Finite Group Databasebeta

← #44548829

Subgroups — #44548829

Order 32128.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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#30632224C211yesmaximal, characteristic
2#30631738C211yesmaximal, characteristic, complemented
3#30631709C211yesmaximal, characteristic, complemented
4Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
5#30631416C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7Q8⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#16860793—21no—
9#22548710—21no—
10#22548706C411yescharacteristic
11#22548704C411yescharacteristic
12#22548700—21no—
13#22548691C411yescharacteristic
14#22548690C411yescharacteristic
15C4×D1004—21no—
16C4×D1004C2211yescharacteristic
17#22548672—21no—
18C42×D251C2211yescharacteristic
19C4⋊(C251⋊C8)C2211yescharacteristic
20Dic251⋊C8—21no—
21Dic251⋊C8—21no—
22(C2×D251)⋊C8—21no—
23C4×C251⋊C8C2211yescharacteristic
24C4×Q8×C251C2211yescharacteristic
25C4⋊C8×C251C2211yescharacteristic
26C4×C8×C251C2211yescharacteristic
27C2×D1004—21no—
28C2×D1004C2×C411yescharacteristic
29C2×C4×D251—21no—
30C2×C4×D251C2×C411yescharacteristic
31C2×C4×D251D411yescharacteristic
32C2×C4×D251D411yescharacteristic
33#16860793—42no—
34C2×C251⋊C8—21no—
35C2×C251⋊C8—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251C2×C411yescharacteristic
38Dic251⋊C4—21no—
39Dic251⋊C4—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45C4×Dic251C2×C411yescharacteristic
46C4×Dic251D411yescharacteristic
47C4×Dic251D411yescharacteristic
48C2×Q8×C251C2×C411yescharacteristic
49C2×C8×C251—21no—
50C2×C8×C251—21no—
51C4⋊C4×C251—21no—
52C4⋊C4×C251C2×C411yescharacteristic
53C42×C251—21no—
54C42×C251C2311yescharacteristic
55C22×D251—21no—
56C22×D251C22⋊C411yescharacteristic
57D1004—21no—
58D1004—21no—
59D1004—42no—
60C2×Dic251—21no—
61C2×Dic251C22⋊C411yescharacteristic
62C2×Dic251C22⋊C411yescharacteristic
63C2×Dic251C22⋊C411yescharacteristic
64C4×D251—21no—
65C4×D251—21no—
66C4×D251—21no—
67C4×D251—21no—
68C4×D251—41no—
69C4×D251—41no—
70C4×D251—41no—
71C251⋊C8—41no—
72C251⋊C8—41no—
73Q8×C251—21no—
74Q8×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—21no—
77C2×C4×C251C22×C411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C8×C251—41no—
81C8×C251—41no—
82C22×C251C2×C22⋊C411yescharacteristic
83C2×D251—21no—
84C2×D251—21no—
85C2×D251—21no—
86C2×D251—41no—
87C2×D251—41no—
88C2×D251—41no—
89C4×C251—21no—
90C4×C251—41no—
91C4×C251—41no—
92C4×C251#13111yescharacteristic
93C4×C251#13111yescharacteristic
94C4×C251AGL(1,ℤ/8)11yescharacteristic
95C4×C251#13711yescharacteristic
96Dic251—21no—
97Dic251—21no—
98Dic251—21no—
99Dic251—41no—
100C2×C251#40811yescharacteristic
101C2×C251#41711yescharacteristic
102C2×C251#42111yescharacteristic
103D251—41no—
104D251—81no—
105C251#149511yescharacteristic, complemented, Sylow
106#1495—2511nomaximal, Sylow
107#529—2511no—
108#431—2511no—
109#423—2511no—
110Q8⋊C8—2511no—
111Q8⋊C8D25111yescharacteristic, complemented
112D4⋊C8—2511no—
113D4⋊C8—2511no—
114C2×M16—5021no—
115#126—5021no—
116#125—2511no—
117#124—2511no—
118#123—5021no—
119C22⋊Q8—2511no—
120#121—2511no—
121C4×Q8C2×D25111yescharacteristic
122C4×D4—5021no—
123C4×D4—2511no—
124#117—5021no—
125C2×C42—2511no—
126C4⋊C8—5021no—
127C4⋊C8—5021no—
128C4⋊C8—2511no—
129C4⋊C8C2×D25111yescharacteristic
130C22⋊C8—5021no—
131C4×C8—2511no—
132C4×C8C2×D25111yescharacteristic
133C2×Q8C4×D25111yescharacteristic
134C2×D4—5021no—
135C2×D4—2511no—
136C22×C4—5021no—
137C22×C4—2511no—
138C22×C4—2511no—
139C22×C4—2511no—
140M16—10042no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—5021no—
144C2×C8—5021no—
145C4⋊C4—21no—
146C4⋊C4—5021no—
147C4⋊C4—5021no—
148C4⋊C4—5021no—
149C4⋊C4—2511no—
150C4⋊C4C4×D25111yescharacteristic
151C22⋊C4—5021no—
152C22⋊C4—5021no—
153C22⋊C4—5021no—
154C22⋊C4—5021no—
155C22⋊C4—5021no—
156C42—21no—
157C42—2511no—
158C42—2511no—
159C42—2511no—
160C42C22×D25111yescharacteristic
161C23—5021no—
162C23—2511no—
163Q8—21no—
164Q8—21no—
165D4—10042no—
166D4—5021no—
167D4—5021no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—10041no—
171C2×C4—10041no—
172C2×C4—10041no—
173C2×C4—5021no—
174C2×C4—5021no—
175C2×C4—5021no—
176C2×C4—5021no—
177C2×C4—5021no—
178C2×C4—2511no—
179C2×C4—2511no—
180C2×C4—2511no—
181C2×C4C2×C4×D25111yescharacteristic
182C2×C4D4×D25111yescharacteristic
183C2×C4D4×D25111yescharacteristic
184C8—41no—
185C8—41no—
186C8—10041no—
187C8—10041no—
188C22—10041no—
189C22—10041no—
190C22—10041no—
191C22—5021no—
192C22—5021no—
193C22—5021no—
194C22C22⋊C4×D25111yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—10041no—
199C4—5021no—
200C4—5021no—
201C4—5021no—
202C4#2254872111yescharacteristic
203C4#2254872311yescharacteristic
204C4#2254874211yescharacteristic
205C4#2254874711yescharacteristic
206C2—20081no—
207C2—10041no—
208C2#3063165211yescharacteristic
209C2#3063168711yescharacteristic
210C2#3063170211yescharacteristic
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC42C22×D251
FittingQ8⋊C8×C251C2
socleC22×C251C2×C22⋊C4
prime radical#22548690C4
cosocleC42×C251C23
hypercenterQ8⋊C8D251
hypocenterC251#1495
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #30631738C2semidirect
3. #30631709C2semidirect
4. Q8⋊(C251⋊C8)C2semidirect
5. #30631416C2semidirect
6. #30631414C2semidirect
7. Q8⋊C8×C251C2semidirect
105. C251#1495semidirect
111. Q8⋊C8D251semidirect

Maximal subgroups

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

subgroupindex
#306322242
#306317382
#306317092
Q8⋊(C251⋊C8)2
#306314162
#306314142
Q8⋊C8×C2512
#1495251