Finite Group Databasebeta

← #37550478

Subgroups — #37550478

Order 23168.

506 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#26042348C211yesmaximal, characteristic, complemented
2#26042346C211yesmaximal, characteristic, complemented
3#26042320C212yesmaximal, complemented
4#26042318C212yesmaximal, complemented
5SD16×#10624382C212yesmaximal, complemented
6D8×#10624382C212yesmaximal, complemented
7#26042290C211yesmaximal, characteristic, complemented
8#26042269C211yesmaximal, characteristic, complemented
9C2×#19153878C211yesmaximal, characteristic, complemented
10#26042245C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×D181C211yesmaximal, characteristic
12C2×D181⋊C8C2211yescharacteristic
13Q8×#10624382C2212yescomplemented
14D4×#10624382—21no—
15D4×#10624382—21no—
16D4×#10624382—22no—
17D4×#10624382C2212yes—
18D4×#10624382C2212yes—
19D4×#10624382C2212yescomplemented
20#19153895—21no—
21#19153895C2211yescharacteristic
22#19153895C2211yescharacteristic
23C2×C4⋊#10624382—21no—
24C2×C4⋊#10624382C2211yescharacteristic
25C2×C22⋊#10624382—22no—
26#19153885C2212yescomplemented
27#19153884C2212yescomplemented
28#19153879C2211yescharacteristic, complemented
29#19153879C2211yescharacteristic, complemented
30#19153878C2212yes—
31#19153878C2212yes—
32#19153878C2211yescharacteristic, complemented
33#19153878C2211yescharacteristic, complemented
34#19153870C2212yescomplemented
35C8×#10624382C2212yescomplemented
36#41×D181C2211yescharacteristic
37C2×D4×D181C2211yescharacteristic
38#19153844C411yescharacteristic, complemented
39#19153843C412yescomplemented
40#19153842C411yescharacteristic, complemented
41#19153841C411yescharacteristic, complemented
42#19153840C412yescomplemented
43SD16×D181C2212yes—
44D8×D181C2212yes—
45M16×D181C2211yescharacteristic
46AGL(1,ℤ/8)×C181C411yescharacteristic, complemented
47C22×#10624382—22no—
48C22×#10624382—22no—
49C22×#10624382—22no—
50C22×#10624382—21no—
51D181⋊C8—21no—
52D181⋊C8C2311yescharacteristic
53D181⋊C8C2311yescharacteristic
54C2×#13089685—21no—
55C4⋊#10624382—22no—
56C4⋊#10624382—21no—
57C4⋊#10624382D411yescharacteristic
58C4⋊#10624382D411yescharacteristic
59C4⋊#10624382C2312yes—
60C4⋊#10624382C2311yescharacteristic
61C4⋊#10624382C2311yescharacteristic
62C22⋊#10624382—22no—
63C22⋊#10624382—22no—
64C22⋊#10624382—22no—
65C22⋊#10624382—21no—
66C22⋊#10624382D412yes—
67C4×#10624382—22no—
68C4×#10624382—21no—
69C4×#10624382C2312yes—
70C23×D181—21no—
71#15668450C2×C411yescharacteristic, complemented
72#15668449—21no—
73#15668449C2×C411yescharacteristic, complemented
74#15668448—21no—
75#15668448C2×C411yescharacteristic, complemented
76Q8×D181C2311yescharacteristic
77C2×D724C2×C411yescharacteristic, complemented
78C2×C22⋊D181—21no—
79D4×D181—21no—
80D4×D181—21no—
81D4×D181—21no—
82D4×D181C2×C412yescomplemented
83D4×D181C2311yescharacteristic
84D4×D181C2312yes—
85C2×C4×D181—21no—
86C2×C4×D181C2311yescharacteristic
87#15668438C2×C412yescomplemented
88#15668437C2×C412yescomplemented
89#15668436C2×C412yescomplemented
90D1448C2×C412yescomplemented
91#15668434C2×C412yescomplemented
92#15668434C2×C412yescomplemented
93#15668433C2×C411yescharacteristic, complemented
94#15668432C2×C412yescomplemented
95C8×D181C2312yes—
96#41×C181C2×C411yescharacteristic, complemented
97C2×D4×C181C2×C411yescharacteristic, complemented
98SD16×C181C2×C412yescomplemented
99D8×C181C2×C412yescomplemented
100M16×C181C2×C411yescharacteristic, complemented
101C2×#10624382—21no—
102C2×#10624382—21no—
103C2×#10624382—22no—
104C2×#10624382—82no—
105C2×#10624382—82no—
106C2×#10624382—82no—
107C2×#10624382—81no—
108C2×#10624382C2×D412yes—
109C2×#10624382C2×D412yes—
110#13089685—21no—
111#13089685—21no—
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—22no—
115C22×D181—22no—
116C22×D181—81no—
117C22×D181—81no—
118C22×D181C2×D411yescharacteristic
119Dic362—21no—
120Dic362C22×C411yescharacteristic
121D724—21no—
122D724—21no—
123D724C22×C411yescharacteristic
124D724C22×C412yes—
125C22⋊D181—21no—
126C22⋊D181—21no—
127C22⋊D181—22no—
128C22⋊D181—22no—
129C22⋊D181—21no—
130C2×Dic181—21no—
131C2×Dic181#4111yescharacteristic
132C4×D181—21no—
133C4×D181—21no—
134C4×D181—21no—
135C4×D181C22×C411yescharacteristic
136C4×D181C2×D411yescharacteristic
137C4×D181#4111yescharacteristic
138C4×D181C2411yescharacteristic
139#13089678C22×C412yes—
140C23×C181—21no—
141Q8×C181C22×C411yescharacteristic
142D4×C181—21no—
143D4×C181—21no—
144D4×C181C22×C411yescharacteristic
145D4×C181C22×C412yes—
146C2×C4×C181—21no—
147C2×C4×C181C22×C411yescharacteristic
148C8×C181C22×C412yes—
149C22×C181—21no—
150C22×C181—22no—
151C22×C181—81no—
152C22×C181C4×D411yescharacteristic
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—22no—
157C2×D181—42no—
158C2×D181—41no—
159C2×D181—41no—
160C2×D181—42no—
161C2×D181—82no—
162C2×D181—81no—
163C2×D181C4×D411yescharacteristic
164C2×D181C22×D411yescharacteristic
165#10624382—22no—
166#10624382—22no—
167#10624382—41no—
168#10624382—42no—
169#10624382—41no—
170C4×C181—21no—
171C4×C181C4×D411yescharacteristic
172C4×C181C23×C411yescharacteristic
173Dic181—21no—
174Dic181C4×D411yescharacteristic
175Dic181C2×#4111yescharacteristic
176C2×C181—21no—
177C2×C181—42no—
178C2×C181—41no—
179C2×C181C2×C4×D411yescharacteristic
180D181—21no—
181D181—41no—
182D181—42no—
183D181#57611yescharacteristic
184D181#57711yescharacteristic
185C181#294611yescharacteristic, complemented, Sylow
186#2946—1811nomaximal, Sylow
187C2×AGL(1,ℤ/8)—1811no—
188#520—1811no—
189#518—1811no—
190#442—1812no—
191#440—1812no—
192C4×SD16—1812no—
193C4×D8—1812no—
194#428—1811no—
195#416—1811no—
196C2×#102—1811no—
197#405—1811no—
198C2×#41—1811no—
199C22×D4—1811no—
200AGL(1,ℤ/8)—1811no—
201AGL(1,ℤ/8)—1811no—
202AGL(1,ℤ/8)—1812no—
203AGL(1,ℤ/8)—1812no—
204AGL(1,ℤ/8)—1811no—
205AGL(1,ℤ/8)#1062438211yescharacteristic, complemented
206C2×SD16—1812no—
207C2×D8—1812no—
208C2×M16—1811no—
209C22×C8—1811no—
210C4×Q8—1812no—
211C4×D4—3621no—
212C4×D4—3621no—
213C4×D4—3622no—
214C4×D4—1812no—
215C4×D4—1812no—
216C4×D4—1812no—
217#117—3621no—
218#117—1811no—
219#117—1811no—
220C2×C4⋊C4—3621no—
221C2×C4⋊C4—1811no—
222C2×C22⋊C4—3622no—
223#107—1812no—
224#106—1812no—
225#103—1811no—
226#103—1811no—
227#102—1812no—
228#102—1812no—
229#102—1811no—
230#102—1811no—
231#97—1812no—
232C4×C8—1812no—
233C24—3621no—
234#41—3621no—
235#41—3621no—
236#41—1811no—
237#41—1811no—
238#41—1811no—
239#41C2×#1062438211yescharacteristic, complemented
240C2×Q8—1811no—
241C2×D4—3621no—
242C2×D4—3621no—
243C2×D4—3621no—
244C2×D4—3621no—
245C2×D4—1811no—
246C2×D4—1812no—
247C2×D4—1812no—
248C2×D4—1811no—
249C2×D4C2×#1062438211yescharacteristic, complemented
250C22×C4—3621no—
251C22×C4—3621no—
252C22×C4—3622no—
253C22×C4—3622no—
254C22×C4—3622no—
255C22×C4—1811no—
256SD16—1812no—
257SD16—1812no—
258SD16—1812no—
259SD16C2×#1062438212yescomplemented
260D8—1812no—
261D8—1812no—
262D8—1812no—
263D8C2×#1062438212yescomplemented
264M16—1811no—
265M16—1812no—
266M16C2×#1062438211yescharacteristic, complemented
267C2×C8—3621no—
268C2×C8—3621no—
269C2×C8—1811no—
270C2×C8—1811no—
271C2×C8—1812no—
272C4⋊C4—3621no—
273C4⋊C4—3622no—
274C4⋊C4—1812no—
275C4⋊C4—1811no—
276C4⋊C4—1811no—
277C4⋊C4—1811no—
278C4⋊C4—1811no—
279C22⋊C4—3621no—
280C22⋊C4—3622no—
281C22⋊C4—3622no—
282C22⋊C4—3622no—
283C22⋊C4—1812no—
284C42—3622no—
285C42—3621no—
286C42—1812no—
287C23—21no—
288C23—3622no—
289C23—3621no—
290C23—3622no—
291C23—3621no—
292C23—14481no—
293C23—14481no—
294C23—1811no—
295Q8—3621no—
296Q8—1811no—
297Q8C22×#1062438211yescharacteristic
298D4—21no—
299D4—21no—
300D4—3621no—
301D4—3621no—
302D4—3621no—
303D4—3622no—
304D4—3622no—
305D4—3621no—
306D4—3621no—
307D4—1811no—
308D4—1812no—
309D4C22×#1062438211yescharacteristic
310D4C22×#1062438212yes—
311C2×C4—21no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—3622no—
319C2×C4—14482no—
320C2×C4—14482no—
321C2×C4—14482no—
322C2×C4—14481no—
323C2×C4—1812no—
324C2×C4—1812no—
325C2×C4—1811no—
326C2×C4—1811no—
327C2×C4—1811no—
328C2×C4—1811no—
329C2×C4—1811no—
330C2×C4C22×#1062438211yescharacteristic
331C8—3621no—
332C8—3621no—
333C8—1812no—
334C8C22×#1062438212yes—
335C22—21no—
336C22—22no—
337C22—81no—
338C22—3621no—
339C22—3621no—
340C22—3621no—
341C22—3622no—
342C22—14481no—
343C22—14482no—
344C22—1811no—
345C22—1811no—
346C22—7241no—
347C22—7242no—
348C22—7242no—
349C22—7241no—
350C22D4×#1062438211yescharacteristic
351C4—21no—
352C4—3622no—
353C4—3622no—
354C4—3621no—
355C4—1811no—
356C4—1811no—
357C4—7241no—
358C4—7242no—
359C4—7241no—
360C4D4×#1062438211yescharacteristic
361C4C23×#1062438211yescharacteristic
362C2—21no—
363C2—42no—
364C2—41no—
365C2—3621no—
366C2—1811no—
367C2—1811no—
368C2—7242no—
369C2—7241no—
370C2C2×D4×#1062438211yescharacteristic
subgroupG/H
centerC2C2×D4×#10624382
commutatorC4×C181C23×C4
FrattiniC4C23×#10624382
FittingAGL(1,ℤ/8)×C181C4
socleC2×C181C2×C4×D4
prime radicalAGL(1,ℤ/8)×D181C2
cosocleC4×D181C24
hypercenterAGL(1,ℤ/8)#10624382
hypocenterC181#2946
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26042348C2semidirect
2. #26042346C2semidirect
3. #26042320C2semidirect
4. #26042318C2semidirect
5. SD16×#10624382C2semidirect
6. D8×#10624382C2semidirect
7. #26042290C2semidirect
8. #26042269C2semidirect
9. C2×#19153878C2semidirect
10. #26042245C2semidirect
13. Q8×#10624382C22semidirect
19. D4×#10624382C22semidirect
26. #19153885C22semidirect
27. #19153884C22semidirect
28. #19153879C22semidirect
29. #19153879C22semidirect
32. #19153878C22semidirect
33. #19153878C22semidirect
34. #19153870C22semidirect
35. C8×#10624382C22semidirect
38. #19153844C4semidirect
39. #19153843C4semidirect
40. #19153842C4semidirect
41. #19153841C4semidirect
42. #19153840C4semidirect
46. AGL(1,ℤ/8)×C181C4semidirect
71. #15668450C2×C4semidirect
73. #15668449C2×C4semidirect
75. #15668448C2×C4semidirect
77. C2×D724C2×C4semidirect
82. D4×D181C2×C4semidirect
87. #15668438C2×C4semidirect
88. #15668437C2×C4semidirect
89. #15668436C2×C4semidirect
90. D1448C2×C4semidirect
91. #15668434C2×C4semidirect
92. #15668434C2×C4semidirect
93. #15668433C2×C4semidirect
94. #15668432C2×C4semidirect
96. #41×C181C2×C4semidirect
97. C2×D4×C181C2×C4semidirect
98. SD16×C181C2×C4semidirect
99. D8×C181C2×C4semidirect
100. M16×C181C2×C4semidirect
185. C181#2946semidirect
205. AGL(1,ℤ/8)#10624382semidirect
239. #41C2×#10624382semidirect
249. C2×D4C2×#10624382semidirect
259. SD16C2×#10624382semidirect
263. D8C2×#10624382semidirect
266. M16C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423482
#260423462
#260423202
#260423182
SD16×#106243822
D8×#106243822
#260422902
#260422692
C2×#191538782
#260422452
AGL(1,ℤ/8)×D1812
#2946181