Finite Group Databasebeta

← #27151271

Subgroups — #27151271

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×#15690836C211yesmaximal, characteristic, complemented
2C2×#15690830C211yesmaximal, characteristic, complemented
3#20261215C211yesmaximal, characteristic, complemented
4#20260954C212yesmaximal, complemented
5#20260947C212yesmaximal, complemented
6#20260932C212yesmaximal, complemented
7#20260927C212yesmaximal, complemented
8#20260762C211yesmaximal, characteristic, complemented
9#20260736C211yesmaximal, characteristic, complemented
10C2×#15690731C211yesmaximal, characteristic, complemented
11#535×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13#15690836—24no—
14#15690830—24no—
15C2×#13095501C2211yescharacteristic, complemented
16C2×#13095499C2212yes—
17C2×#13095499C2211yescharacteristic, complemented
18C2×#13095498C2211yescharacteristic, complemented
19C2×#13095496C2212yes—
20C2×#13095496C2211yescharacteristic, complemented
21C2×#13095495C2211yescharacteristic
22C22×#11715895C2211yescharacteristic
23C4×Dic194C2212yescomplemented
24C4×D388C2212yescomplemented
25#15690770—22no—
26#15690768—21no—
27C2×C4⋊Dic97C2211yescharacteristic
28C4⋊C4×D97—21no—
29C4⋊#11715895C2212yescomplemented
30C4⋊#11715895C2212yescomplemented
31#15690734C2212yescomplemented
32#15690731C2212yes—
33#15690731C2212yes—
34#15690731C2212yescomplemented
35C2×#41×C97C2211yescharacteristic
36C22×D4×C97C2211yescharacteristic
37#127×C97C2212yes—
38#124×C97C2212yes—
39#121×C97—21no—
40#121×C97—21no—
41#120×C97—22no—
42#117×C97C2211yescharacteristic
43#13095511—22no—
44#13095511D414yescomplemented
45C2×Dic194C2311yescharacteristic
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—21no—
48C22×Dic97—21no—
49C2×C4×D97—21no—
50C2×C4×D97—22no—
51#13095501—22no—
52#13095499—24no—
53#13095499—22no—
54#13095498—22no—
55#13095496—24no—
56#13095496—22no—
57#13095495—22no—
58C2×#11715895—22no—
59C2×#11715895C2311yescharacteristic
60C2×#11715895C2311yescharacteristic
61C2×#11715895C2312yes—
62C4⋊Dic97—21no—
63C4⋊Dic97C2312yes—
64#13095490—22no—
65#13095488—22no—
66C4×Dic97—22no—
67C24×C97—21no—
68#41×C97—22no—
69#41×C97—22no—
70C2×Q8×C97C2311yescharacteristic
71C2×D4×C97—21no—
72C2×D4×C97—22no—
73C2×D4×C97—21no—
74C2×D4×C97—22no—
75C2×D4×C97—22no—
76C2×D4×C97—22no—
77C2×D4×C97C2312yes—
78C2×D4×C97C2311yescharacteristic
79C22×C4×C97—21no—
80C22×C4×C97C2311yescharacteristic
81C4⋊C4×C97D411yescharacteristic
82C4⋊C4×C97D411yescharacteristic
83C22⋊C4×C97—22no—
84C22⋊C4×C97—22no—
85C22⋊C4×C97D412yes—
86C42×C97C2312yes—
87C23×C97—22no—
88C23×C97—21no—
89C23×C97—42no—
90C23×C97—81no—
91C23×C97—81no—
92C23×C97C2×D411yescharacteristic
93C22×D97—21no—
94Q8×C97—21no—
95Q8×C97—21no—
96D4×C97—22no—
97D4×C97—21no—
98D4×C97—22no—
99D4×C97—21no—
100D4×C97—42no—
101D4×C97—42no—
102D4×C97—82no—
103D4×C97—82no—
104C2×C4×C97—21no—
105C2×C4×C97—42no—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D412yes—
108C2×C4×C97C2×D412yes—
109C2×C4×C97C2×D411yescharacteristic
110C2×C4×C97#4112yes—
111C2×C4×C97C2411yescharacteristic
112C22⋊D97—24no—
113C2×Dic97—21no—
114C2×Dic97—22no—
115C2×Dic97—81no—
116D388—21no—
117D388C2×D412yes—
118C4×D97—24no—
119C4×D97—82no—
120Dic194—21no—
121Dic194C2×D412yes—
122#11715895—21no—
123#11715895—21no—
124#11715895—22no—
125C22×C97—21no—
126C22×C97—21no—
127C22×C97—41no—
128C22×C97—42no—
129C22×C97—41no—
130C22×C97—42no—
131C22×C97—41no—
132C22×C97—42no—
133C22×C97—84no—
134C22×C97#12112yes—
135C22×C97C22×D411yescharacteristic
136C2×D97—22no—
137C2×D97—41no—
138C2×D97—41no—
139C4×C97—22no—
140C4×C97—22no—
141C4×C97—41no—
142C4×C97#12112yes—
143C4×C97C22×D411yescharacteristic
144C4×C97C2×#4111yescharacteristic
145Dic97—22no—
146Dic97—42no—
147C2×C97—22no—
148C2×C97—81no—
149C2×C97—82no—
150C2×C97C2×#12111yescharacteristic
151C2×C97#57611yescharacteristic
152C2×C97#57711yescharacteristic
153D97—42no—
154#3045—971nomaximal, Sylow
155C97#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—971no—
157C2×#135—971no—
158#535D9711yescharacteristic, complemented
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)—1944no—
171#135—1944no—
172C2×Q16—971no—
173C2×SD16—971no—
174C2×SD16—971no—
175C2×SD16—972no—
176C2×D8—971no—
177C2×D8—972no—
178C2×M16—971no—
179C22×C8—971no—
180#127C2×D9712yes—
181#124C2×D9712yes—
182#121—21no—
183#121—21no—
184#120—22no—
185C4×Q8—972no—
186C4×D4—972no—
187C4×D4—1942no—
188#117—1941no—
189#117C2×D9711yescharacteristic
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—22no—
201#41—974no—
202#41—1942no—
203C2×Q8—971no—
204C2×Q8C22×D9711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—971no—
212C2×D4—1941no—
213C2×D4C22×D9712yes—
214C2×D4C22×D9711yescharacteristic
215C22×C4—21no—
216C22×C4—1941no—
217C22×C4—1942no—
218C22×C4—1941no—
219C22×C4C22×D9711yescharacteristic
220Q16—1942no—
221SD16—1942no—
222SD16—1944no—
223SD16—1942no—
224D8—1942no—
225D8—1944no—
226M16—1942no—
227C2×C8—971no—
228C2×C8—971no—
229C2×C8—972no—
230C2×C8—1942no—
231C4⋊C4—972no—
232C4⋊C4—1942no—
233C4⋊C4—1941no—
234C4⋊C4C22⋊D9711yescharacteristic
235C4⋊C4C22⋊D9711yescharacteristic
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—1942no—
239C22⋊C4C22⋊D9712yes—
240C42—1942no—
241C42C22×D9712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—1941no—
248C23C2×C22⋊D9711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—972no—
252Q8—1941no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—42no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—972no—
262D4—1944no—
263D4—1941no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—7762no—
267C2×C4—7761no—
268C2×C4—1942no—
269C2×C4—1941no—
270C2×C4—1944no—
271C2×C4D4×D9712yes—
272C2×C4C2×C22⋊D9712yes—
273C2×C4C2×C22⋊D9712yes—
274C2×C4C2×C22⋊D9711yescharacteristic
275C2×C4#1309551012yes—
276C2×C4C23×D9711yescharacteristic
277C8—1941no—
278C8—1941no—
279C8—1942no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—84no—
289C22—3881no—
290C22—3881no—
291C22—1942no—
292C22#1569078112yes—
293C22C22×C22⋊D9711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—3882no—
298C4—1942no—
299C4#1569078112yes—
300C4C2×D4×D9711yescharacteristic
301C4C2×#1309551011yescharacteristic
302C2—22no—
303C2—82no—
304C2—81no—
305C2—3882no—
306C2C2×#1569078111yescharacteristic
307C2#2026153811yescharacteristic
308C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#535×C97C2
socleC22×C97C22×D4
hypercenter#535D97
hypocenterC97#3045
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C2×#15690830C2semidirect
3. #20261215C2semidirect
4. #20260954C2semidirect
5. #20260947C2semidirect
6. #20260932C2semidirect
7. #20260927C2semidirect
8. #20260762C2semidirect
9. #20260736C2semidirect
10. C2×#15690731C2semidirect
11. #535×C97C2semidirect
15. C2×#13095501C22semidirect
17. C2×#13095499C22semidirect
18. C2×#13095498C22semidirect
20. C2×#13095496C22semidirect
23. C4×Dic194C22semidirect
24. C4×D388C22semidirect
29. C4⋊#11715895C22semidirect
30. C4⋊#11715895C22semidirect
31. #15690734C22semidirect
34. #15690731C22semidirect
44. #13095511D4semidirect
155. C97#3045semidirect
158. #535D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C2×#156908302
#202612152
#202609542
#202609472
#202609322
#202609272
#202607622
#202607362
C2×#156907312
#535×C972
#304597