Finite Group Databasebeta

← #27142176

Subgroups — #27142176

Order 12416.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#20261242C211yesmaximal, characteristic
2C2×C42×D97C211yesmaximal, characteristic, complemented
3C2×#15690737C211yesmaximal, characteristic, complemented
4#20260698C211yesmaximal, characteristic, complemented
5#20260496C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97—21no—
9C22×C4×D97C2211yescharacteristic
10C2×#13095495C2211yescharacteristic
11#15690791—21no—
12#15690786C412yescomplemented
13#15690781C412yescomplemented
14#15690778—21no—
15C4×C22⋊D97—22no—
16#15690769C2211yescharacteristic
17C2×C4×Dic97—21no—
18C2×C4×Dic97C2211yescharacteristic
19C42×D97—22no—
20C42×D97—22no—
21#15690737—22no—
22#15690722—22no—
23C2×#41×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25C2×C42×C97C2211yescharacteristic
26#104×C97—22no—
27C23×D97D411yescharacteristic
28C2×C22⋊D97—22no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C2×C4×D97—24no—
32C2×C4×D97—28no—
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97C2×C412yes—
38C2×C4×D97D412yes—
39#13095495—21no—
40#13095495—21no—
41C2×#11715895—21no—
42C4⋊Dic97C2×C412yescomplemented
43#13095490—22no—
44C22⋊Dic97—21no—
45C22⋊Dic97—22no—
46#13095488—22no—
47C4×Dic97—24no—
48C4×Dic97—21no—
49C4×Dic97—21no—
50C4×Dic97D412yescomplemented
51#41×C97—21no—
52#41×C97—21no—
53#41×C97—21no—
54#41×C97—21no—
55C2×Q8×C97C2×C411yescharacteristic, complemented
56C2×D4×C97—21no—
57C2×D4×C97C2×C411yescharacteristic, complemented
58C22×C4×C97—21no—
59C22×C4×C97—21no—
60C22×C4×C97C2311yescharacteristic
61M16×C97—21no—
62M16×C97—21no—
63C2×C8×C97—21no—
64C42×C97—21no—
65C42×C97D412yescomplemented
66C23×C97—21no—
67C23×C97C2×D411yescharacteristic
68C22×D97—21no—
69C22×D97—21no—
70C22×D97—24no—
71C22×D97C22⋊C412yes—
72Q8×C97—21no—
73Q8×C97—21no—
74D4×C97—21no—
75D4×C97—21no—
76D4×C97—41no—
77D4×C97—41no—
78C2×C4×C97—24no—
79C2×C4×C97—22no—
80C2×C4×C97—21no—
81C2×C4×C97—41no—
82C2×C4×C97—41no—
83C2×C4×C97C22×C411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97#4111yescharacteristic
88C2×C4×C97#4111yescharacteristic
89C22⋊D97—44no—
90C2×Dic97—24no—
91C2×Dic97—21no—
92C2×Dic97—22no—
93C2×Dic97—21no—
94C2×Dic97—22no—
95C2×Dic97C22⋊C412yes—
96C4×D97—216no—
97C4×D97—22no—
98C4×D97—22no—
99C4×D97—22no—
100C4×D97—22no—
101C8×C97—41no—
102#11715895—41no—
103C22×C97—22no—
104C22×C97—41no—
105C22×C97—42no—
106C22×C97C2×C22⋊C411yescharacteristic
107C22×C97#12011yescharacteristic
108C22×C97#12311yescharacteristic
109C2×D97—24no—
110C2×D97—24no—
111C2×D97—22no—
112C2×D97—22no—
113C2×D97#10414yescomplemented
114C4×C97—24no—
115C4×C97—41no—
116C4×C97C4×D411yescharacteristic
117C4×C97C4×D411yescharacteristic
118C4×C97#12111yescharacteristic
119C4×C97#12111yescharacteristic
120Dic97—24no—
121Dic97—22no—
122Dic97—42no—
123C2×C97—21no—
124C2×C97—21no—
125C2×C97—81no—
126C2×C97#38611yescharacteristic
127C2×C97C2×#10412yes—
128D97—24no—
129#1896—971nomaximal, Sylow
130C97#189611yescharacteristic, complemented, Sylow
131#525—971no—
132C22×C42—971no—
133C2×#104—971no—
134C2×#104D9711yescharacteristic, complemented
135#407—971no—
136#339—971no—
137#339—971no—
138C2×#41C2×D9711yescharacteristic
139C23×C4—971no—
140C23×C4—1941no—
141C2×M16—971no—
142C2×M16C2×D9711yescharacteristic
143#123—1941no—
144C22⋊Q8—972no—
145#121—972no—
146#120—1941no—
147C4×D4—1942no—
148#117—971no—
149C2×C42—971no—
150C2×C42—1942no—
151C2×C42—1941no—
152C2×C42—1942no—
153C2×C42C2×D9711yescharacteristic
154#104—22no—
155#104—1942no—
156C22⋊C8—1942no—
157C24—971no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8C4×D9711yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—1942no—
165C2×D4C4×D9711yescharacteristic, complemented
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—971no—
169C22×C4—972no—
170C22×C4—972no—
171C22×C4—1944no—
172C22×C4—1948no—
173C22×C4—1941no—
174C22×C4—1941no—
175C22×C4—1941no—
176C22×C4—1941no—
177C22×C4—1941no—
178C22×C4C22×D9711yescharacteristic
179M16—21no—
180M16—21no—
181M16—1941no—
182M16—1941no—
183C2×C8—21no—
184C2×C8—1941no—
185C4⋊C4—972no—
186C4⋊C4—1942no—
187C22⋊C4—1942no—
188C22⋊C4—1942no—
189C22⋊C4—1941no—
190C42—21no—
191C42—972no—
192C42—1944no—
193C42—1941no—
194C42—1941no—
195C42C22⋊D9712yescomplemented
196C23—21no—
197C23—972no—
198C23—1944no—
199C23—1941no—
200C23—1941no—
201C23D4×D9711yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—3884no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—972no—
215C2×C4—19416no—
216C2×C4—1942no—
217C2×C4—1944no—
218C2×C4—1942no—
219C2×C4—1941no—
220C2×C4—1941no—
221C2×C4—1942no—
222C2×C4—1942no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4C2×C4×D9711yescharacteristic
226C2×C4D4×D9711yescharacteristic
227C2×C4D4×D9711yescharacteristic
228C2×C4C2×C22⋊D9711yescharacteristic
229C2×C4#1309551011yescharacteristic
230C2×C4#1309551111yescharacteristic
231C8—41no—
232C8—3881no—
233C22—22no—
234C22—41no—
235C22—42no—
236C22—974no—
237C22—1944no—
238C22—1944no—
239C22—1942no—
240C22—1942no—
241C22C22⋊C4×D9711yescharacteristic
242C22#1569077811yescharacteristic
243C22#1569079211yescharacteristic
244C4—24no—
245C4—41no—
246C4—3882no—
247C4—1944no—
248C4—1942no—
249C4#1569077011yescharacteristic
250C4C4×C22⋊D9711yescharacteristic
251C4#1569078111yescharacteristic
252C4#1569078311yescharacteristic
253C2—21no—
254C2—21no—
255C2—81no—
256C2—1944no—
257C2#2026061111yescharacteristic
258C2#104×D9712yes—
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261242C2
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1896
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×C42×D97C2semidirect
3. C2×#15690737C2semidirect
4. #20260698C2semidirect
5. #20260496C2semidirect
6. #20260495C2semidirect
7. C2×#104×C97C2semidirect
12. #15690786C4semidirect
13. #15690781C4semidirect
42. C4⋊Dic97C2×C4semidirect
50. C4×Dic97D4semidirect
55. C2×Q8×C97C2×C4semidirect
57. C2×D4×C97C2×C4semidirect
65. C42×C97D4semidirect
113. C2×D97#104semidirect
130. C97#1896semidirect
134. C2×#104D97semidirect
162. C2×Q8C4×D97semidirect
165. C2×D4C4×D97semidirect
195. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612422
C2×C42×D972
C2×#156907372
#202606982
#202604962
#202604952
C2×#104×C972
#189697