Finite Group Databasebeta

← #27151412

Subgroups — #27151412

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×#15690817C211yesmaximal, characteristic, complemented
3#20261313C211yesmaximal, characteristic, complemented
4#20260986C212yesmaximal, complemented
5#20260979C212yesmaximal, complemented
6#20260969C212yesmaximal, complemented
7#20260962C212yesmaximal, complemented
8#20260778C211yesmaximal, characteristic, complemented
9#20260729C211yesmaximal, characteristic, complemented
10C2×#15690731C211yesmaximal, characteristic, complemented
11#535×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13#15690836—24no—
14C2×#13095499C2212yes—
15C2×#13095496C2212yes—
16#15690817—24no—
17C2×#13095495C2211yescharacteristic
18C2×#13095495C2212yescomplemented
19C22×#11715895C2211yescharacteristic
20C22×#11715895C2212yescomplemented
21#15690789C2212yescomplemented
22#15690786—21no—
23#15690784C2212yescomplemented
24#15690782—22no—
25#15690781—21no—
26C2×C4⋊Dic97C2211yescharacteristic
27#15690743C2212yescomplemented
28#15690741C2212yescomplemented
29#15690733C2211yescharacteristic, complemented
30#15690733C2211yescharacteristic, complemented
31#15690731C2212yes—
32#15690731C2212yes—
33#15690730C2211yescharacteristic, complemented
34#15690730C2211yescharacteristic, complemented
35C2×#41×C97C2211yescharacteristic
36C22×D4×C97C2211yescharacteristic
37#127×C97—21no—
38#124×C97—21no—
39#121×C97C2212yes—
40#121×C97C2212yes—
41#120×C97—22no—
42#117×C97C2211yescharacteristic
43#13095511—22no—
44#13095511—22no—
45C2×Dic194C2311yescharacteristic
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—21no—
48C2×C22⋊D97—22no—
49C22×Dic97—21no—
50C2×C4×D97—21no—
51#13095499—24no—
52#13095496—24no—
53#13095495—24no—
54#13095495D414yescomplemented
55C2×#11715895—22no—
56C2×#11715895—24no—
57C2×#11715895C2311yescharacteristic
58C2×#11715895C2311yescharacteristic
59C2×#11715895C2312yes—
60C4⋊Dic97—21no—
61C4⋊Dic97C2312yes—
62#13095490—22no—
63C22⋊Dic97—22no—
64#13095488—22no—
65C24×C97—21no—
66#41×C97—22no—
67#41×C97—22no—
68C2×Q8×C97D411yescharacteristic
69C2×D4×C97—22no—
70C2×D4×C97—22no—
71C2×D4×C97—22no—
72C2×D4×C97—21no—
73C2×D4×C97—22no—
74C2×D4×C97D411yescharacteristic
75C2×D4×C97D412yes—
76C2×D4×C97C2312yes—
77C22×C4×C97C2312yes—
78C22×C4×C97C2311yescharacteristic
79C4⋊C4×C97C2311yescharacteristic
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C22⋊C4×C97—22no—
83C22⋊C4×C97—22no—
84C42×C97—21no—
85C23×C97—22no—
86C23×C97—42no—
87C23×C97—81no—
88C23×C97—81no—
89C23×C97C2×D412yes—
90C23×C97C2×D411yescharacteristic
91C22×D97—21no—
92Q8×C97—22no—
93D4×C97—24no—
94D4×C97—22no—
95D4×C97—22no—
96D4×C97—22no—
97D4×C97—42no—
98D4×C97—82no—
99D4×C97—82no—
100C2×C4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97#4112yes—
108C2×C4×C97C2411yescharacteristic
109C22⋊D97—42no—
110C22⋊D97—82no—
111C2×Dic97—22no—
112C2×Dic97—21no—
113C2×Dic97—81no—
114D388—21no—
115D388—21no—
116C4×D97—42no—
117Dic194—21no—
118Dic194—21no—
119#11715895—21no—
120#11715895—21no—
121#11715895C2×D414yes—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—22no—
125C22×C97—22no—
126C22×C97—22no—
127C22×C97—42no—
128C22×C97—42no—
129C22×C97—42no—
130C22×C97—84no—
131C22×C97#12112yes—
132C22×C97C22×D411yescharacteristic
133C2×D97—41no—
134C2×D97—41no—
135C2×D97—41no—
136C4×C97—22no—
137C4×C97—41no—
138C4×C97—41no—
139C4×C97#12112yes—
140C4×C97C22×D411yescharacteristic
141C4×C97C2×#4111yescharacteristic
142Dic97—41no—
143Dic97—42no—
144C2×C97—22no—
145C2×C97—42no—
146C2×C97—82no—
147C2×C97C2×#12111yescharacteristic
148C2×C97#57611yescharacteristic
149C2×C97#57711yescharacteristic
150D97—81no—
151#3062—971nomaximal, Sylow
152C97#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—971no—
154C2×#131—971no—
155#537—971no—
156#535D9711yescharacteristic, complemented
157#469—972no—
158#468—972no—
159#466—972no—
160#465—972no—
161#428—971no—
162#416—971no—
163C2×#102—971no—
164C2×#41—971no—
165C2×#41C2×D9711yescharacteristic
166C22×D4C2×D9711yescharacteristic
167AGL(1,ℤ/8)—1944no—
168C2×SD16—972no—
169C2×D8—972no—
170#131—1944no—
171C2×M16—971no—
172C2×M16—972no—
173C22×C8—971no—
174C22×C8—972no—
175#127—21no—
176#124—21no—
177C22⋊Q8—972no—
178C22⋊Q8—1941no—
179#121—972no—
180#121—1941no—
181#121—1942no—
182#121C2×D9712yes—
183#121C2×D9712yes—
184#120—22no—
185#117C2×D9711yescharacteristic
186C2×C4⋊C4—971no—
187#107—972no—
188#106—972no—
189#103—971no—
190#103—971no—
191#102—972no—
192#102—972no—
193#102—971no—
194#102—971no—
195C24—21no—
196#41—22no—
197#41—22no—
198#41—1942no—
199#41—1942no—
200C2×Q8—971no—
201C2×Q8C22⋊D9711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—971no—
208C2×D4—1942no—
209C2×D4—1941no—
210C2×D4C22⋊D9712yes—
211C2×D4C22⋊D9711yescharacteristic
212C2×D4C22×D9712yes—
213C22×C4—1941no—
214C22×C4—1941no—
215C22×C4C22×D9712yes—
216C22×C4C22×D9711yescharacteristic
217SD16—1944no—
218D8—1944no—
219M16—974no—
220M16—1944no—
221C2×C8—972no—
222C2×C8—971no—
223C2×C8—971no—
224C2×C8—1944no—
225C2×C8—1942no—
226C4⋊C4—972no—
227C4⋊C4—1942no—
228C4⋊C4—1941no—
229C4⋊C4C22×D9711yescharacteristic
230C4⋊C4C22×D9711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C42—21no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—1941no—
242C23C2×C22⋊D9712yes—
243C23C2×C22⋊D9711yescharacteristic
244Q8—22no—
245Q8—1941no—
246Q8—1941no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—42no—
252D4—82no—
253D4—82no—
254D4—7762no—
255D4—3882no—
256D4—1941no—
257D4—1941no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—7761no—
263C2×C4—3882no—
264C2×C4—1942no—
265C2×C4—1941no—
266C2×C4D4×D9712yes—
267C2×C4C2×C22⋊D9712yes—
268C2×C4C2×C22⋊D9711yescharacteristic
269C2×C4#1309551012yes—
270C2×C4C23×D9711yescharacteristic
271C8—974no—
272C8—1941no—
273C8—1941no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—84no—
283C22—3881no—
284C22—3881no—
285C22—3881no—
286C22#1569078212yes—
287C22C22×C22⋊D9711yescharacteristic
288C4—22no—
289C4—41no—
290C4—41no—
291C4—3881no—
292C4—3882no—
293C4#1569078212yes—
294C4C2×D4×D9711yescharacteristic
295C4C2×#1309551011yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—7761no—
300C2C2×#1569078211yescharacteristic
301C2#2026153811yescharacteristic
302C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#535×C97C2
socleC22×C97C22×D4
hypercenter#535D97
hypocenterC97#3062
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C2×#15690817C2semidirect
3. #20261313C2semidirect
4. #20260986C2semidirect
5. #20260979C2semidirect
6. #20260969C2semidirect
7. #20260962C2semidirect
8. #20260778C2semidirect
9. #20260729C2semidirect
10. C2×#15690731C2semidirect
11. #535×C97C2semidirect
18. C2×#13095495C22semidirect
20. C22×#11715895C22semidirect
21. #15690789C22semidirect
23. #15690784C22semidirect
27. #15690743C22semidirect
28. #15690741C22semidirect
29. #15690733C22semidirect
30. #15690733C22semidirect
33. #15690730C22semidirect
34. #15690730C22semidirect
54. #13095495D4semidirect
152. C97#3062semidirect
156. #535D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C2×#156908172
#202613132
#202609862
#202609792
#202609692
#202609622
#202607782
#202607292
C2×#156907312
#535×C972
#306297