Finite Group Databasebeta

← #45664662

Subgroups — #45664662

Order 32896.

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

normal subgroups only

This group has 338 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×Dic1028C211yesmaximal, characteristic
2C2×#22551557C211yesmaximal, characteristic
3C2×#22551515C211yesmaximal, characteristic
4C2×#22551515C211yesmaximal, characteristic
5#30656642C211yesmaximal, characteristic
6#30656491C211yesmaximal, characteristic
7#340×C257C211yesmaximal, characteristic
8C22×Dic514C2211yescharacteristic
9C22×Dic514C2211yescharacteristic
10C2×Dic1028—22no—
11C2×Dic1028—22no—
12C2×Dic1028C414yescomplemented
13C22×#14523339C2211yescharacteristic
14#22551557—22no—
15#22551557—22no—
16C2×#16861597—21no—
17C2×C4×Dic257—21no—
18C2×C4×Dic257C2211yescharacteristic
19#22551515—22no—
20#22551515—22no—
21#22551499—21no—
22#22551499—21no—
23C22×C8×C257C2211yescharacteristic
24C2×C4⋊C4×C257C2211yescharacteristic
25C2×C4⋊C4×C257C2211yescharacteristic
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×Dic514—21no—
30C2×Dic514—21no—
31C2×Dic514—21no—
32C2×Dic514—21no—
33C2×Dic514—21no—
34C2×Dic514C2×C412yes—
35C2×Dic514C2×C412yes—
36C2×Dic514D412yes—
37C2×Dic514D412yes—
38C22×Dic257—21no—
39C22×Dic257—21no—
40C22×Dic257D411yescharacteristic
41C22×Dic257D411yescharacteristic
42Dic1028—28no—
43C2×#14523339—21no—
44C2×#14523339—21no—
45C2×#14523339—21no—
46#16861597—24no—
47C4×Dic257—24no—
48C4×Dic257—21no—
49C4×Dic257—21no—
50C22×C4×C257—21no—
51C22×C4×C257—21no—
52C22×C4×C257C2311yescharacteristic
53C2×C8×C257—21no—
54C2×C8×C257—21no—
55C2×C8×C257C2×C412yes—
56C4⋊C4×C257—21no—
57C4⋊C4×C257—21no—
58C4⋊C4×C257—21no—
59C4⋊C4×C257D412yes—
60Dic514—22no—
61Dic514—22no—
62Dic514—22no—
63Dic514—22no—
64Dic514—22no—
65Dic514—41no—
66Dic514—41no—
67Dic514C22⋊C414yes—
68C2×Dic257—22no—
69C2×Dic257—21no—
70C2×Dic257—22no—
71C2×Dic257—22no—
72C2×Dic257—21no—
73C2×Dic257—22no—
74C2×Dic257—21no—
75C2×Dic257—21no—
76C2×Dic257—21no—
77C2×Dic257—41no—
78C2×Dic257—41no—
79C2×Dic257Q1612yescomplemented
80#14523339—41no—
81C23×C257C2×D411yescharacteristic
82C2×C4×C257—22no—
83C2×C4×C257—41no—
84C2×C4×C257—41no—
85C2×C4×C257—41no—
86C2×C4×C257—41no—
87C2×C4×C257—41no—
88C2×C4×C257C22×C411yescharacteristic
89C2×C4×C257C2×D411yescharacteristic
90C2×C4×C257C2×D411yescharacteristic
91C2×C4×C257C2×D411yescharacteristic
92C2×C4×C257#4111yescharacteristic
93C2×C4×C257#4111yescharacteristic
94C8×C257—22no—
95C22×C257C4×D411yescharacteristic
96C22×C257#12011yescharacteristic
97C22×C257#12111yescharacteristic
98C22×C257C2×Q1611yescharacteristic
99C22×C257#13511yescharacteristic
100C22×C257#13711yescharacteristic
101C22×C257#13711yescharacteristic
102C4×C257—42no—
103C4×C257—81no—
104C4×C257C2×C22⋊C411yescharacteristic
105C4×C257C4×D411yescharacteristic
106C4×C257#12111yescharacteristic
107C4×C257#12311yescharacteristic
108Dic257—24no—
109Dic257—22no—
110Dic257—42no—
111Dic257—41no—
112C2×C257#38611yescharacteristic
113C2×C257C4×Q1611yescharacteristic
114C2×C257#44111yescharacteristic
115C2×C257#45111yescharacteristic
116C2×C257#45211yescharacteristic
117C2×C257#46211yescharacteristic
118C2×C257#46411yescharacteristic
119C257#187811yescharacteristic, complemented, Sylow
120#1878—2571nomaximal, Sylow
121C22×Q16—2571no—
122C2×C4×Q8—2571no—
123C2×#103—2571no—
124C2×#103—2571no—
125#391—2571no—
126#340D25711yescharacteristic
127#336—2571no—
128C22×Q8—2571no—
129C22×Q8—2571no—
130C2×Q16—2574no—
131C2×Q16—5142no—
132C2×Q16—5142no—
133C22×C8—2571no—
134C22×C8C2×D25711yescharacteristic
135C4×Q8—5142no—
136C4×Q8—5142no—
137C2×C4⋊C4—5141no—
138C2×C4⋊C4C2×D25711yescharacteristic
139C2×C4⋊C4C2×D25711yescharacteristic
140C2×C42—2571no—
141C2×C42—5141no—
142#103—5142no—
143#103—5142no—
144#95—5141no—
145#95—5141no—
146C2×Q8—2572no—
147C2×Q8—2572no—
148C2×Q8—2572no—
149C2×Q8—2572no—
150C2×Q8—5141no—
151C2×Q8—5141no—
152C2×Q8—5141no—
153C2×Q8—5141no—
154C2×Q8—5141no—
155C2×Q8—5141no—
156C2×Q8—5141no—
157C2×Q8—5141no—
158C22×C4—2571no—
159C22×C4—2571no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—5141no—
163C22×C4—5141no—
164C22×C4C22×D25711yescharacteristic
165Q16—5148no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—5141no—
169C2×C8—5141no—
170C2×C8—5141no—
171C2×C8C4×D25712yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—5144no—
176C4⋊C4D102812yes—
177C42—5144no—
178C42—5141no—
179C42—5141no—
180C23D4×D25711yescharacteristic
181Q8—2574no—
182Q8—5142no—
183Q8—5142no—
184Q8—5142no—
185Q8—5142no—
186Q8—5142no—
187Q8—10281no—
188Q8—10281no—
189C2×C4—2572no—
190C2×C4—22no—
191C2×C4—5142no—
192C2×C4—5141no—
193C2×C4—5142no—
194C2×C4—5142no—
195C2×C4—5141no—
196C2×C4—5142no—
197C2×C4—5141no—
198C2×C4—5141no—
199C2×C4—5141no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—10281no—
206C2×C4—10281no—
207C2×C4C2×C4×D25711yescharacteristic
208C2×C4D4×D25711yescharacteristic
209C2×C4D4×D25711yescharacteristic
210C2×C4C2×D102811yescharacteristic
211C2×C4#1686161811yescharacteristic
212C2×C4#1686161911yescharacteristic
213C8—22no—
214C8—10281no—
215C22#2255155411yescharacteristic
216C22#2255156211yescharacteristic
217C22#2255156511yescharacteristic
218C22Q16×D25711yescharacteristic
219C22#2255161111yescharacteristic
220C22#2255162111yescharacteristic
221C22#2255162111yescharacteristic
222C4—5144no—
223C4—5142no—
224C4—42no—
225C4—10282no—
226C4—10281no—
227C4—81no—
228C4C22⋊C4×D25711yescharacteristic
229C4C4×D102811yescharacteristic
230C4#2255156711yescharacteristic
231C4#2255157711yescharacteristic
232C2#3065661511yescharacteristic
233C2#3065683111yescharacteristic
234C2#3065684311yescharacteristic
235C2#3065689411yescharacteristic
236C2#3065690011yescharacteristic
237C2#3065694611yescharacteristic
238C2#3065695811yescharacteristic
subgroupG/H
centerC23D4×D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#340×C257C2
socle, prime radicalC23×C257C2×D4
cosocleC22×C4×C257C23
hypercenter#340D257
hypocenterC257#1878
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
12. C2×Dic1028C4semidirect
79. C2×Dic257Q16semidirect
119. C257#1878semidirect

Maximal subgroups

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

subgroupindex
C22×Dic10282
C2×#225515572
C2×#225515152
#306566422
#306564912
#340×C2572
#1878257