Finite Group Databasebeta

← #44559898

Subgroups — #44559898

Order 32128.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#22548745C211yesmaximal, characteristic
2C22×Dic1004C211yesmaximal, characteristic, complemented
3#30632258C211yesmaximal, characteristic
4#30632164C211yesmaximal, characteristic, complemented
5#30631908C214yesmaximal, complemented
6#30631896C214yesmaximal, complemented
7#30631713C211yesmaximal, characteristic, complemented
8#30631695C211yesmaximal, characteristic, complemented
9#416×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12#22548745—24no—
13C2×Dic1004—24no—
14C2×Dic1004C2214yes—
15C2×#16860800C2212yes—
16C2×#16860798C2212yes—
17C2×#16860794C2211yescharacteristic
18#22548713C2212yes—
19#22548702C2212yes—
20#22548696—22no—
21#22548694—22no—
22C4×Dic502—22no—
23#22548680C2214yes—
24#22548674C2211yescharacteristic
25#22548671—22no—
26Dic251⋊C8C2214yes—
27#22548638C2214yes—
28C2×#41×C251C2211yescharacteristic
29C22×C8×C251C2211yescharacteristic
30#117×C251C2211yescharacteristic
31#103×C251C2212yes—
32#102×C251C2212yes—
33C2×Dic502—21no—
34C2×Dic502—21no—
35C2×Dic502—22no—
36C2×Dic502—22no—
37C2×Dic502—22no—
38C2×Dic502D414yes—
39C2×Dic502C2312yes—
40C2×Dic502C2312yes—
41C22×Dic251—21no—
42C22×Dic251—21no—
43Dic1004—28no—
44#16860800—24no—
45#16860798—24no—
46#16860794—22no—
47C2×C251⋊C8C2312yes—
48C4⋊Dic251D412yes—
49Dic251⋊C4—24no—
50Dic251⋊C4—24no—
51C22⋊Dic251—24no—
52C22⋊Dic251D412yes—
53C4×Dic251—24no—
54C4×Dic251C2312yes—
55#41×C251—22no—
56#41×C251—22no—
57C2×Q8×C251C2311yescharacteristic
58C2×D4×C251—21no—
59C2×D4×C251C2311yescharacteristic
60C22×C4×C251—21no—
61C22×C4×C251C2311yescharacteristic
62C2×C8×C251—22no—
63C2×C8×C251C2312yes—
64C4⋊C4×C251C2312yes—
65C22⋊C4×C251—21no—
66C42×C251—21no—
67Dic502—24no—
68Dic502—22no—
69Dic502—22no—
70Dic502—22no—
71Dic502—42no—
72Dic502C2×D414yes—
73C2×Dic251—24no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—42no—
77C2×Dic251C2×D414yes—
78C251⋊C8—22no—
79C23×C251—21no—
80C23×C251C2×D411yescharacteristic
81Q8×C251—21no—
82Q8×C251—21no—
83D4×C251—21no—
84D4×C251—21no—
85D4×C251—42no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—42no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D412yes—
91C2×C4×C251#4112yes—
92C2×C4×C251C2411yescharacteristic
93C8×C251—22no—
94C22×C251—22no—
95C22×C251—41no—
96C22×C251—42no—
97C22×C251#12112yes—
98C22×C251C22×D411yescharacteristic
99C4×C251—41no—
100C4×C251—42no—
101C4×C251#12112yes—
102C4×C251C22×D411yescharacteristic
103C4×C251C2×#4111yescharacteristic
104Dic251—24no—
105Dic251—24no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251—81no—
109C2×C251C2×#12111yescharacteristic
110C2×C251#57812yes—
111C251#304411yescharacteristic, complemented, Sylow
112#3044—2511nomaximal, Sylow
113C2×#137—2511no—
114C22×Q16—2511no—
115#536—2511no—
116#519—2511no—
117#464—2514no—
118#462—2514no—
119#424—2511no—
120#419—2511no—
121#416D25111yescharacteristic
122C2×#41C2×D25111yescharacteristic
123C22×Q8—2511no—
124C22×Q8—2511no—
125#137—5024no—
126C2×Q16—5024no—
127C2×Q16—2512no—
128C2×Q16—2514no—
129C2×SD16—2512no—
130C2×M16—2511no—
131C22×C8C2×D25111yescharacteristic
132C4⋊Q8—2512no—
133#124—2512no—
134#123—5022no—
135C22⋊Q8—5022no—
136C4×Q8—5022no—
137C4×Q8—2514no—
138#117—5022no—
139#117—2511no—
140#117C2×D25111yescharacteristic
141C4⋊C8—2514no—
142#103—2514no—
143#103C2×D25112yes—
144#102C2×D25112yes—
145#41—22no—
146#41—22no—
147C2×Q8—5022no—
148C2×Q8—5022no—
149C2×Q8—5021no—
150C2×Q8—5021no—
151C2×Q8—5022no—
152C2×Q8—2514no—
153C2×Q8—2512no—
154C2×Q8—2512no—
155C2×Q8C22×D25111yescharacteristic
156C2×D4—21no—
157C2×D4C22×D25111yescharacteristic
158C22×C4—21no—
159C22×C4—5021no—
160C22×C4—5021no—
161C22×C4C22×D25111yescharacteristic
162Q16—5028no—
163Q16—5024no—
164SD16—5024no—
165M16—5022no—
166C2×C8—22no—
167C2×C8—2512no—
168C2×C8C22×D25112yes—
169C4⋊C4—5024no—
170C4⋊C4—5024no—
171C4⋊C4—2512no—
172C4⋊C4C22×D25112yes—
173C22⋊C4—21no—
174C22⋊C4—5024no—
175C22⋊C4—2512no—
176C42—21no—
177C42—5024no—
178C42—2512no—
179C23—21no—
180C23D4×D25111yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—10042no—
184Q8—5022no—
185Q8—5022no—
186Q8—5022no—
187Q8—5024no—
188Q8—2514no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—10042no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5024no—
199C2×C4—2514no—
200C2×C4D4×D25112yes—
201C2×C4D4×D25111yescharacteristic
202C2×C4#1686081012yes—
203C2×C4C23×D25111yescharacteristic
204C8—22no—
205C8—5022no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2254868812yes—
210C22C2×D4×D25111yescharacteristic
211C4—42no—
212C4—41no—
213C4—10042no—
214C4—5024no—
215C4—5024no—
216C4#2254868812yes—
217C4C2×D4×D25111yescharacteristic
218C4C2×#1686081011yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2254868811yescharacteristic
222C2#3063250112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
prime radicalC2×#41×C251C22
hypercenter#416D251
hypocenterC251#3044
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic1004C2semidirect
4. #30632164C2semidirect
5. #30631908C2semidirect
6. #30631896C2semidirect
7. #30631713C2semidirect
8. #30631695C2semidirect
111. C251#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
C22×Dic10042
#306322582
#306321642
#306319082
#306318962
#306317132
#306316952
#416×C2512
#3044251