Finite Group Databasebeta

← #27158493

Subgroups — #27158493

Order 12416.

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#20261804C211yesmaximal, characteristic, complemented
2#20261802C211yesmaximal, characteristic, complemented
3#20261776C212yesmaximal, complemented
4#20261774C212yesmaximal, complemented
5SD16×#87604C212yesmaximal, complemented
6D8×#87604C212yesmaximal, complemented
7#20261746C211yesmaximal, characteristic, complemented
8#20261725C211yesmaximal, characteristic, complemented
9C2×#15690872C211yesmaximal, characteristic, complemented
10#20261701C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×D97C211yesmaximal, characteristic
12C2×#13095519C2211yescharacteristic
13Q8×#87604C2212yescomplemented
14D4×#87604—21no—
15D4×#87604—21no—
16D4×#87604—22no—
17D4×#87604C2212yes—
18D4×#87604C2212yes—
19D4×#87604C2212yescomplemented
20#15690889—21no—
21#15690889C2211yescharacteristic
22#15690889C2211yescharacteristic
23C2×C4⋊#87604—21no—
24C2×C4⋊#87604C2211yescharacteristic
25C2×C22⋊#87604—22no—
26#15690879C2212yescomplemented
27#15690878C2212yescomplemented
28#15690873C2211yescharacteristic, complemented
29#15690873C2211yescharacteristic, complemented
30#15690872C2212yes—
31#15690872C2212yes—
32#15690872C2211yescharacteristic, complemented
33#15690872C2211yescharacteristic, complemented
34#15690864C2212yescomplemented
35C8×#87604C2212yescomplemented
36#41×D97C2211yescharacteristic
37C2×D4×D97C2211yescharacteristic
38#15690838C411yescharacteristic, complemented
39#15690837C412yescomplemented
40#15690836C411yescharacteristic, complemented
41#15690835C411yescharacteristic, complemented
42#15690834C412yescomplemented
43SD16×D97C2212yes—
44D8×D97C2212yes—
45M16×D97C2211yescharacteristic
46AGL(1,ℤ/8)×C97C411yescharacteristic, complemented
47C22×#87604—21no—
48C22×#87604—22no—
49C22×#87604—22no—
50C22×#87604—22no—
51#13095519—21no—
52#13095519C2311yescharacteristic
53#13095519C2311yescharacteristic
54C2×#11715897—21no—
55C4⋊#87604—22no—
56C4⋊#87604—21no—
57C4⋊#87604D411yescharacteristic
58C4⋊#87604D411yescharacteristic
59C4⋊#87604C2312yes—
60C4⋊#87604C2311yescharacteristic
61C4⋊#87604C2311yescharacteristic
62C22⋊#87604—21no—
63C22⋊#87604—22no—
64C22⋊#87604—22no—
65C22⋊#87604—22no—
66C22⋊#87604D412yes—
67C4×#87604—22no—
68C4×#87604—21no—
69C4×#87604C2312yes—
70C23×D97—21no—
71#13095512C2×C411yescharacteristic, complemented
72#13095511—21no—
73#13095511C2×C411yescharacteristic, complemented
74#13095510—21no—
75#13095510C2×C411yescharacteristic, complemented
76Q8×D97C2311yescharacteristic
77C2×D388C2×C411yescharacteristic, complemented
78C2×C22⋊D97—21no—
79D4×D97—21no—
80D4×D97—21no—
81D4×D97—21no—
82D4×D97C2×C412yescomplemented
83D4×D97C2311yescharacteristic
84D4×D97C2312yes—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87#13095500C2×C412yescomplemented
88#13095499C2×C412yescomplemented
89#13095498C2×C412yescomplemented
90D776C2×C412yescomplemented
91#13095496C2×C412yescomplemented
92#13095496C2×C412yescomplemented
93#13095495C2×C411yescharacteristic, complemented
94#13095494C2×C412yescomplemented
95C8×D97C2312yes—
96#41×C97C2×C411yescharacteristic, complemented
97C2×D4×C97C2×C411yescharacteristic, complemented
98SD16×C97C2×C412yescomplemented
99D8×C97C2×C412yescomplemented
100M16×C97C2×C411yescharacteristic, complemented
101C23×C97—21no—
102C22×D97—21no—
103C22×D97—21no—
104C22×D97—22no—
105C22×D97—22no—
106C22×D97—81no—
107C22×D97—81no—
108C22×D97C2×D411yescharacteristic
109C2×#87604—21no—
110C2×#87604—21no—
111C2×#87604—22no—
112C2×#87604—82no—
113C2×#87604—82no—
114C2×#87604—82no—
115C2×#87604—81no—
116C2×#87604C2×D412yes—
117C2×#87604C2×D412yes—
118Q8×C97C22×C411yescharacteristic
119D4×C97—21no—
120D4×C97—21no—
121D4×C97C22×C411yescharacteristic
122D4×C97C22×C412yes—
123C2×C4×C97—21no—
124C2×C4×C97C22×C411yescharacteristic
125C22⋊D97—21no—
126C22⋊D97—21no—
127C22⋊D97—22no—
128C22⋊D97—22no—
129C22⋊D97—21no—
130C2×Dic97—21no—
131C2×Dic97#4111yescharacteristic
132D388—21no—
133D388—21no—
134D388C22×C411yescharacteristic
135D388C22×C412yes—
136C4×D97—21no—
137C4×D97—21no—
138C4×D97—21no—
139C4×D97C22×C411yescharacteristic
140C4×D97C2×D411yescharacteristic
141C4×D97#4111yescharacteristic
142C4×D97C2411yescharacteristic
143Dic194—21no—
144Dic194C22×C411yescharacteristic
145#11715897—21no—
146#11715897—21no—
147C8×C97C22×C412yes—
148#11715895C22×C412yes—
149C22×C97—21no—
150C22×C97—22no—
151C22×C97—81no—
152C22×C97C4×D411yescharacteristic
153C2×D97—21no—
154C2×D97—21no—
155C2×D97—21no—
156C2×D97—22no—
157C2×D97—41no—
158C2×D97—42no—
159C2×D97—42no—
160C2×D97—41no—
161C2×D97—82no—
162C2×D97—81no—
163C2×D97C4×D411yescharacteristic
164C2×D97C22×D411yescharacteristic
165#87604—22no—
166#87604—22no—
167#87604—41no—
168#87604—42no—
169#87604—41no—
170C4×C97—21no—
171C4×C97C4×D411yescharacteristic
172C4×C97C23×C411yescharacteristic
173Dic97—21no—
174Dic97C4×D411yescharacteristic
175Dic97C2×#4111yescharacteristic
176C2×C97—21no—
177C2×C97—42no—
178C2×C97—41no—
179C2×C97C2×C4×D411yescharacteristic
180D97—21no—
181D97—41no—
182D97—42no—
183D97#57611yescharacteristic
184D97#57711yescharacteristic
185#2946—971nomaximal, Sylow
186C97#294611yescharacteristic, complemented, Sylow
187C2×AGL(1,ℤ/8)—971no—
188#520—971no—
189#518—971no—
190#442—972no—
191#440—972no—
192C4×SD16—972no—
193C4×D8—972no—
194#428—971no—
195#416—971no—
196C2×#102—971no—
197#405—971no—
198C2×#41—971no—
199C22×D4—971no—
200AGL(1,ℤ/8)—971no—
201AGL(1,ℤ/8)—971no—
202AGL(1,ℤ/8)—972no—
203AGL(1,ℤ/8)—972no—
204AGL(1,ℤ/8)—971no—
205AGL(1,ℤ/8)#8760411yescharacteristic, complemented
206C2×SD16—972no—
207C2×D8—972no—
208C2×M16—971no—
209C22×C8—971no—
210C4×Q8—972no—
211C4×D4—972no—
212C4×D4—972no—
213C4×D4—972no—
214C4×D4—1942no—
215C4×D4—1941no—
216C4×D4—1941no—
217#117—971no—
218#117—971no—
219#117—1941no—
220C2×C4⋊C4—971no—
221C2×C4⋊C4—1941no—
222C2×C22⋊C4—1942no—
223#107—972no—
224#106—972no—
225#103—971no—
226#103—971no—
227#102—972no—
228#102—972no—
229#102—971no—
230#102—971no—
231#97—972no—
232C4×C8—972no—
233C24—1941no—
234#41—971no—
235#41—971no—
236#41—971no—
237#41—1941no—
238#41—1941no—
239#41C2×#8760411yescharacteristic, complemented
240C2×Q8—971no—
241C2×D4—972no—
242C2×D4—971no—
243C2×D4—972no—
244C2×D4—971no—
245C2×D4—1941no—
246C2×D4—1941no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4C2×#8760411yescharacteristic, complemented
250C22×C4—971no—
251C22×C4—1941no—
252C22×C4—1942no—
253C22×C4—1942no—
254C22×C4—1942no—
255C22×C4—1941no—
256SD16—972no—
257SD16—972no—
258SD16—972no—
259SD16C2×#8760412yescomplemented
260D8—972no—
261D8—972no—
262D8—972no—
263D8C2×#8760412yescomplemented
264M16—971no—
265M16—972no—
266M16C2×#8760411yescharacteristic, complemented
267C2×C8—972no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—1941no—
271C2×C8—1941no—
272C4⋊C4—971no—
273C4⋊C4—971no—
274C4⋊C4—972no—
275C4⋊C4—971no—
276C4⋊C4—971no—
277C4⋊C4—1941no—
278C4⋊C4—1942no—
279C22⋊C4—972no—
280C22⋊C4—1942no—
281C22⋊C4—1942no—
282C22⋊C4—1942no—
283C22⋊C4—1941no—
284C42—972no—
285C42—1941no—
286C42—1942no—
287C23—21no—
288C23—7761no—
289C23—7761no—
290C23—971no—
291C23—1942no—
292C23—1941no—
293C23—1942no—
294C23—1941no—
295Q8—971no—
296Q8—1941no—
297Q8C22×#8760411yescharacteristic
298D4—21no—
299D4—21no—
300D4—971no—
301D4—972no—
302D4—1941no—
303D4—1942no—
304D4—1942no—
305D4—1941no—
306D4—1941no—
307D4—1941no—
308D4—1941no—
309D4C22×#8760411yescharacteristic
310D4C22×#8760412yes—
311C2×C4—21no—
312C2×C4—7761no—
313C2×C4—7762no—
314C2×C4—7762no—
315C2×C4—7762no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—971no—
319C2×C4—971no—
320C2×C4—971no—
321C2×C4—972no—
322C2×C4—972no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1942no—
329C2×C4—1941no—
330C2×C4C22×#8760411yescharacteristic
331C8—972no—
332C8—1941no—
333C8—1941no—
334C8C22×#8760412yes—
335C22—21no—
336C22—22no—
337C22—81no—
338C22—7761no—
339C22—7762no—
340C22—971no—
341C22—971no—
342C22—3881no—
343C22—3882no—
344C22—3882no—
345C22—3881no—
346C22—1941no—
347C22—1941no—
348C22—1941no—
349C22—1942no—
350C22D4×#8760411yescharacteristic
351C4—21no—
352C4—971no—
353C4—971no—
354C4—3882no—
355C4—3881no—
356C4—3881no—
357C4—1942no—
358C4—1942no—
359C4—1941no—
360C4D4×#8760411yescharacteristic
361C4C23×#8760411yescharacteristic
362C2—21no—
363C2—42no—
364C2—41no—
365C2—971no—
366C2—971no—
367C2—3882no—
368C2—3881no—
369C2—1941no—
370C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC2C2×D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingAGL(1,ℤ/8)×C97C4
socleC2×C97C2×C4×D4
prime radicalAGL(1,ℤ/8)×D97C2
cosocleC4×D97C24
hypercenterAGL(1,ℤ/8)#87604
hypocenterC97#2946
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20261804C2semidirect
2. #20261802C2semidirect
3. #20261776C2semidirect
4. #20261774C2semidirect
5. SD16×#87604C2semidirect
6. D8×#87604C2semidirect
7. #20261746C2semidirect
8. #20261725C2semidirect
9. C2×#15690872C2semidirect
10. #20261701C2semidirect
13. Q8×#87604C22semidirect
19. D4×#87604C22semidirect
26. #15690879C22semidirect
27. #15690878C22semidirect
28. #15690873C22semidirect
29. #15690873C22semidirect
32. #15690872C22semidirect
33. #15690872C22semidirect
34. #15690864C22semidirect
35. C8×#87604C22semidirect
38. #15690838C4semidirect
39. #15690837C4semidirect
40. #15690836C4semidirect
41. #15690835C4semidirect
42. #15690834C4semidirect
46. AGL(1,ℤ/8)×C97C4semidirect
71. #13095512C2×C4semidirect
73. #13095511C2×C4semidirect
75. #13095510C2×C4semidirect
77. C2×D388C2×C4semidirect
82. D4×D97C2×C4semidirect
87. #13095500C2×C4semidirect
88. #13095499C2×C4semidirect
89. #13095498C2×C4semidirect
90. D776C2×C4semidirect
91. #13095496C2×C4semidirect
92. #13095496C2×C4semidirect
93. #13095495C2×C4semidirect
94. #13095494C2×C4semidirect
96. #41×C97C2×C4semidirect
97. C2×D4×C97C2×C4semidirect
98. SD16×C97C2×C4semidirect
99. D8×C97C2×C4semidirect
100. M16×C97C2×C4semidirect
186. C97#2946semidirect
205. AGL(1,ℤ/8)#87604semidirect
239. #41C2×#87604semidirect
249. C2×D4C2×#87604semidirect
259. SD16C2×#87604semidirect
263. D8C2×#87604semidirect
266. M16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618042
#202618022
#202617762
#202617742
SD16×#876042
D8×#876042
#202617462
#202617252
C2×#156908722
#202617012
AGL(1,ℤ/8)×D972
#294697