Finite Group Databasebeta

← #44559877

Subgroups — #44559877

Order 32128.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548741C211yesmaximal, characteristic
2C22×#16860798C211yesmaximal, characteristic, complemented
3#30632263C211yesmaximal, characteristic, complemented
4#30631902C214yesmaximal, complemented
5#30631892C214yesmaximal, complemented
6#30631714C211yesmaximal, characteristic, complemented
7#30631695C211yesmaximal, characteristic, complemented
8#30631681C211yesmaximal, characteristic, complemented
9#518×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12#22548741—24no—
13C2×#16860798—24no—
14C2×#16860798C2214yes—
15C2×#16860798C2212yes—
16C2×#16860795C2212yes—
17C2×#16860794C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic
19#22548715C2212yescomplemented
20#22548703C2212yescomplemented
21#22548699—22no—
22#22548695—22no—
23C4⋊(C251⋊C8)C2214yescomplemented
24#22548638C2212yescomplemented
25#22548638C2214yescomplemented
26#22548635C2212yescomplemented
27C22×D4×C251C2211yescharacteristic
28C4×D4×C251—22no—
29C4×D4×C251C2214yes—
30#117×C251C2211yescharacteristic
31#117×C251C2211yescharacteristic
32C2×C22⋊C4×C251—22no—
33#16860810—22no—
34#16860810—22no—
35C2×Dic502—21no—
36C2×Dic502—22no—
37C2×Dic502—22no—
38C2×Dic502C2312yes—
39C2×Dic502C2311yescharacteristic
40C2×D1004C2311yescharacteristic
41C2×C22⋊D251—21no—
42C22×Dic251—21no—
43C2×C4×D251—21no—
44#16860798—24no—
45#16860798—28no—
46#16860795—24no—
47#16860794—22no—
48C2×C251⋊C8—22no—
49C2×C251⋊C8C2312yes—
50C2×C251⋊C8C2312yes—
51C4⋊Dic251—24no—
52(C2×D251)⋊C4—24no—
53C24×C251—21no—
54C2×D4×C251—21no—
55C2×D4×C251—22no—
56C2×D4×C251D414yes—
57C2×D4×C251C2312yes—
58C22×C4×C251—24no—
59C22×C4×C251C2311yescharacteristic
60C4⋊C4×C251D412yes—
61C4⋊C4×C251C2312yes—
62C22⋊C4×C251—21no—
63C22⋊C4×C251—24no—
64C22⋊C4×C251D412yes—
65C42×C251—21no—
66C42×C251C2312yes—
67C22×D251—21no—
68Dic502—22no—
69Dic502—21no—
70Dic502—22no—
71Dic502—21no—
72Dic502—42no—
73D1004—21no—
74D1004—21no—
75C22⋊D251—42no—
76C2×Dic251—22no—
77C2×Dic251—21no—
78C2×Dic251—42no—
79C4×D251—42no—
80C251⋊C8—22no—
81C251⋊C8—22no—
82C23×C251—22no—
83C23×C251—24no—
84C23×C251—82no—
85C23×C251C2×D411yescharacteristic
86D4×C251—24no—
87D4×C251—22no—
88D4×C251C2×D414yes—
89C2×C4×C251—22no—
90C2×C4×C251—84no—
91C2×C4×C251C2×D414yes—
92C2×C4×C251C2×D411yescharacteristic
93C2×C4×C251C2×D412yes—
94C2×C4×C251#4112yes—
95C2×C4×C251C2411yescharacteristic
96C22×C251—22no—
97C22×C251—24no—
98C22×C251—44no—
99C22×C251—84no—
100C22×C251#12112yes—
101C22×C251C22×D411yescharacteristic
102C2×D251—41no—
103C2×D251—42no—
104C4×C251—24no—
105C4×C251—42no—
106C4×C251#12112yes—
107C4×C251C22×D411yescharacteristic
108C4×C251C2×#4111yescharacteristic
109Dic251—41no—
110Dic251—42no—
111C2×C251—22no—
112C2×C251—44no—
113C2×C251C2×#12111yescharacteristic
114C2×C251#57712yes—
115D251—81no—
116C251#304211yescharacteristic, complemented, Sylow
117#3042—2511nomaximal, Sylow
118C2×AGL(1,ℤ/8)—2511no—
119C22×SD16—2511no—
120#536—2511no—
121#518D25111yescharacteristic, complemented
122#463—2514no—
123#461—2514no—
124#424—2511no—
125#419—2511no—
126#416—2511no—
127C2×#41—2511no—
128C22×Q8—2511no—
129C22×D4C2×D25111yescharacteristic
130AGL(1,ℤ/8)—5024no—
131C2×SD16—5024no—
132C2×SD16—2514no—
133C2×SD16—2512no—
134C2×D8—2512no—
135C2×M16—2511no—
136C22×C8—2511no—
137C4⋊Q8—2512no—
138#124—2512no—
139#123—5022no—
140C22⋊Q8—5022no—
141C4×D4—22no—
142C4×D4C2×D25114yes—
143#117C2×D25111yescharacteristic
144#117C2×D25111yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—2514no—
147#103—2512no—
148#103—2514no—
149#102—2512no—
150C24—21no—
151#41—5022no—
152#41—5022no—
153C2×Q8—5022no—
154C2×Q8—5022no—
155C2×Q8—5021no—
156C2×Q8—2512no—
157C2×Q8—2511no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—5021no—
161C2×D4—2511no—
162C2×D4D100414yes—
163C2×D4C22×D25112yes—
164C22×C4—24no—
165C22×C4—5021no—
166C22×C4—5021no—
167C22×C4C22×D25111yescharacteristic
168SD16—5028no—
169SD16—5024no—
170D8—5024no—
171M16—5022no—
172C2×C8—5022no—
173C2×C8—2512no—
174C2×C8—2512no—
175C4⋊C4—5024no—
176C4⋊C4C22⋊D25112yes—
177C4⋊C4C22×D25112yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—5024no—
181C22⋊C4C22⋊D25112yes—
182C42—21no—
183C42C22×D25112yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—5021no—
188C23C2×C22⋊D25111yescharacteristic
189Q8—10042no—
190Q8—5022no—
191Q8—5021no—
192Q8—5022no—
193Q8—5021no—
194D4—22no—
195D4—24no—
196D4—10042no—
197D4—5021no—
198D4—5021no—
199D4C2×D100414yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—10042no—
203C2×C4—10042no—
204C2×C4—5022no—
205C2×C4—5021no—
206C2×C4C2×C22⋊D25114yes—
207C2×C4C2×C22⋊D25111yescharacteristic
208C2×C4C2×D100412yes—
209C2×C4#1686081012yes—
210C2×C4C23×D25111yescharacteristic
211C8—5022no—
212C8—5022no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—10042no—
218C22—10041no—
219C22#2254868912yes—
220C22C22×C22⋊D25111yescharacteristic
221C4—24no—
222C4—42no—
223C4—10041no—
224C4—10042no—
225C4#2254868912yes—
226C4C22×D100411yescharacteristic
227C4C2×#1686081011yescharacteristic
228C2—22no—
229C2—44no—
230C2—20081no—
231C2C2×#2254868911yescharacteristic
232C2#3063249812yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#518×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548741C2
hypercenter#518D251
hypocenterC251#3042
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#16860798C2semidirect
3. #30632263C2semidirect
4. #30631902C2semidirect
5. #30631892C2semidirect
6. #30631714C2semidirect
7. #30631695C2semidirect
8. #30631681C2semidirect
9. #518×C251C2semidirect
19. #22548715C22semidirect
20. #22548703C22semidirect
23. C4⋊(C251⋊C8)C22semidirect
24. #22548638C22semidirect
25. #22548638C22semidirect
26. #22548635C22semidirect
116. C251#3042semidirect
121. #518D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C22×#168607982
#306322632
#306319022
#306318922
#306317142
#306316952
#306316812
#518×C2512
#3042251