Finite Group Databasebeta

← #27153467

Subgroups — #27153467

Order 12416.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690831C212yesmaximal, complemented
2#20261407C211yesmaximal, characteristic, complemented
3#20261389C211yesmaximal, characteristic, complemented
4#20261108C211yesmaximal, characteristic, complemented
5#20260965C212yesmaximal, complemented
6#20260945C211yesmaximal, characteristic, complemented
7#20260930C211yesmaximal, characteristic, complemented
8#20260900C212yesmaximal, complemented
9#20260881C212yesmaximal, complemented
10C8×D388C211yesmaximal, characteristic, complemented
11C4×SD16×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×#13095510C2212yes—
14#15690831—28no—
15C2×#13095501C2212yes—
16C2×#13095500C2212yes—
17C2×#13095500C2212yes—
18C2×#13095496C2212yes—
19C2×C8×D97C2212yes—
20#15690810C2211yescharacteristic
21#15690807—21no—
22#15690807C2211yescharacteristic
23#15690794—22no—
24#15690790C2212yes—
25#15690785—22no—
26#15690785C2212yes—
27#15690784—21no—
28C4×D388—21no—
29C4×D388C2211yescharacteristic
30C2×C4⋊Dic97—21no—
31C4⋊#11715895C2211yescharacteristic
32#15690733C2212yes—
33#15690730C2212yes—
34#15690722C2212yes—
35C2×SD16×C97C2211yescharacteristic
36C4×Q8×C97C2211yescharacteristic
37C4×D4×C97C2211yescharacteristic
38#106×C97C2211yescharacteristic
39#103×C97C2211yescharacteristic
40#102×C97C2211yescharacteristic
41C4×C8×C97C2211yescharacteristic
42#13095512—24no—
43#13095512—42no—
44#13095510—24no—
45#13095510—42no—
46C2×Dic194C2312yes—
47C2×D388—22no—
48C2×D388—21no—
49C2×D388—22no—
50C2×D388—21no—
51C2×D388D411yescharacteristic
52C2×D388C2312yes—
53C2×C22⋊D97—22no—
54C22×Dic97—22no—
55C2×C4×D97—22no—
56C2×C4×D97C2312yes—
57#13095501—24no—
58#13095500—24no—
59#13095500—42no—
60#13095496—24no—
61C2×#11715895C2312yes—
62C8×D97—24no—
63C4⋊Dic97—21no—
64C4⋊Dic97—21no—
65C4⋊Dic97—22no—
66C4⋊Dic97D411yescharacteristic
67#13095488—22no—
68#13095488—22no—
69#13095488D412yes—
70C2×Q8×C97C2311yescharacteristic
71C2×D4×C97C2311yescharacteristic
72C22×C4×C97—21no—
73SD16×C97D414yescomplemented
74C2×C8×C97C2311yescharacteristic
75C2×C8×C97C2311yescharacteristic
76C4⋊C4×C97—21no—
77C4⋊C4×C97C2311yescharacteristic
78C4⋊C4×C97C2311yescharacteristic
79C22⋊C4×C97—21no—
80C42×C97—21no—
81C42×C97C2311yescharacteristic
82C23×C97—21no—
83C22×D97—22no—
84C22×D97C2×D412yes—
85Q8×C97—21no—
86Q8×C97C2×D412yes—
87D4×C97—21no—
88D4×C97C2×D412yes—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—42no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—44no—
97C2×Dic97—22no—
98C2×Dic97—44no—
99C2×Dic97C2×D412yes—
100D388—22no—
101D388—22no—
102D388—22no—
103D388—44no—
104D388—42no—
105D388—81no—
106D388—82no—
107C4×D97—22no—
108C4×D97—22no—
109C4×D97—44no—
110Dic194—22no—
111Dic194—22no—
112C8×C97—21no—
113C8×C97C2×D412yes—
114#11715895—22no—
115C22×C97—22no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97C22×D411yescharacteristic
119C2×D97—22no—
120C2×D97—22no—
121C2×D97—22no—
122C2×D97—42no—
123C2×D97—42no—
124C2×D97—42no—
125C4×C97—22no—
126C4×C97—21no—
127C4×C97—41no—
128C4×C97—41no—
129C4×C97#13512yescomplemented
130C4×C97C22×D411yescharacteristic
131C4×C97H(5,𝔽2)11yescharacteristic
132Dic97—22no—
133Dic97—42no—
134C2×C97—42no—
135C2×C97D4211yescharacteristic
136C2×C97C2×#13511yescharacteristic
137C2×C97#57711yescharacteristic
138D97—42no—
139D97—82no—
140#3284—971nomaximal, Sylow
141C97#328411yescharacteristic, complemented, Sylow
142C2×#135—972no—
143#550—971no—
144#547—971no—
145#492—971no—
146#465—972no—
147#462—971no—
148#459—971no—
149#452—972no—
150#449—972no—
151C4×SD16D9711yescharacteristic, complemented
152C8×D4—971no—
153C2×#41—972no—
154C2×#41—972no—
155#135—1948no—
156C2×Q16—972no—
157C2×SD16—972no—
158C2×SD16—972no—
159C2×SD16C2×D9711yescharacteristic
160C2×D8—972no—
161C22×C8—972no—
162C4⋊Q8—971no—
163#127—971no—
164#127—1941no—
165#123—1942no—
166C22⋊Q8—972no—
167#121—972no—
168#121—1942no—
169#121—1941no—
170C4×Q8C2×D9711yescharacteristic
171C4×D4—971no—
172C4×D4—1941no—
173C4×D4C2×D9711yescharacteristic
174C2×C4⋊C4—1941no—
175#106C2×D9711yescharacteristic
176C4⋊C8—971no—
177#103—972no—
178#103C2×D9711yescharacteristic
179#102—972no—
180#102C2×D9711yescharacteristic
181C22⋊C8—972no—
182C4×C8C2×D9711yescharacteristic
183#41—3882no—
184#41—3882no—
185#41—1944no—
186#41—1944no—
187C2×Q8—972no—
188C2×Q8C22×D9711yescharacteristic
189C2×D4—971no—
190C2×D4—972no—
191C2×D4—1942no—
192C2×D4—1942no—
193C2×D4—1942no—
194C2×D4—1941no—
195C2×D4—1941no—
196C2×D4C22×D9711yescharacteristic
197C22×C4—21no—
198C22×C4—972no—
199C22×C4—1942no—
200C22×C4—1942no—
201Q16—1944no—
202SD16—3882no—
203SD16—1944no—
204SD16D38814yescomplemented
205D8—1944no—
206C2×C8—972no—
207C2×C8—1944no—
208C2×C8C22×D9711yescharacteristic
209C2×C8C22×D9711yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—971no—
212C4⋊C4—1942no—
213C4⋊C4—1941no—
214C4⋊C4—1941no—
215C4⋊C4C22×D9711yescharacteristic
216C4⋊C4C22×D9711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—972no—
219C22⋊C4—1942no—
220C22⋊C4—1942no—
221C42—21no—
222C42C22×D9711yescharacteristic
223C23—21no—
224C23—972no—
225C23—1942no—
226Q8—21no—
227Q8—1942no—
228Q8—1942no—
229Q8C2×D38812yes—
230D4—21no—
231D4—7761no—
232D4—7762no—
233D4—3884no—
234D4—3884no—
235D4—3882no—
236D4—1942no—
237D4—1942no—
238D4—1942no—
239D4C2×D38812yes—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—972no—
245C2×C4—3884no—
246C2×C4—3884no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4D4×D9711yescharacteristic
251C2×C4D4×D9711yescharacteristic
252C2×C4C23×D9711yescharacteristic
253C8—21no—
254C8—1942no—
255C8C2×D38812yes—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—3882no—
260C22—3882no—
261C22—3882no—
262C22—1942no—
263C22—1942no—
264C22—1942no—
265C22C2×D4×D9711yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—3882no—
271C4—1942no—
272C4#1569083112yescomplemented
273C4C22×D38811yescharacteristic
274C4#1569085611yescharacteristic
275C2—42no—
276C2—7762no—
277C2—3882no—
278C2D4×D38811yescharacteristic
279C2C2×#1569083111yescharacteristic
280C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
hypercenterC4×SD16D97
hypocenterC97#3284
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690831C2semidirect
2. #20261407C2semidirect
3. #20261389C2semidirect
4. #20261108C2semidirect
5. #20260965C2semidirect
6. #20260945C2semidirect
7. #20260930C2semidirect
8. #20260900C2semidirect
9. #20260881C2semidirect
10. C8×D388C2semidirect
11. C4×SD16×C97C2semidirect
73. SD16×C97D4semidirect
129. C4×C97#135semidirect
141. C97#3284semidirect
151. C4×SD16D97semidirect
204. SD16D388semidirect
272. C4#15690831semidirect

Maximal subgroups

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

subgroupindex
C2×#156908312
#202614072
#202613892
#202611082
#202609652
#202609452
#202609302
#202609002
#202608812
C8×D3882
C4×SD16×C972
#328497