Finite Group Databasebeta

← #44549533

Subgroups — #44549533

Order 32128.

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

normal subgroups only

This group has 228 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#30632038C211yesmaximal, characteristic, complemented
3#30631983C211yesmaximal, characteristic, complemented
4#30631977C211yesmaximal, characteristic
5#30631434C211yesmaximal, characteristic, complemented
6#30631419C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic
8C22×Dic502—21no—
9#22548715C2211yescharacteristic
10#22548715C2211yescharacteristic
11#22548714—21no—
12#22548694—21no—
13#22548692—22no—
14C4×Dic502C2211yescharacteristic
15D4×Dic251—21no—
16#22548649—21no—
17#22548646—21no—
18C4⋊(C251⋊C8)C2211yescharacteristic
19#22548640—21no—
20#22548640—22no—
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—
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251D412yes—
37C4⋊Dic251D411yescharacteristic
38Dic251⋊C4—22no—
39Dic251⋊C4—21no—
40C22⋊Dic251—22no—
41C4×Dic251—21no—
42C2×D4×C251D411yescharacteristic
43C22×C4×C251—21no—
44C2×C8×C251—21no—
45C2×C8×C251—21no—
46C4⋊C4×C251D411yescharacteristic
47C22⋊C4×C251—21no—
48C42×C251C2311yescharacteristic
49Dic502—21no—
50Dic502—44no—
51Dic502—41no—
52Dic502—41no—
53Dic502—42no—
54Dic502SD1612yes—
55C2×Dic251—21no—
56C2×Dic251—21no—
57C2×Dic251—22no—
58C2×Dic251—44no—
59C2×Dic251—41no—
60C251⋊C8—41no—
61C23×C251—21no—
62D4×C251—21no—
63D4×C251Q1612yes—
64C2×C4×C251—21no—
65C2×C4×C251—41no—
66C2×C4×C251—41no—
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C8×C251—41no—
71C8×C251—41no—
72C22×C251—22no—
73C22×C251—41no—
74C22×C251—41no—
75C22×C251#12011yescharacteristic
76C4×C251—21no—
77C4×C251—41no—
78C4×C251C2×SD1611yescharacteristic
79C4×C251C2×Q1611yescharacteristic
80C4×C251AGL(1,ℤ/8)11yescharacteristic
81C4×C251#13711yescharacteristic
82Dic251—22no—
83Dic251—42no—
84Dic251—41no—
85Dic251—81no—
86C2×C251—42no—
87C2×C251#45011yescharacteristic
88C2×C251#45111yescharacteristic
89C2×C251#45411yescharacteristic
90C251#163711yescharacteristic, complemented, Sylow
91#1637—2511nomaximal, Sylow
92D4×Q8—2511no—
93#487—2511no—
94#477—2511no—
95#476—2511no—
96#332—2511no—
97Q8⋊C8—2511no—
98D4⋊C8D25111yescharacteristic
99C22×Q8—5021no—
100C4⋊Q8—5021no—
101C4⋊Q8—2511no—
102C4⋊Q8—2511no—
103C22⋊Q8—5022no—
104C22⋊Q8—5021no—
105C4×Q8—2511no—
106C4×D4—5021no—
107C4×D4C2×D25111yescharacteristic
108#107—5021no—
109#106—5021no—
110C4⋊C8—2511no—
111C4⋊C8C2×D25111yescharacteristic
112#103—5021no—
113#103—5022no—
114#102—5021no—
115C4×C8C2×D25111yescharacteristic
116C2×Q8—10042no—
117C2×Q8—10042no—
118C2×Q8—5022no—
119C2×Q8—5021no—
120C2×Q8—5021no—
121C2×Q8—2511no—
122C2×D4D100411yescharacteristic
123C22×C4—21no—
124C22×C4—5022no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—5021no—
128C4⋊C4—5022no—
129C4⋊C4—5021no—
130C4⋊C4—5021no—
131C4⋊C4—5021no—
132C4⋊C4—5021no—
133C4⋊C4—2512no—
134C4⋊C4—2511no—
135C4⋊C4D100411yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—5022no—
138C42—5021no—
139C42C22×D25111yescharacteristic
140C23—21no—
141Q8—10042no—
142Q8—10044no—
143Q8—10041no—
144Q8—10041no—
145Q8—5021no—
146Q8—2512no—
147D4—21no—
148D4Dic100412yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—10041no—
153C2×C4—10044no—
154C2×C4—5021no—
155C2×C4—5022no—
156C2×C4—5021no—
157C2×C4D4×D25111yescharacteristic
158C2×C4D4×D25111yescharacteristic
159C2×C4C2×D100411yescharacteristic
160C8—41no—
161C8—41no—
162C8—10041no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#2254868511yescharacteristic
167C4—21no—
168C4—41no—
169C4—20081no—
170C4—10042no—
171C4—10041no—
172C4—5022no—
173C4SD16×D25111yescharacteristic
174C4C2×Dic100411yescharacteristic
175C4#2254873911yescharacteristic
176C4#2254874811yescharacteristic
177C2—42no—
178C2#3063183911yescharacteristic
179C2#3063184511yescharacteristic
180C2#3063186111yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingD4⋊C8×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenterD4⋊C8D251
hypocenterC251#1637
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. #30632038C2semidirect
3. #30631983C2semidirect
5. #30631434C2semidirect
6. #30631419C2semidirect
90. C251#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5022
#306320382
#306319832
#306319772
#306314342
#306314192
D4⋊C8×C2512
#1637251