Finite Group Databasebeta

← #27154868

Subgroups — #27154868

Order 12416.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15690829C211yesmaximal, characteristic, complemented
2#135×D97C211yesmaximal, characteristic, complemented
3#20261178C211yesmaximal, characteristic, complemented
4#20261175C211yesmaximal, characteristic, complemented
5#20261174C212yesmaximal, complemented
6#20261171C211yesmaximal, characteristic, complemented
7#20261169C212yesmaximal, complemented
8#20261166C212yesmaximal, complemented
9#20261164C212yesmaximal, complemented
10M32×D97C211yesmaximal, characteristic, complemented
11#509×C97C211yesmaximal, characteristic, complemented
12C2×#13095510—21no—
13#41×D97—21no—
14#15690833C2211yescharacteristic, complemented
15#15690832C2211yescharacteristic
16#15690831—21no—
17#15690830—21no—
18#15690829—21no—
19#15690829—21no—
20#15690829C2212yes—
21#15690829C2211yescharacteristic, complemented
22#15690829C2212yescomplemented
23C2×Dic388C2211yescharacteristic, complemented
24Q16×D97C2211yescharacteristic, complemented
25C2×#13095499—21no—
26SD16×D97—21no—
27D8×D97C2211yescharacteristic, complemented
28C2×C8×D97C2211yescharacteristic
29Dic776C2212yescomplemented
30#15690756C2212yescomplemented
31#15690755C2212yescomplemented
32#15690754C2212yescomplemented
33#15690754C2212yescomplemented
34#15690751C2212yescomplemented
35#15690750C2211yescharacteristic, complemented
36#15690749C2212yescomplemented
37C16×D97C2212yescomplemented
38#135×C97C2211yescharacteristic
39C2×D8×C97C2211yescharacteristic, complemented
40SD32×C97C2212yescomplemented
41D16×C97C2212yescomplemented
42M32×C97C2211yescharacteristic, complemented
43#13095512—21no—
44#13095511—21no—
45#13095511—41no—
46#13095510—22no—
47#13095510—22no—
48#13095510—21no—
49#13095510—41no—
50#13095510—41no—
51#13095510—41no—
52C2×Dic194—21no—
53Q8×D97—21no—
54C2×C22⋊D97—41no—
55D4×D97—21no—
56D4×D97—41no—
57C22×Dic97—41no—
58C2×C4×D97—41no—
59C2×C4×D97D411yescharacteristic
60Dic388—21no—
61Dic388C2311yescharacteristic
62Dic388C2312yes—
63#13095501—21no—
64#13095500—21no—
65#13095499—21no—
66#13095499—22no—
67#13095499—22no—
68#13095498—21no—
69D776C2311yescharacteristic
70#13095496—21no—
71C2×#11715895D411yescharacteristic
72C8×D97D411yescharacteristic
73C8×D97D411yescharacteristic
74C8×D97C2311yescharacteristic
75C8×D97C2311yescharacteristic
76#13095486C2312yes—
77#41×C97—21no—
78C2×D4×C97—21no—
79Q16×C97C2311yescharacteristic
80SD16×C97—21no—
81D8×C97—21no—
82D8×C97C2311yescharacteristic
83D8×C97C2312yes—
84C2×C8×C97C2311yescharacteristic
85C16×C97C2312yes—
86C23×C97—41no—
87C22×D97—41no—
88C22×D97D811yescharacteristic
89Q8×C97—21no—
90D4×C97—22no—
91D4×C97—21no—
92D4×C97—41no—
93D4×C97—41no—
94C2×C4×C97—41no—
95C2×C4×C97C2×D411yescharacteristic
96C22⋊D97—41no—
97C22⋊D97—41no—
98C22⋊D97—42no—
99C22⋊D97—42no—
100C22⋊D97—41no—
101C2×Dic97—41no—
102C2×Dic97—41no—
103C2×Dic97—42no—
104C2×Dic97—42no—
105C2×Dic97D811yescharacteristic
106D388—21no—
107D388—41no—
108C4×D97—41no—
109C4×D97—41no—
110C4×D97—41no—
111C4×D97D811yescharacteristic
112C4×D97D811yescharacteristic
113C4×D97C2×D411yescharacteristic
114C4×D97C2×D411yescharacteristic
115Dic194—22no—
116Dic194—21no—
117Dic194—41no—
118Dic194—41no—
119C8×C97C2×D411yescharacteristic
120C8×C97C2411yescharacteristic
121#11715895C2×D411yescharacteristic
122#11715895C2×D411yescharacteristic
123C22×C97—42no—
124C22×C97—41no—
125C22×C97—82no—
126C22×C97C2×D811yescharacteristic
127C2×D97—41no—
128C2×D97—41no—
129C2×D97—82no—
130C2×D97C2×D811yescharacteristic
131C2×D97C2×D811yescharacteristic
132C4×C97—41no—
133C4×C97C2×D811yescharacteristic
134C4×C97C22×D411yescharacteristic
135Dic97—41no—
136Dic97—42no—
137Dic97C2×D811yescharacteristic
138Dic97C2×D811yescharacteristic
139C2×C97—21no—
140C2×C97—81no—
141C2×C97—82no—
142C2×C97C22×D811yescharacteristic
143D97—21no—
144D97—21no—
145D97—81no—
146#3414—971nomaximal, Sylow
147C97#341411yescharacteristic, complemented, Sylow
148C2×#135—971no—
149C2×#135—971no—
150#510—971no—
151#510—972no—
152#510—971no—
153#509—971no—
154#509—972no—
155#509D9711yescharacteristic, complemented
156#508—972no—
157#508—972no—
158C2×M32—971no—
159C2×#41—1941no—
160C2×#41—1941no—
161#135—971no—
162#135—972no—
163#135—972no—
164#135—971no—
165#135—971no—
166#135—1941no—
167#135—1941no—
168#135—1941no—
169#135—1941no—
170#135C2×D9711yescharacteristic
171C2×Q16—971no—
172C2×Q16—971no—
173C2×SD16—1941no—
174C2×SD16—1941no—
175C2×D8—971no—
176C2×D8C2×D9711yescharacteristic, complemented
177C22×C8—971no—
178Q32—972no—
179Q32—972no—
180SD32—972no—
181SD32—972no—
182SD32—972no—
183SD32C2×D9712yescomplemented
184D16—972no—
185D16C2×D9712yescomplemented
186M32—971no—
187M32—972no—
188M32C2×D9711yescharacteristic, complemented
189C2×C16—972no—
190#41—21no—
191#41—3881no—
192#41—3881no—
193#41—3881no—
194#41—3881no—
195#41—1942no—
196#41—1942no—
197#41—1941no—
198#41—1941no—
199#41—1941no—
200C2×Q8—1941no—
201C2×Q8—1941no—
202C2×D4—21no—
203C2×D4—3881no—
204C2×D4—3881no—
205C2×D4—1941no—
206C22×C4—971no—
207C22×C4—3881no—
208C22×C4—3881no—
209Q16—971no—
210Q16—972no—
211Q16—1941no—
212Q16—1941no—
213Q16C22×D9711yescharacteristic
214SD16—21no—
215SD16—1942no—
216SD16—1942no—
217SD16—1941no—
218SD16—1941no—
219SD16—1941no—
220D8—21no—
221D8—971no—
222D8—1941no—
223D8C22×D9712yes—
224D8C22×D9711yescharacteristic
225C2×C8—971no—
226C2×C8—971no—
227C2×C8—971no—
228C2×C8—971no—
229C2×C8—971no—
230C2×C8C22×D9711yescharacteristic
231C16—972no—
232C16C22×D9712yes—
233C23—41no—
234C23—971no—
235C23—3881no—
236Q8—21no—
237Q8—3881no—
238Q8—3881no—
239Q8—1942no—
240Q8—1941no—
241D4—22no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—3881no—
246D4—3881no—
247D4—3881no—
248D4—3882no—
249D4—3882no—
250D4—3881no—
251D4—1941no—
252C2×C4—41no—
253C2×C4—971no—
254C2×C4—971no—
255C2×C4—971no—
256C2×C4—971no—
257C2×C4—971no—
258C2×C4—3881no—
259C2×C4—3881no—
260C2×C4—3881no—
261C2×C4—3882no—
262C2×C4—3882no—
263C2×C4—3881no—
264C2×C4—3881no—
265C2×C4D4×D9711yescharacteristic
266C8—971no—
267C8—971no—
268C8D4×D9711yescharacteristic
269C8C23×D9711yescharacteristic
270C22—41no—
271C22—42no—
272C22—82no—
273C22—7762no—
274C22—971no—
275C22—971no—
276C22—3881no—
277C22—3881no—
278C22D8×D9711yescharacteristic
279C4—41no—
280C4—971no—
281C4—971no—
282C4—3881no—
283C4—3882no—
284C4D8×D9711yescharacteristic
285C4C2×D4×D9711yescharacteristic
286C2—21no—
287C2—82no—
288C2—81no—
289C2—7761no—
290C2—1941no—
291C2—1941no—
292C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC2C2×D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#509×C97C2
socleC2×C97C22×D8
hypercenter#509D97
hypocenterC97#3414
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. C2×#15690829C2semidirect
2. #135×D97C2semidirect
3. #20261178C2semidirect
4. #20261175C2semidirect
5. #20261174C2semidirect
6. #20261171C2semidirect
7. #20261169C2semidirect
8. #20261166C2semidirect
9. #20261164C2semidirect
10. M32×D97C2semidirect
11. #509×C97C2semidirect
14. #15690833C22semidirect
21. #15690829C22semidirect
22. #15690829C22semidirect
23. C2×Dic388C22semidirect
24. Q16×D97C22semidirect
27. D8×D97C22semidirect
29. Dic776C22semidirect
30. #15690756C22semidirect
31. #15690755C22semidirect
32. #15690754C22semidirect
33. #15690754C22semidirect
34. #15690751C22semidirect
35. #15690750C22semidirect
36. #15690749C22semidirect
37. C16×D97C22semidirect
39. C2×D8×C97C22semidirect
40. SD32×C97C22semidirect
41. D16×C97C22semidirect
42. M32×C97C22semidirect
147. C97#3414semidirect
155. #509D97semidirect
176. C2×D8C2×D97semidirect
183. SD32C2×D97semidirect
185. D16C2×D97semidirect
188. M32C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908292
#135×D972
#202611782
#202611752
#202611742
#202611712
#202611692
#202611662
#202611642
M32×D972
#509×C972
#341497