Finite Group Databasebeta

← #44548889

Subgroups — #44548889

Order 32128.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#30632223C211yesmaximal, characteristic
2#30631741C211yesmaximal, characteristic, complemented
3#30631538C211yesmaximal, characteristic, complemented
4#30631424C211yesmaximal, characteristic, complemented
5#30631422C211yesmaximal, characteristic
6#30631420C211yesmaximal, characteristic
7#328×C251C211yesmaximal, characteristic, complemented
8#22548710—21no—
9#22548706C411yescharacteristic, complemented
10#22548704C411yescharacteristic, complemented
11#22548699—21no—
12#22548689C2211yescharacteristic
13Q8×Dic251C411yescharacteristic, complemented
14#22548677C411yescharacteristic, complemented
15(C22⋊D251)⋊C4—21no—
16#22548673—21no—
17C2×Dic251⋊C4—21no—
18C2×C4×Dic251—21no—
19C2×C4×Dic251C2211yescharacteristic
20Dic251⋊C8—21no—
21C22⋊(C251⋊C8)C2211yescharacteristic
22#22548625—21no—
23C251⋊#95C2211yescharacteristic
24C22⋊Q8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C2×D1004C2×C411yescharacteristic
28C2×C22⋊D251—21no—
29C22×Dic251—21no—
30C22×Dic251D411yescharacteristic
31C22×Dic251D411yescharacteristic
32C2×C4×D251—21no—
33C2×C251⋊C8—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251C2×C411yescharacteristic
37Dic251⋊C4—24no—
38Dic251⋊C4—21no—
39(C2×D251)⋊C4—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43C4×Dic251—21no—
44C4×Dic251—42no—
45C4×Dic251C2×C411yescharacteristic
46C4×Dic251C2×C411yescharacteristic
47C4×Dic251D411yescharacteristic
48C4×Dic251D411yescharacteristic
49C2×Q8×C251C2×C411yescharacteristic
50C22×C4×C251—21no—
51C22×C4×C251C2311yescharacteristic
52C2×C8×C251—21no—
53C4⋊C4×C251—21no—
54C4⋊C4×C251C2×C411yescharacteristic
55C22⋊C4×C251—21no—
56C22×D251—21no—
57D1004—41no—
58C22⋊D251—41no—
59C22⋊D251—41no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—21no—
65C2×Dic251—41no—
66C2×Dic251—41no—
67C2×Dic251C22⋊C411yescharacteristic
68C2×Dic251C22⋊C411yescharacteristic
69C2×Dic251C22⋊C411yescharacteristic
70C2×Dic251C22⋊C411yescharacteristic
71C4×D251—42no—
72C251⋊C8—41no—
73C23×C251C2×D411yescharacteristic
74Q8×C251—41no—
75C2×C4×C251—22no—
76C2×C4×C251—21no—
77C2×C4×C251—21no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251—41no—
81C2×C4×C251C22×C411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C8×C251—41no—
84C22×C251—21no—
85C22×C251—21no—
86C22×C251C2×C22⋊C411yescharacteristic
87C22×C251AGL(1,ℤ/8)11yescharacteristic
88C22×C251#13711yescharacteristic
89C2×D251—41no—
90C2×D251—41no—
91C2×D251—41no—
92C4×C251—21no—
93C4×C251—21no—
94C4×C251—42no—
95C4×C251—41no—
96C4×C251—41no—
97Dic251—21no—
98Dic251—21no—
99Dic251—21no—
100Dic251—42no—
101Dic251—41no—
102Dic251#10412yes—
103C2×C251—21no—
104C2×C251—21no—
105C2×C251#41011yescharacteristic
106C2×C251#41711yescharacteristic
107C2×C251C2×#10411yescharacteristic
108D251—81no—
109C251#150911yescharacteristic, complemented, Sylow
110#1509—2511nomaximal, Sylow
111#529—2511no—
112#432—2511no—
113C4×C4⋊C4—2511no—
114#328—2511no—
115#328D25111yescharacteristic, complemented
116#327—2511no—
117#327—2511no—
118#126—5021no—
119#125—2511no—
120#124—2511no—
121#123—5021no—
122C22⋊Q8C2×D25111yescharacteristic
123#121—2511no—
124C4×Q8—2511no—
125C4×D4—5021no—
126C4×D4—2511no—
127#117—5021no—
128C2×C4⋊C4—5021no—
129C2×C42—5021no—
130C2×C42—2511no—
131C4⋊C8—5021no—
132C22⋊C8—2511no—
133C22⋊C8C2×D25111yescharacteristic
134#97—5021no—
135#95—2511no—
136#95C2×D25111yescharacteristic
137C2×Q8C4×D25111yescharacteristic
138C2×D4—5021no—
139C2×D4—2511no—
140C22×C4—21no—
141C22×C4—5021no—
142C22×C4—5021no—
143C22×C4—2511no—
144C22×C4—2511no—
145C22×C4C22×D25111yescharacteristic
146C2×C8—21no—
147C2×C8—5021no—
148C4⋊C4—21no—
149C4⋊C4—5024no—
150C4⋊C4—5021no—
151C4⋊C4—5021no—
152C4⋊C4—5021no—
153C4⋊C4—2511no—
154C4⋊C4C4×D25111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—5021no—
157C22⋊C4—5021no—
158C22⋊C4—5021no—
159C22⋊C4—5021no—
160C42—10042no—
161C42—5021no—
162C42—2511no—
163C42—2511no—
164C42—2511no—
165C42—2511no—
166C23—5021no—
167C23D4×D25111yescharacteristic
168Q8—41no—
169D4—10041no—
170D4—10041no—
171D4—10041no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—10042no—
179C2×C4—10041no—
180C2×C4—10041no—
181C2×C4—5022no—
182C2×C4—5021no—
183C2×C4—5021no—
184C2×C4—5021no—
185C2×C4—5022no—
186C2×C4—2511no—
187C2×C4—2511no—
188C2×C4—2511no—
189C2×C4—2511no—
190C2×C4C2×C4×D25111yescharacteristic
191C2×C4D4×D25111yescharacteristic
192C8—41no—
193C8—10041no—
194C22—21no—
195C22—21no—
196C22—10041no—
197C22—10041no—
198C22—10041no—
199C22C22⋊C4×D25111yescharacteristic
200C22#2254874211yescharacteristic
201C22#2254874711yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—10042no—
208C4—10041no—
209C4—5021no—
210C4—5021no—
211C4—5021no—
212C4—2512no—
213C2—21no—
214C2—21no—
215C2—20081no—
216C2#3063165911yescharacteristic
217C2#3063168711yescharacteristic
218C2#104×D25111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#328×C251C2
socleC22×C251C2×C22⋊C4
prime radical#22548689C22
cosocleC22×C4×C251C23
hypercenter#328D251
hypocenterC251#1509
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #30631741C2semidirect
3. #30631538C2semidirect
4. #30631424C2semidirect
7. #328×C251C2semidirect
9. #22548706C4semidirect
10. #22548704C4semidirect
13. Q8×Dic251C4semidirect
14. #22548677C4semidirect
109. C251#1509semidirect
115. #328D251semidirect

Maximal subgroups

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

subgroupindex
#306322232
#306317412
#306315382
#306314242
#306314222
#306314202
#328×C2512
#1509251