Finite Group Databasebeta

← #44549365

Subgroups — #44549365

Order 32128.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632286C211yesmaximal, characteristic
2#30632015C211yesmaximal, characteristic, complemented
3#30632001C211yesmaximal, characteristic
4#30631619C211yesmaximal, characteristic, complemented
5#30631420C211yesmaximal, characteristic
6#30631412C211yesmaximal, characteristic, complemented
7#328×C251C211yesmaximal, characteristic, complemented
8#22548710—21no—
9#22548706—21no—
10#22548700—21no—
11#22548699—21no—
12#22548691—21no—
13#22548689C2211yescharacteristic
14(C22⋊D251)⋊C4—21no—
15C2×C4⋊Dic251—21no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18#22548649—21no—
19#22548646—21no—
20#22548639—21no—
21#22548637—21no—
22C22⋊(C251⋊C8)C2211yescharacteristic
23C251⋊#95—21no—
24C22⋊Q8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C2×D1004D411yescharacteristic
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⋊Dic251—21no—
37C4⋊Dic251—21no—
38C4⋊Dic251—42no—
39C4⋊Dic251D411yescharacteristic
40Dic251⋊C4—21no—
41Dic251⋊C4—21no—
42Dic251⋊C4—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45(C2×D251)⋊C4—21no—
46(C2×D251)⋊C4—21no—
47C4×Dic251—21no—
48C2×Q8×C251D411yescharacteristic
49C22×C4×C251—21no—
50C22×C4×C251C2311yescharacteristic
51C2×C8×C251—21no—
52C4⋊C4×C251—21no—
53C4⋊C4×C251D411yescharacteristic
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—21no—
63C2×Dic251—21no—
64C2×Dic251—41no—
65C2×Dic251—41no—
66C2×Dic251—41no—
67C2×Dic251—41no—
68C4×D251—81no—
69C251⋊C8—41no—
70C23×C251C2×D411yescharacteristic
71Q8×C251—41no—
72C2×C4×C251—22no—
73C2×C4×C251—21no—
74C2×C4×C251—21no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251C2×D411yescharacteristic
79C2×C4×C251C2×D411yescharacteristic
80C8×C251—41no—
81C22×C251—21no—
82C22×C251—21no—
83C22×C251#12011yescharacteristic
84C22×C251#13511yescharacteristic
85C22×C251AGL(1,ℤ/8)11yescharacteristic
86C2×D251—41no—
87C2×D251—41no—
88C2×D251—41no—
89C4×C251—21no—
90C4×C251—21no—
91C4×C251—42no—
92C4×C251—41no—
93C4×C251—41no—
94Dic251—41no—
95Dic251—41no—
96Dic251—41no—
97Dic251—81no—
98C2×C251—21no—
99C2×C251—21no—
100C2×C251#44911yescharacteristic
101C2×C251#45611yescharacteristic
102C2×C251#45811yescharacteristic
103D251—81no—
104C251#160911yescharacteristic, complemented, Sylow
105#1609—2511nomaximal, Sylow
106#539—2511no—
107#483—2511no—
108#480—2511no—
109#398—2511no—
110#328D25111yescharacteristic, complemented
111#327—2511no—
112#324—2511no—
113#126—5021no—
114#125—5021no—
115#123—5021no—
116#123—5021no—
117C22⋊Q8—5021no—
118C22⋊Q8C2×D25111yescharacteristic
119#121—2511no—
120C4×D4—5021no—
121C2×C4⋊C4—5021no—
122C2×C4⋊C4—2511no—
123C2×C4⋊C4—2511no—
124#107—5021no—
125#106—5021no—
126#103—5021no—
127#102—5021no—
128C22⋊C8—2511no—
129C22⋊C8C2×D25111yescharacteristic
130#95—5021no—
131#95C2×D25111yescharacteristic
132C2×Q8D100411yescharacteristic
133C2×D4—5021no—
134C2×D4—2511no—
135C22×C4—21no—
136C22×C4—5021no—
137C22×C4—5021no—
138C22×C4—2511no—
139C22×C4—2511no—
140C22×C4C22×D25111yescharacteristic
141C2×C8—21no—
142C2×C8—5021no—
143C4⋊C4—21no—
144C4⋊C4—10042no—
145C4⋊C4—5021no—
146C4⋊C4—5021no—
147C4⋊C4—5021no—
148C4⋊C4—5021no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C4⋊C4—5021no—
152C4⋊C4—2511no—
153C4⋊C4D100411yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—5021no—
156C22⋊C4—5021no—
157C22⋊C4—5021no—
158C22⋊C4—5021no—
159C42—5021no—
160C23—5021no—
161C23D4×D25111yescharacteristic
162Q8—41no—
163D4—10041no—
164D4—10041no—
165D4—10041no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—20081no—
173C2×C4—10041no—
174C2×C4—10041no—
175C2×C4—10041no—
176C2×C4—10041no—
177C2×C4—5021no—
178C2×C4—5021no—
179C2×C4—5021no—
180C2×C4—5021no—
181C2×C4—5021no—
182C2×C4D4×D25111yescharacteristic
183C2×C4C2×D100411yescharacteristic
184C8—41no—
185C8—10041no—
186C22—21no—
187C22—21no—
188C22—10041no—
189C22—10041no—
190C22—10041no—
191C22#2254868511yescharacteristic
192C22#2254873811yescharacteristic
193C22#2254874211yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—20081no—
200C4—10041no—
201C4—10041no—
202C4—10041no—
203C2—21no—
204C2—21no—
205C2—20081no—
206C2#3063183511yescharacteristic
207C2#3063187111yescharacteristic
208C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#328×C251C2
socleC22×C251#120
prime radical#22548689C22
hypercenter#328D251
hypocenterC251#1609
LCSG ▷ C22×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. #30632015C2semidirect
4. #30631619C2semidirect
6. #30631412C2semidirect
7. #328×C251C2semidirect
104. C251#1609semidirect
110. #328D251semidirect

Maximal subgroups

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

subgroupindex
#306322862
#306320152
#306320012
#306316192
#306314202
#306314122
#328×C2512
#1609251