Finite Group Databasebeta

← #37545558

Subgroups — #37545558

Order 23168.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041969C211yesmaximal, characteristic, complemented
2#26041925C211yesmaximal, characteristic
3#26041634C211yesmaximal, characteristic, complemented
4#26041610C211yesmaximal, characteristic, complemented
5#26041604C211yesmaximal, characteristic, complemented
6#26041580C211yesmaximal, characteristic, complemented
7#26041520C211yesmaximal, characteristic, complemented
8#26041508C211yesmaximal, characteristic, complemented
9#26041502C211yesmaximal, characteristic, complemented
10#26041443C211yesmaximal, characteristic, complemented
11#26041432C211yesmaximal, characteristic, complemented
12C4×#15668438C211yesmaximal, characteristic, complemented
13#26041276C211yesmaximal, characteristic, complemented
14C2×#19153738C211yesmaximal, characteristic, complemented
15C8×D4×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×Dic724C2211yescharacteristic, complemented
19C2×#15668438C2211yescharacteristic
20C2×#15668438C2211yescharacteristic
21#19153809—21no—
22#19153808C2211yescharacteristic, complemented
23#19153806—21no—
24#19153804C2211yescharacteristic
25#19153801—21no—
26#19153797—21no—
27#19153795C2211yescharacteristic
28#19153795C2211yescharacteristic
29#19153795C2211yescharacteristic, complemented
30#19153790C2211yescharacteristic
31#19153789—21no—
32#19153784—21no—
33C4×Dic362C2211yescharacteristic, complemented
34C4×D724C2211yescharacteristic
35#19153775C2211yescharacteristic, complemented
36#19153772—21no—
37C2×C4⋊Dic181C2211yescharacteristic
38C22⋊C4×D181—21no—
39#19153750C2211yescharacteristic, complemented
40#19153748C2211yescharacteristic, complemented
41#19153748C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153741C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153738C2211yescharacteristic
50#19153738C2211yescharacteristic
51#19153738C2211yescharacteristic
52C22×C8×C181C2211yescharacteristic
53C22×C8×C181C2211yescharacteristic, complemented
54C4×D4×C181C2211yescharacteristic
55C4⋊C8×C181C2211yescharacteristic, complemented
56C22⋊C8×C181C2211yescharacteristic
57C22⋊C8×C181C2211yescharacteristic, complemented
58C4×C8×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60#15668449—22no—
61#15668449—41no—
62C2×Dic362C2311yescharacteristic
63C2×Dic362C2311yescharacteristic
64C2×D724—22no—
65C2×D724—21no—
66C2×D724—41no—
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74Dic724—41no—
75#15668438—22no—
76#15668438—41no—
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428—21no—
87#15668428—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96C4×Dic181—21no—
97C4×Dic181—21no—
98C2×D4×C181D411yescharacteristic
99C22×C4×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101C2×C8×C181—41no—
102C2×C8×C181—41no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C22⋊C4×C181D411yescharacteristic
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181—21no—
113C22×D181—42no—
114C22×D181—42no—
115C22×D181—81no—
116Dic362—21no—
117Dic362—21no—
118Dic362—21no—
119Dic362—21no—
120D724—21no—
121D724—21no—
122D724—21no—
123D724—42no—
124D724#4112yes—
125C22⋊D181—42no—
126C22⋊D181—81no—
127C2×Dic181—21no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C2×Dic181—21no—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181—42no—
134C4×D181—42no—
135C4×D181—81no—
136C23×C181C2×D411yescharacteristic
137C23×C181C2×D411yescharacteristic
138D4×C181—22no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2411yescharacteristic
148C8×C181—21no—
149C8×C181—21no—
150C8×C181—21no—
151C8×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181#13512yes—
158C22×C181C22×D411yescharacteristic
159C2×D181—22no—
160C2×D181—41no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—84no—
166C4×C181—21no—
167C4×C181—21no—
168C4×C181—21no—
169C4×C181—21no—
170C4×C181C2×#4111yescharacteristic
171C4×C181H(5,𝔽2)11yescharacteristic
172Dic181—41no—
173Dic181—41no—
174Dic181—41no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178C2×C181—22no—
179C2×C181—41no—
180C2×C181#54611yescharacteristic
181C2×C181C2×#13511yescharacteristic
182C2×C181#57711yescharacteristic
183D181—42no—
184D181—81no—
185C181#329611yescharacteristic, complemented, Sylow
186#3296—1811nomaximal, Sylow
187#552—1811no—
188#546—1811no—
189#488—1811no—
190#483—1811no—
191#482—1811no—
192#478—1811no—
193#467—1811no—
194#465—1811no—
195#464—1811no—
196#452—1811no—
197#450—1811no—
198C4×SD16—1811no—
199C8×D4D18111yescharacteristic, complemented
200#416—1811no—
201C2×#102—1811no—
202C2×#41—1811no—
203C22×D4—1811no—
204C2×Q16—1811no—
205C2×SD16—1811no—
206C2×SD16—1811no—
207C22×C8C2×D18111yescharacteristic
208C22×C8C2×D18111yescharacteristic, complemented
209#126—3621no—
210#125—3621no—
211#125—1811no—
212#124—1811no—
213#123—3621no—
214#123—3621no—
215C22⋊Q8—1811no—
216C22⋊Q8—1811no—
217C22⋊Q8—1811no—
218#121—3621no—
219#121—1811no—
220#120—3621no—
221C4×Q8—1811no—
222C4×D4—1811no—
223C4×D4C2×D18111yescharacteristic
224#117—3621no—
225#117—1811no—
226C2×C4⋊C4—1811no—
227C2×C22⋊C4—3621no—
228#107—1811no—
229#106—1811no—
230#106—1811no—
231C4⋊C8C2×D18111yescharacteristic, complemented
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103—1811no—
236#102—1811no—
237#102—1811no—
238#102—1811no—
239#102—1811no—
240#102—1811no—
241#102—1811no—
242C22⋊C8C2×D18111yescharacteristic
243C22⋊C8C2×D18111yescharacteristic, complemented
244C4×C8C2×D18111yescharacteristic, complemented
245C24—3621no—
246#41—3622no—
247#41—7241no—
248C2×Q8—1811no—
249C2×Q8—1811no—
250C2×D4—3622no—
251C2×D4—3621no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—1811no—
255C2×D4—1811no—
256C2×D4—7241no—
257C2×D4D72411yescharacteristic
258C22×C4—3621no—
259C22×C4—3621no—
260C22×C4—3621no—
261C22×C4C22×D18111yescharacteristic
262C22×C4C22×D18111yescharacteristic
263Q16—7241no—
264SD16—3622no—
265SD16—7241no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D18111yescharacteristic
269C2×C8C22×D18111yescharacteristic
270C2×C8C22×D18111yescharacteristic
271C2×C8C22×D18111yescharacteristic
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—3621no—
275C4⋊C4—3621no—
276C4⋊C4—3621no—
277C4⋊C4—3621no—
278C4⋊C4—3621no—
279C4⋊C4—1811no—
280C4⋊C4—1811no—
281C4⋊C4—1811no—
282C4⋊C4—1811no—
283C4⋊C4D72411yescharacteristic
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—3621no—
287C22⋊C4—3621no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4D72411yescharacteristic
293C22⋊C4D72411yescharacteristic
294C42—3621no—
295C42—3621no—
296C42C22×D18111yescharacteristic
297C23—3621no—
298C23—3621no—
299C23—14481no—
300C23—7242no—
301C23—7242no—
302C23C2×D72411yescharacteristic
303C23C2×D72411yescharacteristic
304Q8—3621no—
305Q8—3621no—
306Q8—3621no—
307Q8—3621no—
308D4—22no—
309D4—3621no—
310D4—3621no—
311D4—3621no—
312D4—14481no—
313D4—1812no—
314D4—7242no—
315D4—7242no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3621no—
321C2×C4—3621no—
322C2×C4—3621no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—14481no—
327C2×C4—7242no—
328C2×C4—7242no—
329C2×C4C2×D72411yescharacteristic
330C2×C4C2×D72411yescharacteristic
331C2×C4C2×D72411yescharacteristic
332C2×C4C2×D72411yescharacteristic
333C2×C4C23×D18111yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3622no—
344C22—14484no—
345C22—7241no—
346C22—7241no—
347C22—7241no—
348C22—7241no—
349C22—7241no—
350C22#1915383812yes—
351C22C22×D72411yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—7241no—
357C4—7241no—
358C4—7241no—
359C4—7241no—
360C4—7241no—
361C4—7241no—
362C4#41×D18111yescharacteristic
363C4#1915386111yescharacteristic
364C2—22no—
365C2—41no—
366C2—14481no—
367C2—7242no—
368C2#2604192411yescharacteristic
369C2C2×#1915383811yescharacteristic
370C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC8×D4×C181C2
socleC22×C181C22×D4
prime radical#26041925C2
hypercenterC8×D4D181
hypocenterC181#3296
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26041969C2semidirect
3. #26041634C2semidirect
4. #26041610C2semidirect
5. #26041604C2semidirect
6. #26041580C2semidirect
7. #26041520C2semidirect
8. #26041508C2semidirect
9. #26041502C2semidirect
10. #26041443C2semidirect
11. #26041432C2semidirect
12. C4×#15668438C2semidirect
13. #26041276C2semidirect
14. C2×#19153738C2semidirect
15. C8×D4×C181C2semidirect
18. C2×Dic724C22semidirect
22. #19153808C22semidirect
29. #19153795C22semidirect
33. C4×Dic362C22semidirect
35. #19153775C22semidirect
39. #19153750C22semidirect
40. #19153748C22semidirect
41. #19153748C22semidirect
42. #19153741C22semidirect
43. #19153741C22semidirect
44. #19153741C22semidirect
45. #19153741C22semidirect
53. C22×C8×C181C22semidirect
55. C4⋊C8×C181C22semidirect
57. C22⋊C8×C181C22semidirect
58. C4×C8×C181C22semidirect
185. C181#3296semidirect
199. C8×D4D181semidirect
208. C22×C8C2×D181semidirect
231. C4⋊C8C2×D181semidirect
243. C22⋊C8C2×D181semidirect
244. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419692
#260419252
#260416342
#260416102
#260416042
#260415802
#260415202
#260415082
#260415022
#260414432
#260414322
C4×#156684382
#260412762
C2×#191537382
C8×D4×C1812
#3296181