Finite Group Databasebeta

← #37546738

Subgroups — #37546738

Order 23168.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041998C211yesmaximal, characteristic, complemented
2#26041949C211yesmaximal, characteristic, complemented
3#26041667C211yesmaximal, characteristic, complemented
4#26041650C211yesmaximal, characteristic, complemented
5#26041492C211yesmaximal, characteristic, complemented
6#26041491C211yesmaximal, characteristic, complemented
7#26041482C211yesmaximal, characteristic, complemented
8#26041477C211yesmaximal, characteristic, complemented
9#26041476C211yesmaximal, characteristic, complemented
10#26041472C211yesmaximal, characteristic, complemented
11#26041407C211yesmaximal, characteristic, complemented
12#26041381C211yesmaximal, characteristic, complemented
13#26041380C211yesmaximal, characteristic, complemented
14D8×Dic181C211yesmaximal, characteristic, complemented
15#496×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668438C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C2×#15668434C2211yescharacteristic
22#19153814C2211yescharacteristic
23#19153812C2211yescharacteristic
24#19153812C2211yescharacteristic
25#19153809—21no—
26#19153809—21no—
27#19153803C2211yescharacteristic
28#19153793—21no—
29#19153788—21no—
30#19153788—21no—
31#19153787—21no—
32Q8×Dic181C2211yescharacteristic
33C4×Dic362C2211yescharacteristic
34C4×Dic362C2211yescharacteristic
35C4×D724C2211yescharacteristic
36D4×Dic181C2211yescharacteristic
37D4×Dic181C2211yescharacteristic
38D4×Dic181C2211yescharacteristic
39#19153772—21no—
40#19153750C2211yescharacteristic
41C4⋊#13089678C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153737C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54C2×D8×C181C2211yescharacteristic
55C2×D8×C181C2211yescharacteristic
56#127×C181C2211yescharacteristic
57#124×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668448—44no—
60C2×Dic362D411yescharacteristic
61C2×Dic362C2311yescharacteristic
62C2×D724C2311yescharacteristic
63C2×C22⋊D181—21no—
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C22×Dic181—21no—
68C22×Dic181—21no—
69C22×Dic181—21no—
70C2×C4×D181—21no—
71#15668438—22no—
72#15668437—41no—
73#15668437—41no—
74#15668436—41no—
75#15668434—41no—
76C2×#13089678C2311yescharacteristic
77C2×#13089678C2311yescharacteristic
78C4⋊Dic181—21no—
79C4⋊Dic181D411yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428D411yescharacteristic
86#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C4×Dic181C2311yescharacteristic
97C2×Q8×C181C2311yescharacteristic
98C2×D4×C181—21no—
99C2×D4×C181—21no—
100C2×D4×C181C2311yescharacteristic
101C2×D4×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103SD16×C181—22no—
104SD16×C181—41no—
105D8×C181—22no—
106D8×C181—41no—
107C2×C8×C181C2311yescharacteristic
108C2×C8×C181C2311yescharacteristic
109C22⋊C4×C181—21no—
110C22⋊C4×C181—21no—
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113Dic362—22no—
114Dic362—21no—
115Dic362—21no—
116D724—21no—
117D724—21no—
118C22⋊D181—42no—
119C22⋊D181—42no—
120C22⋊D181—81no—
121C22⋊D181—81no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181—42no—
126C2×Dic181—42no—
127C2×Dic181—42no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C2×Dic181C2×D411yescharacteristic
131C2×Dic181C2×D411yescharacteristic
132C4×D181—42no—
133#13089678—21no—
134#13089678—21no—
135C23×C181—21no—
136C23×C181—21no—
137C23×C181—21no—
138Q8×C181—21no—
139Q8×C181—21no—
140D4×C181—21no—
141D4×C181—21no—
142D4×C181—21no—
143D4×C181—21no—
144D4×C181—21no—
145D4×C181—21no—
146D4×C181—42no—
147D4×C181—81no—
148C2×C4×C181—21no—
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2411yescharacteristic
152C8×C181—21no—
153C8×C181#4112yescomplemented
154C22×C181—41no—
155C22×C181—41no—
156C22×C181—41no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181C22×D411yescharacteristic
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C4×C181—21no—
168C4×C181—21no—
169C4×C181—41no—
170C4×C181C2×#4111yescharacteristic
171C4×C181H(5,𝔽2)11yescharacteristic
172Dic181—21no—
173Dic181—21no—
174Dic181—21no—
175Dic181—41no—
176Dic181—41no—
177Dic181—41no—
178Dic181AGL(1,ℤ/8)12yes—
179C2×C181—81no—
180C2×C181—81no—
181C2×C181—81no—
182C2×C181#55011yescharacteristic
183C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C181#57711yescharacteristic
185D181—81no—
186C181#339711yescharacteristic, complemented, Sylow
187#3397—1811nomaximal, Sylow
188#556—1811no—
189#550—1811no—
190#496—1811no—
191#496D18111yescharacteristic, complemented
192#492—1811no—
193#463—1811no—
194#463—1811no—
195#461—1811no—
196#460—1811no—
197#460—1811no—
198#459—1811no—
199#446—1811no—
200#440—1811no—
201#440—1811no—
202C4×D8—1811no—
203C2×#41—3621no—
204C2×SD16—1811no—
205C2×SD16—1811no—
206C2×SD16—1811no—
207C2×SD16—1811no—
208C2×SD16C2×D18111yescharacteristic
209C2×SD16C2×D18111yescharacteristic
210C2×D8—1811no—
211C2×D8C2×D18111yescharacteristic
212C2×D8C2×D18111yescharacteristic
213C4⋊Q8—1811no—
214#127—1811no—
215#127—1811no—
216#127C2×D18111yescharacteristic
217#126—3621no—
218#126—3621no—
219#124—1811no—
220#124C2×D18111yescharacteristic
221C22⋊Q8—3621no—
222#121—3621no—
223#121—3621no—
224#121—3621no—
225C4×Q8—1811no—
226C4×Q8—1811no—
227C4×Q8—1811no—
228C4×D4—1811no—
229C4×D4—1811no—
230C4×D4—1811no—
231C4×D4—1811no—
232#117—3621no—
233#107—1811no—
234C4⋊C8—1811no—
235C4⋊C8—1811no—
236C4⋊C8—1811no—
237#103—1811no—
238#103—1811no—
239#102—1811no—
240#102—1811no—
241#102—1811no—
242#102—1811no—
243#97—1811no—
244#97C2×D18111yescharacteristic
245C4×C8—1811no—
246#41—7244no—
247C2×Q8—1811no—
248C2×Q8—1811no—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3621no—
253C2×D4—3621no—
254C2×D4—3621no—
255C2×D4—3621no—
256C2×D4—1811no—
257C2×D4C22×D18111yescharacteristic
258C2×D4C22×D18111yescharacteristic
259C2×D4C22×D18111yescharacteristic
260C22×C4—3621no—
261C22×C4—3621no—
262C22×C4—3621no—
263C22×C4—3621no—
264SD16—22no—
265SD16—41no—
266SD16—3622no—
267SD16—7241no—
268SD16—7241no—
269SD16—7241no—
270D8—22no—
271D8—41no—
272D8—7241no—
273C2×C8—1811no—
274C2×C8—1811no—
275C2×C8C22×D18111yescharacteristic
276C2×C8C22×D18111yescharacteristic
277C4⋊C4—3621no—
278C4⋊C4—3621no—
279C4⋊C4—3621no—
280C4⋊C4—3621no—
281C4⋊C4—1811no—
282C4⋊C4—1811no—
283C4⋊C4—1811no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—3621no—
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C42—3621no—
295C42—3621no—
296C42—1811no—
297C42—1811no—
298C42C22×D18111yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—3621no—
303Q8—21no—
304Q8—21no—
305Q8—3622no—
306Q8—3621no—
307Q8—3621no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—81no—
316D4—3621no—
317D4—3621no—
318D4—14481no—
319D4—14481no—
320D4—7242no—
321D4—7242no—
322C2×C4—21no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—1811no—
327C2×C4—1811no—
328C2×C4—1811no—
329C2×C4—1811no—
330C2×C4—7242no—
331C2×C4—7242no—
332C2×C4—7242no—
333C2×C4—7242no—
334C2×C4D4×D18111yescharacteristic
335C2×C4D4×D18111yescharacteristic
336C2×C4C23×D18111yescharacteristic
337C8—21no—
338C8—3621no—
339C8—3621no—
340C8#1566844812yescomplemented
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—7241no—
351C22—7241no—
352C22—7241no—
353C22C2×D4×D18111yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—3621no—
358C4—3621no—
359C4—3621no—
360C4—1812no—
361C4—7241no—
362C4—7241no—
363C4—7241no—
364C4C2×#1566844811yescharacteristic
365C4#1915386111yescharacteristic
366C2—81no—
367C2—81no—
368C2—81no—
369C2—14481no—
370C2#2604194911yescharacteristic
371C2AGL(1,ℤ/8)×D18111yescharacteristic
372C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC22×C181C22×D4
hypercenter#496D181
hypocenterC181#3397
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26041998C2semidirect
2. #26041949C2semidirect
3. #26041667C2semidirect
4. #26041650C2semidirect
5. #26041492C2semidirect
6. #26041491C2semidirect
7. #26041482C2semidirect
8. #26041477C2semidirect
9. #26041476C2semidirect
10. #26041472C2semidirect
11. #26041407C2semidirect
12. #26041381C2semidirect
13. #26041380C2semidirect
14. D8×Dic181C2semidirect
15. #496×C181C2semidirect
153. C8×C181#41semidirect
186. C181#3397semidirect
191. #496D181semidirect
340. C8#15668448semidirect

Maximal subgroups

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

subgroupindex
#260419982
#260419492
#260416672
#260416502
#260414922
#260414912
#260414822
#260414772
#260414762
#260414722
#260414072
#260413812
#260413802
D8×Dic1812
#496×C1812
#3397181