Finite Group Databasebeta

← #44548909

Subgroups — #44548909

Order 32128.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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
1#30632198C211yesmaximal, characteristic
2#30631547C211yesmaximal, characteristic, complemented
3#30631425C212yesmaximal, complemented
4#30631422C212yesmaximal, complemented
5#407×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C411yescharacteristic, complemented
7#22548698—21no—
8#22548694C411yescharacteristic, complemented
9#22548694C2211yescharacteristic
10#22548689C411yescharacteristic, complemented
11#22548689C2211yescharacteristic
12#22548684—21no—
13C4×C22⋊D251—21no—
14C4×C22⋊D251—21no—
15#22548674C411yescharacteristic, complemented
16C2×C4⋊Dic251—21no—
17C2×C22⋊Dic251—21no—
18C251⋊#95C2212yes—
19C23×C4×C251C2211yescharacteristic
20C2×M16×C251—21no—
21C22⋊C8×C251—21no—
22C22⋊C8×C251C2212yes—
23#16860810—42no—
24C2×Dic502C2×C411yescharacteristic
25C2×D1004C2×C411yescharacteristic
26C2×C22⋊D251—21no—
27C2×C22⋊D251—21no—
28C2×C22⋊D251—21no—
29C22×Dic251—22no—
30C2×C4×D251—21no—
31C4⋊Dic251—42no—
32C4⋊Dic251C2×C411yescharacteristic
33C4⋊Dic251C2×C411yescharacteristic
34Dic251⋊C4—21no—
35Dic251⋊C4—21no—
36C22⋊Dic251—21no—
37C22⋊Dic251—21no—
38C22⋊Dic251—21no—
39C22⋊Dic251—42no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42C4×Dic251—21no—
43C24×C251D411yescharacteristic
44C22×C4×C251—42no—
45C22×C4×C251C2×C411yescharacteristic
46C22×C4×C251C2×C411yescharacteristic
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251D411yescharacteristic
50C22×C4×C251C2311yescharacteristic
51M16×C251—42no—
52C2×C8×C251—22no—
53C22×D251—21no—
54Dic502—41no—
55D1004—41no—
56C22⋊D251—42no—
57C22⋊D251—42no—
58C22⋊D251—41no—
59C22⋊D251—41no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—21no—
63C2×Dic251—42no—
64C2×Dic251—42no—
65C2×Dic251—42no—
66C4×D251—42no—
67C23×C251—21no—
68C23×C251—21no—
69C23×C251—41no—
70C23×C251—41no—
71C23×C251C22⋊C411yescharacteristic
72C23×C251C22⋊C411yescharacteristic
73C23×C251C2×D411yescharacteristic
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251C22⋊C411yescharacteristic
81C2×C4×C251C22⋊C411yescharacteristic
82C2×C4×C251C22×C411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C8×C251—42no—
85C22×C251—21no—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251—21no—
89C22×C251—21no—
90C22×C251—21no—
91C22×C251—21no—
92C22×C251—21no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251C2×C22⋊C411yescharacteristic
96C22×C251AGL(1,ℤ/8)11yescharacteristic
97C22×C251#13711yescharacteristic
98C2×D251—41no—
99C2×D251—41no—
100C2×D251—41no—
101C4×C251—21no—
102C4×C251—21no—
103C4×C251—21no—
104C4×C251#9912yes—
105Dic251—41no—
106Dic251—41no—
107Dic251—41no—
108Dic251—82no—
109C2×C251—21no—
110C2×C251—21no—
111C2×C251—41no—
112C2×C251—41no—
113C2×C251C2×#9911yescharacteristic
114C2×C251#41711yescharacteristic
115C2×C251#42111yescharacteristic
116D251—81no—
117C251#151311yescharacteristic, complemented, Sylow
118#1513—2511nomaximal, Sylow
119#525—2511no—
120#407D25111yescharacteristic, complemented
121#380—2511no—
122#328—2512no—
123#327—2512no—
124C2×#41—2511no—
125C23×C4C2×D25111yescharacteristic
126C2×M16—21no—
127#123—5021no—
128C22⋊Q8—2511no—
129C22⋊Q8—2511no—
130#121—2511no—
131#121—2511no—
132#120—5021no—
133C4×D4—5021no—
134C4×D4—5021no—
135#117—2511no—
136C2×C4⋊C4—5021no—
137C2×C22⋊C4—5021no—
138C22⋊C8—21no—
139C22⋊C8C2×D25112yes—
140#95—2512no—
141C24D100411yescharacteristic
142#41—10042no—
143C2×Q8—2511no—
144C2×D4—5021no—
145C2×D4—5021no—
146C2×D4—5021no—
147C2×D4—2511no—
148C22×C4—42no—
149C22×C4—5022no—
150C22×C4—5021no—
151C22×C4C4×D25111yescharacteristic
152C22×C4C4×D25111yescharacteristic
153C22×C4C22⋊D25111yescharacteristic
154C22×C4C22⋊D25111yescharacteristic
155C22×C4D100411yescharacteristic
156C22×C4C22×D25111yescharacteristic
157M16—42no—
158C2×C8—22no—
159C4⋊C4—10042no—
160C4⋊C4—5021no—
161C4⋊C4—5021no—
162C4⋊C4—2511no—
163C4⋊C4—2511no—
164C22⋊C4—10042no—
165C22⋊C4—5021no—
166C22⋊C4—5021no—
167C22⋊C4—5021no—
168C22⋊C4—5021no—
169C22⋊C4—5021no—
170C42—5021no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—5021no—
176C23(C2×D251)⋊C411yescharacteristic
177C23(C2×D251)⋊C411yescharacteristic
178C23C2×D100411yescharacteristic
179Q8—10041no—
180D4—10041no—
181D4—10042no—
182D4—10042no—
183D4—10041no—
184D4—10041no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—10042no—
192C2×C4—10042no—
193C2×C4—10042no—
194C2×C4—10042no—
195C2×C4—5021no—
196C2×C4—5021no—
197C2×C4—5021no—
198C2×C4(C2×D251)⋊C411yescharacteristic
199C2×C4(C2×D251)⋊C411yescharacteristic
200C2×C4C2×C4×D25111yescharacteristic
201C2×C4C2×C22⋊D25111yescharacteristic
202C8—42no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—10041no—
214C22—10041no—
215C22—10041no—
216C22C2×(C2×D251)⋊C411yescharacteristic
217C22#2254874311yescharacteristic
218C22#2254874811yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—20082no—
223C4—10041no—
224C4—10041no—
225C4—10041no—
226C4#2254862912yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—20081no—
232C2C2×#2254862911yescharacteristic
233C2#3063168911yescharacteristic
234C2#3063170311yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1513
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #30631547C2semidirect
3. #30631425C2semidirect
4. #30631422C2semidirect
5. #407×C251C2semidirect
6. C2×#16860810C4semidirect
8. #22548694C4semidirect
10. #22548689C4semidirect
15. #22548674C4semidirect
117. C251#1513semidirect
120. #407D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
#306315472
#306314252
#306314222
#407×C2512
#1513251