Finite Group Databasebeta

← #44548902

Subgroups — #44548902

Order 32128.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#30631551C211yesmaximal, 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×Dic251C2211yescharacteristic
19Dic251⋊C8—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic
21#22548625—21no—
22C251⋊#95—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—21no—
38Dic251⋊C4—42no—
39(C2×D251)⋊C4—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43C4×Dic251—21no—
44C4×Dic251C2×C411yescharacteristic
45C4×Dic251C2×C411yescharacteristic
46C4×Dic251D411yescharacteristic
47C4×Dic251D411yescharacteristic
48C2×Q8×C251C2×C411yescharacteristic
49C22×C4×C251—21no—
50C22×C4×C251C2311yescharacteristic
51C2×C8×C251—21no—
52C4⋊C4×C251—21no—
53C4⋊C4×C251C2×C411yescharacteristic
54C22⋊C4×C251—21no—
55C22×D251—21no—
56D1004—41no—
57C22⋊D251—41no—
58C22⋊D251—41no—
59C2×Dic251—21no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—41no—
63C2×Dic251—41no—
64C2×Dic251—41no—
65C2×Dic251—41no—
66C2×Dic251C22⋊C411yescharacteristic
67C2×Dic251C22⋊C411yescharacteristic
68C2×Dic251C22⋊C411yescharacteristic
69C2×Dic251C22⋊C411yescharacteristic
70C4×D251—42no—
71C251⋊C8—41no—
72C23×C251C2×D411yescharacteristic
73Q8×C251—41no—
74C2×C4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251C22×C411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C8×C251—41no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251C2×C22⋊C411yescharacteristic
86C22×C251AGL(1,ℤ/8)11yescharacteristic
87C22×C251#13711yescharacteristic
88C2×D251—41no—
89C2×D251—41no—
90C2×D251—41no—
91C4×C251—21no—
92C4×C251—21no—
93C4×C251—41no—
94C4×C251—41no—
95C4×C251—81no—
96Dic251—21no—
97Dic251—21no—
98Dic251—21no—
99Dic251—21no—
100Dic251—41no—
101Dic251—81no—
102C2×C251—21no—
103C2×C251—21no—
104C2×C251#41011yescharacteristic
105C2×C251#41711yescharacteristic
106C2×C251#42111yescharacteristic
107D251—81no—
108C251#151211yescharacteristic, complemented, Sylow
109#1512—2511nomaximal, Sylow
110#529—2511no—
111#432—2511no—
112#382—2511no—
113#328—2511no—
114#328D25111yescharacteristic, complemented
115#327—2511no—
116#327—2511no—
117#126—5021no—
118#125—2511no—
119#124—2511no—
120#123—5021no—
121C22⋊Q8C2×D25111yescharacteristic
122#121—2511no—
123C4×Q8—2511no—
124C4×D4—5021no—
125C4×D4—2511no—
126#117—5021no—
127C2×C4⋊C4—5021no—
128C2×C42—2511no—
129C4⋊C8—5021no—
130C22⋊C8—2511no—
131C22⋊C8C2×D25111yescharacteristic
132#97—5021no—
133#95—5021no—
134#95—2511no—
135#95C2×D25111yescharacteristic
136C2×Q8C4×D25111yescharacteristic
137C2×D4—5021no—
138C2×D4—2511no—
139C22×C4—21no—
140C22×C4—5021no—
141C22×C4—5021no—
142C22×C4—2511no—
143C22×C4—2511no—
144C22×C4C22×D25111yescharacteristic
145C2×C8—21no—
146C2×C8—5021no—
147C4⋊C4—21no—
148C4⋊C4—10042no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C4⋊C4—5021no—
152C4⋊C4—2511no—
153C4⋊C4C4×D25111yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—5021no—
156C22⋊C4—5021no—
157C22⋊C4—5021no—
158C22⋊C4—5021no—
159C42—5021no—
160C42—2511no—
161C42—2511no—
162C42—2511no—
163C42—2511no—
164C23—5021no—
165C23D4×D25111yescharacteristic
166Q8—41no—
167D4—10041no—
168D4—10041no—
169D4—10041no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—10042no—
177C2×C4—10041no—
178C2×C4—10041no—
179C2×C4—10041no—
180C2×C4—10041no—
181C2×C4—5021no—
182C2×C4—5021no—
183C2×C4—5021no—
184C2×C4—2511no—
185C2×C4—2511no—
186C2×C4—2511no—
187C2×C4—2511no—
188C2×C4C2×C4×D25111yescharacteristic
189C2×C4D4×D25111yescharacteristic
190C8—41no—
191C8—10041no—
192C22—21no—
193C22—21no—
194C22—10041no—
195C22—10041no—
196C22—10041no—
197C22C22⋊C4×D25111yescharacteristic
198C22#2254874211yescharacteristic
199C22#2254874711yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—81no—
205C4—20081no—
206C4—10041no—
207C4—5021no—
208C4—5021no—
209C4—5021no—
210C4—5021no—
211C2—21no—
212C2—21no—
213C2—20081no—
214C2#3063165911yescharacteristic
215C2#3063168711yescharacteristic
216C2#3063170211yescharacteristic
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#1512
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. #30631551C2semidirect
4. #30631424C2semidirect
7. #328×C251C2semidirect
9. #22548706C4semidirect
10. #22548704C4semidirect
13. Q8×Dic251C4semidirect
14. #22548677C4semidirect
108. C251#1512semidirect
114. #328D251semidirect

Maximal subgroups

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

subgroupindex
#306322232
#306317412
#306315512
#306314242
#306314222
#306314202
#328×C2512
#1512251