Finite Group Databasebeta

← #45665421

Subgroups — #45665421

Order 32896.

370 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 93 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
2C22×#16861608C211yesmaximal, characteristic
3C2×#22551590C211yesmaximal, characteristic
4C2×#22551515C211yesmaximal, characteristic
5#30656642C211yesmaximal, characteristic
6#30656491C211yesmaximal, characteristic
7C2×#103×C257C211yesmaximal, characteristic
8C22×Dic514C2211yescharacteristic
9C22×Dic514C2211yescharacteristic
10C2×Dic1028—22no—
11C2×Dic1028—22no—
12C2×Dic1028—22no—
13C2×#16861608—22no—
14C2×#16861608—22no—
15C2×#16861608—22no—
16C22×#14523339C2211yescharacteristic
17#22551590—22no—
18#22551590—22no—
19C2×#16861597—22no—
20C2×C4×Dic257C2211yescharacteristic
21#22551515—22no—
22#22551499—22no—
23C22×Q8×C257C2211yescharacteristic
24C22×C8×C257C2211yescharacteristic
25C2×C4⋊C4×C257C2211yescharacteristic
26#103×C257—22no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×Dic514—21no—
30C2×Dic514—21no—
31C2×Dic514—21no—
32C2×Dic514—21no—
33C2×Dic514—21no—
34C2×Dic514—21no—
35C2×Dic514—21no—
36C2×Dic514—21no—
37C2×Dic514D412yes—
38C2×Dic514D412yes—
39C22×Dic257—21no—
40C22×Dic257—21no—
41C22×Dic257D412yes—
42Dic1028—28no—
43#16861608—28no—
44C2×#14523339—21no—
45C2×#14523339—21no—
46C2×#14523339D412yes—
47#16861597—28no—
48C4×Dic257—21no—
49C4×Dic257—21no—
50C2×Q8×C257—21no—
51C2×Q8×C257—21no—
52C2×Q8×C257—21no—
53C2×Q8×C257—21no—
54C2×Q8×C257—21no—
55C2×Q8×C257D412yes—
56C22×C4×C257—21no—
57C22×C4×C257—21no—
58C22×C4×C257C2311yescharacteristic
59C2×C8×C257—21no—
60C2×C8×C257—21no—
61C2×C8×C257D412yes—
62C4⋊C4×C257—21no—
63C4⋊C4×C257—21no—
64Dic514—22no—
65Dic514—22no—
66Dic514—22no—
67Dic514—22no—
68Dic514—41no—
69Dic514—41no—
70Dic514—41no—
71Dic514—41no—
72C2×Dic257—22no—
73C2×Dic257—22no—
74C2×Dic257—22no—
75C2×Dic257—22no—
76C2×Dic257—41no—
77C2×Dic257—41no—
78C2×Dic257—41no—
79C2×Dic257—41no—
80C2×Dic257Q1614yescomplemented
81#14523339—22no—
82C23×C257C2×D411yescharacteristic
83Q8×C257—22no—
84Q8×C257—22no—
85Q8×C257—41no—
86Q8×C257—41no—
87C2×C4×C257—22no—
88C2×C4×C257—41no—
89C2×C4×C257—41no—
90C2×C4×C257—41no—
91C2×C4×C257—42no—
92C2×C4×C257C2×D411yescharacteristic
93C2×C4×C257C2×D411yescharacteristic
94C2×C4×C257C2×D411yescharacteristic
95C2×C4×C257C2×D411yescharacteristic
96C2×C4×C257C2×D411yescharacteristic
97C2×C4×C257#4111yescharacteristic
98C8×C257—22no—
99C22×C257#12011yescharacteristic
100C22×C257#12111yescharacteristic
101C22×C257#12711yescharacteristic
102C22×C257C2×Q1612yes—
103C22×C257#13712yes—
104C4×C257—42no—
105C4×C257—81no—
106C4×C257#12011yescharacteristic
107C4×C257#12011yescharacteristic
108C4×C257#12111yescharacteristic
109C4×C257#12111yescharacteristic
110Dic257—24no—
111Dic257—42no—
112Dic257—42no—
113C2×C257#39211yescharacteristic
114C2×C257#45112yes—
115C2×C257#46212yes—
116C2×C257#49411yescharacteristic
117C2×C257#49711yescharacteristic
118C257#201611yescharacteristic, complemented, Sylow
119#2016—2571nomaximal, Sylow
120C22×Q16—2571no—
121C22×Q16—2571no—
122C2×C4⋊Q8—2571no—
123C2×#103—2571no—
124C2×#103D25711yescharacteristic
125#391—2571no—
126#336—2571no—
127C22×Q8—2571no—
128C22×Q8—2571no—
129C22×Q8C2×D25711yescharacteristic
130C2×Q16—5142no—
131C2×Q16—5142no—
132C2×Q16—5142no—
133C2×Q16—5142no—
134C2×Q16—5142no—
135C2×Q16—5142no—
136C22×C8—2571no—
137C22×C8C2×D25711yescharacteristic
138C4⋊Q8—5142no—
139C4⋊Q8—5142no—
140C2×C4⋊C4—5142no—
141C2×C4⋊C4C2×D25711yescharacteristic
142C2×C42—2571no—
143#103—22no—
144#103—5142no—
145#95—5142no—
146C2×Q8—2572no—
147C2×Q8—2572no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—5141no—
154C2×Q8—5141no—
155C2×Q8—5141no—
156C2×Q8—5141no—
157C2×Q8—5141no—
158C2×Q8—5141no—
159C2×Q8—5141no—
160C2×Q8—5141no—
161C2×Q8—5141no—
162C2×Q8—5141no—
163C2×Q8D102812yes—
164C22×C4—2572no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—5141no—
168C22×C4—5141no—
169C22×C4C22×D25711yescharacteristic
170Q16—5148no—
171Q16—5148no—
172C2×C8—2572no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—5141no—
176C2×C8—5141no—
177C2×C8C22⋊D25712yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—5148no—
181C42—5141no—
182C42—5141no—
183C23D4×D25711yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—5142no—
187Q8—5142no—
188Q8—5142no—
189Q8—5142no—
190Q8—41no—
191Q8—41no—
192Q8—10281no—
193Q8—10281no—
194Q8—10281no—
195Q8—10281no—
196C2×C4—2574no—
197C2×C4—22no—
198C2×C4—5142no—
199C2×C4—5142no—
200C2×C4—5142no—
201C2×C4—5142no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—10281no—
207C2×C4—10281no—
208C2×C4—10281no—
209C2×C4—10281no—
210C2×C4D4×D25711yescharacteristic
211C2×C4D4×D25711yescharacteristic
212C2×C4D4×D25711yescharacteristic
213C2×C4C2×C22⋊D25711yescharacteristic
214C2×C4C2×D102811yescharacteristic
215C2×C4#1686161811yescharacteristic
216C8—22no—
217C8—5142no—
218C22#2255156211yescharacteristic
219C22#2255156511yescharacteristic
220C22#2255158811yescharacteristic
221C22Q16×D25712yes—
222C22#2255162112yes—
223C4—5144no—
224C4—42no—
225C4—10282no—
226C4—10282no—
227C4—81no—
228C4#2255156011yescharacteristic
229C4#2255156211yescharacteristic
230C4#2255156511yescharacteristic
231C4#2255156611yescharacteristic
232C2#3065664611yescharacteristic
233C2#3065689412yes—
234C2#3065694612yes—
235C2#3065711511yescharacteristic
236C2#3065712811yescharacteristic
subgroupG/H
centerC23D4×D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
FittingC2×#103×C257C2
socle, prime radicalC23×C257C2×D4
hypercenterC2×#103D257
hypocenterC257#2016
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
80. C2×Dic257Q16semidirect
118. C257#2016semidirect

Maximal subgroups

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

subgroupindex
C22×Dic10282
C22×#168616082
C2×#225515902
C2×#225515152
#306566422
#306564912
C2×#103×C2572
#2016257