Finite Group Databasebeta

← #37545867

Subgroups — #37545867

Order 23168.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041979C211yesmaximal, characteristic, complemented
2D4×D724C211yesmaximal, characteristic, complemented
3#26041644C211yesmaximal, characteristic, complemented
4#26041605C211yesmaximal, characteristic, complemented
5#26041599C211yesmaximal, characteristic, complemented
6#26041579C211yesmaximal, characteristic, complemented
7#26041531C211yesmaximal, characteristic, complemented
8#26041514C211yesmaximal, characteristic, complemented
9#26041474C211yesmaximal, characteristic, complemented
10#26041411C211yesmaximal, characteristic, complemented
11#26041411C211yesmaximal, characteristic, complemented
12#26041352C211yesmaximal, characteristic, complemented
13#26041282C211yesmaximal, characteristic, complemented
14#102×D181C211yesmaximal, characteristic, complemented
15#442×C181C211yesmaximal, characteristic, complemented
16C22×D724—21no—
17C2×D4×D181C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19C2×#15668434C2211yescharacteristic, complemented
20C2×#15668434C2211yescharacteristic, complemented
21C2×#15668432C2211yescharacteristic, complemented
22C2×C8×D181C2211yescharacteristic, complemented
23#19153813C2211yescharacteristic
24#19153809—21no—
25#19153808C2211yescharacteristic, complemented
26#19153799—21no—
27#19153791C2211yescharacteristic
28#19153791C2211yescharacteristic
29#19153790—21no—
30#19153789—21no—
31#19153787C2211yescharacteristic, complemented
32#19153787C2211yescharacteristic, complemented
33#19153786—21no—
34#19153786—21no—
35C4×D724C2211yescharacteristic
36C4×C22⋊D181—21no—
37#19153774C2211yescharacteristic, complemented
38C4⋊C4×D181C2211yescharacteristic, complemented
39#19153749C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41C4⋊#13089678C2211yescharacteristic, complemented
42#19153738C2211yescharacteristic, complemented
43#19153738C2211yescharacteristic, complemented
44#19153738C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic, complemented
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic, complemented
49#19153736C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic, complemented
51#19153728C2211yescharacteristic, complemented
52C2×D8×C181C2211yescharacteristic
53C4×D4×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic, complemented
55#106×C181C2211yescharacteristic, complemented
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic, complemented
58#97×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C23×D181—21no—
61C2×D724—21no—
62C2×D724—21no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—44no—
66C2×D724D411yescharacteristic
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73D4×D181—22no—
74D4×D181—22no—
75D4×D181—41no—
76D4×D181—41no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79C2×C4×D181C2311yescharacteristic
80#15668434—41no—
81#15668434—41no—
82#15668432—41no—
83C2×#13089678C2311yescharacteristic
84C2×#13089678C2311yescharacteristic
85C8×D181—41no—
86C4⋊Dic181D411yescharacteristic
87C4⋊Dic181C2311yescharacteristic
88C4⋊Dic181C2311yescharacteristic
89#15668428—21no—
90#15668428—21no—
91#15668428—21no—
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426D411yescharacteristic
98#15668426D411yescharacteristic
99C4×Dic181—21no—
100C2×D4×C181C2311yescharacteristic
101C2×D4×C181C2311yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181—21no—
104D8×C181—22no—
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C22⋊C4×C181—21no—
110C22⋊C4×C181—21no—
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—42no—
115C22×D181—42no—
116C22×D181—42no—
117C22×D181—42no—
118C22×D181—81no—
119C22×D181—81no—
120C22×D181C2×D411yescharacteristic
121C22×D181C2×D411yescharacteristic
122D724—22no—
123D724—21no—
124D724—21no—
125D724—21no—
126D724—21no—
127D724—42no—
128D724—42no—
129D724—42no—
130D724—81no—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C22⋊D181—42no—
134C22⋊D181—81no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181C2×D411yescharacteristic
138C2×Dic181C2×D411yescharacteristic
139C4×D181—21no—
140C4×D181—21no—
141C4×D181—21no—
142C4×D181—21no—
143C4×D181—81no—
144#13089678—21no—
145#13089678—21no—
146C23×C181—21no—
147C23×C181—21no—
148D4×C181—21no—
149D4×C181—21no—
150D4×C181—21no—
151D4×C181#4112yes—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—42no—
155C2×C4×C181—81no—
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2411yescharacteristic
159C8×C181—21no—
160C8×C181—21no—
161C22×C181—22no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181C22×D411yescharacteristic
168C2×D181—21no—
169C2×D181—21no—
170C2×D181—21no—
171C2×D181—21no—
172C2×D181—21no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—41no—
178C2×D181—41no—
179C2×D181—84no—
180C2×D181—84no—
181C2×D181AGL(1,ℤ/8)12yes—
182C4×C181—21no—
183C4×C181—21no—
184C4×C181—41no—
185C4×C181—41no—
186C4×C181C2×#4111yescharacteristic
187C4×C181H(5,𝔽2)11yescharacteristic
188Dic181—21no—
189Dic181—21no—
190Dic181—41no—
191Dic181—41no—
192C2×C181—42no—
193C2×C181—81no—
194C2×C181#54611yescharacteristic
195C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
196C2×C181#57611yescharacteristic
197D181—22no—
198D181—41no—
199D181—81no—
200D181—81no—
201C181#331811yescharacteristic, complemented, Sylow
202#3318—1811nomaximal, Sylow
203#553—1811no—
204D42—1811no—
205#490—1811no—
206#482—1811no—
207#481—1811no—
208#478—1811no—
209#469—1811no—
210#466—1811no—
211#459—1811no—
212#447—1811no—
213#447—1811no—
214#442D18111yescharacteristic, complemented
215#435—1811no—
216#418—1811no—
217C2×#102—1811no—
218C22×D4—3621no—
219C22×D4—1811no—
220C22×D4—1811no—
221C2×D8—1811no—
222C2×D8—1811no—
223C2×D8C2×D18111yescharacteristic
224C2×M16—1811no—
225C22×C8—1811no—
226#127—1811no—
227#126—3621no—
228#125—1811no—
229#123—3621no—
230#121—3621no—
231#121—3621no—
232#121—1811no—
233#121—1811no—
234#121—1811no—
235#121—1811no—
236#120—3621no—
237#120—3621no—
238C4×D4—3621no—
239C4×D4—1811no—
240C4×D4C2×D18111yescharacteristic
241C4×D4C2×D18111yescharacteristic, complemented
242#117—1811no—
243C2×C4⋊C4—1811no—
244#107—1811no—
245#106—1811no—
246#106C2×D18111yescharacteristic, complemented
247C4⋊C8—1811no—
248#102—1811no—
249#102—1811no—
250#102—1811no—
251#102—1811no—
252#102—1811no—
253#102—1811no—
254#102—1811no—
255#102—1811no—
256#102C2×D18111yescharacteristic
257#102C2×D18111yescharacteristic, complemented
258C22⋊C8—1811no—
259C22⋊C8—1811no—
260#97C2×D18111yescharacteristic, complemented
261C24—3621no—
262C24—3621no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3622no—
266C2×D4—3622no—
267C2×D4—3621no—
268C2×D4—3621no—
269C2×D4—3621no—
270C2×D4—3621no—
271C2×D4—3621no—
272C2×D4—3621no—
273C2×D4—1811no—
274C2×D4—1811no—
275C2×D4—1811no—
276C2×D4—7244no—
277C2×D4—7241no—
278C2×D4—7241no—
279C2×D4C22×D18111yescharacteristic
280C2×D4C22×D18111yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—3621no—
284C22×C4—1811no—
285C22×C4—1811no—
286D8—22no—
287D8—7241no—
288D8—7241no—
289M16—7241no—
290C2×C8—1811no—
291C2×C8—1811no—
292C2×C8—7241no—
293C2×C8C22×D18111yescharacteristic
294C2×C8C22×D18111yescharacteristic
295C4⋊C4—3621no—
296C4⋊C4—3621no—
297C4⋊C4—3621no—
298C4⋊C4—1811no—
299C4⋊C4—1811no—
300C4⋊C4—1811no—
301C4⋊C4C22×D18111yescharacteristic
302C4⋊C4C22×D18111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4—1811no—
311C22⋊C4—1811no—
312C42—3621no—
313C42C22×D18111yescharacteristic
314C23—21no—
315C23—21no—
316C23—3621no—
317C23—3621no—
318C23—14481no—
319C23—14481no—
320C23—1811no—
321C23—1811no—
322C23—7242no—
323C23—7242no—
324C23—7242no—
325C23—7242no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—3622no—
330D4—3621no—
331D4—3621no—
332D4—3621no—
333D4—3621no—
334D4—14481no—
335D4—14481no—
336D4—7242no—
337D4—7242no—
338D4—7242no—
339D4—7242no—
340D4—7242no—
341D4—7242no—
342D4#1566844912yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—3621no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—3621no—
351C2×C4—3621no—
352C2×C4—3621no—
353C2×C4—14481no—
354C2×C4—1811no—
355C2×C4—1811no—
356C2×C4D4×D18111yescharacteristic
357C2×C4D4×D18111yescharacteristic
358C2×C4C23×D18111yescharacteristic
359C8—21no—
360C8—21no—
361C8—3621no—
362C8—3621no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—3621no—
370C22—3621no—
371C22—3621no—
372C22—3621no—
373C22—3621no—
374C22—14484no—
375C22—14484no—
376C22—1812no—
377C22—7241no—
378C22—7241no—
379C22—7241no—
380C22—7241no—
381C22—7241no—
382C22—7241no—
383C22C2×D4×D18111yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—3621no—
389C4—3621no—
390C4—7241no—
391C4—7241no—
392C4C2×#1566844911yescharacteristic
393C4#1915386211yescharacteristic
394C2—42no—
395C2—81no—
396C2—3622no—
397C2—14481no—
398C2—14481no—
399C2—7241no—
400C2#2604192511yescharacteristic
401C2AGL(1,ℤ/8)×D18111yescharacteristic
402C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#442×C181C2
socleC22×C181C22×D4
hypercenter#442D181
hypocenterC181#3318
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #26041979C2semidirect
2. D4×D724C2semidirect
3. #26041644C2semidirect
4. #26041605C2semidirect
5. #26041599C2semidirect
6. #26041579C2semidirect
7. #26041531C2semidirect
8. #26041514C2semidirect
9. #26041474C2semidirect
10. #26041411C2semidirect
11. #26041411C2semidirect
12. #26041352C2semidirect
13. #26041282C2semidirect
14. #102×D181C2semidirect
15. #442×C181C2semidirect
19. C2×#15668434C22semidirect
20. C2×#15668434C22semidirect
21. C2×#15668432C22semidirect
22. C2×C8×D181C22semidirect
25. #19153808C22semidirect
31. #19153787C22semidirect
32. #19153787C22semidirect
37. #19153774C22semidirect
38. C4⋊C4×D181C22semidirect
39. #19153749C22semidirect
40. #19153747C22semidirect
41. C4⋊#13089678C22semidirect
42. #19153738C22semidirect
43. #19153738C22semidirect
44. #19153738C22semidirect
45. #19153738C22semidirect
48. #19153736C22semidirect
49. #19153736C22semidirect
50. #19153728C22semidirect
51. #19153728C22semidirect
54. C4×D4×C181C22semidirect
55. #106×C181C22semidirect
57. #102×C181C22semidirect
58. #97×C181C22semidirect
201. C181#3318semidirect
214. #442D181semidirect
241. C4×D4C2×D181semidirect
246. #106C2×D181semidirect
257. #102C2×D181semidirect
260. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419792
D4×D7242
#260416442
#260416052
#260415992
#260415792
#260415312
#260415142
#260414742
#260414112
#260413522
#260412822
#102×D1812
#442×C1812
#3318181