Finite Group Databasebeta

← #44549625

Subgroups — #44549625

Order 32128.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#30632352C211yesmaximal, characteristic
2#30631969C211yesmaximal, characteristic, complemented
3#30631889C211yesmaximal, characteristic
4#30631433C211yesmaximal, characteristic, complemented
5#30631419C211yesmaximal, characteristic, complemented
6#30631416C211yesmaximal, characteristic
7#498×C251C211yesmaximal, characteristic, complemented
8C2×Q8×D251—21no—
9C2×#16860797—21no—
10#22548714—21no—
11#22548693—22no—
12#22548691—21no—
13C4×Dic502C2211yescharacteristic
14C4×D1004C2211yescharacteristic
15#22548677—21no—
16#22548646—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18C4⋊(C251⋊C8)C2211yescharacteristic
19#22548639—21no—
20#22548638—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4⋊Q8×C251C2211yescharacteristic
23#106×C251—22no—
24C4×C8×C251C2211yescharacteristic
25C2×Dic502—22no—
26C2×Dic502D411yescharacteristic
27Q8×D251—42no—
28Q8×D251—42no—
29C2×D1004D411yescharacteristic
30C2×C4×D251—22no—
31C2×C4×D251—21no—
32#16860797—42no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251D411yescharacteristic
36C4⋊Dic251D411yescharacteristic
37Dic251⋊C4—22no—
38Dic251⋊C4—21no—
39(C2×D251)⋊C4—22no—
40(C2×D251)⋊C4—21no—
41C4×Dic251—21no—
42C2×Q8×C251—21no—
43C2×Q8×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251—21no—
48C4⋊C4×C251D412yes—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—21no—
52Dic502—44no—
53Dic502SD1612yescomplemented
54D1004—21no—
55D1004SD1612yes—
56C2×Dic251—21no—
57C2×Dic251—21no—
58C2×Dic251—21no—
59C4×D251—44no—
60C4×D251—41no—
61C4×D251—41no—
62C251⋊C8—41no—
63C251⋊C8—41no—
64Q8×C251—41no—
65Q8×C251—41no—
66Q8×C251—41no—
67Q8×C251—41no—
68C2×C4×C251—22no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—22no—
74C22×C251#12011yescharacteristic
75C2×D251—22no—
76C2×D251—41no—
77C2×D251—41no—
78C4×C251—21no—
79C4×C251—42no—
80C4×C251—81no—
81C4×C251C2×SD1611yescharacteristic
82C4×C251C2×SD1611yescharacteristic
83C4×C251AGL(1,ℤ/8)11yescharacteristic
84C4×C251#13711yescharacteristic
85Dic251—22no—
86Dic251—41no—
87Dic251—41no—
88C2×C251#44811yescharacteristic
89C2×C251#45011yescharacteristic
90C2×C251#45611yescharacteristic
91D251—42no—
92C251#165111yescharacteristic, complemented, Sylow
93#1651—2511nomaximal, Sylow
94D4×Q8—2511no—
95#498D25111yescharacteristic, complemented
96#475—2511no—
97#461—2511no—
98#332—2511no—
99Q8⋊C8—2511no—
100D4⋊C8—2511no—
101C22×Q8—5021no—
102C2×SD16—5021no—
103C4⋊Q8—5021no—
104C4⋊Q8C2×D25111yescharacteristic
105C4⋊Q8C2×D25111yescharacteristic
106C22⋊Q8—5021no—
107C22⋊Q8—5022no—
108C4×Q8—2511no—
109C4×D4—5021no—
110C4×D4—2511no—
111#106—22no—
112#106—5021no—
113C4⋊C8—2511no—
114C4⋊C8—2511no—
115#103—5021no—
116#103—5021no—
117C4×C8C2×D25111yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—10042no—
121C2×Q8—10042no—
122C2×Q8—5022no—
123C2×Q8—2511no—
124C2×D4—2511no—
125C22×C4—5022no—
126C22×C4—5021no—
127SD16—10042no—
128C2×C8—21no—
129C2×C8—5021no—
130C2×C8—5021no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—5022no—
135C4⋊C4—5021no—
136C4⋊C4—2511no—
137C4⋊C4—2511no—
138C4⋊C4C22⋊D25112yes—
139C22⋊C4—5022no—
140C22⋊C4—5021no—
141C42—5021no—
142C42C22×D25111yescharacteristic
143C23—5021no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—10044no—
149Q8—5021no—
150Q8—2512no—
151D4—5021no—
152D4—2512no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—10044no—
156C2×C4—10041no—
157C2×C4—10041no—
158C2×C4—5021no—
159C2×C4—5021no—
160C2×C4—5021no—
161C2×C4D4×D25111yescharacteristic
162C2×C4D4×D25111yescharacteristic
163C2×C4C2×C22⋊D25111yescharacteristic
164C8—22no—
165C8—10041no—
166C8—10041no—
167C22—10041no—
168C22—10041no—
169C22—5022no—
170C22#2254868311yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—10041no—
175C4—10041no—
176C4—5022no—
177C4SD16×D25111yescharacteristic
178C4SD16×D25111yescharacteristic
179C4#2254874211yescharacteristic
180C4#2254874411yescharacteristic
181C2—10042no—
182C2#3063182411yescharacteristic
183C2#3063183611yescharacteristic
184C2#3063186911yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#498×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#498D251
hypocenterC251#1651
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #30631969C2semidirect
4. #30631433C2semidirect
5. #30631419C2semidirect
7. #498×C251C2semidirect
53. Dic502SD16semidirect
92. C251#1651semidirect
95. #498D251semidirect

Maximal subgroups

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

subgroupindex
#306323522
#306319692
#306318892
#306314332
#306314192
#306314162
#498×C2512
#1651251