Finite Group Databasebeta

← #27157151

Subgroups — #27157151

Order 12416.

286 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261706C211yesmaximal, characteristic, complemented
2#20261677C211yesmaximal, characteristic, complemented
3#20261589C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#20261242C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×C4⋊#87604—21no—
8C2×C22⋊#87604—21no—
9#15690867C2212yes—
10C22⋊#11715897—21no—
11(C2×#87604)⋊C4C2212yes—
12C22×C4×D97C2211yescharacteristic
13#15690791—21no—
14#15690786C411yescharacteristic, complemented
15#15690786C2211yescharacteristic
16#15690781C411yescharacteristic, complemented
17#15690781C2211yescharacteristic
18#15690778—21no—
19C4×C22⋊D97—21no—
20C4×C22⋊D97—21no—
21#15690769C411yescharacteristic, complemented
22C2×#41×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095521—42no—
25C2×#11715897—22no—
26C4⋊#87604—42no—
27C22⋊#87604—42no—
28C23×D97D411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C22×Dic97D411yescharacteristic
32C2×C4×D97—42no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97C2311yescharacteristic
37C4⋊Dic97C2×C411yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40#13095490—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44#13095488—21no—
45#13095488—21no—
46C4×Dic97—21no—
47#41×C97—42no—
48C2×Q8×C97C2×C411yescharacteristic
49C2×D4×C97—21no—
50C2×D4×C97C2×C411yescharacteristic
51C22×C4×C97—21no—
52C22×C4×C97C2×C411yescharacteristic
53C23×C97—21no—
54C23×C97C22⋊C411yescharacteristic
55C22×D97—21no—
56C22×D97—21no—
57C22×D97—41no—
58C22×D97—41no—
59C22×D97C22⋊C411yescharacteristic
60C22×D97C2×D411yescharacteristic
61C2×#87604—42no—
62C2×#87604—42no—
63C2×#87604—42no—
64Q8×C97—41no—
65D4×C97—41no—
66D4×C97—42no—
67C2×C4×C97—22no—
68C2×C4×C97—21no—
69C2×C4×C97—42no—
70C2×C4×C97C22⋊C411yescharacteristic
71C2×C4×C97C22×C411yescharacteristic
72C22⋊D97—42no—
73C22⋊D97—41no—
74C22⋊D97—41no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—41no—
78C2×Dic97C22⋊C411yescharacteristic
79C2×Dic97C2×D411yescharacteristic
80C4×D97—22no—
81C4×D97—22no—
82C4×D97—41no—
83C4×D97—41no—
84#11715897—42no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—41no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97C2×C22⋊C411yescharacteristic
92C2×D97—21no—
93C2×D97—21no—
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—21no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97AGL(1,ℤ/8)11yescharacteristic
100C2×D97#13711yescharacteristic
101#87604—82no—
102C4×C97—21no—
103C4×C97—41no—
104C4×C97#9912yes—
105Dic97—21no—
106Dic97—21no—
107Dic97—41no—
108Dic97—41no—
109C2×C97—41no—
110C2×C97—81no—
111C2×C97C2×#9911yescharacteristic
112C2×C97#41711yescharacteristic
113C2×C97#42111yescharacteristic
114D97—21no—
115D97—21no—
116D97—41no—
117#1513—971nomaximal, Sylow
118C97#151311yescharacteristic, complemented, Sylow
119#525—971no—
120#407—971no—
121#380—971no—
122#328—972no—
123#327—972no—
124C2×#41#8760411yescharacteristic, complemented
125C23×C4—971no—
126C2×M16—1941no—
127#123—1941no—
128C22⋊Q8—971no—
129C22⋊Q8—971no—
130#121—971no—
131#121—971no—
132#120—1941no—
133C4×D4—1941no—
134C4×D4—1941no—
135#117—971no—
136C2×C4⋊C4—1941no—
137C2×C22⋊C4—1941no—
138C22⋊C8—972no—
139C22⋊C8—1941no—
140#95—972no—
141C24—971no—
142#41—42no—
143C2×Q8C2×#8760411yescharacteristic
144C2×D4—21no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4C2×#8760411yescharacteristic
148C22×C4—21no—
149C22×C4—971no—
150C22×C4—971no—
151C22×C4—971no—
152C22×C4—971no—
153C22×C4—971no—
154C22×C4—3882no—
155C22×C4—1942no—
156C22×C4C2×#8760411yescharacteristic
157M16—3882no—
158C2×C8—1942no—
159C4⋊C4—971no—
160C4⋊C4—971no—
161C4⋊C4—3882no—
162C4⋊C4—1941no—
163C4⋊C4—1941no—
164C22⋊C4—3882no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C42—1941no—
171C23—21no—
172C23—971no—
173C23—971no—
174C23—3881no—
175C23—3881no—
176C23—1941no—
177C23—1941no—
178C23C22⋊#8760411yescharacteristic
179Q8—41no—
180D4—41no—
181D4—42no—
182D4—3881no—
183D4—3881no—
184D4—3882no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—42no—
188C2×C4—971no—
189C2×C4—971no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—3882no—
193C2×C4—3882no—
194C2×C4—3882no—
195C2×C4—3881no—
196C2×C4—1941no—
197C2×C4—1941no—
198C2×C4—1942no—
199C2×C4—1942no—
200C2×C4C22⋊#8760411yescharacteristic
201C2×C4C22×#8760411yescharacteristic
202C8—3882no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—971no—
210C22—971no—
211C22—3882no—
212C22—3882no—
213C22—1941no—
214C22—1941no—
215C22—1941no—
216C22—1941no—
217C22—1941no—
218C22C2×C22⋊#8760411yescharacteristic
219C4—21no—
220C4—41no—
221C4—7762no—
222C4—3881no—
223C4—3881no—
224C4—1941no—
225C4—1941no—
226C4#1569086812yes—
227C2—41no—
228C2—81no—
229C2—3881no—
230C2—1941no—
231C2—1941no—
232C2C2×#1569086811yescharacteristic
233C2#2026172611yescharacteristic
234C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×#41×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261242C2
cosocleC2×C4×D97C23
hypercenterC2×#41#87604
hypocenterC97#1513
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261706C2semidirect
2. #20261677C2semidirect
3. #20261589C2semidirect
4. #20261586C2semidirect
14. #15690786C4semidirect
16. #15690781C4semidirect
21. #15690769C4semidirect
22. C2×#41×C97C4semidirect
118. C97#1513semidirect
124. C2×#41#87604semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202616772
#202615892
#202615862
#202612422
#151397