Finite Group Databasebeta

← #44549514

Subgroups — #44549514

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#30632341C211yesmaximal, characteristic
2#30632037C211yesmaximal, characteristic, complemented
3#30631974C211yesmaximal, characteristic, complemented
4#30631434C211yesmaximal, characteristic, complemented
5D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#476×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9#22548715C2211yescharacteristic
10#22548699—22no—
11#22548695—21no—
12#22548689—21no—
13C4×D1004C2211yescharacteristic
14C2×C4⋊Dic251—21no—
15#22548646—21no—
16C4⋊(C251⋊C8)C2211yescharacteristic
17#22548639—22no—
18#22548635—21no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×D4×C251C2211yescharacteristic
22#106×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25#16860809—42no—
26#16860809—42no—
27C2×Dic502—21no—
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—22no—
30C22×Dic251—22no—
31C2×C4×D251—21no—
32C2×C251⋊C8—21no—
33C2×C251⋊C8—21no—
34C4⋊Dic251—22no—
35C4⋊Dic251—21no—
36C4⋊Dic251D412yes—
37C4⋊Dic251D411yescharacteristic
38(C2×D251)⋊C4—22no—
39(C2×D251)⋊C4—21no—
40C2×Q8×C251—21no—
41C2×D4×C251D411yescharacteristic
42C22×C4×C251—21no—
43C2×C8×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251D411yescharacteristic
47C22⋊C4×C251—21no—
48C42×C251C2311yescharacteristic
49C22×D251—21no—
50Dic502—41no—
51Dic502—41no—
52D1004—21no—
53D1004—21no—
54C22⋊D251—44no—
55C2×Dic251—21no—
56C2×Dic251—22no—
57C2×Dic251—44no—
58C4×D251—41no—
59C4×D251—41no—
60C251⋊C8—41no—
61C251⋊C8—41no—
62C23×C251—21no—
63Q8×C251—42no—
64D4×C251—21no—
65D4×C251SD1612yes—
66C2×C4×C251—21no—
67C2×C4×C251—41no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—41no—
74C22×C251—22no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251#12011yescharacteristic
78C2×D251—41no—
79C2×D251—41no—
80C2×D251—41no—
81C4×C251—21no—
82C4×C251—41no—
83C4×C251—81no—
84C4×C251C2×SD1611yescharacteristic
85C4×C251#13511yescharacteristic
86C4×C251AGL(1,ℤ/8)11yescharacteristic
87C4×C251#13711yescharacteristic
88Dic251—41no—
89Dic251—42no—
90C2×C251—42no—
91C2×C251#45011yescharacteristic
92C2×C251#45211yescharacteristic
93C2×C251#45411yescharacteristic
94D251—81no—
95C251#163511yescharacteristic, complemented, Sylow
96#1635—2511nomaximal, Sylow
97#547—2511no—
98#487—2511no—
99#476—2511no—
100#476D25111yescharacteristic, complemented
101#332—2511no—
102D4⋊C8—2511no—
103D4⋊C8—2511no—
104C2×#41—5021no—
105C4⋊Q8—2511no—
106C4⋊Q8C2×D25111yescharacteristic
107#123—5022no—
108C22⋊Q8—5021no—
109#121—5021no—
110C4×D4—2511no—
111C4×D4C2×D25111yescharacteristic
112C2×C4⋊C4—5021no—
113#106—21no—
114#106—5021no—
115C4⋊C8—2511no—
116C4⋊C8C2×D25111yescharacteristic
117#103—5022no—
118#102—21no—
119#102—5021no—
120C4×C8—2511no—
121#41—10042no—
122#41—10042no—
123C2×Q8—21no—
124C2×Q8—5021no—
125C2×D4—5022no—
126C2×D4—2511no—
127C2×D4C22⋊D25111yescharacteristic
128C22×C4—21no—
129C22×C4—5022no—
130C22×C4—5021no—
131C2×C8—21no—
132C2×C8—5021no—
133C2×C8—5021no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—5022no—
137C4⋊C4—5021no—
138C4⋊C4—2512no—
139C4⋊C4—2511no—
140C4⋊C4C22⋊D25111yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—5022no—
143C22⋊C4—5021no—
144C42C22×D25111yescharacteristic
145C23—21no—
146C23—5021no—
147Q8—42no—
148Q8—10041no—
149Q8—10041no—
150D4—21no—
151D4—10044no—
152D4—5021no—
153D4—5021no—
154D4#1686079712yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—10044no—
160C2×C4—10041no—
161C2×C4—10041no—
162C2×C4—5021no—
163C2×C4—5022no—
164C2×C4D4×D25111yescharacteristic
165C2×C4D4×D25111yescharacteristic
166C2×C4C2×C22⋊D25111yescharacteristic
167C8—41no—
168C8—10041no—
169C8—10041no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—10041no—
174C22—10041no—
175C22—10041no—
176C22#2254868311yescharacteristic
177C4—21no—
178C4—41no—
179C4—81no—
180C4—10041no—
181C4—10042no—
182C4C2×#1686079711yescharacteristic
183C4#2254873611yescharacteristic
184C4#2254874011yescharacteristic
185C4#2254874411yescharacteristic
186C2—42no—
187C2—20081no—
188C2#3063183711yescharacteristic
189C2#3063184611yescharacteristic
190C2#3063185811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#476×C251C2
socleC22×C251#120
prime radical#30632341C2
hypercenter#476D251
hypocenterC251#1635
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #30632037C2semidirect
3. #30631974C2semidirect
4. #30631434C2semidirect
5. D4⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #476×C251C2semidirect
95. C251#1635semidirect
100. #476D251semidirect

Maximal subgroups

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

subgroupindex
#306323412
#306320372
#306319742
#306314342
D4⋊(C251⋊C8)2
#306314142
#476×C2512
#1635251