Finite Group Databasebeta

← #44549592

Subgroups — #44549592

Order 32128.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632402C211yesmaximal, characteristic
2#30632039C211yesmaximal, characteristic, complemented
3#30631889C211yesmaximal, characteristic, complemented
4#30631435C211yesmaximal, characteristic, complemented
5D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#474×C251C211yesmaximal, characteristic, complemented
8C2×#16860797—21no—
9#22548715C2211yescharacteristic
10#22548701—22no—
11#22548689—21no—
12#22548686—21no—
13C4×D1004C2211yescharacteristic
14#22548674—21no—
15#22548673—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548639—21no—
18#22548638—22no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×D4×C251C2211yescharacteristic
22#107×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25C2×Dic502D412yes—
26C2×D1004D411yescharacteristic
27C2×C22⋊D251—22no—
28C2×C4×D251—21no—
29#16860797—42no—
30C2×C251⋊C8—21no—
31C2×C251⋊C8—21no—
32C4⋊Dic251—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251D411yescharacteristic
35C22⋊Dic251—22no—
36(C2×D251)⋊C4—22no—
37(C2×D251)⋊C4—21no—
38C4×Dic251—22no—
39C2×Q8×C251—21no—
40C2×D4×C251D411yescharacteristic
41C22×C4×C251—21no—
42C2×C8×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—21no—
47C42×C251C2311yescharacteristic
48C22×D251—21no—
49Dic502—42no—
50D1004—21no—
51D1004—21no—
52C22⋊D251—82no—
53C2×Dic251—21no—
54C2×Dic251—22no—
55C4×D251—41no—
56C4×D251—41no—
57C251⋊C8—41no—
58C251⋊C8—41no—
59C23×C251—21no—
60Q8×C251—41no—
61Q8×C251—41no—
62D4×C251—21no—
63D4×C251—21no—
64C2×C4×C251—21no—
65C2×C4×C251—41no—
66C2×C4×C251—41no—
67C2×C4×C251—41no—
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C8×C251—41no—
72C22×C251—41no—
73C22×C251—41no—
74C22×C251—41no—
75C22×C251#12011yescharacteristic
76C2×D251—41no—
77C2×D251—41no—
78C2×D251—41no—
79C4×C251—21no—
80C4×C251—41no—
81C4×C251—81no—
82C4×C251#13511yescharacteristic
83C4×C251#13511yescharacteristic
84C4×C251AGL(1,ℤ/8)11yescharacteristic
85C4×C251#13711yescharacteristic
86Dic251—41no—
87Dic251—42no—
88C2×C251—81no—
89C2×C251#44911yescharacteristic
90C2×C251#45211yescharacteristic
91C2×C251#45511yescharacteristic
92D251—81no—
93C251#164611yescharacteristic, complemented, Sylow
94#1646—2511nomaximal, Sylow
95#555—2511no—
96#487—2511no—
97#474D25111yescharacteristic, complemented
98#461—2511no—
99#333—2511no—
100D4⋊C8—2511no—
101D4⋊C8—2511no—
102C2×SD16—5021no—
103C4⋊Q8—2511no—
104C4⋊Q8C2×D25111yescharacteristic
105#124—5022no—
106#121—5021no—
107#121—5021no—
108C4×D4—2511no—
109C4×D4C2×D25111yescharacteristic
110#117—5021no—
111#117—5021no—
112#107—21no—
113C4⋊C8—2511no—
114C4⋊C8C2×D25111yescharacteristic
115#103—5022no—
116#103—5021no—
117#102—21no—
118C4×C8—2511no—
119C2×Q8—21no—
120C2×Q8—2512no—
121C2×D4—5022no—
122C2×D4—2511no—
123C2×D4C22⋊D25111yescharacteristic
124C22×C4—21no—
125C22×C4—5021no—
126SD16—10042no—
127C2×C8—21no—
128C2×C8—5021no—
129C2×C8—5021no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—5021no—
133C4⋊C4—5021no—
134C4⋊C4—2511no—
135C4⋊C4C22⋊D25111yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—5022no—
138C22⋊C4—5022no—
139C22⋊C4—5021no—
140C42—5022no—
141C42C22×D25111yescharacteristic
142C23—21no—
143C23—5021no—
144Q8—41no—
145Q8—41no—
146Q8—10042no—
147D4—21no—
148D4—21no—
149D4—20082no—
150D4—5021no—
151D4—5021no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—10041no—
157C2×C4—10041no—
158C2×C4—5021no—
159C2×C4—5022no—
160C2×C4D4×D25111yescharacteristic
161C2×C4D4×D25111yescharacteristic
162C2×C4C2×C22⋊D25111yescharacteristic
163C8—41no—
164C8—10041no—
165C8—10041no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—10041no—
170C22—10041no—
171C22—10041no—
172C22#2254868311yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—10041no—
177C4—10042no—
178C4#2254873411yescharacteristic
179C4#2254873511yescharacteristic
180C4#2254873911yescharacteristic
181C4#2254874611yescharacteristic
182C2—81no—
183C2—20081no—
184C2#3063182911yescharacteristic
185C2#3063184711yescharacteristic
186C2#3063186311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#474×C251C2
socleC22×C251#120
prime radical#30632402C2
hypercenter#474D251
hypocenterC251#1646
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #30632039C2semidirect
3. #30631889C2semidirect
4. #30631435C2semidirect
5. D4⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #474×C251C2semidirect
93. C251#1646semidirect
97. #474D251semidirect

Maximal subgroups

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

subgroupindex
#306324022
#306320392
#306318892
#306314352
D4⋊(C251⋊C8)2
#306314142
#474×C2512
#1646251