Finite Group Databasebeta

← #27151279

Subgroups — #27151279

Order 12416.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690830C211yesmaximal, characteristic, complemented
2#20261302C211yesmaximal, characteristic, complemented
3#20261218C211yesmaximal, characteristic, complemented
4#20260955C212yesmaximal, complemented
5#20260948C212yesmaximal, complemented
6#20260933C212yesmaximal, complemented
7#20260928C212yesmaximal, complemented
8#20260763C211yesmaximal, characteristic, complemented
9#20260737C211yesmaximal, characteristic, complemented
10C2×#15690731C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13#15690830—24no—
14C2×#13095501C2211yescharacteristic, complemented
15C2×#13095499C2211yescharacteristic, complemented
16C2×#13095498C2211yescharacteristic, complemented
17C2×#13095496C2211yescharacteristic, complemented
18C22×#11715895C2211yescharacteristic
19#15690806C2212yes—
20#15690797C2212yes—
21#15690781—21no—
22#15690781—21no—
23#15690779—22no—
24Q8×Dic97C2212yescomplemented
25D4×Dic97—22no—
26D4×Dic97C2212yescomplemented
27#15690769—21no—
28#15690769C2211yescharacteristic
29C2×C4⋊Dic97—21no—
30C2×C4⋊Dic97C2211yescharacteristic
31#15690739C2212yescomplemented
32#15690739C2212yescomplemented
33#15690735C2212yescomplemented
34#15690732C2212yescomplemented
35#15690731C2212yes—
36#15690731C2212yes—
37C2×#41×C97C2211yescharacteristic
38C22×D4×C97C2211yescharacteristic
39AGL(1,ℤ/8)×C97—24no—
40C2×SD16×C97C2212yes—
41C2×D8×C97C2212yes—
42C2×M16×C97C2211yescharacteristic
43#13095511—22no—
44#13095511—22no—
45C2×Dic194C2311yescharacteristic
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—21no—
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C22×Dic97—21no—
51C22×Dic97—22no—
52C2×C4×D97—21no—
53#13095501—22no—
54#13095499—22no—
55#13095498—22no—
56#13095496—22no—
57C2×#11715895—22no—
58C2×#11715895C2311yescharacteristic
59C2×#11715895C2311yescharacteristic
60C4⋊Dic97—21no—
61C4⋊Dic97—22no—
62C4⋊Dic97D411yescharacteristic
63C4⋊Dic97D411yescharacteristic
64C4⋊Dic97C2312yes—
65C22⋊Dic97—22no—
66C22⋊Dic97—22no—
67C22⋊Dic97—22no—
68C22⋊Dic97D412yes—
69C4×Dic97—22no—
70C4×Dic97C2312yes—
71C24×C97—21no—
72#41×C97—22no—
73#41×C97D414yescomplemented
74C2×Q8×C97C2311yescharacteristic
75C2×D4×C97—21no—
76C2×D4×C97—22no—
77C2×D4×C97—21no—
78C2×D4×C97—22no—
79C2×D4×C97C2312yes—
80C2×D4×C97C2311yescharacteristic
81C22×C4×C97—21no—
82C22×C4×C97C2311yescharacteristic
83SD16×C97—24no—
84D8×C97—24no—
85M16×C97—22no—
86C2×C8×C97C2312yes—
87C23×C97—21no—
88C23×C97—22no—
89C23×C97—42no—
90C23×C97—81no—
91C23×C97—81no—
92C23×C97C2×D411yescharacteristic
93C22×D97—21no—
94Q8×C97—21no—
95Q8×C97C2×D412yes—
96D4×C97—24no—
97D4×C97—22no—
98D4×C97—21no—
99D4×C97—22no—
100D4×C97—42no—
101D4×C97C2×D412yes—
102C2×C4×C97—21no—
103C2×C4×C97—24no—
104C2×C4×C97C2×D411yescharacteristic
105C2×C4×C97C2×D412yes—
106C2×C4×C97#4112yes—
107C2×C4×C97C2411yescharacteristic
108C22⋊D97—42no—
109C22⋊D97—82no—
110C22⋊D97—82no—
111C2×Dic97—22no—
112C2×Dic97—21no—
113C2×Dic97—82no—
114C2×Dic97—81no—
115C2×Dic97C2×D412yes—
116C2×Dic97C2×D412yes—
117D388—21no—
118D388—21no—
119C4×D97—42no—
120Dic194—21no—
121Dic194—21no—
122C8×C97—22no—
123#11715895—21no—
124#11715895—21no—
125C22×C97—21no—
126C22×C97—21no—
127C22×C97—22no—
128C22×C97—41no—
129C22×C97—41no—
130C22×C97—42no—
131C22×C97—42no—
132C22×C97—42no—
133C22×C97—84no—
134C22×C97#12112yes—
135C22×C97C22×D411yescharacteristic
136C2×D97—41no—
137C2×D97—41no—
138C2×D97—41no—
139C4×C97—22no—
140C4×C97#12112yes—
141C4×C97C22×D411yescharacteristic
142C4×C97C2×#4111yescharacteristic
143Dic97—22no—
144Dic97—22no—
145Dic97—41no—
146Dic97—42no—
147C2×C97—22no—
148C2×C97—42no—
149C2×C97—82no—
150C2×C97C2×#12111yescharacteristic
151C2×C97#57611yescharacteristic
152C2×C97#57711yescharacteristic
153D97—81no—
154#3045—971nomaximal, Sylow
155C97#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
157C2×#135—971no—
158#535—971no—
159#520—971no—
160#464—972no—
161#463—972no—
162#460—972no—
163#459—972no—
164#424—971no—
165#417—971no—
166C2×#102—971no—
167C2×#41—971no—
168C2×#41C2×D9711yescharacteristic
169C22×D4C2×D9711yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—1944no—
172C2×Q16—971no—
173C2×SD16—971no—
174C2×SD16—971no—
175C2×SD16C2×D9712yes—
176C2×D8—971no—
177C2×D8C2×D9712yes—
178C2×M16C2×D9711yescharacteristic
179C22×C8—971no—
180#127—972no—
181#124—972no—
182#121—1941no—
183#121—1941no—
184#120—1942no—
185C4×Q8—972no—
186C4×D4—972no—
187C4×D4—1942no—
188#117—971no—
189#117—1941no—
190C2×C4⋊C4—971no—
191C2×C4⋊C4—1941no—
192C4⋊C8—972no—
193C4⋊C8—972no—
194#103—972no—
195#102—972no—
196#102—972no—
197#102—972no—
198C24—21no—
199#41—22no—
200#41—1942no—
201#41—1942no—
202#41C22⋊D9714yescomplemented
203C2×Q8—971no—
204C2×Q8C22×D9711yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—971no—
210C2×D4—1942no—
211C2×D4—1942no—
212C2×D4—1941no—
213C2×D4C22×D9712yes—
214C2×D4C22×D9711yescharacteristic
215C22×C4—21no—
216C22×C4—1942no—
217C22×C4—1941no—
218C22×C4—1941no—
219C22×C4C22×D9711yescharacteristic
220Q16—1942no—
221SD16—24no—
222SD16—1942no—
223SD16—1942no—
224D8—24no—
225D8—1942no—
226M16—22no—
227C2×C8—971no—
228C2×C8—971no—
229C2×C8—1942no—
230C2×C8C22×D9712yes—
231C4⋊C4—971no—
232C4⋊C4—971no—
233C4⋊C4—972no—
234C4⋊C4—1942no—
235C4⋊C4—1941no—
236C22⋊C4—972no—
237C22⋊C4—1942no—
238C22⋊C4—1942no—
239C22⋊C4—1942no—
240C42—972no—
241C42—1942no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—1941no—
248C23D4×D9711yescharacteristic
249Q8—21no—
250Q8—1941no—
251Q8—1941no—
252Q8C2×C22⋊D9712yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—7762no—
259D4—7762no—
260D4—3882no—
261D4—1941no—
262D4—1941no—
263D4C2×C22⋊D9712yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—7761no—
267C2×C4—7762no—
268C2×C4—972no—
269C2×C4—972no—
270C2×C4—3882no—
271C2×C4—1942no—
272C2×C4—1941no—
273C2×C4D4×D9711yescharacteristic
274C2×C4C2×C22⋊D9712yes—
275C2×C4#1309551012yes—
276C2×C4C23×D9711yescharacteristic
277C8—22no—
278C8—1941no—
279C8—1941no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—3881no—
290C22—3881no—
291C22—3881no—
292C22#1569078212yes—
293C22C2×D4×D9711yescharacteristic
294C4—22no—
295C4—3881no—
296C4—3882no—
297C4—1942no—
298C4—1942no—
299C4#1569078212yes—
300C4C22×C22⋊D9711yescharacteristic
301C4C2×#1309551011yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—7761no—
306C2C2×#1569078211yescharacteristic
307C2#2026153911yescharacteristic
308C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3045
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15690830C2semidirect
2. #20261302C2semidirect
3. #20261218C2semidirect
4. #20260955C2semidirect
5. #20260948C2semidirect
6. #20260933C2semidirect
7. #20260928C2semidirect
8. #20260763C2semidirect
9. #20260737C2semidirect
10. C2×#15690731C2semidirect
11. C2×AGL(1,ℤ/8)×C97C2semidirect
14. C2×#13095501C22semidirect
15. C2×#13095499C22semidirect
16. C2×#13095498C22semidirect
17. C2×#13095496C22semidirect
24. Q8×Dic97C22semidirect
26. D4×Dic97C22semidirect
31. #15690739C22semidirect
32. #15690739C22semidirect
33. #15690735C22semidirect
34. #15690732C22semidirect
73. #41×C97D4semidirect
155. C97#3045semidirect
156. C2×AGL(1,ℤ/8)D97semidirect
202. #41C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908302
#202613022
#202612182
#202609552
#202609482
#202609332
#202609282
#202607632
#202607372
C2×#156907312
C2×AGL(1,ℤ/8)×C972
#304597