Finite Group Databasebeta

← #44549599

Subgroups — #44549599

Order 32128.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632342C211yesmaximal, characteristic
2#30632090C211yesmaximal, characteristic, complemented
3#30631889C211yesmaximal, characteristic, complemented
4#30631432C211yesmaximal, characteristic, complemented
5D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#474×C251C211yesmaximal, characteristic, complemented
8C2×#16860810—21no—
9C2×#16860797—21no—
10#22548715C2211yescharacteristic
11#22548697—22no—
12#22548694—21no—
13#22548686—21no—
14C4×D1004C2211yescharacteristic
15C4⋊C4×D251—21no—
16#22548648—22no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548639—21no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×D4×C251C2211yescharacteristic
22#107×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25#16860810—42no—
26#16860810—42no—
27C2×Dic502—21no—
28C2×D1004D411yescharacteristic
29C2×C22⋊D251—22no—
30C2×C4×D251—22no—
31C2×C4×D251—21no—
32#16860797—42no—
33C2×C251⋊C8—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37C4⋊Dic251D412yes—
38Dic251⋊C4—22no—
39C22⋊Dic251—22no—
40(C2×D251)⋊C4—21no—
41C2×Q8×C251—21no—
42C2×D4×C251D411yescharacteristic
43C22×C4×C251—21no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—41no—
52Dic502—41no—
53D1004—21no—
54D1004D812yes—
55C22⋊D251—44no—
56C2×Dic251—21no—
57C2×Dic251—22no—
58C4×D251—44no—
59C4×D251—41no—
60C4×D251—41no—
61C251⋊C8—22no—
62C251⋊C8—41no—
63C23×C251—21no—
64Q8×C251—41no—
65Q8×C251—41no—
66D4×C251—21no—
67D4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—41no—
70C2×C4×C251—41no—
71C2×C4×C251—41no—
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C8×C251—41no—
76C22×C251—41no—
77C22×C251—41no—
78C22×C251—41no—
79C22×C251#12011yescharacteristic
80C2×D251—22no—
81C2×D251—41no—
82C2×D251—41no—
83C4×C251—21no—
84C4×C251—41no—
85C4×C251—81no—
86C4×C251C2×D811yescharacteristic
87C4×C251#13511yescharacteristic
88C4×C251AGL(1,ℤ/8)11yescharacteristic
89C4×C251#13711yescharacteristic
90Dic251—42no—
91Dic251—41no—
92C2×C251—81no—
93C2×C251#44711yescharacteristic
94C2×C251#45211yescharacteristic
95C2×C251#45611yescharacteristic
96D251—42no—
97C251#164711yescharacteristic, complemented, Sylow
98#1647—2511nomaximal, Sylow
99#547—2511no—
100#500—2511no—
101#474D25111yescharacteristic, complemented
102#461—2511no—
103#332—2511no—
104D4⋊C8—2511no—
105D4⋊C8—2511no—
106C2×#41—5021no—
107C2×SD16—5021no—
108C4⋊Q8—2511no—
109C4⋊Q8C2×D25111yescharacteristic
110#123—5022no—
111C22⋊Q8—5021no—
112#121—5021no—
113C4×D4—2511no—
114C4×D4C2×D25111yescharacteristic
115C2×C4⋊C4—5021no—
116#107—21no—
117#107—5022no—
118C4⋊C8—2511no—
119C4⋊C8C2×D25111yescharacteristic
120#103—5021no—
121#102—21no—
122C4×C8—2511no—
123#41—10042no—
124#41—10042no—
125C2×Q8—21no—
126C2×Q8—5021no—
127C2×D4—5022no—
128C2×D4—2511no—
129C2×D4C22⋊D25111yescharacteristic
130C22×C4—21no—
131C22×C4—5022no—
132C22×C4—5021no—
133SD16—10042no—
134C2×C8—21no—
135C2×C8—5021no—
136C2×C8—5021no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—5022no—
140C4⋊C4—5021no—
141C4⋊C4—2511no—
142C4⋊C4—2512no—
143C4⋊C4C22⋊D25111yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—5022no—
146C22⋊C4—5021no—
147C42C22×D25111yescharacteristic
148C23—21no—
149C23—5021no—
150Q8—41no—
151Q8—41no—
152Q8—10041no—
153Q8—10041no—
154D4—21no—
155D4—21no—
156D4—10044no—
157D4—5021no—
158D4—2512no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—10044no—
164C2×C4—10041no—
165C2×C4—10041no—
166C2×C4—5021no—
167C2×C4—5022no—
168C2×C4D4×D25111yescharacteristic
169C2×C4D4×D25111yescharacteristic
170C2×C4C2×C22⋊D25111yescharacteristic
171C8—41no—
172C8—10041no—
173C8—5022no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—10041no—
178C22—10041no—
179C22—5022no—
180C22#2254868311yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—10042no—
185C4—10041no—
186C4D8×D25111yescharacteristic
187C4#2254873511yescharacteristic
188C4#2254873911yescharacteristic
189C4#2254874611yescharacteristic
190C2—81no—
191C2—10042no—
192C2#3063181911yescharacteristic
193C2#3063184711yescharacteristic
194C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#474×C251C2
socleC22×C251#120
prime radical#30632342C2
hypercenter#474D251
hypocenterC251#1647
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. #30632090C2semidirect
3. #30631889C2semidirect
4. #30631432C2semidirect
5. D4⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #474×C251C2semidirect
97. C251#1647semidirect
101. #474D251semidirect

Maximal subgroups

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

subgroupindex
#306323422
#306320902
#306318892
#306314322
D4⋊(C251⋊C8)2
#306314142
#474×C2512
#1647251