Finite Group Databasebeta

← #37544706

Subgroups — #37544706

Order 23168.

444 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041826C211yesmaximal, characteristic, complemented
2#124×D181C211yesmaximal, characteristic, complemented
3#26041667C211yesmaximal, characteristic, complemented
4#26041665C211yesmaximal, characteristic, complemented
5#26041640C211yesmaximal, characteristic, complemented
6#26041637C211yesmaximal, characteristic, complemented
7#26041495C211yesmaximal, characteristic, complemented
8#26041494C211yesmaximal, characteristic, complemented
9#26041491C211yesmaximal, characteristic, complemented
10#26041419C211yesmaximal, characteristic, complemented
11#26041416C211yesmaximal, characteristic, complemented
12#26041414C211yesmaximal, characteristic, complemented
13#26041411C211yesmaximal, characteristic, complemented
14#26041335C211yesmaximal, characteristic, complemented
15#463×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×Q8×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19C2×#15668438C2211yescharacteristic, complemented
20C2×#15668437C2211yescharacteristic, complemented
21C2×#15668436C2211yescharacteristic, complemented
22C2×D1448C2211yescharacteristic
23C2×#15668434C2211yescharacteristic
24C2×#15668434C2211yescharacteristic
25#19153815C2211yescharacteristic, complemented
26#19153812C2211yescharacteristic, complemented
27#19153805C2211yescharacteristic
28#19153804C2211yescharacteristic
29#19153803C2211yescharacteristic
30#19153802—21no—
31#19153802—21no—
32#19153802—21no—
33#19153791C2211yescharacteristic
34#19153789—21no—
35#19153787C2211yescharacteristic
36C4×D724C2211yescharacteristic, complemented
37C4×C22⋊D181—21no—
38C22⋊C4×D181—21no—
39C22⋊C4×D181—21no—
40C42×D181C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153728C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153726C2211yescharacteristic, complemented
51#19153726C2211yescharacteristic, complemented
52C2×SD16×C181C2211yescharacteristic, complemented
53C2×D8×C181C2211yescharacteristic
54#124×C181C2211yescharacteristic
55C4×D4×C181C2211yescharacteristic, complemented
56C4⋊C8×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58#102×C181C2211yescharacteristic
59C23×D181—21no—
60#15668449—44no—
61C2×Dic362—21no—
62C2×Dic362—21no—
63C2×Dic362C2311yescharacteristic
64Q8×D181—41no—
65Q8×D181—41no—
66C2×D724—21no—
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73D4×D181—41no—
74D4×D181—41no—
75C2×C4×D181—21no—
76C2×C4×D181—21no—
77C2×C4×D181D411yescharacteristic
78C2×C4×D181D411yescharacteristic
79C2×C4×D181C2311yescharacteristic
80#15668438—41no—
81#15668437—41no—
82#15668436—41no—
83D1448—41no—
84#15668434—41no—
85#15668434—41no—
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C4⋊Dic181C2311yescharacteristic
89#15668428—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92C22⋊Dic181—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99C4×Dic181D411yescharacteristic
100C4×Dic181D411yescharacteristic
101C4×Dic181C2311yescharacteristic
102C2×Q8×C181C2311yescharacteristic
103C2×D4×C181C2311yescharacteristic
104C2×D4×C181C2311yescharacteristic
105C22×C4×C181—21no—
106SD16×C181—41no—
107D8×C181—41no—
108C2×C8×C181C2311yescharacteristic
109C2×C8×C181C2311yescharacteristic
110C4⋊C4×C181C2311yescharacteristic
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C22⋊C4×C181—21no—
114C42×C181C2311yescharacteristic
115C22×D181—21no—
116C22×D181—21no—
117C22×D181—42no—
118C22×D181—42no—
119C22×D181—42no—
120C22×D181C2×D411yescharacteristic
121Dic362—21no—
122Dic362—21no—
123Dic362—42no—
124Dic362—42no—
125D724—21no—
126D724—21no—
127D724—21no—
128D724—21no—
129D724—42no—
130C22⋊D181—42no—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C22⋊D181—42no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181C2×D411yescharacteristic
137C2×Dic181C2×D411yescharacteristic
138C2×Dic181C2×D411yescharacteristic
139C4×D181—22no—
140C4×D181—21no—
141C4×D181—21no—
142C4×D181—41no—
143C4×D181—42no—
144C4×D181—42no—
145#13089678—21no—
146#13089678—21no—
147C23×C181—21no—
148C23×C181—21no—
149Q8×C181—21no—
150Q8×C181—21no—
151D4×C181—21no—
152D4×C181—21no—
153D4×C181—21no—
154D4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181—42no—
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2×D411yescharacteristic
160C2×C4×C181C2411yescharacteristic
161C8×C181—21no—
162C8×C181—21no—
163C22×C181—41no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181C22×D411yescharacteristic
170C2×D181—21no—
171C2×D181—21no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—41no—
178C2×D181—84no—
179C2×D181AGL(1,ℤ/8)12yes—
180C4×C181—21no—
181C4×C181—21no—
182C4×C181—41no—
183C4×C181—41no—
184C4×C181H(5,𝔽2)11yescharacteristic
185C4×C181H(5,𝔽2)11yescharacteristic
186Dic181—21no—
187Dic181—21no—
188Dic181—21no—
189Dic181—41no—
190Dic181—41no—
191C2×C181—81no—
192C2×C181—81no—
193C2×C181#53511yescharacteristic
194C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
195C2×C181#57511yescharacteristic
196D181—22no—
197D181—81no—
198D181—81no—
199C181#321311yescharacteristic, complemented, Sylow
200#3213—1811nomaximal, Sylow
201#532—1811no—
202C2×#124—1811no—
203#496—1811no—
204#496—1811no—
205#489—1811no—
206#489—1811no—
207#463—1811no—
208#463—1811no—
209#463—1811no—
210#463D18111yescharacteristic, complemented
211#448—1811no—
212#448—1811no—
213#447—1811no—
214#447—1811no—
215#432—1811no—
216C2×#41—3621no—
217C22×Q8—1811no—
218C22×D4—1811no—
219C2×SD16—1811no—
220C2×SD16—1811no—
221C2×SD16—1811no—
222C2×SD16C2×D18111yescharacteristic, complemented
223C2×D8—1811no—
224C2×D8—1811no—
225C2×D8—1811no—
226C2×D8C2×D18111yescharacteristic
227C4⋊Q8—1811no—
228#127—1811no—
229#124—3621no—
230#124—3621no—
231#124—3621no—
232#124—1811no—
233#124—1811no—
234#124—1811no—
235#124C2×D18111yescharacteristic
236#121—3621no—
237#121—1811no—
238#121—1811no—
239C4×D4—3621no—
240C4×D4—1811no—
241C4×D4C2×D18111yescharacteristic, complemented
242C2×C22⋊C4—3621no—
243C2×C22⋊C4—3621no—
244C2×C42—1811no—
245C4⋊C8—1811no—
246C4⋊C8C2×D18111yescharacteristic, complemented
247#103—1811no—
248#103—1811no—
249#103—1811no—
250#103C2×D18111yescharacteristic, complemented
251#102—1811no—
252#102—1811no—
253#102—1811no—
254#102C2×D18111yescharacteristic
255C22⋊C8—1811no—
256C22⋊C8—1811no—
257#97—1811no—
258#97—1811no—
259C24—3621no—
260#41—7244no—
261C2×Q8—3621no—
262C2×Q8—3621no—
263C2×Q8—1811no—
264C2×Q8—7241no—
265C2×Q8—7241no—
266C2×Q8C22×D18111yescharacteristic
267C2×D4—3621no—
268C2×D4—3621no—
269C2×D4—3621no—
270C2×D4—3621no—
271C2×D4—3621no—
272C2×D4—1811no—
273C2×D4—1811no—
274C2×D4—7241no—
275C2×D4—7241no—
276C2×D4C22×D18111yescharacteristic
277C2×D4C22×D18111yescharacteristic
278C22×C4—21no—
279C22×C4—3621no—
280C22×C4—3621no—
281C22×C4—1811no—
282C22×C4—1811no—
283C22×C4—1811no—
284SD16—41no—
285SD16—7241no—
286SD16—7241no—
287SD16—7241no—
288D8—41no—
289D8—7241no—
290D8—7241no—
291D8—7241no—
292C2×C8—1811no—
293C2×C8—1811no—
294C2×C8C22×D18111yescharacteristic
295C2×C8C22×D18111yescharacteristic
296C4⋊C4—3621no—
297C4⋊C4—1811no—
298C4⋊C4C22×D18111yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4—3621no—
311C42—1811no—
312C42—1811no—
313C42—1811no—
314C42C22×D18111yescharacteristic
315C23—21no—
316C23—21no—
317C23—3621no—
318C23—3621no—
319C23—1811no—
320C23—7242no—
321C23—7242no—
322C23—7242no—
323Q8—21no—
324Q8—21no—
325Q8—3621no—
326Q8—3621no—
327Q8—7242no—
328Q8—7242no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—3621no—
334D4—3621no—
335D4—3621no—
336D4—3621no—
337D4—7242no—
338D4—7242no—
339D4—7242no—
340D4—7242no—
341D4—7242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—3622no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—1811no—
351C2×C4—1811no—
352C2×C4—1811no—
353C2×C4—7241no—
354C2×C4—7242no—
355C2×C4—7242no—
356C2×C4D4×D18111yescharacteristic
357C2×C4D4×D18111yescharacteristic
358C2×C4C23×D18111yescharacteristic
359C8—21no—
360C8—21no—
361C8—3621no—
362C8—3621no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—3621no—
370C22—3621no—
371C22—14484no—
372C22—1812no—
373C22—7241no—
374C22—7241no—
375C22—7241no—
376C22—7241no—
377C22—7241no—
378C22—7241no—
379C22C2×D4×D18111yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—3621no—
385C4—3621no—
386C4—3621no—
387C4—7241no—
388C4—7241no—
389C4#1915386111yescharacteristic
390C4#1915386211yescharacteristic
391C2—81no—
392C2—81no—
393C2—3622no—
394C2—14481no—
395C2—14481no—
396C2#2604184511yescharacteristic
397C2AGL(1,ℤ/8)×D18111yescharacteristic
398C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#463×C181C2
socleC22×C181C22×D4
hypercenter#463D181
hypocenterC181#3213
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. #26041826C2semidirect
2. #124×D181C2semidirect
3. #26041667C2semidirect
4. #26041665C2semidirect
5. #26041640C2semidirect
6. #26041637C2semidirect
7. #26041495C2semidirect
8. #26041494C2semidirect
9. #26041491C2semidirect
10. #26041419C2semidirect
11. #26041416C2semidirect
12. #26041414C2semidirect
13. #26041411C2semidirect
14. #26041335C2semidirect
15. #463×C181C2semidirect
19. C2×#15668438C22semidirect
20. C2×#15668437C22semidirect
21. C2×#15668436C22semidirect
25. #19153815C22semidirect
26. #19153812C22semidirect
36. C4×D724C22semidirect
41. C4⋊#13089678C22semidirect
42. #19153741C22semidirect
43. #19153740C22semidirect
44. #19153739C22semidirect
50. #19153726C22semidirect
51. #19153726C22semidirect
52. C2×SD16×C181C22semidirect
55. C4×D4×C181C22semidirect
56. C4⋊C8×C181C22semidirect
57. #103×C181C22semidirect
199. C181#3213semidirect
210. #463D181semidirect
222. C2×SD16C2×D181semidirect
241. C4×D4C2×D181semidirect
246. C4⋊C8C2×D181semidirect
250. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418262
#124×D1812
#260416672
#260416652
#260416402
#260416372
#260414952
#260414942
#260414912
#260414192
#260414162
#260414142
#260414112
#260413352
#463×C1812
#3213181