Finite Group Databasebeta

← #27151452

Subgroups — #27151452

Order 12416.

452 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#20261533C212yesmaximal, complemented
2C2×#15690816C211yesmaximal, characteristic, complemented
3#20261007C212yesmaximal, complemented
4#20261002C212yesmaximal, complemented
5#20260995C214yesmaximal, complemented
6C2×#15690746C211yesmaximal, characteristic, complemented
7#20260719C212yesmaximal, complemented
8C2×#135×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2211yescharacteristic
10#15690841C2214yes—
11#15690840—22no—
12#15690836C2214yes—
13#15690835—22no—
14#15690830—24no—
15#15690816—21no—
16#15690816—21no—
17#15690816C2212yes—
18#15690816C2212yes—
19C2×#13095495C2211yescharacteristic
20C2×#13095495C2212yes—
21C2×#13095494—21no—
22C2×C8×D97—21no—
23#15690746C2212yescomplemented
24#15690746C2212yescomplemented
25#15690729C2214yescomplemented
26#15690727C2214yescomplemented
27C2×#41×C97C2212yes—
28#135×C97—21no—
29#135×C97—21no—
30#135×C97—21no—
31#135×C97—21no—
32C2×Q16×C97C2211yescharacteristic, complemented
33C2×SD16×C97C2212yescomplemented
34C2×D8×C97C2211yescharacteristic, complemented
35C22×C8×C97C2211yescharacteristic
36#13095511—21no—
37#13095511—21no—
38#13095511D412yes—
39#13095511C2312yes—
40C2×Dic194D411yescharacteristic
41C2×D388D411yescharacteristic
42C2×C22⋊D97—21no—
43C2×C4×D97—21no—
44#13095501—24no—
45#13095499—24no—
46#13095498—24no—
47#13095496—24no—
48#13095495—21no—
49#13095495—22no—
50#13095495C2312yes—
51#13095495C2314yes—
52#13095494—22no—
53#13095494—22no—
54C2×#11715895—21no—
55C2×#11715895—22no—
56C8×D97—22no—
57C8×D97—22no—
58#41×C97—22no—
59#41×C97—22no—
60#41×C97—22no—
61#41×C97—22no—
62C2×Q8×C97C2312yes—
63C2×D4×C97—22no—
64C2×D4×C97C2312yes—
65C22×C4×C97—22no—
66C22×C4×C97C2311yescharacteristic
67Q16×C97—21no—
68Q16×C97—21no—
69SD16×C97—22no—
70SD16×C97—22no—
71D8×C97—21no—
72D8×C97—21no—
73C2×C8×C97D411yescharacteristic
74C2×C8×C97D411yescharacteristic
75C2×C8×C97D411yescharacteristic
76C2×C8×C97D411yescharacteristic
77C2×C8×C97C2311yescharacteristic
78C2×C8×C97C2311yescharacteristic
79C23×C97—22no—
80C23×C97#4111yescharacteristic
81C22×D97—21no—
82Q8×C97—22no—
83Q8×C97—22no—
84D4×C97—22no—
85D4×C97—22no—
86D4×C97—42no—
87D4×C97—42no—
88C2×C4×C97—22no—
89C2×C4×C97—42no—
90C2×C4×C97—42no—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97#4111yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—22no—
98C22⋊D97—22no—
99C2×Dic97—21no—
100D388—21no—
101D388C2×D412yes—
102C4×D97—22no—
103C4×D97—22no—
104Dic194—21no—
105Dic194C2×D412yes—
106C8×C97C2×D411yescharacteristic
107C8×C97C2×D411yescharacteristic
108C8×C97C2×D411yescharacteristic
109C8×C97C2×D411yescharacteristic
110#11715895—22no—
111#11715895—24no—
112C22×C97—41no—
113C22×C97—42no—
114C22×C97—82no—
115C22×C97#12111yescharacteristic
116C22×C97#12111yescharacteristic
117C22×C97C2×#4111yescharacteristic
118C2×D97—22no—
119C2×D97—42no—
120C4×C97—42no—
121C4×C97#12111yescharacteristic
122C4×C97#12111yescharacteristic
123C4×C97C22×D411yescharacteristic
124C4×C97C22×D411yescharacteristic
125Dic97—22no—
126C2×C97—21no—
127C2×C97—21no—
128C2×C97—21no—
129C2×C97—82no—
130C2×C97C2×#12111yescharacteristic
131D97—42no—
132#3067—971nomaximal, Sylow
133C97#306711yescharacteristic, complemented, Sylow
134#575—972no—
135C2×#135D9711yescharacteristic, complemented
136C2×#131—971no—
137#473—972no—
138#472—972no—
139#471—974no—
140C2×#108—971no—
141#413—972no—
142C2×#41—971no—
143C2×#41C2×D9712yes—
144#137—974no—
145#137—1942no—
146AGL(1,ℤ/8)—974no—
147AGL(1,ℤ/8)—1942no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—1944no—
153C2×Q16C2×D9711yescharacteristic, complemented
154C2×SD16C2×D9712yescomplemented
155C2×D8C2×D9711yescharacteristic, complemented
156#131—972no—
157#131—972no—
158#131—1941no—
159#131—1941no—
160C2×M16—971no—
161C2×M16—972no—
162C2×M16—1941no—
163C22×C8—1941no—
164C22×C8C2×D9711yescharacteristic
165#108—972no—
166#108—972no—
167#101—974no—
168#100—974no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—972no—
174#41—972no—
175#41—1941no—
176#41—1941no—
177C2×Q8—971no—
178C2×Q8C22×D9712yes—
179C2×D4—22no—
180C2×D4—971no—
181C2×D4—1941no—
182C2×D4C22×D9712yes—
183C22×C4—22no—
184C22×C4—1941no—
185C22×C4C22×D9711yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—1944no—
189SD16—22no—
190SD16—22no—
191SD16—1944no—
192SD16—1944no—
193D8—21no—
194D8—21no—
195D8—1944no—
196M16—972no—
197M16—974no—
198M16—1942no—
199M16—1941no—
200M16—1942no—
201M16—1942no—
202C2×C8—1942no—
203C2×C8—1941no—
204C2×C8—1942no—
205C2×C8—1942no—
206C2×C8C22⋊D9711yescharacteristic
207C2×C8C22⋊D9711yescharacteristic
208C2×C8C22⋊D9711yescharacteristic
209C2×C8C22⋊D9711yescharacteristic
210C2×C8C22×D9711yescharacteristic
211C2×C8C22×D9711yescharacteristic
212C23—22no—
213C23—1941no—
214C23#1309551011yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—972no—
218Q8—1941no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—972no—
224D4—1942no—
225D4—1942no—
226D4—1941no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—1941no—
231C2×C4—1942no—
232C2×C4—1942no—
233C2×C4D4×D9711yescharacteristic
234C2×C4D4×D9711yescharacteristic
235C2×C4C2×C22⋊D9711yescharacteristic
236C2×C4C2×C22⋊D9711yescharacteristic
237C2×C4#1309551011yescharacteristic
238C2×C4C23×D9711yescharacteristic
239C8—1944no—
240C8—1942no—
241C8C2×C22⋊D9711yescharacteristic
242C8C2×C22⋊D9711yescharacteristic
243C8C2×C22⋊D9711yescharacteristic
244C8C2×C22⋊D9711yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—3882no—
249C22—1942no—
250C22#1569078111yescharacteristic
251C22#1569078111yescharacteristic
252C22C2×#1309551011yescharacteristic
253C4—42no—
254C4—1942no—
255C4#1569078111yescharacteristic
256C4#1569078111yescharacteristic
257C4C2×D4×D9711yescharacteristic
258C4C22×C22⋊D9711yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—3882no—
264C2C2×#1569078111yescharacteristic
subgroupG/H
centerC4#15690781
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#135×C97C2
socleC2×C97C2×#121
hypercenterC2×#135D97
hypocenterC97#3067
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #20261533C2semidirect
2. C2×#15690816C2semidirect
3. #20261007C2semidirect
4. #20261002C2semidirect
5. #20260995C2semidirect
6. C2×#15690746C2semidirect
7. #20260719C2semidirect
8. C2×#135×C97C2semidirect
23. #15690746C22semidirect
24. #15690746C22semidirect
25. #15690729C22semidirect
26. #15690727C22semidirect
32. C2×Q16×C97C22semidirect
33. C2×SD16×C97C22semidirect
34. C2×D8×C97C22semidirect
133. C97#3067semidirect
135. C2×#135D97semidirect
153. C2×Q16C2×D97semidirect
154. C2×SD16C2×D97semidirect
155. C2×D8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
C2×#156908162
#202610072
#202610022
#202609952
C2×#156907462
#202607192
C2×#135×C972
#306797