484no—
406C22—7241no—
407C22—7241no—
408C22—7241no—
409C22—7241no—
410C22—7241no—
411C22C2×D4×D18111yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3621no—
417C4—3621no—
418C4—7241no—
419C4—7241no—
420C4C2×D4×D18111yescharacteristic
421C4#1915386211yescharacteristic
422C2—42no—
423C2—81no—
424C2—14481no—
425C2—7241no—
426C2—7241no—
427C2—7242no—
428C2#2604191311yescharacteristic
429C2#2604207611yescharacteristic
430C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#463×C181C2
socleC22×C181C22×D4
hypercenter#463D181
hypocenterC181#3273
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×#19153840C2semidirect
3. #26041924C2semidirect
4. #26041922C2semidirect
5. #26041661C2semidirect
6. #26041528C2semidirect
7. #26041524C2semidirect
8. #26041495C2semidirect
9. #26041444C2semidirect
10. #26041438C2semidirect
11. #26041414C2semidirect
12. #26041411C2semidirect
13. C4×#15668434C2semidirect
14. #26041356C2semidirect
15. #463×C181C2semidirect
18. C2×D4×D181C22semidirect
19. C2×D4×D181C22semidirect
22. C2×#15668439C22semidirect
23. C2×#15668438C22semidirect
24. C2×#15668437C22semidirect
25. C2×#15668436C22semidirect
29. C2×#15668432C22semidirect
30. C2×#15668432C22semidirect
31. #19153804C22semidirect
34. #19153796C22semidirect
35. #19153792C22semidirect
43. C4×D724C22semidirect
46. #19153749C22semidirect
47. #19153741C22semidirect
48. #19153738C22semidirect
49. #19153737C22semidirect
50. #19153736C22semidirect
51. #19153728C22semidirect
52. #19153728C22semidirect
53. C4×#13089678C22semidirect
54. C2×SD16×C181C22semidirect
56. #124×C181C22semidirect
57. C4×D4×C181C22semidirect
58. C4⋊C8×C181C22semidirect
59. #103×C181C22semidirect
60. #102×C181C22semidirect
92. #15668434D4semidirect
215. C181#3273semidirect
225. #463D181semidirect
242. C2×SD16C2×D181semidirect
250. #124C2×D181semidirect
263. C4×D4C2×D181semidirect
267. C4⋊C8C2×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×#191538432
C2×#191538402
#260419242
#260419222
#260416612
#260415282
#260415242
#260414952
#260414442
#260414382
#260414142
#260414112
C4×#156684342
#260413562
#463×C1812
#3273181