Finite Group Databasebeta

← #45664672

Subgroups — #45664672

Order 32896.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#16861606C211yesmaximal, characteristic
2C2×#22551557C211yesmaximal, characteristic, complemented
3C2×#22551517C211yesmaximal, characteristic, complemented
4#30656640C211yesmaximal, characteristic
5#30656499C211yesmaximal, characteristic, complemented
6#30656491C211yesmaximal, characteristic, complemented
7C2×#102×C257C211yesmaximal, characteristic
8C22×Dic514C2211yescharacteristic
9C2×#16861606—22no—
10C2×#16861606—22no—
11C2×#16861606C414yescomplemented
12C22×#14523339C2211yescharacteristic
13#22551557—22no—
14#22551557—22no—
15C2×C4⋊Dic257C2211yescharacteristic
16C2×#16861597—21no—
17C2×C22⋊Dic257—21no—
18C2×C22⋊Dic257—21no—
19C2×C4×Dic257—21no—
20C2×C4×Dic257C2211yescharacteristic
21#22551517—22no—
22C22×D4×C257C2211yescharacteristic
23C22×C8×C257C2211yescharacteristic
24C2×C4⋊C4×C257C2211yescharacteristic
25#102×C257—22no—
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×Dic514C2×C412yes—
30C2×Dic514D412yes—
31C2×Dic514D412yes—
32C22×Dic257—21no—
33C22×Dic257—21no—
34C22×Dic257D411yescharacteristic
35C22×Dic257D411yescharacteristic
36#16861606—28no—
37C2×#14523339—21no—
38C2×#14523339—21no—
39C2×#14523339C2×C412yes—
40C4⋊Dic257—21no—
41C4⋊Dic257—21no—
42#16861597—24no—
43C22⋊Dic257—42no—
44C22⋊Dic257—42no—
45C4×Dic257—24no—
46C4×Dic257—21no—
47C4×Dic257—21no—
48C24×C257—21no—
49C2×D4×C257—21no—
50C2×D4×C257—21no—
51C2×D4×C257—21no—
52C2×D4×C257—21no—
53C2×D4×C257—21no—
54C2×D4×C257C2×C412yes—
55C22×C4×C257—21no—
56C22×C4×C257C2311yescharacteristic
57C2×C8×C257—21no—
58C2×C8×C257—21no—
59C2×C8×C257—21no—
60C4⋊C4×C257—21no—
61C4⋊C4×C257D412yes—
62Dic514—22no—
63Dic514—22no—
64Dic514—22no—
65Dic514C22⋊C414yescomplemented
66C2×Dic257—22no—
67C2×Dic257—21no—
68C2×Dic257—22no—
69C2×Dic257—22no—
70C2×Dic257—21no—
71C2×Dic257—21no—
72C2×Dic257—21no—
73C2×Dic257—21no—
74C2×Dic257—41no—
75C2×Dic257—41no—
76C2×Dic257—41no—
77C2×Dic257SD1612yescomplemented
78#14523339—22no—
79C23×C257—22no—
80C23×C257—41no—
81C23×C257—41no—
82C23×C257—41no—
83C23×C257—41no—
84C23×C257—41no—
85C23×C257—41no—
86C23×C257C2×D411yescharacteristic
87D4×C257—22no—
88D4×C257—22no—
89D4×C257—41no—
90D4×C257—41no—
91C2×C4×C257—22no—
92C2×C4×C257—41no—
93C2×C4×C257—41no—
94C2×C4×C257C22×C411yescharacteristic
95C2×C4×C257C2×D411yescharacteristic
96C2×C4×C257C2×D411yescharacteristic
97C2×C4×C257C2×D411yescharacteristic
98C2×C4×C257#4111yescharacteristic
99C2×C4×C257#4111yescharacteristic
100C8×C257—41no—
101C22×C257—42no—
102C22×C257—42no—
103C22×C257—42no—
104C22×C257—81no—
105C22×C257—81no—
106C22×C257—81no—
107C22×C257—81no—
108C22×C257C4×D411yescharacteristic
109C22×C257#12011yescharacteristic
110C22×C257#12111yescharacteristic
111C22×C257C2×SD1611yescharacteristic
112C22×C257#13511yescharacteristic
113C22×C257AGL(1,ℤ/8)11yescharacteristic
114C22×C257#13711yescharacteristic
115C4×C257—42no—
116C4×C257C2×C22⋊C411yescharacteristic
117C4×C257C4×D411yescharacteristic
118C4×C257#12111yescharacteristic
119C4×C257#12311yescharacteristic
120Dic257—24no—
121Dic257—22no—
122Dic257—41no—
123Dic257—81no—
124C2×C257—82no—
125C2×C257#38611yescharacteristic
126C2×C257C4×SD1611yescharacteristic
127C2×C257#44011yescharacteristic
128C2×C257#44811yescharacteristic
129C2×C257#44911yescharacteristic
130C2×C257#46011yescharacteristic
131C2×C257#46411yescharacteristic
132C257#188011yescharacteristic, complemented, Sylow
133#1880—2571nomaximal, Sylow
134C22×SD16—2571no—
135C2×C4×Q8—2571no—
136C2×#103—2571no—
137C2×#102D25711yescharacteristic
138#390—2571no—
139#340—2571no—
140#336—2571no—
141C22×Q8—2571no—
142C22×D4C2×D25711yescharacteristic
143C2×SD16—2574no—
144C2×SD16—5142no—
145C2×SD16—5142no—
146C22×C8—2571no—
147C22×C8C2×D25711yescharacteristic
148C4×Q8—5142no—
149C4×Q8—5142no—
150C2×C4⋊C4—2571no—
151C2×C4⋊C4—5141no—
152C2×C4⋊C4C2×D25711yescharacteristic
153C2×C22⋊C4—5141no—
154C2×C22⋊C4—5141no—
155C2×C42—2571no—
156C2×C42—5141no—
157#103—5142no—
158#102—22no—
159C24—21no—
160C2×Q8—2572no—
161C2×Q8—2572no—
162C2×Q8—2572no—
163C2×Q8—5141no—
164C2×Q8—5141no—
165C2×Q8—5141no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C4×D25712yes—
172C22×C4—2571no—
173C22×C4—2571no—
174C22×C4—21no—
175C22×C4—5141no—
176C22×C4—5141no—
177C22×C4C22×D25711yescharacteristic
178SD16—5148no—
179C2×C8—2572no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—5141no—
184C2×C8—5141no—
185C4⋊C4—21no—
186C4⋊C4—5144no—
187C4⋊C4—5141no—
188C4⋊C4—5141no—
189C4⋊C4C22⋊D25712yes—
190C22⋊C4—10282no—
191C22⋊C4—10282no—
192C42—5144no—
193C42—5141no—
194C42—5141no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23D4×D25711yescharacteristic
203Q8—2574no—
204Q8—5142no—
205Q8—5142no—
206Q8—5142no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—2572no—
212C2×C4—22no—
213C2×C4—5142no—
214C2×C4—5141no—
215C2×C4—5142no—
216C2×C4—5142no—
217C2×C4—5141no—
218C2×C4—5141no—
219C2×C4—5141no—
220C2×C4—5141no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—10281no—
224C2×C4—10281no—
225C2×C4—10281no—
226C2×C4C2×C4×D25711yescharacteristic
227C2×C4D4×D25711yescharacteristic
228C2×C4D4×D25711yescharacteristic
229C2×C4C2×C22⋊D25711yescharacteristic
230C2×C4#1686161711yescharacteristic
231C2×C4#1686161811yescharacteristic
232C8—5142no—
233C8—41no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22#2255155211yescharacteristic
242C22#2255156011yescharacteristic
243C22#2255156511yescharacteristic
244C22SD16×D25711yescharacteristic
245C22#2255161111yescharacteristic
246C22#2255161611yescharacteristic
247C22#2255162111yescharacteristic
248C4—5144no—
249C4—5142no—
250C4—42no—
251C4—10281no—
252C4—20561no—
253C4C22⋊C4×D25711yescharacteristic
254C4C4×C22⋊D25711yescharacteristic
255C4#2255156311yescharacteristic
256C4#2255157411yescharacteristic
257C2—82no—
258C2#3065661311yescharacteristic
259C2#3065682811yescharacteristic
260C2#3065683411yescharacteristic
261C2#3065687411yescharacteristic
262C2#3065687911yescharacteristic
263C2#3065693611yescharacteristic
264C2#3065695811yescharacteristic
subgroupG/H
centerC23D4×D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingC2×#102×C257C2
socleC23×C257C2×D4
prime radicalC22×D4×C257C22
cosocleC22×C4×C257C23
hypercenterC2×#102D257
hypocenterC257#1880
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#22551557C2semidirect
3. C2×#22551517C2semidirect
5. #30656499C2semidirect
6. #30656491C2semidirect
11. C2×#16861606C4semidirect
65. Dic514C22⋊C4semidirect
77. C2×Dic257SD16semidirect
132. C257#1880semidirect

Maximal subgroups

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

subgroupindex
C22×#168616062
C2×#225515572
C2×#225515172
#306566402
#306564992
#306564912
C2×#102×C2572
#1880257