Finite Group Databasebeta

← #44549612

Subgroups — #44549612

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
1Q8×D1004C211yesmaximal, characteristic
2#30632090C211yesmaximal, characteristic, complemented
3#30631964C211yesmaximal, characteristic
4#30631434C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic
7#462×C251C211yesmaximal, characteristic, complemented
8C2×Q8×D251—21no—
9#22548715—21no—
10#22548715C2211yescharacteristic
11#22548695—22no—
12#22548695—21no—
13C4×D1004—21no—
14C4×D1004C2211yescharacteristic
15#22548648—21no—
16#22548648—22no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548635—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×Q16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502—22no—
26C2×Dic502—21no—
27Q8×D251—42no—
28Q8×D251—42no—
29C2×D1004D411yescharacteristic
30C2×C4×D251—22no—
31C2×C4×D251—21no—
32C2×C251⋊C8—21no—
33C2×C251⋊C8—21no—
34C4⋊Dic251—22no—
35C4⋊Dic251—21no—
36C4⋊Dic251D411yescharacteristic
37C4⋊Dic251D412yes—
38(C2×D251)⋊C4—22no—
39(C2×D251)⋊C4—21no—
40C2×Q8×C251—21no—
41C2×Q8×C251D411yescharacteristic
42Q16×C251—42no—
43C2×C8×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—44no—
52Dic502—41no—
53Dic502—41no—
54D1004—21no—
55D1004Q1612yescomplemented
56C2×Dic251—22no—
57C2×Dic251—21no—
58C4×D251—44no—
59C4×D251—41no—
60C4×D251—41no—
61C251⋊C8—22no—
62C251⋊C8—41no—
63Q8×C251—21no—
64Q8×C251—41no—
65Q8×C251—41no—
66Q8×C251D812yes—
67C2×C4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—41no—
74C22×C251#12011yescharacteristic
75C2×D251—22no—
76C2×D251—41no—
77C2×D251—41no—
78C4×C251—22no—
79C4×C251—21no—
80C4×C251—41no—
81C4×C251—81no—
82C4×C251C2×D811yescharacteristic
83C4×C251C2×Q1611yescharacteristic
84C4×C251AGL(1,ℤ/8)11yescharacteristic
85C4×C251#13711yescharacteristic
86Dic251—42no—
87Dic251—41no—
88C2×C251#44711yescharacteristic
89C2×C251#45111yescharacteristic
90C2×C251#45611yescharacteristic
91D251—42no—
92C251#164911yescharacteristic, complemented, Sylow
93#1649—2511nomaximal, Sylow
94D4×Q8—2511no—
95#500—2511no—
96#474—2511no—
97#462D25111yescharacteristic, complemented
98#332—2511no—
99Q8⋊C8—2511no—
100D4⋊C8—2511no—
101C22×Q8—5021no—
102C2×Q16—21no—
103C4⋊Q8—5021no—
104C4⋊Q8—2511no—
105C4⋊Q8C2×D25111yescharacteristic
106C22⋊Q8—5022no—
107C22⋊Q8—5021no—
108C4×Q8C2×D25111yescharacteristic
109C4×D4—5021no—
110C4×D4—2511no—
111#107—5021no—
112#107—5022no—
113C4⋊C8—2511no—
114C4⋊C8C2×D25111yescharacteristic
115#103—21no—
116#102—5021no—
117C4×C8—2511no—
118C2×Q8—21no—
119C2×Q8—10042no—
120C2×Q8—10042no—
121C2×Q8—5022no—
122C2×Q8—5021no—
123C2×Q8C22⋊D25111yescharacteristic
124C2×D4—2511no—
125C22×C4—5022no—
126C22×C4—5021no—
127Q16—42no—
128C2×C8—21no—
129C2×C8—5021no—
130C2×C8—5021no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—5022no—
135C4⋊C4—5021no—
136C4⋊C4—2511no—
137C4⋊C4—2512no—
138C4⋊C4C22⋊D25111yescharacteristic
139C22⋊C4—5022no—
140C22⋊C4—5021no—
141C42—21no—
142C42C22×D25111yescharacteristic
143C23—5021no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—10044no—
148Q8—10041no—
149Q8—10041no—
150Q8#1686079512yes—
151D4—5021no—
152D4—2512no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—10044no—
157C2×C4—10041no—
158C2×C4—10041no—
159C2×C4—5022no—
160C2×C4—5021no—
161C2×C4D4×D25111yescharacteristic
162C2×C4D4×D25111yescharacteristic
163C2×C4C2×C22⋊D25111yescharacteristic
164C8—41no—
165C8—10041no—
166C8—5022no—
167C22—10041no—
168C22—10041no—
169C22—5022no—
170C22#2254868311yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—10042no—
176C4—10041no—
177C4C2×#1686079511yescharacteristic
178C4Q16×D25111yescharacteristic
179C4#2254874011yescharacteristic
180C4#2254874711yescharacteristic
181C2—10042no—
182C2#3063182011yescharacteristic
183C2#3063184111yescharacteristic
184C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#462×C251C2
socleC22×C251#120
prime radicalC2×D1004D4
hypercenter#462D251
hypocenterC251#1649
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #30632090C2semidirect
4. #30631434C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
7. #462×C251C2semidirect
55. D1004Q16semidirect
92. C251#1649semidirect
97. #462D251semidirect

Maximal subgroups

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

subgroupindex
Q8×D10042
#306320902
#306319642
#306314342
Q8⋊(C251⋊C8)2
#306314142
#462×C2512
#1649251