Finite Group Databasebeta

← #44558827

Subgroups — #44558827

Order 32128.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22548762C211yesmaximal, characteristic
2#30632164C212yesmaximal, complemented
3#30631661C211yesmaximal, characteristic, complemented
4#30631660C212yesmaximal, complemented
5#30631659C218yesmaximal, complemented
6#410×C251C211yesmaximal, characteristic, complemented
7#22548762—24no—
8#22548762—24no—
9C2×#16860810C412yescomplemented
10C2×#16860810C2211yescharacteristic
11C2×#16860809C412yescomplemented
12C2×#16860809C2214yes—
13C22×Dic502C412yescomplemented
14C22×Dic502C2211yescharacteristic
15C2×Q8×D251C412yescomplemented
16C4×Dic502—24no—
17#22548680—24no—
18#22548674C2212yes—
19#22548671C2218yes—
20#22548630C2214yes—
21#22548629C2218yes—
22C2×#41×C251C2211yescharacteristic
23#117×C251C2212yes—
24#99×C251C2214yes—
25#16860810—24no—
26#16860810—24no—
27#16860810—22no—
28#16860810—22no—
29#16860809—28no—
30#16860809—216no—
31C2×Dic502—24no—
32C2×Dic502—24no—
33C2×Dic502C2×C414yes—
34C2×Dic502C2×C411yescharacteristic
35C2×Dic502C2×C412yes—
36C2×Dic502C2×C412yes—
37C2×Dic502D414yes—
38C2×Dic502D414yes—
39Q8×D251—28no—
40C2×D1004C2×C411yescharacteristic
41C2×C22⋊D251C2×C414yes—
42C2×C22⋊D251C2312yes—
43C22×Dic251C2×C414yes—
44C22×Dic251C2312yes—
45C2×C4×D251C2×C412yes—
46C2×C4×D251C2×C414yes—
47C4⋊Dic251—22no—
48Dic251⋊C4—28no—
49C22⋊Dic251C2314yes—
50C4×Dic251—28no—
51C4×Dic251—22no—
52#41×C251—22no—
53#41×C251—22no—
54C2×Q8×C251C2×C411yescharacteristic
55C2×D4×C251C2×C411yescharacteristic
56C2×D4×C251C2312yes—
57C22×C4×C251C2×C412yes—
58C22×C4×C251C2311yescharacteristic
59C4⋊C4×C251—22no—
60C22⋊C4×C251C2314yes—
61C42×C251—22no—
62C22×D251C22×C412yes—
63Dic502—21no—
64Dic502—21no—
65Dic502—28no—
66Dic502C22⋊C418yes—
67Dic502C22⋊C414yes—
68Dic502C22⋊C414yes—
69D1004—21no—
70D1004—21no—
71C22⋊D251—24no—
72C22⋊D251—28no—
73C2×Dic251—28no—
74C2×Dic251—24no—
75C2×Dic251C22×C412yes—
76C2×Dic251C22×C414yes—
77C2×Dic251C2×D418yes—
78C4×D251—24no—
79C4×D251—28no—
80C23×C251C22×C412yes—
81C23×C251C2411yescharacteristic
82Q8×C251—22no—
83D4×C251—24no—
84D4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251—24no—
88C2×C4×C251C22×C412yes—
89C2×C4×C251C22×C411yescharacteristic
90C2×C4×C251C22×C411yescharacteristic
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2×D412yes—
93C22×C251—22no—
94C22×C251—41no—
95C22×C251—42no—
96C22×C251C23×C411yescharacteristic
97C22×C251C22×D412yes—
98C2×D251—22no—
99C2×D251—42no—
100C4×C251—22no—
101C4×C251—44no—
102C4×C251C2×C22⋊C411yescharacteristic
103C4×C251C2×C22⋊C411yescharacteristic
104C4×C251C2×C22⋊C412yes—
105Dic251—22no—
106Dic251—44no—
107Dic251C2×C22⋊C418yes—
108C2×C251—22no—
109C2×C251—21no—
110C2×C251—42no—
111C2×C251C22×C22⋊C411yescharacteristic
112D251—42no—
113C251#288411yescharacteristic, complemented, Sylow
114#2884—2511nomaximal, Sylow
115C2×2−1+4—2511no—
116#519—2512no—
117#410—2511no—
118#410—2512no—
119#410—2518no—
120#410D25111yescharacteristic, complemented
1212−1+4—5024no—
1222−1+4—5024no—
123C2×#41—2512no—
124C2×#41—2514no—
125C2×#41—2512no—
126C2×#41—2511no—
127C2×#41C2×D25111yescharacteristic
128C22×Q8—2512no—
129C22×Q8—2512no—
130C22×Q8—2511no—
131C4×Q8—5024no—
132C4×Q8—5024no—
133#117—2512no—
134#117—2518no—
135#117C2×D25112yes—
136#99—2514no—
137#99—2518no—
138#99C2×D25114yes—
139#41—22no—
140#41—22no—
141#41—5024no—
142#41—5024no—
143#41—5028no—
144#41—50216no—
145#41—5022no—
146#41—5022no—
147C2×Q8—5024no—
148C2×Q8—5024no—
149C2×Q8—5028no—
150C2×Q8—2511no—
151C2×Q8—2514no—
152C2×Q8—2512no—
153C2×Q8—2512no—
154C2×Q8—2514no—
155C2×Q8—2514no—
156C2×Q8C4×D25111yescharacteristic
157C2×D4—2514no—
158C2×D4—2512no—
159C2×D4—2511no—
160C2×D4C4×D25111yescharacteristic
161C2×D4C22×D25112yes—
162C22×C4—2514no—
163C22×C4—2514no—
164C22×C4—2512no—
165C22×C4—2512no—
166C22×C4C4×D25112yes—
167C22×C4C22×D25111yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—5028no—
170C4⋊C4—5022no—
171C22⋊C4—2514no—
172C22⋊C4C22×D25114yes—
173C42—22no—
174C42—5028no—
175C42—5022no—
176C23—2512no—
177C23C2×C4×D25112yes—
178C23C23×D25111yescharacteristic
179Q8—22no—
180Q8—5028no—
181Q8—5021no—
182Q8—5021no—
183Q8—2514no—
184Q8—2514no—
185Q8—2518no—
186D4—24no—
187D4—22no—
188D4—5024no—
189D4—5028no—
190D4—5021no—
191D4—5021no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—5028no—
196C2×C4—5024no—
197C2×C4—5028no—
198C2×C4—5024no—
199C2×C4—2512no—
200C2×C4—2514no—
201C2×C4—2518no—
202C2×C4C2×C4×D25112yes—
203C2×C4C2×C4×D25111yescharacteristic
204C2×C4C2×C4×D25111yescharacteristic
205C2×C4D4×D25112yes—
206C2×C4D4×D25112yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—10042no—
211C22—5022no—
212C22C22×C4×D25111yescharacteristic
213C22C2×D4×D25112yes—
214C4—22no—
215C4—44no—
216C4—10044no—
217C4—5022no—
218C4—2518no—
219C4C22⋊C4×D25111yescharacteristic
220C4C22⋊C4×D25111yescharacteristic
221C4C22⋊C4×D25112yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—10042no—
226C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC23C23×D251
Fitting#410×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548762C2
cosocleC23×C251C24
hypercenter#410D251
hypocenterC251#2884
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #30632164C2semidirect
3. #30631661C2semidirect
4. #30631660C2semidirect
5. #30631659C2semidirect
6. #410×C251C2semidirect
9. C2×#16860810C4semidirect
11. C2×#16860809C4semidirect
13. C22×Dic502C4semidirect
15. C2×Q8×D251C4semidirect
113. C251#2884semidirect
120. #410D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487622
#306321642
#306316612
#306316602
#306316592
#410×C2512
#2884251