Finite Group Databasebeta

← #37545216

Subgroups — #37545216

Order 23168.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153843C211yesmaximal, characteristic, complemented
2#26041940C211yesmaximal, characteristic, complemented
3D4×C22⋊D181C211yesmaximal, characteristic, complemented
4#26041665C211yesmaximal, characteristic, complemented
5#26041522C211yesmaximal, characteristic, complemented
6#26041515C211yesmaximal, characteristic, complemented
7#26041496C211yesmaximal, characteristic, complemented
8#26041477C211yesmaximal, characteristic, complemented
9#26041428C211yesmaximal, characteristic, complemented
10#26041423C211yesmaximal, characteristic, complemented
11#26041418C211yesmaximal, characteristic, complemented
12#26041413C211yesmaximal, characteristic, complemented
13#26041380C211yesmaximal, characteristic, complemented
14#26041350C211yesmaximal, characteristic, complemented
15C22×SD16×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic, complemented
17C22×C22⋊D181—21no—
18C2×D4×D181C2211yescharacteristic
19#19153843—24no—
20C2×#15668436C2211yescharacteristic, complemented
21C2×#15668436C2211yescharacteristic, complemented
22C2×D1448C2211yescharacteristic, complemented
23C2×#15668434C2211yescharacteristic
24C2×#15668434C2211yescharacteristic, complemented
25C2×#15668432C2211yescharacteristic, complemented
26#19153812C2211yescharacteristic
27#19153805—21no—
28#19153805C2211yescharacteristic, complemented
29#19153792—21no—
30#19153792C2211yescharacteristic, complemented
31#19153790—21no—
32#19153790C2211yescharacteristic
33#19153790C2211yescharacteristic, complemented
34#19153788—21no—
35#19153787C2211yescharacteristic
36#19153785—21no—
37#19153784—21no—
38Q8×Dic181C2211yescharacteristic, complemented
39D4×Dic181C2211yescharacteristic
40C4×C22⋊D181—21no—
41C4×C22⋊D181C2211yescharacteristic
42#19153750C2211yescharacteristic, complemented
43#19153745C2211yescharacteristic, complemented
44#19153740C2211yescharacteristic, complemented
45#19153740C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic, complemented
47#19153738C2211yescharacteristic, complemented
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic, complemented
50C22⋊#13089678C2211yescharacteristic, complemented
51#19153728C2211yescharacteristic, complemented
52#19153726C2211yescharacteristic, complemented
53C22×Q8×C181C2211yescharacteristic, complemented
54C22×D4×C181C2211yescharacteristic
55C2×SD16×C181—24no—
56C2×SD16×C181C2211yescharacteristic
57C2×SD16×C181C2211yescharacteristic
58C2×SD16×C181C2211yescharacteristic, complemented
59C2×SD16×C181C2211yescharacteristic, complemented
60C22×C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668450—22no—
63#15668450—41no—
64C2×D724—21no—
65C2×D724C2311yescharacteristic
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—44no—
74C2×C22⋊D181D411yescharacteristic
75D4×D181—22no—
76D4×D181—41no—
77C22×Dic181—21no—
78C2×C4×D181—21no—
79C2×C4×D181C2311yescharacteristic
80#15668436—22no—
81#15668436—41no—
82D1448—22no—
83#15668434—22no—
84#15668434—41no—
85#15668432—22no—
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C4⋊Dic181—21no—
89C4⋊Dic181C2311yescharacteristic
90C4⋊Dic181C2311yescharacteristic
91#15668428—21no—
92#15668428D411yescharacteristic
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181D411yescharacteristic
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426D411yescharacteristic
102C4×Dic181—21no—
103C4×Dic181C2311yescharacteristic
104C24×C181—21no—
105C2×Q8×C181—21no—
106C2×Q8×C181—22no—
107C2×Q8×C181—41no—
108C2×Q8×C181C2311yescharacteristic
109C2×Q8×C181C2311yescharacteristic
110C2×D4×C181—22no—
111C2×D4×C181—21no—
112C2×D4×C181—41no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181—21no—
116C22×C4×C181C2311yescharacteristic
117SD16×C181—22no—
118SD16×C181—22no—
119SD16×C181—22no—
120SD16×C181D414yescomplemented
121C2×C8×C181—22no—
122C2×C8×C181C2311yescharacteristic
123C2×C8×C181C2311yescharacteristic
124C22×D181—21no—
125C22×D181—21no—
126C22×D181—42no—
127C22×D181—42no—
128C22×D181—81no—
129C22×D181C2×D411yescharacteristic
130D724—21no—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—42no—
135C22⋊D181—22no—
136C22⋊D181—42no—
137C22⋊D181—42no—
138C22⋊D181—42no—
139C22⋊D181—42no—
140C22⋊D181—42no—
141C22⋊D181—81no—
142C2×Dic181—21no—
143C2×Dic181—21no—
144C2×Dic181—42no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C2×Dic181C2×D411yescharacteristic
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—42no—
151#13089678—21no—
152#13089678—21no—
153C23×C181—21no—
154C23×C181—21no—
155C23×C181—42no—
156C23×C181—42no—
157C23×C181—81no—
158C23×C181C2×D411yescharacteristic
159Q8×C181—21no—
160Q8×C181—21no—
161Q8×C181—21no—
162Q8×C181—42no—
163Q8×C181C2×D412yescomplemented
164D4×C181—21no—
165D4×C181—21no—
166D4×C181—21no—
167D4×C181—42no—
168D4×C181C2×D412yes—
169C2×C4×C181—21no—
170C2×C4×C181—21no—
171C2×C4×C181—21no—
172C2×C4×C181—21no—
173C2×C4×C181—42no—
174C2×C4×C181C2×D411yescharacteristic
175C2×C4×C181C2411yescharacteristic
176C8×C181—21no—
177C8×C181C2×D412yescomplemented
178C22×C181—21no—
179C22×C181—21no—
180C22×C181—22no—
181C22×C181—41no—
182C22×C181—41no—
183C22×C181—41no—
184C22×C181—41no—
185C22×C181—41no—
186C22×C181—84no—
187C22×C181AGL(1,ℤ/8)12yescomplemented
188C22×C181C22×D411yescharacteristic
189C2×D181—21no—
190C2×D181—21no—
191C2×D181—21no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—41no—
195C2×D181—41no—
196C2×D181—41no—
197C2×D181—41no—
198C2×D181—84no—
199C4×C181—21no—
200C4×C181—22no—
201C4×C181—41no—
202C4×C181C22×D411yescharacteristic
203C4×C181H(5,𝔽2)11yescharacteristic
204Dic181—21no—
205Dic181—21no—
206Dic181—21no—
207Dic181—41no—
208Dic181—41no—
209C2×C181—22no—
210C2×C181—42no—
211C2×C181—81no—
212C2×C181D4211yescharacteristic
213C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C181#57711yescharacteristic
215D181—41no—
216D181—81no—
217D181—81no—
218C181#326511yescharacteristic, complemented, Sylow
219#3265—1811nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1811no—
221C22×SD16D18111yescharacteristic, complemented
222#548—1811no—
223D42—1811no—
224#496—1811no—
225#468—1811no—
226#466—1811no—
227#463—1811no—
228#460—1811no—
229#450—1811no—
230#449—1811no—
231#448—1811no—
232#447—1811no—
233#440—1811no—
234#435—1811no—
235C2×#41—1811no—
236C22×Q8C2×D18111yescharacteristic, complemented
237C22×D4—3621no—
238C22×D4—1811no—
239C22×D4C2×D18111yescharacteristic
240AGL(1,ℤ/8)—3624no—
241C2×SD16—24no—
242C2×SD16—1811no—
243C2×SD16—1811no—
244C2×SD16C2×D18111yescharacteristic
245C2×SD16C2×D18111yescharacteristic
246C2×SD16C2×D18111yescharacteristic, complemented
247C2×SD16C2×D18111yescharacteristic, complemented
248C2×D8—1811no—
249C2×D8—1811no—
250C2×D8—1811no—
251C2×M16—1811no—
252C22×C8C2×D18111yescharacteristic, complemented
253#127—1811no—
254#124—3621no—
255#124—1811no—
256C22⋊Q8—3621no—
257C22⋊Q8—1811no—
258#121—3621no—
259#121—3621no—
260#121—1811no—
261#121—1811no—
262#121—1811no—
263#120—3621no—
264#120—3621no—
265C4×Q8—1811no—
266C4×D4—3621no—
267C4×D4—1811no—
268C4×D4—1811no—
269#107—1811no—
270C4⋊C8—1811no—
271#103—1811no—
272#103—1811no—
273#102—1811no—
274#102—1811no—
275#102—1811no—
276#102—1811no—
277C22⋊C8—1811no—
278C22⋊C8—1811no—
279#97—1811no—
280C24—21no—
281C24—3621no—
282#41—3622no—
283#41—7241no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D18111yescharacteristic
288C2×Q8C22×D18111yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3621no—
293C2×D4—3622no—
294C2×D4—3621no—
295C2×D4—3621no—
296C2×D4—3621no—
297C2×D4—3621no—
298C2×D4—3621no—
299C2×D4—3621no—
300C2×D4—1811no—
301C2×D4—1811no—
302C2×D4—1811no—
303C2×D4—7241no—
304C2×D4—7244no—
305C2×D4C22×D18111yescharacteristic
306C2×D4C22×D18111yescharacteristic
307C22×C4—21no—
308C22×C4—3621no—
309C22×C4—3621no—
310C22×C4—1811no—
311C22×C4C22×D18111yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—22no—
315SD16—3622no—
316SD16—7241no—
317SD16C22⋊D18114yescomplemented
318D8—3622no—
319D8—3622no—
320D8—7241no—
321M16—3622no—
322C2×C8—22no—
323C2×C8—1811no—
324C2×C8—1811no—
325C2×C8C22×D18111yescharacteristic
326C2×C8C22×D18111yescharacteristic
327C4⋊C4—3621no—
328C4⋊C4—3621no—
329C4⋊C4—1811no—
330C4⋊C4—1811no—
331C4⋊C4—1811no—
332C22⋊C4—3621no—
333C22⋊C4—3621no—
334C22⋊C4—3621no—
335C22⋊C4—3621no—
336C22⋊C4—3621no—
337C22⋊C4—3621no—
338C22⋊C4—3621no—
339C22⋊C4—1811no—
340C22⋊C4—1811no—
341C42—3621no—
342C42—1811no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—3621no—
349C23—3621no—
350C23—14481no—
351C23—1811no—
352C23—7242no—
353C23—7242no—
354C23D4×D18111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8C2×C22⋊D18112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—3622no—
365D4—3621no—
366D4—3621no—
367D4—3621no—
368D4—3621no—
369D4—14481no—
370D4—7242no—
371D4—7242no—
372D4—7242no—
373D4—7242no—
374D4—7242no—
375D4—7242no—
376D4C2×C22⋊D18112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—3621no—
383C2×C4—3621no—
384C2×C4—3621no—
385C2×C4—3621no—
386C2×C4—1811no—
387C2×C4—1811no—
388C2×C4—1811no—
389C2×C4—7242no—
390C2×C4—7242no—
391C2×C4D4×D18111yescharacteristic
392C2×C4C23×D18111yescharacteristic
393C8—21no—
394C8—3621no—
395C8—3621no—
396C8C2×C22⋊D18112yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—84no—
406C22—3621no—
407C22—3621no—
408C22—3621no—
409C22—14484no—
410C22—7241no—
411C22—7241no—
412C22—7241no—
413C22—7241no—
414C22—7241no—
415C22—7241no—
416C22#1915384312yescomplemented
417C22C2×D4×D18111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—3621no—
422C4—3621no—
423C4—3621no—
424C4—7241no—
425C4—7241no—
426C4C22×C22⋊D18111yescharacteristic
427C4#1915386111yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—14481no—
432C2—14481no—
433C2—7241no—
434C2D4×C22⋊D18111yescharacteristic
435C2C2×#1915384311yescharacteristic
436C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC22×SD16×C181C2
socleC22×C181C22×D4
hypercenterC22×SD16D181
hypocenterC181#3265
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153843C2semidirect
2. #26041940C2semidirect
3. D4×C22⋊D181C2semidirect
4. #26041665C2semidirect
5. #26041522C2semidirect
6. #26041515C2semidirect
7. #26041496C2semidirect
8. #26041477C2semidirect
9. #26041428C2semidirect
10. #26041423C2semidirect
11. #26041418C2semidirect
12. #26041413C2semidirect
13. #26041380C2semidirect
14. #26041350C2semidirect
15. C22×SD16×C181C2semidirect
16. C2×#15668450C22semidirect
20. C2×#15668436C22semidirect
21. C2×#15668436C22semidirect
22. C2×D1448C22semidirect
24. C2×#15668434C22semidirect
25. C2×#15668432C22semidirect
28. #19153805C22semidirect
30. #19153792C22semidirect
33. #19153790C22semidirect
38. Q8×Dic181C22semidirect
42. #19153750C22semidirect
43. #19153745C22semidirect
44. #19153740C22semidirect
45. #19153740C22semidirect
46. #19153738C22semidirect
47. #19153738C22semidirect
49. #19153737C22semidirect
50. C22⋊#13089678C22semidirect
51. #19153728C22semidirect
52. #19153726C22semidirect
53. C22×Q8×C181C22semidirect
58. C2×SD16×C181C22semidirect
59. C2×SD16×C181C22semidirect
60. C22×C8×C181C22semidirect
120. SD16×C181D4semidirect
163. Q8×C181C2×D4semidirect
177. C8×C181C2×D4semidirect
187. C22×C181AGL(1,ℤ/8)semidirect
218. C181#3265semidirect
221. C22×SD16D181semidirect
236. C22×Q8C2×D181semidirect
246. C2×SD16C2×D181semidirect
247. C2×SD16C2×D181semidirect
252. C22×C8C2×D181semidirect
317. SD16C22⋊D181semidirect
359. Q8C2×C22⋊D181semidirect
396. C8C2×C22⋊D181semidirect
416. C22#19153843semidirect

Maximal subgroups

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

subgroupindex
C2×#191538432
#260419402
D4×C22⋊D1812
#260416652
#260415222
#260415152
#260414962
#260414772
#260414282
#260414232
#260414182
#260414132
#260413802
#260413502
C22×SD16×C1812
#3265181