Finite Group Databasebeta

← #44548801

Subgroups — #44548801

Order 32128.

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

normal subgroups only

This group has 266 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
1C4×#16860809C211yesmaximal, characteristic
2#30631709C211yesmaximal, characteristic, complemented
3C4×#16860793C211yesmaximal, characteristic, complemented
4#30631419C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#16860809C411yescharacteristic
9C2×#16860793—21no—
10C2×#16860793—21no—
11C4×Dic502C2211yescharacteristic
12#22548680C411yescharacteristic
13D4×Dic251C411yescharacteristic
14C4×C22⋊D251—21no—
15(C22⋊D251)⋊C4—21no—
16#22548673C411yescharacteristic
17#22548671—21no—
18C2×C4×Dic251—21no—
19C42×D251C2211yescharacteristic
20C4⋊(C251⋊C8)C2211yescharacteristic
21Dic251⋊C8—21no—
22#22548625—21no—
23C4×C251⋊C8C2211yescharacteristic
24C4×D4×C251C2211yescharacteristic
25C4⋊C8×C251C2211yescharacteristic
26C4×C8×C251C2211yescharacteristic
27#16860809—22no—
28#16860809—22no—
29C2×Dic502C2×C411yescharacteristic
30C2×C22⋊D251—21no—
31C22×Dic251—21no—
32C22×Dic251—21no—
33C2×C4×D251C2×C411yescharacteristic
34C2×C4×D251D411yescharacteristic
35C2×C4×D251D411yescharacteristic
36#16860793—24no—
37#16860793—42no—
38C2×C251⋊C8—21no—
39C2×C251⋊C8—21no—
40C4⋊Dic251C2×C411yescharacteristic
41Dic251⋊C4—21no—
42Dic251⋊C4—21no—
43C22⋊Dic251—21no—
44(C2×D251)⋊C4—21no—
45C4×Dic251—21no—
46C4×Dic251—21no—
47C4×Dic251—21no—
48C4×Dic251C2×C411yescharacteristic
49C4×Dic251D411yescharacteristic
50C4×Dic251D411yescharacteristic
51C2×D4×C251C2×C411yescharacteristic
52C22×C4×C251—21no—
53C2×C8×C251—21no—
54C2×C8×C251—21no—
55C4⋊C4×C251C2×C411yescharacteristic
56C22⋊C4×C251—21no—
57C42×C251C2311yescharacteristic
58C22×D251C22⋊C411yescharacteristic
59Dic502—21no—
60Dic502M1612yescomplemented
61C22⋊D251—24no—
62C2×Dic251—24no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—42no—
66C2×Dic251C22⋊C411yescharacteristic
67C2×Dic251C22⋊C411yescharacteristic
68C2×Dic251C22⋊C411yescharacteristic
69C4×D251—21no—
70C4×D251—21no—
71C4×D251—21no—
72C4×D251—21no—
73C4×D251—22no—
74C251⋊C8—22no—
75C251⋊C8—41no—
76C23×C251—21no—
77D4×C251—21no—
78D4×C251M1612yescomplemented
79C2×C4×C251—21no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251C22×C411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C8×C251—22no—
86C8×C251—41no—
87C22×C251—22no—
88C22×C251—41no—
89C22×C251—41no—
90C22×C251C2×C22⋊C411yescharacteristic
91C2×D251—21no—
92C2×D251—21no—
93C2×D251—21no—
94C4×C251—21no—
95C4×C251—41no—
96C4×C251C2×M1611yescharacteristic
97C4×C251C2×M1611yescharacteristic
98C4×C251AGL(1,ℤ/8)11yescharacteristic
99C4×C251#13711yescharacteristic
100Dic251—21no—
101Dic251—22no—
102Dic251—21no—
103Dic251—41no—
104Dic251#10412yes—
105C2×C251—42no—
106C2×C251#40711yescharacteristic
107C2×C251#41711yescharacteristic
108C2×C251C2×#10411yescharacteristic
109D251—41no—
110C251#148911yescharacteristic, complemented, Sylow
111#1489—2511nomaximal, Sylow
112C4×#41—2511no—
113#423—2511no—
114C4×M16—2511no—
115Q8⋊C8—2511no—
116Q8⋊C8—2511no—
117D4⋊C8—2511no—
118D4⋊C8D25111yescharacteristic, complemented
119C2×#41—2511no—
120C2×M16—5021no—
121C2×M16—5021no—
122C4×Q8—2511no—
123C4×Q8—2511no—
124C4×D4—5021no—
125C4×D4—5021no—
126C4×D4—2511no—
127C4×D4C2×D25111yescharacteristic
128#117—5021no—
129#117—2511no—
130C2×C42—5021no—
131C2×C42—2511no—
132C4⋊C8—5021no—
133C4⋊C8—2511no—
134C4⋊C8C2×D25111yescharacteristic
135#97—5021no—
136C4×C8—2511no—
137C4×C8C2×D25111yescharacteristic
138#41—5022no—
139#41—5022no—
140C2×Q8—2511no—
141C2×D4—5021no—
142C2×D4C4×D25111yescharacteristic
143C22×C4—21no—
144C22×C4—5021no—
145C22×C4—5021no—
146C22×C4—2511no—
147C22×C4—2511no—
148C22×C4—2511no—
149M16—10042no—
150M16—5024no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—5021no—
154C2×C8—5021no—
155C4⋊C4—5021no—
156C4⋊C4—5021no—
157C4⋊C4—2511no—
158C4⋊C4C4×D25111yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—5021no—
161C22⋊C4—5021no—
162C42—5021no—
163C42—5021no—
164C42—5021no—
165C42—2511no—
166C42—2511no—
167C42—2511no—
168C42C22×D25111yescharacteristic
169C23—21no—
170C23—2511no—
171Q8—5021no—
172Q8—2512no—
173D4—21no—
174D4—5024no—
175D4#1686079312yescomplemented
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—10042no—
180C2×C4—5022no—
181C2×C4—5021no—
182C2×C4—5024no—
183C2×C4—5021no—
184C2×C4—5021no—
185C2×C4—5021no—
186C2×C4—5021no—
187C2×C4—5021no—
188C2×C4—2511no—
189C2×C4—2511no—
190C2×C4—2511no—
191C2×C4C2×C4×D25111yescharacteristic
192C2×C4D4×D25111yescharacteristic
193C2×C4D4×D25111yescharacteristic
194C8—22no—
195C8—41no—
196C8—10041no—
197C8—5022no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—5021no—
202C22—5021no—
203C22—5021no—
204C22C22⋊C4×D25111yescharacteristic
205C4—21no—
206C4—41no—
207C4—10041no—
208C4—5021no—
209C4—5021no—
210C4—5022no—
211C4—2512no—
212C4M16×D25111yescharacteristic
213C4C2×#1686079311yescharacteristic
214C4#2254873911yescharacteristic
215C4#2254874411yescharacteristic
216C2—42no—
217C2—10041no—
218C2#3063164911yescharacteristic
219C2#3063168511yescharacteristic
220C2#104×D25111yescharacteristic
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
FittingD4⋊C8×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC2×#16860809C4
cosocleC42×C251C23
hypercenterD4⋊C8D251
hypocenterC251#1489
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30631709C2semidirect
3. C4×#16860793C2semidirect
4. #30631419C2semidirect
5. #30631417C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
7. D4⋊C8×C251C2semidirect
60. Dic502M16semidirect
78. D4×C251M16semidirect
110. C251#1489semidirect
118. D4⋊C8D251semidirect
175. D4#16860793semidirect

Maximal subgroups

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

subgroupindex
C4×#168608092
#306317092
C4×#168607932
#306314192
#306314172
D4⋊(C251⋊C8)2
D4⋊C8×C2512
#1489251