Finite Group Databasebeta

← #37545686

Subgroups — #37545686

Order 23168.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041929C211yesmaximal, characteristic, complemented
2#26041923C211yesmaximal, characteristic, complemented
3#26041637C211yesmaximal, characteristic, complemented
4#26041610C211yesmaximal, characteristic, complemented
5#26041596C211yesmaximal, characteristic, complemented
6#26041567C211yesmaximal, characteristic, complemented
7#26041529C211yesmaximal, characteristic, complemented
8#26041506C211yesmaximal, characteristic, complemented
9#26041492C211yesmaximal, characteristic, complemented
10#26041421C211yesmaximal, characteristic, complemented
11#26041391C211yesmaximal, characteristic, complemented
12#26041347C211yesmaximal, characteristic, complemented
13#26041277C211yesmaximal, characteristic, complemented
14C2×#19153737C211yesmaximal, characteristic, complemented
15#447×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668448C2211yescharacteristic
18C2×#15668437C2211yescharacteristic, complemented
19C2×#15668432C2211yescharacteristic, complemented
20C22×#13089678C2211yescharacteristic
21#19153815C2211yescharacteristic, complemented
22#19153803C2211yescharacteristic
23#19153798—21no—
24#19153797—21no—
25#19153797—21no—
26#19153795C2211yescharacteristic
27#19153793—21no—
28#19153788—21no—
29#19153787C2211yescharacteristic
30#19153787C2211yescharacteristic, complemented
31#19153785—21no—
32D4×Dic181C2211yescharacteristic
33D4×Dic181C2211yescharacteristic, complemented
34C4×C22⋊D181C2211yescharacteristic
35C2×C4⋊Dic181C2211yescharacteristic
36C4⋊C4×D181C2211yescharacteristic, complemented
37C2×C22⋊Dic181—21no—
38#19153748C2211yescharacteristic, complemented
39#19153748C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153737C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic, complemented
51#19153726C2211yescharacteristic, complemented
52C22×D4×C181C2211yescharacteristic
53C2×D8×C181C2211yescharacteristic
54C2×D8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#102×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668448—22no—
60#15668448—41no—
61#15668448—44no—
62C2×Dic362—21no—
63C2×Dic362C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181D411yescharacteristic
68C22×Dic181—21no—
69C22×Dic181—21no—
70C22×Dic181—21no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73C2×C4×D181C2311yescharacteristic
74#15668437—41no—
75#15668432—41no—
76C2×#13089678—41no—
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C4⋊Dic181—21no—
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181D411yescharacteristic
95#15668426D411yescharacteristic
96C4×Dic181C2311yescharacteristic
97C24×C181—21no—
98C2×D4×C181—22no—
99C2×D4×C181—21no—
100C2×D4×C181—21no—
101C2×D4×C181—41no—
102C2×D4×C181C2311yescharacteristic
103C2×D4×C181C2311yescharacteristic
104C2×D4×C181C2311yescharacteristic
105C22×C4×C181C2311yescharacteristic
106D8×C181—22no—
107D8×C181—41no—
108C2×C8×C181C2311yescharacteristic
109C2×C8×C181C2311yescharacteristic
110C4⋊C4×C181C2311yescharacteristic
111C22⋊C4×C181—21no—
112C22×D181C2×D411yescharacteristic
113Dic362—21no—
114Dic362—21no—
115Dic362—42no—
116C22⋊D181—22no—
117C22⋊D181—42no—
118C22⋊D181—42no—
119C22⋊D181—81no—
120C2×Dic181—21no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—42no—
125C2×Dic181—42no—
126C2×Dic181—42no—
127C2×Dic181—81no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C2×Dic181C2×D411yescharacteristic
131C4×D181—21no—
132C4×D181—21no—
133C4×D181—42no—
134#13089678—21no—
135#13089678—21no—
136C23×C181—21no—
137C23×C181—21no—
138C23×C181—21no—
139C23×C181—42no—
140C23×C181—42no—
141C23×C181—81no—
142C23×C181C2×D411yescharacteristic
143D4×C181—21no—
144D4×C181—21no—
145D4×C181—21no—
146D4×C181—21no—
147D4×C181—21no—
148D4×C181—42no—
149D4×C181—42no—
150D4×C181#4112yescomplemented
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181C2×D411yescharacteristic
155C2×C4×C181C2411yescharacteristic
156C8×C181—21no—
157C8×C181—21no—
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—22no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181—41no—
168C22×C181—41no—
169C22×C181—84no—
170C22×C181AGL(1,ℤ/8)12yes—
171C22×C181C22×D411yescharacteristic
172C2×D181—21no—
173C2×D181—21no—
174C2×D181—21no—
175C4×C181—21no—
176C4×C181—41no—
177C4×C181C2×#4111yescharacteristic
178C4×C181H(5,𝔽2)11yescharacteristic
179Dic181—21no—
180Dic181—21no—
181Dic181—21no—
182Dic181—41no—
183Dic181—41no—
184Dic181—41no—
185Dic181—41no—
186C2×C181—22no—
187C2×C181—42no—
188C2×C181—81no—
189C2×C181—81no—
190C2×C181#54611yescharacteristic
191C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C181#57711yescharacteristic
193D181—41no—
194C181#330611yescharacteristic, complemented, Sylow
195#3306—1811nomaximal, Sylow
196#547—1811no—
197#546—1811no—
198#489—1811no—
199#483—1811no—
200#481—1811no—
201#476—1811no—
202#469—1811no—
203#465—1811no—
204#463—1811no—
205#449—1811no—
206#447D18111yescharacteristic, complemented
207#442—1811no—
208#435—1811no—
209#417—1811no—
210C2×#102—1811no—
211C2×#41—3621no—
212C2×#41—1811no—
213C22×D4C2×D18111yescharacteristic
214C2×SD16—1811no—
215C2×D8C2×D18111yescharacteristic
216C2×D8C2×D18111yescharacteristic
217C2×M16—1811no—
218C22×C8—1811no—
219C4⋊Q8—1811no—
220#124—1811no—
221#123—3621no—
222#123—3621no—
223#123—3621no—
224C22⋊Q8—3621no—
225C22⋊Q8—1811no—
226#121—3621no—
227#121—1811no—
228#121—1811no—
229#121C2×D18111yescharacteristic
230#120—3621no—
231C4×D4—1811no—
232C4×D4—1811no—
233C4×D4—1811no—
234C2×C4⋊C4—1811no—
235C2×C4⋊C4—1811no—
236C2×C22⋊C4—3621no—
237#106—1811no—
238#106—1811no—
239#106—1811no—
240C4⋊C8—1811no—
241#103—1811no—
242#103—1811no—
243#103—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248#102—1811no—
249#102C2×D18111yescharacteristic
250#102C2×D18111yescharacteristic
251C22⋊C8—1811no—
252C22⋊C8C2×D18111yescharacteristic
253#97—1811no—
254C24—21no—
255#41—3622no—
256#41—7241no—
257#41—7244no—
258C2×Q8—3621no—
259C2×Q8—1811no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—3621no—
267C2×D4—1811no—
268C2×D4C22×D18111yescharacteristic
269C2×D4C22×D18111yescharacteristic
270C2×D4C22×D18111yescharacteristic
271C22×C4—3621no—
272C22×C4—3621no—
273C22×C4—3621no—
274C22×C4—3621no—
275C22×C4—3621no—
276C22×C4—1811no—
277C22×C4C22×D18111yescharacteristic
278SD16—7241no—
279D8—22no—
280D8—41no—
281M16—7241no—
282C2×C8—1811no—
283C2×C8—1811no—
284C2×C8—7241no—
285C2×C8C22×D18111yescharacteristic
286C2×C8C22×D18111yescharacteristic
287C4⋊C4—3621no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—1811no—
292C4⋊C4—1811no—
293C4⋊C4—1811no—
294C4⋊C4—1811no—
295C4⋊C4C22×D18111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—3621no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—1811no—
305C22⋊C4—1811no—
306C42—1811no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—1811no—
314C23D4×D18111yescharacteristic
315Q8—3621no—
316Q8—3621no—
317Q8—7242no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—3622no—
326D4—14481no—
327D4—7242no—
328D4—7242no—
329D4#1566844812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—3621no—
336C2×C4—3621no—
337C2×C4—3621no—
338C2×C4—3621no—
339C2×C4—14481no—
340C2×C4—1811no—
341C2×C4—1811no—
342C2×C4—1811no—
343C2×C4—7242no—
344C2×C4—7242no—
345C2×C4—7242no—
346C2×C4—7242no—
347C2×C4D4×D18111yescharacteristic
348C2×C4C23×D18111yescharacteristic
349C8—21no—
350C8—21no—
351C8—3621no—
352C8—3621no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—3621no—
366C22—3621no—
367C22—3621no—
368C22#1915384012yes—
369C22C2×D4×D18111yescharacteristic
370C4—21no—
371C4—41no—
372C4—3621no—
373C4—3621no—
374C4—3621no—
375C4—7241no—
376C4—7241no—
377C4—7241no—
378C4—7241no—
379C4C2×#1566844811yescharacteristic
380C4#1915386111yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—7241no—
386C2#2604191611yescharacteristic
387C2C2×#1915384011yescharacteristic
388C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181#3306
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26041929C2semidirect
2. #26041923C2semidirect
3. #26041637C2semidirect
4. #26041610C2semidirect
5. #26041596C2semidirect
6. #26041567C2semidirect
7. #26041529C2semidirect
8. #26041506C2semidirect
9. #26041492C2semidirect
10. #26041421C2semidirect
11. #26041391C2semidirect
12. #26041347C2semidirect
13. #26041277C2semidirect
14. C2×#19153737C2semidirect
15. #447×C181C2semidirect
18. C2×#15668437C22semidirect
19. C2×#15668432C22semidirect
21. #19153815C22semidirect
30. #19153787C22semidirect
33. D4×Dic181C22semidirect
36. C4⋊C4×D181C22semidirect
38. #19153748C22semidirect
39. #19153748C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153741C22semidirect
43. #19153741C22semidirect
44. #19153739C22semidirect
49. #19153737C22semidirect
50. #19153728C22semidirect
51. #19153726C22semidirect
150. D4×C181#41semidirect
194. C181#3306semidirect
206. #447D181semidirect
329. D4#15668448semidirect

Maximal subgroups

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

subgroupindex
#260419292
#260419232
#260416372
#260416102
#260415962
#260415672
#260415292
#260415062
#260414922
#260414212
#260413912
#260413472
#260412772
C2×#191537372
#447×C1812
#3306181