Finite Group Databasebeta

← #44549626

Subgroups — #44549626

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#30632083C211yesmaximal, characteristic, complemented
3#30631969C211yesmaximal, characteristic, complemented
4#30631432C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic
7#461×C251C211yesmaximal, characteristic
8C22×Dic502—21no—
9#22548715C2211yescharacteristic
10#22548714—21no—
11#22548694—21no—
12#22548692—22no—
13C4×Dic502C2211yescharacteristic
14D4×Dic251—21no—
15#22548646—21no—
16#22548646—22no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548638—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×SD16×C251—21no—
21C4⋊Q8×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25C2×Dic502—22no—
26C2×Dic502—21no—
27C2×Dic502—42no—
28C2×Dic502—42no—
29C2×Dic502D411yescharacteristic
30C22×Dic251—22no—
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251D411yescharacteristic
35C4⋊Dic251D412yes—
36Dic251⋊C4—22no—
37Dic251⋊C4—21no—
38C22⋊Dic251—22no—
39C4×Dic251—21no—
40C2×Q8×C251—21no—
41C2×D4×C251D411yescharacteristic
42C22×C4×C251—21no—
43SD16×C251—42no—
44C2×C8×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—21no—
49C42×C251C2311yescharacteristic
50Dic502—21no—
51Dic502—44no—
52Dic502—41no—
53Dic502—41no—
54Dic502SD1612yescomplemented
55C2×Dic251—21no—
56C2×Dic251—21no—
57C2×Dic251—22no—
58C2×Dic251—44no—
59C251⋊C8—22no—
60C251⋊C8—41no—
61C23×C251—21no—
62Q8×C251—41no—
63Q8×C251—41no—
64D4×C251—21no—
65D4×C251SD1612yes—
66C2×C4×C251—21no—
67C2×C4×C251—41no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C8×C251—41no—
74C22×C251—22no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251#12011yescharacteristic
78C4×C251—21no—
79C4×C251—41no—
80C4×C251—81no—
81C4×C251C2×SD1611yescharacteristic
82C4×C251C2×SD1611yescharacteristic
83C4×C251AGL(1,ℤ/8)11yescharacteristic
84C4×C251#13711yescharacteristic
85Dic251—22no—
86Dic251—42no—
87Dic251—41no—
88C2×C251—42no—
89C2×C251#44811yescharacteristic
90C2×C251#45011yescharacteristic
91C2×C251#45611yescharacteristic
92C251#165111yescharacteristic, complemented, Sylow
93#1651—2511nomaximal, Sylow
94D4×Q8—2511no—
95#498—2511no—
96#475—2511no—
97#461D25111yescharacteristic
98#332—2511no—
99Q8⋊C8—2511no—
100D4⋊C8—2511no—
101C22×Q8—5021no—
102C2×SD16—21no—
103C4⋊Q8—5021no—
104C4⋊Q8—2511no—
105C4⋊Q8C2×D25111yescharacteristic
106C22⋊Q8—5022no—
107C22⋊Q8—5021no—
108C4×Q8—2511no—
109C4×D4—5021no—
110C4×D4C2×D25111yescharacteristic
111#106—5021no—
112#106—5022no—
113C4⋊C8—2511no—
114C4⋊C8C2×D25111yescharacteristic
115#103—21no—
116#103—5021no—
117C4×C8—2511no—
118C2×Q8—21no—
119C2×Q8—10042no—
120C2×Q8—10042no—
121C2×Q8—5022no—
122C2×Q8—5021no—
123C2×Q8—2511no—
124C2×D4C22⋊D25111yescharacteristic
125C22×C4—21no—
126C22×C4—5022no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—5021no—
130C2×C8—5021no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—5022no—
134C4⋊C4—5021no—
135C4⋊C4—5021no—
136C4⋊C4—2511no—
137C4⋊C4—2512no—
138C4⋊C4C22⋊D25111yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—5022no—
141C42—5021no—
142C42C22×D25111yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—10044no—
147Q8—10041no—
148Q8—10041no—
149Q8—5021no—
150Q8—2512no—
151D4—21no—
152D4#1686079812yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—10044no—
158C2×C4—5021no—
159C2×C4—5022no—
160C2×C4—5021no—
161C2×C4D4×D25111yescharacteristic
162C2×C4D4×D25111yescharacteristic
163C2×C4C2×C22⋊D25111yescharacteristic
164C8—41no—
165C8—10041no—
166C8—5022no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2254868311yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—10042no—
175C4—10041no—
176C4—5022no—
177C4SD16×D25111yescharacteristic
178C4C2×#1686079811yescharacteristic
179C4#2254874211yescharacteristic
180C4#2254874511yescharacteristic
181C2—42no—
182C2#3063182511yescharacteristic
183C2#3063183611yescharacteristic
184C2#3063186811yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#461×C251C2
socleC22×C251#120
prime radicalC2×D4×C251D4
hypercenter#461D251
hypocenterC251#1651
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #30632083C2semidirect
3. #30631969C2semidirect
4. #30631432C2semidirect
5. #30631417C2semidirect
54. Dic502SD16semidirect
92. C251#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5022
#306320832
#306319692
#306314322
#306314172
D4⋊(C251⋊C8)2
#461×C2512
#1651251