Finite Group Databasebeta

← #37543091

Subgroups — #37543091

Order 23168.

736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042116C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D181C212yesmaximal, complemented
3#26041847C211yesmaximal, characteristic, complemented
4#26041454C211yesmaximal, characteristic, complemented
5#26041452C211yesmaximal, characteristic, complemented
6#26041451C211yesmaximal, characteristic, complemented
7#26041448C212yesmaximal, complemented
8#26041445C212yesmaximal, complemented
9#26041294C212yesmaximal, complemented
10#26041262C211yesmaximal, characteristic, complemented
11#454×C181C211yesmaximal, characteristic, complemented
12#19153864—22no—
13#19153864C2211yescharacteristic, complemented
14#19153862—22no—
15#19153862—22no—
16#19153862C2212yescomplemented
17C2×#15668450—22no—
18C2×#15668450C2211yescharacteristic
19#41×D181—22no—
20#41×D181—22no—
21#41×D181C2212yes—
22C2×D4×D181C2211yescharacteristic, complemented
23#19153844C2212yescomplemented
24#19153843—22no—
25#19153842C2212yescomplemented
26#19153841C2212yescomplemented
27#19153840—22no—
28SD16×D181—22no—
29D8×D181—22no—
30M16×D181C2212yescomplemented
31#19153813C2211yescharacteristic, complemented
32#19153812C2211yescharacteristic, complemented
33#19153805C2211yescharacteristic, complemented
34#19153789—22no—
35#19153786—22no—
36#19153773C2211yescharacteristic, complemented
37#19153744C2212yescomplemented
38#19153743C2212yescomplemented
39#19153742C2212yescomplemented
40#19153735C2211yescharacteristic, complemented
41#19153732C2212yescomplemented
422−1+4×C181C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C181C2212yescomplemented
44#124×C181C2211yescharacteristic, complemented
45#104×C181C2212yescomplemented
46#101×C181C2211yescharacteristic, complemented
47C23×D181—21no—
48#15668450—22no—
49#15668450—24no—
50#15668450—22no—
51#15668450—22no—
52#15668450—22no—
53#15668450—22no—
54#15668450D412yescomplemented
55#15668449—24no—
56#15668449—24no—
57#15668449—22no—
58#15668449—22no—
59#15668449—22no—
60#15668449C2312yes—
61#15668448—22no—
62#15668448—22no—
63#15668448D412yescomplemented
64Q8×D181—22no—
65Q8×D181—22no—
66Q8×D181D412yescomplemented
67C2×D724—24no—
68C2×D724—22no—
69C2×D724—22no—
70C2×D724D412yes—
71C2×D724C2311yescharacteristic
72C2×D724C2311yescharacteristic
73C2×C22⋊D181—22no—
74C2×C22⋊D181—21no—
75D4×D181—24no—
76D4×D181—22no—
77D4×D181—24no—
78D4×D181—22no—
79D4×D181—22no—
80D4×D181—22no—
81D4×D181—22no—
82D4×D181D412yescomplemented
83C2×C4×D181—24no—
84C2×C4×D181—22no—
85C2×C4×D181D412yes—
86C2×C4×D181C2311yescharacteristic
87#15668438—22no—
88#15668437—22no—
89#15668436—22no—
90D1448—22no—
91#15668434—22no—
92#15668434—22no—
93#15668433C2312yes—
94#15668432—22no—
95C8×D181—22no—
96#15668428—21no—
97#15668426—21no—
98#15668426—22no—
99C4×Dic181C2311yescharacteristic
100#41×C181—22no—
101#41×C181—22no—
102#41×C181C2312yes—
103C2×Q8×C181—22no—
104C2×Q8×C181C2311yescharacteristic
105C2×D4×C181C2311yescharacteristic
106SD16×C181—22no—
107D8×C181—22no—
108M16×C181C2312yes—
109C22⋊C4×C181—22no—
110C42×C181C2311yescharacteristic
111C22×D181—24no—
112C22×D181—22no—
113C22×D181—21no—
114C22×D181—42no—
115C22×D181—82no—
116C22×D181C2×D412yes—
117C22×D181C2×D411yescharacteristic
118Dic362—22no—
119Dic362—22no—
120Dic362C2×D412yes—
121D724—24no—
122D724—22no—
123D724—22no—
124D724—24no—
125D724—22no—
126D724—22no—
127D724—82no—
128D724C2×D412yes—
129C22⋊D181—24no—
130C22⋊D181—22no—
131C22⋊D181—22no—
132C22⋊D181—22no—
133C22⋊D181—41no—
134C22⋊D181—41no—
135C22⋊D181—82no—
136C2×Dic181—22no—
137C2×Dic181—21no—
138C2×Dic181C2×D411yescharacteristic
139C4×D181—24no—
140C4×D181—24no—
141C4×D181—22no—
142C4×D181—22no—
143C4×D181—22no—
144C4×D181—22no—
145C4×D181—22no—
146C4×D181C2×D412yes—
147C4×D181C2×D412yes—
148#13089678—22no—
149C23×C181—21no—
150Q8×C181—22no—
151Q8×C181—22no—
152Q8×C181C2×D412yes—
153D4×C181—22no—
154D4×C181—22no—
155D4×C181—22no—
156D4×C181C2×D412yes—
157C2×C4×C181—24no—
158C2×C4×C181—22no—
159C2×C4×C181—21no—
160C2×C4×C181C2×D412yes—
161C2×C4×C181C2411yescharacteristic
162C8×C181—22no—
163C22×C181—22no—
164C22×C181—42no—
165C22×C181—41no—
166C22×C181C22×D411yescharacteristic
167C2×D181—22no—
168C2×D181—22no—
169C2×D181—41no—
170C2×D181—42no—
171C2×D181—42no—
172C2×D181—41no—
173C2×D181—82no—
174C2×D181—84no—
175C2×D181—84no—
176C2×D181#12012yes—
177C4×C181—22no—
178C4×C181—22no—
179C4×C181—41no—
180C4×C181C22×D412yes—
181Dic181—22no—
182Dic181—41no—
183Dic181#12011yescharacteristic
184Dic181#12011yescharacteristic
185C2×C181—21no—
186C2×C181—42no—
187C2×C181—81no—
188C2×C181C2×#12011yescharacteristic
189D181—22no—
190D181—42no—
191D181—42no—
192D181—81no—
193C181#302111yescharacteristic, complemented, Sylow
194#3021—1811nomaximal, Sylow
195#585—1811no—
196C2×AGL(1,ℤ/8)—1812no—
197#535—1811no—
198#454—1811no—
199#454—1811no—
200#454—1811no—
201#454D18111yescharacteristic, complemented
202#453—1812no—
203#453—1812no—
204#421—1812no—
205#413—1811no—
2062−1+4—3622no—
2072−1+4—1811no—
2082−1+4C2×D18111yescharacteristic, complemented
209H(5,𝔽2)—3622no—
210H(5,𝔽2)—3622no—
211H(5,𝔽2)—1812no—
212C2×#41—3622no—
213C2×#41—3622no—
214C2×#41—3622no—
215C2×#41—1812no—
216C2×#41—1811no—
217C22×D4—1811no—
218AGL(1,ℤ/8)—3622no—
219AGL(1,ℤ/8)—3622no—
220AGL(1,ℤ/8)—1812no—
221AGL(1,ℤ/8)—1812no—
222AGL(1,ℤ/8)—1812no—
223AGL(1,ℤ/8)C2×D18112yescomplemented
224C2×SD16—3622no—
225C2×D8—3622no—
226C2×M16—1812no—
227#127—1811no—
228#127—1811no—
229#124—1811no—
230#124C2×D18111yescharacteristic, complemented
231#121—3622no—
232#120—3622no—
233#117—1811no—
234#104—1812no—
235#104—1812no—
236#104—1812no—
237#104C2×D18112yescomplemented
238#101—1811no—
239#101C2×D18111yescharacteristic, complemented
240#100—1812no—
241C24—3621no—
242#41—22no—
243#41—22no—
244#41—3624no—
245#41—3624no—
246#41—3622no—
247#41—3622no—
248#41—3622no—
249#41—3622no—
250#41—3624no—
251#41—3622no—
252#41—3622no—
253#41—3622no—
254#41—3622no—
255#41—3622no—
256#41—3622no—
257#41—1812no—
258#41—1812no—
259#41—1812no—
260#41C22×D18112yes—
261C2×Q8—22no—
262C2×Q8—3622no—
263C2×Q8—3622no—
264C2×Q8—1812no—
265C2×Q8C22×D18111yescharacteristic
266C2×D4—3624no—
267C2×D4—3622no—
268C2×D4—3622no—
269C2×D4—3624no—
270C2×D4—3622no—
271C2×D4—3624no—
272C2×D4—3622no—
273C2×D4—3622no—
274C2×D4—3622no—
275C2×D4—3622no—
276C2×D4—3622no—
277C2×D4—3621no—
278C2×D4—1812no—
279C2×D4—1812no—
280C2×D4—1811no—
281C2×D4—1811no—
282C2×D4C22×D18111yescharacteristic
283C22×C4—3624no—
284C22×C4—3622no—
285C22×C4—1812no—
286C22×C4—1811no—
287SD16—22no—
288SD16—3622no—
289SD16—3622no—
290SD16—3622no—
291D8—22no—
292D8—3622no—
293D8—3622no—
294D8—3622no—
295M16—3622no—
296M16—1812no—
297M16C22×D18112yes—
298C2×C8—3622no—
299C4⋊C4—3621no—
300C22⋊C4—22no—
301C22⋊C4—3621no—
302C22⋊C4—3622no—
303C42—1811no—
304C42C22×D18111yescharacteristic
305C23—21no—
306C23—3624no—
307C23—3622no—
308C23—3621no—
309C23—14482no—
310C23—1812no—
311C23—1811no—
312C23—7242no—
313Q8—22no—
314Q8—22no—
315Q8—3622no—
316Q8—3622no—
317Q8—1812no—
318Q8D4×D18112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—3624no—
323D4—3622no—
324D4—3622no—
325D4—3624no—
326D4—3624no—
327D4—3622no—
328D4—3622no—
329D4—3622no—
330D4—3622no—
331D4—3622no—
332D4—14482no—
333D4—14482no—
334D4—1812no—
335D4—7241no—
336D4—7241no—
337D4D4×D18112yes—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—3624no—
342C2×C4—3624no—
343C2×C4—3622no—
344C2×C4—3622no—
345C2×C4—3622no—
346C2×C4—3622no—
347C2×C4—3622no—
348C2×C4—3622no—
349C2×C4—3621no—
350C2×C4—1811no—
351C2×C4—1812no—
352C2×C4—1812no—
353C2×C4D4×D18112yes—
354C2×C4C23×D18111yescharacteristic
355C8—22no—
356C8—3622no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—3622no—
361C22—3622no—
362C22—14482no—
363C22—14484no—
364C22—14484no—
365C22—1812no—
366C22—7241no—
367C22—7242no—
368C22—7242no—
369C22—7241no—
370C22C2×D4×D18111yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—3622no—
375C4—1811no—
376C4—1811no—
377C4—7241no—
378C4C2×D4×D18112yes—
379C2—21no—
380C2—42no—
381C2—81no—
382C2—3622no—
383C2—14481no—
384C2—7242no—
385C2—7242no—
386C2#120×D18111yescharacteristic
subgroupG/H
centerC2#120×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#454×C181C2
socleC2×C181C2×#120
hypercenter#454D181
hypocenterC181#3021
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26042116C2semidirect
2. AGL(1,ℤ/8)×D181C2semidirect
3. #26041847C2semidirect
4. #26041454C2semidirect
5. #26041452C2semidirect
6. #26041451C2semidirect
7. #26041448C2semidirect
8. #26041445C2semidirect
9. #26041294C2semidirect
10. #26041262C2semidirect
11. #454×C181C2semidirect
13. #19153864C22semidirect
16. #19153862C22semidirect
22. C2×D4×D181C22semidirect
23. #19153844C22semidirect
25. #19153842C22semidirect
26. #19153841C22semidirect
30. M16×D181C22semidirect
31. #19153813C22semidirect
32. #19153812C22semidirect
33. #19153805C22semidirect
36. #19153773C22semidirect
37. #19153744C22semidirect
38. #19153743C22semidirect
39. #19153742C22semidirect
40. #19153735C22semidirect
41. #19153732C22semidirect
42. 2−1+4×C181C22semidirect
43. AGL(1,ℤ/8)×C181C22semidirect
44. #124×C181C22semidirect
45. #104×C181C22semidirect
46. #101×C181C22semidirect
54. #15668450D4semidirect
63. #15668448D4semidirect
66. Q8×D181D4semidirect
82. D4×D181D4semidirect
193. C181#3021semidirect
201. #454D181semidirect
208. 2−1+4C2×D181semidirect
223. AGL(1,ℤ/8)C2×D181semidirect
230. #124C2×D181semidirect
237. #104C2×D181semidirect
239. #101C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421162
AGL(1,ℤ/8)×D1812
#260418472
#260414542
#260414522
#260414512
#260414482
#260414452
#260412942
#260412622
#454×C1812
#3021181