Finite Group Databasebeta

← #37545856

Subgroups — #37545856

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
1D4×D724C211yesmaximal, characteristic, complemented
2#26041645C211yesmaximal, characteristic, complemented
3#26041603C211yesmaximal, characteristic, complemented
4#26041597C211yesmaximal, characteristic, complemented
5#26041576C211yesmaximal, characteristic, complemented
6#26041530C211yesmaximal, characteristic, complemented
7#26041513C211yesmaximal, characteristic, complemented
8#26041474C211yesmaximal, characteristic, complemented
9#26041412C211yesmaximal, characteristic, complemented
10#26041412C211yesmaximal, characteristic, complemented
11#26041389C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041283C211yesmaximal, characteristic, complemented
14C2×#19153736C211yesmaximal, characteristic, complemented
15#553×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×D4×D181—21no—
19C2×#15668434C2211yescharacteristic
20C2×#15668434C2211yescharacteristic, complemented
21C2×#15668434C2211yescharacteristic, complemented
22C2×#15668433C2211yescharacteristic, complemented
23C22×#13089678C2211yescharacteristic, complemented
24#19153813C2211yescharacteristic
25#19153791—21no—
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153786—21no—
29#19153786—21no—
30C4×D724C2211yescharacteristic
31#19153749C2211yescharacteristic, complemented
32#19153747C2211yescharacteristic, complemented
33#19153747C2211yescharacteristic, complemented
34C4⋊#13089678C2211yescharacteristic, complemented
35#19153737C2211yescharacteristic, complemented
36#19153736C2211yescharacteristic
37#19153736C2211yescharacteristic
38#19153736C2211yescharacteristic
39#19153736C2211yescharacteristic, complemented
40#19153736C2211yescharacteristic, complemented
41#19153736C2211yescharacteristic, complemented
42#19153736C2211yescharacteristic, complemented
43#19153736C2211yescharacteristic, complemented
44#19153736C2211yescharacteristic, complemented
45C22⋊#13089678C2211yescharacteristic, complemented
46C22⋊#13089678C2211yescharacteristic, complemented
47#19153727C2211yescharacteristic, complemented
48#126×C181—21no—
49#125×C181C2211yescharacteristic, complemented
50#123×C181—21no—
51#121×C181—21no—
52#121×C181C2211yescharacteristic, complemented
53#121×C181C2211yescharacteristic, complemented
54C4×D4×C181—21no—
55C4×D4×C181C2211yescharacteristic
56C4×D4×C181C2211yescharacteristic, complemented
57#117×C181C2211yescharacteristic, complemented
58C2×C4⋊C4×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C23×D181—21no—
61C2×D724—22no—
62C2×D724—22no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—21no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724—41no—
69C2×D724C2311yescharacteristic
70C2×D724C2311yescharacteristic
71C2×D724C2311yescharacteristic
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74D4×D181—44no—
75C2×C4×D181—21no—
76#15668434—22no—
77#15668434—41no—
78#15668434—41no—
79#15668433—41no—
80C2×#13089678—41no—
81C2×#13089678C2311yescharacteristic
82C2×#13089678C2311yescharacteristic
83C2×#13089678C2311yescharacteristic
84C2×#13089678C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89C2×D4×C181—21no—
90C2×D4×C181—21no—
91C2×D4×C181D411yescharacteristic
92C2×D4×C181C2311yescharacteristic
93C22×C4×C181—21no—
94C22×C4×C181—21no—
95C22×C4×C181C2311yescharacteristic
96C22×C4×C181C2311yescharacteristic
97C4⋊C4×C181—21no—
98C4⋊C4×C181—21no—
99C4⋊C4×C181—21no—
100C4⋊C4×C181D411yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181—21no—
108C22⋊C4×C181D411yescharacteristic
109C22⋊C4×C181D411yescharacteristic
110C42×C181—21no—
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—21no—
115C22×D181—42no—
116C22×D181—42no—
117C22×D181—42no—
118C22×D181—42no—
119C22×D181—81no—
120C22×D181—81no—
121D724—21no—
122D724—21no—
123D724—21no—
124D724—21no—
125D724—21no—
126D724—42no—
127D724—42no—
128D724—42no—
129D724—81no—
130D724#4112yes—
131C22⋊D181—42no—
132C22⋊D181—42no—
133C2×Dic181—21no—
134C4×D181—42no—
135#13089678—21no—
136#13089678—21no—
137#13089678—21no—
138#13089678—21no—
139C23×C181—21no—
140C23×C181C2×D411yescharacteristic
141C23×C181C2×D411yescharacteristic
142D4×C181—22no—
143D4×C181—21no—
144D4×C181—21no—
145D4×C181—42no—
146D4×C181—81no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—81no—
155C2×C4×C181—81no—
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2×D411yescharacteristic
160C2×C4×C181C2411yescharacteristic
161C22×C181—21no—
162C22×C181—21no—
163C22×C181—21no—
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181AGL(1,ℤ/8)12yes—
170C22×C181C22×D411yescharacteristic
171C2×D181—22no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—41no—
178C2×D181—41no—
179C2×D181—41no—
180C2×D181—84no—
181C2×D181—84no—
182C4×C181—21no—
183C4×C181—21no—
184C4×C181—21no—
185C4×C181—21no—
186C4×C181—41no—
187C4×C181—41no—
188C4×C181—41no—
189C4×C181C2×#4111yescharacteristic
190C4×C181H(5,𝔽2)11yescharacteristic
191Dic181—41no—
192C2×C181—22no—
193C2×C181—41no—
194C2×C181—81no—
195C2×C181#54611yescharacteristic
196C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C181#57611yescharacteristic
198D181—42no—
199D181—81no—
200D181—81no—
201C181#331811yescharacteristic, complemented, Sylow
202#3318—1811nomaximal, Sylow
203#553D18111yescharacteristic, complemented
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#442—1811no—
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×D8—1811no—
224C2×M16—1811no—
225C22×C8—1811no—
226#127—1811no—
227#126—21no—
228#125C2×D18111yescharacteristic, complemented
229#123—21no—
230#121—21no—
231#121—3621no—
232#121—1811no—
233#121—1811no—
234#121C2×D18111yescharacteristic, complemented
235#121C2×D18111yescharacteristic, complemented
236#120—3621no—
237#120—3621no—
238C4×D4—21no—
239C4×D4—1811no—
240C4×D4C2×D18111yescharacteristic
241C4×D4C2×D18111yescharacteristic, complemented
242#117C2×D18111yescharacteristic, complemented
243C2×C4⋊C4C2×D18111yescharacteristic, complemented
244#107—1811no—
245#106—1811no—
246#106—1811no—
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#102—1811no—
257#102—1811no—
258C22⋊C8—1811no—
259C22⋊C8—1811no—
260#97—1811no—
261C24—3621no—
262C24—3621no—
263C2×D4—21no—
264C2×D4—21no—
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—7241no—
277C2×D4—7241no—
278C2×D4—7244no—
279C2×D4C22⋊D18111yescharacteristic
280C2×D4C22×D18111yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—3621no—
284C22×C4C22×D18111yescharacteristic
285C22×C4C22×D18111yescharacteristic
286D8—3622no—
287D8—7241no—
288D8—7241no—
289M16—7241no—
290C2×C8—1811no—
291C2×C8—1811no—
292C2×C8—1811no—
293C2×C8—1811no—
294C2×C8—7241no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—1811no—
299C4⋊C4C22⋊D18111yescharacteristic
300C4⋊C4C22×D18111yescharacteristic
301C4⋊C4C22×D18111yescharacteristic
302C4⋊C4C22×D18111yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4C22⋊D18111yescharacteristic
311C22⋊C4C22⋊D18111yescharacteristic
312C42—21no—
313C42C22×D18111yescharacteristic
314C23—21no—
315C23—3621no—
316C23—3621no—
317C23—3621no—
318C23—14481no—
319C23—14481no—
320C23—7242no—
321C23—7242no—
322C23—7242no—
323C23—7242no—
324C23C2×C22⋊D18111yescharacteristic
325C23C2×C22⋊D18111yescharacteristic
326D4—22no—
327D4—21no—
328D4—21no—
329D4—42no—
330D4—81no—
331D4—3621no—
332D4—3621no—
333D4—3621no—
334D4—3621no—
335D4—3621no—
336D4—14481no—
337D4—1812no—
338D4—7242no—
339D4—7242no—
340D4—7242no—
341D4—7242no—
342D4—7242no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—3621no—
353C2×C4—7242no—
354C2×C4C2×C22⋊D18111yescharacteristic
355C2×C4C2×C22⋊D18111yescharacteristic
356C2×C4C2×C22⋊D18111yescharacteristic
357C2×C4C2×C22⋊D18111yescharacteristic
358C2×C4C23×D18111yescharacteristic
359C8—3621no—
360C8—3621no—
361C8—3621no—
362C8—3621no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—3622no—
372C22—14484no—
373C22—14484no—
374C22—7241no—
375C22—7241no—
376C22—7241no—
377C22—7241no—
378C22—7241no—
379C22—7241no—
380C22—7241no—
381C22—7241no—
382C22#1915384112yes—
383C22C22×C22⋊D18111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—7241no—
392C4#41×D18111yescharacteristic
393C4#1915386211yescharacteristic
394C2—22no—
395C2—41no—
396C2—81no—
397C2—14481no—
398C2—14481no—
399C2—7242no—
400C2#2604191711yescharacteristic
401C2C2×#1915384111yescharacteristic
402C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#553×C181C2
socleC22×C181C22×D4
hypercenter#553D181
hypocenterC181#3318
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. D4×D724C2semidirect
2. #26041645C2semidirect
3. #26041603C2semidirect
4. #26041597C2semidirect
5. #26041576C2semidirect
6. #26041530C2semidirect
7. #26041513C2semidirect
8. #26041474C2semidirect
9. #26041412C2semidirect
10. #26041412C2semidirect
11. #26041389C2semidirect
12. #26041351C2semidirect
13. #26041283C2semidirect
14. C2×#19153736C2semidirect
15. #553×C181C2semidirect
20. C2×#15668434C22semidirect
21. C2×#15668434C22semidirect
22. C2×#15668433C22semidirect
23. C22×#13089678C22semidirect
31. #19153749C22semidirect
32. #19153747C22semidirect
33. #19153747C22semidirect
34. C4⋊#13089678C22semidirect
35. #19153737C22semidirect
39. #19153736C22semidirect
40. #19153736C22semidirect
41. #19153736C22semidirect
42. #19153736C22semidirect
43. #19153736C22semidirect
44. #19153736C22semidirect
45. C22⋊#13089678C22semidirect
46. C22⋊#13089678C22semidirect
47. #19153727C22semidirect
49. #125×C181C22semidirect
52. #121×C181C22semidirect
53. #121×C181C22semidirect
56. C4×D4×C181C22semidirect
57. #117×C181C22semidirect
58. C2×C4⋊C4×C181C22semidirect
201. C181#3318semidirect
203. #553D181semidirect
228. #125C2×D181semidirect
234. #121C2×D181semidirect
235. #121C2×D181semidirect
241. C4×D4C2×D181semidirect
242. #117C2×D181semidirect
243. C2×C4⋊C4C2×D181semidirect

Maximal subgroups

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

subgroupindex
D4×D7242
#260416452
#260416032
#260415972
#260415762
#260415302
#260415132
#260414742
#260414122
#260413892
#260413512
#260412832
C2×#191537362
#553×C1812
#3318181