Finite Group Databasebeta

← #44560755

Subgroups — #44560755

Order 32128.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632235C211yesmaximal, characteristic
2C4⋊Q8×D251C211yesmaximal, characteristic, complemented
3#30632045C214yesmaximal, complemented
4#30632039C211yesmaximal, characteristic, complemented
5#30632038C211yesmaximal, characteristic, complemented
6#30632033C211yesmaximal, characteristic, complemented
7#30631685C214yesmaximal, complemented
8C4×#16860793C211yesmaximal, characteristic, complemented
9#486×C251C211yesmaximal, characteristic, complemented
10C2×#16860809C2212yes—
11C2×Q8×D251—21no—
12C2×#16860793C2212yes—
13#22548715C2211yescharacteristic
14#22548715C2211yescharacteristic
15#22548714—21no—
16#22548714C2212yes—
17#22548713—21no—
18#22548702C2212yes—
19#22548686—21no—
20#22548686—21no—
21D4×Dic251—21no—
22D4×Dic251—21no—
23C4⋊C4×D251—21no—
24C4⋊C4×D251C2212yes—
25C42×D251C2211yescharacteristic
26#22548640C2214yes—
27#22548638C2214yes—
28#22548636C2214yes—
29#22548625C2212yes—
30C4×C251⋊C8C2211yescharacteristic
31C4⋊Q8×C251C2211yescharacteristic
32#127×C251C2211yescharacteristic
33#102×C251C2214yes—
34C4×C8×C251C2211yescharacteristic
35#16860809—28no—
36C2×Dic502—22no—
37C2×Dic502—21no—
38C2×Dic502C2312yes—
39Q8×D251—44no—
40C2×C22⋊D251—22no—
41C22×Dic251—22no—
42C2×C4×D251—22no—
43C2×C4×D251D411yescharacteristic
44C2×C4×D251D411yescharacteristic
45C2×C4×D251C2311yescharacteristic
46#16860793—24no—
47C2×C251⋊C8C2312yes—
48C4⋊Dic251—21no—
49C4⋊Dic251—21no—
50C4⋊Dic251—21no—
51C4⋊Dic251C2312yes—
52Dic251⋊C4—22no—
53Dic251⋊C4—22no—
54C22⋊Dic251—22no—
55C22⋊Dic251—22no—
56C4×Dic251D411yescharacteristic
57C4×Dic251D411yescharacteristic
58C4×Dic251C2311yescharacteristic
59C2×Q8×C251—21no—
60C2×D4×C251—21no—
61C2×D4×C251—21no—
62C2×D4×C251C2312yes—
63C2×C8×C251C2312yes—
64C4⋊C4×C251—21no—
65C4⋊C4×C251C2312yes—
66C42×C251C2311yescharacteristic
67C22×D251C2×D411yescharacteristic
68Dic502—24no—
69Dic502—44no—
70Dic502—42no—
71C22⋊D251—44no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—44no—
75C2×Dic251C2×D411yescharacteristic
76C2×Dic251C2×D411yescharacteristic
77C2×Dic251C2×D411yescharacteristic
78C4×D251—22no—
79C4×D251—22no—
80C4×D251—44no—
81C4×D251#4114yes—
82C251⋊C8—22no—
83C23×C251—22no—
84Q8×C251—42no—
85D4×C251—24no—
86D4×C251—42no—
87D4×C251—42no—
88C2×C4×C251—22no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2411yescharacteristic
92C8×C251—22no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251—42no—
96C22×C251C22×D411yescharacteristic
97C2×D251—21no—
98C2×D251—21no—
99C2×D251#12412yes—
100C4×C251—42no—
101C4×C251AGL(1,ℤ/8)12yes—
102C4×C251#13712yes—
103C4×C251C2×#4112yes—
104Dic251—21no—
105Dic251—21no—
106Dic251—42no—
107Dic251—42no—
108Dic251#12412yes—
109C2×C251—82no—
110C2×C251C2×#12411yescharacteristic
111C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C251C2×#13711yescharacteristic
113D251—22no—
114C251#314111yescharacteristic, complemented, Sylow
115#3141—2511nomaximal, Sylow
116#532—2511no—
117C2×C4⋊Q8—2511no—
118#489—2514no—
119#487—2511no—
120#487—2511no—
121#486—2511no—
122#486D25111yescharacteristic, complemented
123#417—2514no—
124C4×M16—2511no—
125C2×#41—2512no—
126C22×Q8—5021no—
127C2×M16—2512no—
128C4⋊Q8—5021no—
129C4⋊Q8—5021no—
130C4⋊Q8—2512no—
131C4⋊Q8—2511no—
132C4⋊Q8—2511no—
133C4⋊Q8C2×D25111yescharacteristic
134#127C2×D25111yescharacteristic
135#124—2512no—
136#121—5021no—
137#121—5021no—
138C4×D4—5021no—
139C4×D4—5021no—
140C2×C4⋊C4—5021no—
141C2×C4⋊C4—2512no—
142C2×C42—2511no—
143#103—2514no—
144#103—2514no—
145#102—2514no—
146#102C2×D25114yes—
147#97—2512no—
148C4×C8—2511no—
149C4×C8C2×D25111yescharacteristic
150#41—5028no—
151C2×Q8—21no—
152C2×Q8—10044no—
153C2×Q8—5022no—
154C2×Q8—5021no—
155C2×Q8—2512no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—5022no—
159C2×D4C22×D25112yes—
160C22×C4—5022no—
161C22×C4—5022no—
162C22×C4—2511no—
163C22×C4—2511no—
164C22×C4—2511no—
165M16—5024no—
166C2×C8—2512no—
167C2×C8C22×D25112yes—
168C4⋊C4—21no—
169C4⋊C4—5022no—
170C4⋊C4—5022no—
171C4⋊C4—5021no—
172C4⋊C4—5021no—
173C4⋊C4—5021no—
174C4⋊C4—2512no—
175C4⋊C4C22×D25112yes—
176C22⋊C4—5022no—
177C22⋊C4—5022no—
178C42—2511no—
179C42—2511no—
180C42—2511no—
181C42C22×D25111yescharacteristic
182C23—22no—
183C23—2511no—
184Q8—42no—
185Q8—10044no—
186Q8—10042no—
187Q8—5024no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—10044no—
192C2×C4—22no—
193C2×C4—10044no—
194C2×C4—10044no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—2511no—
200C2×C4—2511no—
201C2×C4—2511no—
202C2×C4—2514no—
203C2×C4D4×D25111yescharacteristic
204C2×C4D4×D25111yescharacteristic
205C2×C4C23×D25111yescharacteristic
206C8—22no—
207C8—5022no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—5021no—
212C22—5021no—
213C22—2512no—
214C22C2×D4×D25111yescharacteristic
215C4—42no—
216C4—10042no—
217C4—10042no—
218C4—5021no—
219C4—5021no—
220C4—2512no—
221C4#2254873912yes—
222C4#2254874412yes—
223C4#41×D25112yes—
224C2—82no—
225C2—5022no—
226C2#124×D25111yescharacteristic
227C2C2×#2254873911yescharacteristic
228C2C2×#2254874411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#486×C251C2
socleC22×C251C22×D4
prime radical#30632235C2
hypercenter#486D251
hypocenterC251#3141
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D251C2semidirect
3. #30632045C2semidirect
4. #30632039C2semidirect
5. #30632038C2semidirect
6. #30632033C2semidirect
7. #30631685C2semidirect
8. C4×#16860793C2semidirect
9. #486×C251C2semidirect
114. C251#3141semidirect
122. #486D251semidirect

Maximal subgroups

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

subgroupindex
#306322352
C4⋊Q8×D2512
#306320452
#306320392
#306320382
#306320332
#306316852
C4×#168607932
#486×C2512
#3141251