Finite Group Databasebeta

← #44549617

Subgroups — #44549617

Order 32128.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic502C211yesmaximal, characteristic
2#30632092C211yesmaximal, characteristic, complemented
3#30631967C211yesmaximal, characteristic
4#30631898C211yesmaximal, characteristic, complemented
5#30631434C211yesmaximal, characteristic, complemented
6#30631419C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic
8C22×Dic502—21no—
9C2×Dic1004—21no—
10#22548715C2211yescharacteristic
11#22548715C2211yescharacteristic
12#22548714—21no—
13#22548694—21no—
14#22548692—22no—
15C4×Dic502C2211yescharacteristic
16D4×Dic251—21no—
17#22548649—22no—
18#22548648—21no—
19C4⋊(C251⋊C8)C2211yescharacteristic
20#22548640—21no—
21#22548636—21no—
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25C2×Dic502—22no—
26C2×Dic502—21no—
27C2×Dic502—21no—
28C2×Dic502—42no—
29C2×Dic502—42no—
30C2×Dic502D411yescharacteristic
31C22×Dic251—22no—
32Dic1004—42no—
33C2×C251⋊C8—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251D411yescharacteristic
38C4⋊Dic251D412yes—
39Dic251⋊C4—22no—
40Dic251⋊C4—21no—
41C22⋊Dic251—22no—
42C4×Dic251—21no—
43C2×D4×C251D411yescharacteristic
44C22×C4×C251—21no—
45C2×C8×C251—21no—
46C2×C8×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—21no—
49C42×C251C2311yescharacteristic
50Dic502—21no—
51Dic502—44no—
52Dic502—41no—
53Dic502—41no—
54Dic502—41no—
55Dic502—41no—
56Dic502D812yes—
57C2×Dic251—21no—
58C2×Dic251—21no—
59C2×Dic251—22no—
60C2×Dic251—44no—
61C2×Dic251—41no—
62C251⋊C8—41no—
63C23×C251—21no—
64D4×C251—21no—
65D4×C251Q1612yescomplemented
66C2×C4×C251—21no—
67C2×C4×C251—41no—
68C2×C4×C251—41no—
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C8×C251—22no—
73C8×C251—41no—
74C22×C251—22no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251#12011yescharacteristic
78C4×C251—21no—
79C4×C251—41no—
80C4×C251C2×D811yescharacteristic
81C4×C251C2×Q1611yescharacteristic
82C4×C251AGL(1,ℤ/8)11yescharacteristic
83C4×C251#13711yescharacteristic
84Dic251—22no—
85Dic251—42no—
86Dic251—41no—
87Dic251—81no—
88C2×C251—42no—
89C2×C251#44711yescharacteristic
90C2×C251#45111yescharacteristic
91C2×C251#45611yescharacteristic
92C251#164911yescharacteristic, complemented, Sylow
93#1649—2511nomaximal, Sylow
94D4×Q8—2511no—
95#500—2511no—
96#474—2511no—
97#462—2511no—
98#332—2511no—
99Q8⋊C8—2511no—
100D4⋊C8D25111yescharacteristic
101C22×Q8—5021no—
102C2×Q16—5021no—
103C4⋊Q8—5021no—
104C4⋊Q8—2511no—
105C4⋊Q8—2511no—
106C22⋊Q8—5022no—
107C22⋊Q8—5021no—
108C4×Q8—2511no—
109C4×D4—5021no—
110C4×D4C2×D25111yescharacteristic
111#107—5022no—
112#107—5021no—
113C4⋊C8—2511no—
114C4⋊C8C2×D25111yescharacteristic
115#103—5021no—
116#102—5021no—
117C4×C8C2×D25111yescharacteristic
118C2×Q8—10042no—
119C2×Q8—10042no—
120C2×Q8—5022no—
121C2×Q8—5021no—
122C2×Q8—5021no—
123C2×Q8—2511no—
124C2×D4D100411yescharacteristic
125C22×C4—21no—
126C22×C4—5022no—
127Q16—10042no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—5021no—
131C4⋊C4—5022no—
132C4⋊C4—5021no—
133C4⋊C4—5021no—
134C4⋊C4—5021no—
135C4⋊C4—5021no—
136C4⋊C4—2511no—
137C4⋊C4—2512no—
138C4⋊C4D100411yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—5022no—
141C42—5021no—
142C42C22×D25111yescharacteristic
143C23—21no—
144Q8—10044no—
145Q8—10041no—
146Q8—10041no—
147Q8—10041no—
148Q8—10041no—
149Q8—5021no—
150Q8—2512no—
151D4—21no—
152D4Dic100412yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—10041no—
157C2×C4—10044no—
158C2×C4—5021no—
159C2×C4—5022no—
160C2×C4—5021no—
161C2×C4D4×D25111yescharacteristic
162C2×C4D4×D25111yescharacteristic
163C2×C4C2×D100411yescharacteristic
164C8—22no—
165C8—41no—
166C8—10041no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2254868511yescharacteristic
171C4—21no—
172C4—41no—
173C4—20081no—
174C4—10042no—
175C4—10041no—
176C4—5022no—
177C4D8×D25111yescharacteristic
178C4C2×Dic100411yescharacteristic
179C4#2254874211yescharacteristic
180C4#2254874811yescharacteristic
181C2—42no—
182C2#3063182211yescharacteristic
183C2#3063184511yescharacteristic
184C2#3063187111yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingD4⋊C8×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenterD4⋊C8D251
hypocenterC251#1649
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30632092C2semidirect
4. #30631898C2semidirect
5. #30631434C2semidirect
6. #30631419C2semidirect
65. D4×C251Q16semidirect
92. C251#1649semidirect
152. D4Dic1004semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5022
#306320922
#306319672
#306318982
#306314342
#306314192
D4⋊C8×C2512
#1649251