Finite Group Databasebeta

← #27151099

Subgroups — #27151099

Order 12416.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#15690858C211yesmaximal, characteristic, complemented
2#20261533C212yesmaximal, complemented
3#20260917C214yesmaximal, complemented
4#20260907C214yesmaximal, complemented
5#20260753C212yesmaximal, complemented
6C2×#15690729C211yesmaximal, characteristic, complemented
7#536×C97C211yesmaximal, characteristic, complemented
8#15690858—24no—
9#15690858—22no—
10#15690858C2214yes—
11C2×#13095512—22no—
12C2×#13095511—22no—
13C2×#13095511C2212yes—
14C2×Q8×D97—22no—
15#15690841C2214yescomplemented
16#15690840—22no—
17#15690836C2214yescomplemented
18#15690835—22no—
19#15690830—24no—
20C2×#13095495C2212yes—
21#15690738C2218yescomplemented
22#15690729C2214yescomplemented
23C2×#41×C97C2211yescharacteristic
24C22×Q8×C97C2211yescharacteristic
25C4⋊Q8×C97C2212yescomplemented
26#124×C97C2212yescomplemented
27#123×C97—22no—
28C22⋊Q8×C97—22no—
29#117×C97C2211yescharacteristic
30#13095512—28no—
31#13095512—28no—
32#13095511—28no—
33#13095511—28no—
34#13095511—22no—
35#13095511—22no—
36#13095511D414yes—
37#13095511C2314yes—
38C2×Dic194—22no—
39C2×Dic194D412yes—
40Q8×D97—28no—
41Q8×D97—28no—
42C2×D388—22no—
43C2×D388D412yes—
44C2×C22⋊D97—22no—
45C2×C4×D97—24no—
46C2×C4×D97—22no—
47#13095501—24no—
48#13095499—24no—
49#13095498—24no—
50#13095496—24no—
51#13095495—22no—
52#13095495C2314yes—
53C2×#11715895—22no—
54#41×C97—22no—
55#41×C97—22no—
56C2×Q8×C97—22no—
57C2×Q8×C97—22no—
58C2×Q8×C97D412yes—
59C2×Q8×C97C2311yescharacteristic
60C2×Q8×C97C2312yes—
61C2×D4×C97—21no—
62C2×D4×C97C2311yescharacteristic
63C22×C4×C97—21no—
64C22×C4×C97D412yes—
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—21no—
67C4⋊C4×C97—24no—
68C22⋊C4×C97—21no—
69C22⋊C4×C97—24no—
70C42×C97C2312yes—
71C23×C97—21no—
72C23×C97C2×D411yescharacteristic
73C22×D97—22no—
74Q8×C97—24no—
75Q8×C97—24no—
76Q8×C97—22no—
77D4×C97—22no—
78D4×C97—41no—
79D4×C97—41no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—21no—
84C2×C4×C97—42no—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D412yes—
87C2×C4×C97C2×D414yes—
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—24no—
91C22⋊D97—24no—
92C2×Dic97—22no—
93D388—28no—
94D388—22no—
95D388C2×D414yes—
96C4×D97—216no—
97C4×D97—24no—
98C4×D97—24no—
99Dic194—28no—
100Dic194—22no—
101Dic194C2×D414yes—
102#11715895—24no—
103C22×C97—41no—
104C22×C97—42no—
105C22×C97—41no—
106C22×C97#12011yescharacteristic
107C22×C97#12011yescharacteristic
108C22×C97C22×D411yescharacteristic
109C2×D97—24no—
110C2×D97—44no—
111C4×C97—24no—
112C4×C97—42no—
113C4×C97—41no—
114C4×C97#12012yes—
115C4×C97C22×D412yes—
116Dic97—24no—
117C2×C97—21no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—81no—
121C2×C97C2×#12011yescharacteristic
122D97—44no—
123#3020—971nomaximal, Sylow
124C97#302011yescharacteristic, complemented, Sylow
125C2×2−1+4—971no—
126#575—972no—
127#536D9711yescharacteristic, complemented
128#456—974no—
129#454—974no—
130#421—972no—
131C2×#101—971no—
1322−1+4—974no—
1332−1+4—1944no—
1342−1+4—1942no—
135C2×#41—972no—
136C2×#41—1942no—
137C2×#41—1942no—
138C2×#41C2×D9711yescharacteristic
139C22×Q8—1942no—
140C22×Q8C2×D9711yescharacteristic
141#137—974no—
142#137—1942no—
143AGL(1,ℤ/8)—974no—
144AGL(1,ℤ/8)—1942no—
145#135—1944no—
146C2×M16—972no—
147C4⋊Q8C2×D9712yescomplemented
148#124C2×D9712yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D9711yescharacteristic
152#104—978no—
153#101—974no—
154#41—22no—
155#41—22no—
156#41—974no—
157#41—974no—
158#41—1948no—
159#41—1948no—
160#41—1948no—
161#41—1942no—
162#41—1942no—
163#41—1948no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—972no—
167C2×Q8—1942no—
168C2×Q8—1948no—
169C2×Q8—1948no—
170C2×Q8C22⋊D9712yes—
171C2×Q8C22×D9711yescharacteristic
172C2×Q8C22×D9712yes—
173C2×D4—21no—
174C2×D4—972no—
175C2×D4—1942no—
176C2×D4—1942no—
177C2×D4C22×D9711yescharacteristic
178C22×C4—21no—
179C22×C4—1944no—
180C22×C4—1942no—
181C22×C4C22⋊D9712yes—
182C22×C4C22×D9711yescharacteristic
183Q16—1944no—
184SD16—1944no—
185SD16—1944no—
186D8—1944no—
187M16—974no—
188M16—1942no—
189C2×C8—1942no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D9712yes—
195C23—21no—
196C23—1942no—
197C23C2×C22⋊D9711yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—974no—
202Q8—1948no—
203Q8—1942no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—974no—
208D4—1948no—
209D4—1944no—
210D4—1944no—
211D4—1942no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—19416no—
218C2×C4—1942no—
219C2×C4—1944no—
220C2×C4—1944no—
221C2×C4D4×D9712yes—
222C2×C4D4×D9712yes—
223C2×C4C2×C22⋊D9714yes—
224C2×C4C2×C22⋊D9711yescharacteristic
225C2×C4C23×D9711yescharacteristic
226C8—1944no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—3884no—
231C22—1944no—
232C22#1569077811yescharacteristic
233C22#1569077811yescharacteristic
234C22C22×C22⋊D9711yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—1944no—
239C4#1569077812yes—
240C4C2×D4×D9712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—3884no—
246C2C2×#1569077811yescharacteristic
subgroupG/H
centerC2C2×#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#536×C97C2
socleC2×C97C2×#120
hypercenter#536D97
hypocenterC97#3020
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#15690858C2semidirect
2. #20261533C2semidirect
3. #20260917C2semidirect
4. #20260907C2semidirect
5. #20260753C2semidirect
6. C2×#15690729C2semidirect
7. #536×C97C2semidirect
15. #15690841C22semidirect
17. #15690836C22semidirect
21. #15690738C22semidirect
22. #15690729C22semidirect
25. C4⋊Q8×C97C22semidirect
26. #124×C97C22semidirect
124. C97#3020semidirect
127. #536D97semidirect
147. C4⋊Q8C2×D97semidirect
148. #124C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908582
#202615332
#202609172
#202609072
#202607532
C2×#156907292
#536×C972
#302097