Finite Group Databasebeta

← #44549436

Subgroups — #44549436

Order 32128.

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

normal subgroups only

This group has 236 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#30632397C211yesmaximal, characteristic, complemented
2#30632059C211yesmaximal, characteristic, complemented
3#30631879C211yesmaximal, characteristic
4#30631435C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#460×C251C211yesmaximal, characteristic, complemented
8C2×#16860798—22no—
9C2×#16860795—21no—
10#22548715C2211yescharacteristic
11#22548707—21no—
12#22548693—22no—
13#22548691—21no—
14C4×D1004—21no—
15C4×D1004C2211yescharacteristic
16C4⋊C4×D251—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548636—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×SD16×C251—21no—
21#127×C251C2211yescharacteristic
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25C2×Dic502D412yes—
26C2×D1004D411yescharacteristic
27C2×C4×D251—22no—
28C2×C4×D251—21no—
29#16860798—44no—
30#16860795—42no—
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251D411yescharacteristic
36Dic251⋊C4—22no—
37Dic251⋊C4—22no—
38(C2×D251)⋊C4—22no—
39(C2×D251)⋊C4—21no—
40C2×Q8×C251D411yescharacteristic
41C2×D4×C251—21no—
42C2×D4×C251—21no—
43C2×D4×C251—21no—
44SD16×C251—42no—
45C2×C8×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—42no—
52D1004—21no—
53D1004SD1612yescomplemented
54C2×Dic251—21no—
55C2×Dic251—22no—
56C4×D251—41no—
57C4×D251—41no—
58C4×D251—82no—
59C251⋊C8—22no—
60C251⋊C8—41no—
61C23×C251—41no—
62Q8×C251—21no—
63Q8×C251—21no—
64D4×C251—42no—
65D4×C251—41no—
66D4×C251—41no—
67D4×C251—41no—
68D4×C251—41no—
69C2×C4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251C2×D411yescharacteristic
74C8×C251—41no—
75C22×C251—81no—
76C22×C251—81no—
77C22×C251—81no—
78C22×C251#12011yescharacteristic
79C2×D251—22no—
80C2×D251—41no—
81C2×D251—41no—
82C4×C251—21no—
83C4×C251—41no—
84C4×C251—41no—
85C4×C251C2×SD1611yescharacteristic
86C4×C251#13511yescharacteristic
87C4×C251AGL(1,ℤ/8)11yescharacteristic
88C4×C251#13711yescharacteristic
89Dic251—41no—
90Dic251—42no—
91C2×C251—161no—
92C2×C251#44811yescharacteristic
93C2×C251#44911yescharacteristic
94C2×C251#45311yescharacteristic
95D251—42no—
96C251#162211yescharacteristic, complemented, Sylow
97#1622—2511nomaximal, Sylow
98#554—2511no—
99#492—2511no—
100#460D25111yescharacteristic, complemented
101#459—2511no—
102#333—2511no—
103Q8⋊C8—2511no—
104D4⋊C8—2511no—
105C2×SD16—21no—
106C2×SD16—5022no—
107C2×D8—5021no—
108C4⋊Q8—2511no—
109#127C2×D25111yescharacteristic
110#125—5021no—
111C22⋊Q8—5022no—
112C22⋊Q8—5021no—
113C4×Q8C2×D25111yescharacteristic
114C4×D4—5021no—
115C4×D4—2511no—
116C2×C4⋊C4—5021no—
117C4⋊C8—2511no—
118C4⋊C8C2×D25111yescharacteristic
119#102—21no—
120#102—5021no—
121C4×C8—2511no—
122C2×Q8—2512no—
123C2×Q8C22⋊D25111yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—2511no—
128C22×C4—5022no—
129C22×C4—5021no—
130SD16—42no—
131SD16—10044no—
132D8—10042no—
133C2×C8—21no—
134C2×C8—5021no—
135C2×C8—5021no—
136C4⋊C4—21no—
137C4⋊C4—5022no—
138C4⋊C4—5022no—
139C4⋊C4—5021no—
140C4⋊C4—5021no—
141C4⋊C4—2511no—
142C4⋊C4C22⋊D25111yescharacteristic
143C22⋊C4—5022no—
144C22⋊C4—5021no—
145C42—21no—
146C42C22×D25111yescharacteristic
147C23—41no—
148C23—5021no—
149Q8—21no—
150Q8—21no—
151Q8—10042no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—5021no—
158D4—2512no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—20082no—
162C2×C4—10041no—
163C2×C4—10041no—
164C2×C4—5021no—
165C2×C4—5022no—
166C2×C4D4×D25111yescharacteristic
167C2×C4D4×D25111yescharacteristic
168C2×C4C2×C22⋊D25111yescharacteristic
169C8—41no—
170C8—10041no—
171C8—5022no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—10041no—
176C22—10041no—
177C22—5022no—
178C22#2254868311yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—10041no—
183C4—10042no—
184C4SD16×D25111yescharacteristic
185C4#2254873511yescharacteristic
186C4#2254874111yescharacteristic
187C4#2254874411yescharacteristic
188C2—161no—
189C2—10042no—
190C2#3063182411yescharacteristic
191C2#3063183011yescharacteristic
192C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#460×C251C2
socleC22×C251#120
prime radical#30631879C2
hypercenter#460D251
hypocenterC251#1622
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #30632397C2semidirect
2. #30632059C2semidirect
4. #30631435C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #460×C251C2semidirect
53. D1004SD16semidirect
96. C251#1622semidirect
100. #460D251semidirect

Maximal subgroups

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

subgroupindex
#306323972
#306320592
#306318792
#306314352
Q8⋊(C251⋊C8)2
#306314142
#460×C2512
#1622251