Finite Group Databasebeta

← #44551406

Subgroups — #44551406

Order 32128.

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

normal subgroups only

This group has 406 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
1C2×#22548741C212yesmaximal, complemented
2#30632268C211yesmaximal, characteristic, complemented
3C2×#22548643C212yesmaximal, complemented
4#30631456C211yesmaximal, characteristic, complemented
5#390×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2212yes—
7#22548741—24no—
8#22548741—24no—
9C2×#16860798—22no—
10C2×#16860795—22no—
11C2×#16860794C2212yes—
12#22548695—22no—
13#22548690—22no—
14#22548689—22no—
15#22548643—24no—
16C22×D4×C251C2211yescharacteristic
17C2×C4⋊C4×C251C2211yescharacteristic
18C2×C22⋊C4×C251—22no—
19C2×C42×C251C2211yescharacteristic
20#16860810—22no—
21#16860810—22no—
22#16860810—22no—
23#16860810—22no—
24C2×Dic502D412yes—
25C2×D1004—22no—
26C2×D1004D412yes—
27C2×C22⋊D251—22no—
28C2×C4×D251—22no—
29#16860798—28no—
30#16860795—28no—
31#16860794—22no—
32#16860794D414yescomplemented
33C2×C251⋊C8—22no—
34C4⋊Dic251—22no—
35(C2×D251)⋊C4—24no—
36C24×C251—21no—
37C2×D4×C251—22no—
38C2×D4×C251—22no—
39C2×D4×C251—21no—
40C2×D4×C251D412yescomplemented
41C22×C4×C251—21no—
42C22×C4×C251D412yes—
43C22×C4×C251C2311yescharacteristic
44C4⋊C4×C251—22no—
45C22⋊C4×C251—44no—
46C42×C251—22no—
47C22×D251—22no—
48Dic502—22no—
49Dic502—22no—
50D1004—22no—
51D1004—22no—
52D1004—82no—
53C22⋊D251—42no—
54C22⋊D251—42no—
55C2×Dic251—22no—
56C4×D251—42no—
57C4×D251—42no—
58C251⋊C8—24no—
59C23×C251—22no—
60C23×C251—42no—
61C23×C251—42no—
62C23×C251—41no—
63C23×C251—41no—
64C23×C251C2×D411yescharacteristic
65D4×C251—24no—
66D4×C251—42no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—41no—
70C2×C4×C251—41no—
71C2×C4×C251—42no—
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251#4111yescharacteristic
76C22×C251—21no—
77C22×C251—21no—
78C22×C251—44no—
79C22×C251—42no—
80C22×C251—82no—
81C22×C251—81no—
82C22×C251—81no—
83C22×C251#12011yescharacteristic
84C22×C251#12111yescharacteristic
85C22×C251#12711yescharacteristic
86C2×D251—42no—
87C2×D251—42no—
88C2×D251—42no—
89C4×C251—42no—
90C4×C251—42no—
91C4×C251#12012yes—
92C4×C251#12112yes—
93Dic251—42no—
94C2×C251—21no—
95C2×C251—21no—
96C2×C251—82no—
97C2×C251#39211yescharacteristic
98C2×C251#45311yescharacteristic
99C2×C251#45411yescharacteristic
100D251—82no—
101C251#200611yescharacteristic, complemented, Sylow
102#2006—2511nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2512no—
104#537—2511no—
105C2×#104—2512no—
106#390D25111yescharacteristic, complemented
107#343—2511no—
108C2×#41—2512no—
109C22×D4C2×D25111yescharacteristic
110AGL(1,ℤ/8)—5024no—
111AGL(1,ℤ/8)—5024no—
112C2×SD16—5022no—
113C2×D8—5022no—
114C2×M16—2512no—
115C22⋊Q8—5022no—
116#121—5022no—
117#121—5022no—
118C2×C4⋊C4C2×D25111yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D25111yescharacteristic
121#104—5024no—
122C24—21no—
123#41—5022no—
124#41—5022no—
125#41—5022no—
126#41—5022no—
127C2×Q8—2512no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—5022no—
132C2×D4—5022no—
133C2×D4—2512no—
134C2×D4D100412yescomplemented
135C22×C4—21no—
136C22×C4—5022no—
137C22×C4C22⋊D25112yes—
138C22×C4C22×D25111yescharacteristic
139SD16—5028no—
140D8—5028no—
141M16—5022no—
142M16—2514no—
143C2×C8—5022no—
144C4⋊C4—22no—
145C4⋊C4—5022no—
146C22⋊C4—44no—
147C22⋊C4—5024no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—5022no—
155C23C2×C22⋊D25111yescharacteristic
156Q8—5022no—
157Q8—5022no—
158D4—24no—
159D4—42no—
160D4—20082no—
161D4—10042no—
162D4—10042no—
163D4—5022no—
164D4—5022no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—10042no—
171C2×C4—10042no—
172C2×C4—5022no—
173C2×C4D4×D25112yes—
174C2×C4D4×D25112yes—
175C2×C4C2×D100411yescharacteristic
176C2×C4#1686081011yescharacteristic
177C8—5024no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—10042no—
186C22—10042no—
187C22—10042no—
188C22#2254868311yescharacteristic
189C22#2254868911yescharacteristic
190C22#2254871111yescharacteristic
191C4—42no—
192C4—42no—
193C4—10042no—
194C4#2254868512yes—
195C4#2254868812yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—20082no—
200C2#3063159611yescharacteristic
201C2#3063185411yescharacteristic
202C2#3063185911yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#390×C251C2
socleC22×C251#120
hypercenter#390D251
hypocenterC251#2006
LCSG ▷ C22×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. #30632268C2semidirect
3. C2×#22548643C2semidirect
4. #30631456C2semidirect
5. #390×C251C2semidirect
32. #16860794D4semidirect
40. C2×D4×C251D4semidirect
101. C251#2006semidirect
106. #390D251semidirect
134. C2×D4D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
#306322682
C2×#225486432
#306314562
#390×C2512
#2006251