Finite Group Databasebeta

← #45665404

Subgroups — #45665404

Order 32896.

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

normal subgroups only

This group has 406 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×#16861607C211yesmaximal, characteristic
2C22×#16861605C211yesmaximal, characteristic
3C2×#22551590C211yesmaximal, characteristic, complemented
4C2×#22551515C211yesmaximal, characteristic, complemented
5#30656638C211yesmaximal, characteristic
6#30656491C211yesmaximal, characteristic, complemented
7C2×#103×C257C211yesmaximal, characteristic, complemented
8C22×Dic514C2211yescharacteristic
9C22×D1028C2211yescharacteristic
10C2×#16861607—22no—
11C2×#16861607—22no—
12C2×#16861607—22no—
13C2×#16861605—22no—
14C2×#16861605—22no—
15C2×#16861605—22no—
16C22×#14523339C2211yescharacteristic
17#22551590—22no—
18#22551590—22no—
19C2×#16861597—22no—
20C2×#16861595—22no—
21C2×C4×Dic257C2211yescharacteristic
22#22551515—22no—
23C22×Q8×C257C2211yescharacteristic
24C22×C8×C257C2211yescharacteristic
25C2×C4⋊C4×C257C2211yescharacteristic
26#103×C257—22no—
27C23×D257—21no—
28C2×Dic514—21no—
29C2×Dic514—21no—
30C2×Dic514—21no—
31C2×Dic514—21no—
32C2×Dic514—21no—
33C2×Dic514D412yes—
34C2×D1028—21no—
35C2×D1028—21no—
36C2×D1028—21no—
37C2×D1028—21no—
38C2×D1028—21no—
39C2×D1028D412yes—
40C22×Dic257—21no—
41C22×Dic257D412yes—
42#16861607—28no—
43#16861605—28no—
44C2×#14523339—21no—
45C2×#14523339—21no—
46C2×#14523339D412yes—
47#16861597—28no—
48#16861595—44no—
49C4×Dic257—21no—
50C4×Dic257—21no—
51C2×Q8×C257—21no—
52C2×Q8×C257—21no—
53C2×Q8×C257—21no—
54C2×Q8×C257—21no—
55C2×Q8×C257—21no—
56C2×Q8×C257D412yes—
57C22×C4×C257—21no—
58C22×C4×C257—21no—
59C22×C4×C257C2311yescharacteristic
60C2×C8×C257—21no—
61C2×C8×C257—21no—
62C2×C8×C257D412yes—
63C4⋊C4×C257—21no—
64C4⋊C4×C257—21no—
65C22×D257—22no—
66C22×D257—41no—
67C22×D257—41no—
68C22×D257—42no—
69C22×D257—42no—
70Dic514—22no—
71Dic514—22no—
72Dic514—41no—
73Dic514—41no—
74D1028—22no—
75D1028—22no—
76D1028—41no—
77D1028—41no—
78C2×Dic257—22no—
79C2×Dic257—22no—
80C2×Dic257—22no—
81C2×Dic257—41no—
82C2×Dic257—41no—
83C2×Dic257SD1614yescomplemented
84#14523339—22no—
85C23×C257C2×D411yescharacteristic
86Q8×C257—22no—
87Q8×C257—22no—
88Q8×C257—41no—
89Q8×C257—41no—
90C2×C4×C257—22no—
91C2×C4×C257—41no—
92C2×C4×C257—41no—
93C2×C4×C257—41no—
94C2×C4×C257—42no—
95C2×C4×C257C2×D411yescharacteristic
96C2×C4×C257C2×D411yescharacteristic
97C2×C4×C257C2×D411yescharacteristic
98C2×C4×C257C2×D411yescharacteristic
99C2×C4×C257C2×D411yescharacteristic
100C2×C4×C257#4111yescharacteristic
101C8×C257—22no—
102C22×C257#12011yescharacteristic
103C22×C257#12111yescharacteristic
104C22×C257#12711yescharacteristic
105C22×C257C2×SD1612yes—
106C22×C257#13712yes—
107C2×D257—42no—
108C2×D257—44no—
109C2×D257—82no—
110C2×D257—81no—
111C2×D257—81no—
112C4×C257—42no—
113C4×C257—81no—
114C4×C257#12011yescharacteristic
115C4×C257#12011yescharacteristic
116C4×C257#12111yescharacteristic
117C4×C257#12111yescharacteristic
118Dic257—24no—
119Dic257—42no—
120C2×C257#39211yescharacteristic
121C2×C257#44812yes—
122C2×C257#46112yes—
123C2×C257#49211yescharacteristic
124C2×C257#49711yescharacteristic
125D257—82no—
126C257#201311yescharacteristic, complemented, Sylow
127#2013—2571nomaximal, Sylow
128C22×SD16—2571no—
129C22×SD16—2571no—
130C2×C4⋊Q8—2571no—
131C2×#103—2571no—
132C2×#103D25711yescharacteristic, complemented
133#390—2571no—
134#336—2571no—
135C22×Q8—2571no—
136C22×Q8C2×D25711yescharacteristic
137C22×D4—2571no—
138C2×SD16—5142no—
139C2×SD16—5142no—
140C2×SD16—5142no—
141C2×SD16—5142no—
142C2×SD16—5142no—
143C2×SD16—5142no—
144C22×C8—2571no—
145C22×C8C2×D25711yescharacteristic
146C4⋊Q8—5142no—
147C4⋊Q8—5142no—
148C2×C4⋊C4—5142no—
149C2×C4⋊C4C2×D25711yescharacteristic
150C2×C22⋊C4—5142no—
151C2×C42—2571no—
152#103—22no—
153#103—5142no—
154C24—5141no—
155C2×Q8—2572no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—5141no—
162C2×Q8—5141no—
163C2×Q8—5141no—
164C2×Q8—5141no—
165C2×Q8—5141no—
166C2×Q8D102812yes—
167C2×D4—2572no—
168C2×D4—5141no—
169C2×D4—5141no—
170C2×D4—5141no—
171C2×D4—5141no—
172C2×D4—5141no—
173C22×C4—2572no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—5141no—
177C22×C4C22×D25711yescharacteristic
178SD16—5148no—
179SD16—5148no—
180C2×C8—2572no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—5141no—
184C2×C8—5141no—
185C2×C8C22⋊D25712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—5148no—
189C22⋊C4—10284no—
190C42—5141no—
191C42—5141no—
192C23—5142no—
193C23—10281no—
194C23—10281no—
195C23—10282no—
196C23—10282no—
197C23D4×D25711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—5142no—
201Q8—5142no—
202Q8—41no—
203Q8—41no—
204Q8—10281no—
205Q8—10281no—
206D4—5142no—
207D4—5142no—
208D4—10281no—
209D4—10281no—
210C2×C4—2574no—
211C2×C4—22no—
212C2×C4—5142no—
213C2×C4—5142no—
214C2×C4—5142no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—10281no—
220C2×C4—10281no—
221C2×C4D4×D25711yescharacteristic
222C2×C4D4×D25711yescharacteristic
223C2×C4D4×D25711yescharacteristic
224C2×C4C2×C22⋊D25711yescharacteristic
225C2×C4C2×D102811yescharacteristic
226C2×C4#1686161811yescharacteristic
227C8—22no—
228C8—5142no—
229C22—10282no—
230C22—10284no—
231C22—20561no—
232C22—20561no—
233C22—20562no—
234C22#2255156211yescharacteristic
235C22#2255156511yescharacteristic
236C22#2255158811yescharacteristic
237C22SD16×D25712yes—
238C22#2255162412yes—
239C4—5144no—
240C4—42no—
241C4—10282no—
242C4—81no—
243C4#2255156011yescharacteristic
244C4#2255156211yescharacteristic
245C4#2255156511yescharacteristic
246C4#2255156611yescharacteristic
247C2—20562no—
248C2#3065664611yescharacteristic
249C2#3065687712yes—
250C2#3065694112yes—
251C2#3065710811yescharacteristic
252C2#3065713011yescharacteristic
subgroupG/H
centerC23D4×D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
FittingC2×#103×C257C2
socleC23×C257C2×D4
prime radicalC22×D1028C22
hypercenterC2×#103D257
hypocenterC257#2013
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
3. C2×#22551590C2semidirect
4. C2×#22551515C2semidirect
6. #30656491C2semidirect
7. C2×#103×C257C2semidirect
83. C2×Dic257SD16semidirect
126. C257#2013semidirect
132. C2×#103D257semidirect

Maximal subgroups

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

subgroupindex
C22×#168616072
C22×#168616052
C2×#225515902
C2×#225515152
#306566382
#306564912
C2×#103×C2572
#2013257