Finite Group Databasebeta

← #44549440

Subgroups — #44549440

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#30632060C211yesmaximal, characteristic, complemented
3#30631887C211yesmaximal, characteristic, complemented
4#30631883C211yesmaximal, characteristic
5#30631435C211yesmaximal, characteristic, complemented
6#30631416C211yesmaximal, characteristic, complemented
7Q8⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#16860799—22no—
9C2×#16860797—21no—
10C2×D2008—21no—
11#22548715C2211yescharacteristic
12#22548712C2211yescharacteristic
13#22548707—21no—
14#22548693—22no—
15#22548691—21no—
16C4×D1004—21no—
17C4×D1004C2211yescharacteristic
18C4⋊C4×D251—21no—
19C4⋊(C251⋊C8)C2211yescharacteristic
20#22548637—21no—
21#22548635—21no—
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25C2×Dic502D412yes—
26C2×D1004—21no—
27C2×D1004—21no—
28C2×D1004—21no—
29C2×D1004D411yescharacteristic
30C2×C4×D251—22no—
31C2×C4×D251—21no—
32#16860799—44no—
33#16860797—42no—
34D2008—42no—
35C2×C251⋊C8—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251—21no—
38C4⋊Dic251D411yescharacteristic
39Dic251⋊C4—22no—
40Dic251⋊C4—22no—
41(C2×D251)⋊C4—22no—
42(C2×D251)⋊C4—21no—
43C2×Q8×C251D411yescharacteristic
44C2×C8×C251—21no—
45C2×C8×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51C22×D251—41no—
52Dic502—42no—
53D1004—21no—
54D1004—42no—
55D1004—41no—
56D1004—41no—
57D1004—41no—
58D1004—41no—
59D1004SD1612yescomplemented
60C2×Dic251—21no—
61C2×Dic251—22no—
62C4×D251—41no—
63C4×D251—41no—
64C4×D251—82no—
65C251⋊C8—41no—
66Q8×C251—21no—
67Q8×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—22no—
74C8×C251—41no—
75C22×C251#12011yescharacteristic
76C2×D251—22no—
77C2×D251—41no—
78C2×D251—41no—
79C2×D251—81no—
80C2×D251—81no—
81C2×D251—81no—
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#44811yescharacteristic
92C2×C251#44911yescharacteristic
93C2×C251#45311yescharacteristic
94D251—42no—
95D251—161no—
96C251#162211yescharacteristic, complemented, Sylow
97#1622—2511nomaximal, Sylow
98#554—2511no—
99#492—2511no—
100#460—2511no—
101#459—2511no—
102#333—2511no—
103Q8⋊C8D25111yescharacteristic, complemented
104D4⋊C8—2511no—
105C2×SD16—5022no—
106C2×SD16—5021no—
107C2×D8—5021no—
108C4⋊Q8—2511no—
109#127—2511no—
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—5021no—
120#102—5021no—
121C4×C8C2×D25111yescharacteristic
122C2×Q8—2512no—
123C2×Q8D100411yescharacteristic
124C2×D4—5021no—
125C2×D4—5021no—
126C2×D4—5021no—
127C2×D4—2511no—
128C22×C4—5022no—
129C22×C4—5021no—
130SD16—10044no—
131SD16—10042no—
132D8—10042no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—5021no—
136C4⋊C4—21no—
137C4⋊C4—5022no—
138C4⋊C4—5022no—
139C4⋊C4—5021no—
140C4⋊C4—5021no—
141C4⋊C4—2511no—
142C4⋊C4D100411yescharacteristic
143C22⋊C4—5022no—
144C22⋊C4—5021no—
145C42—21no—
146C42C22×D25111yescharacteristic
147C23—10041no—
148C23—5021no—
149Q8—21no—
150Q8—21no—
151Q8—10042no—
152D4—10042no—
153D4—10041no—
154D4—10041no—
155D4—10041no—
156D4—10041no—
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×D100411yescharacteristic
169C8—22no—
170C8—41no—
171C8—10041no—
172C22—20081no—
173C22—20081no—
174C22—20081no—
175C22—10041no—
176C22—10041no—
177C22—5022no—
178C22#2254868511yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—10041no—
183C4—10042no—
184C4SD16×D25111yescharacteristic
185C4#2254873711yescharacteristic
186C4#2254874311yescharacteristic
187C4#2254874711yescharacteristic
188C2—40161no—
189C2—10042no—
190C2#3063182711yescharacteristic
191C2#3063183411yescharacteristic
192C2#3063185611yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingQ8⋊C8×C251C2
socleC22×C251#120
prime radical#30631883C2
hypercenterQ8⋊C8D251
hypocenterC251#1622
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #30632397C2semidirect
2. #30632060C2semidirect
3. #30631887C2semidirect
5. #30631435C2semidirect
6. #30631416C2semidirect
7. Q8⋊C8×C251C2semidirect
59. D1004SD16semidirect
96. C251#1622semidirect
103. Q8⋊C8D251semidirect

Maximal subgroups

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

subgroupindex
#306323972
#306320602
#306318872
#306318832
#306314352
#306314162
Q8⋊C8×C2512
#1622251