Finite Group Databasebeta

← #41012660

Subgroups — #41012660

Order 27008.

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×D844C211yesmaximal, characteristic
2#27213542C211yesmaximal, characteristic, complemented
3#27213348C211yesmaximal, characteristic
4#27212893C211yesmaximal, characteristic, complemented
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic
7#475×C211C211yesmaximal, characteristic, complemented
8C2×Q8×D211—21no—
9C2×#16809949—21no—
10#20269977—21no—
11#20269977C2211yescharacteristic
12#20269957—22no—
13#20269957—21no—
14C4×D844—21no—
15C4×D844C2211yescharacteristic
16#20269908—22no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18#20269901—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4×Q8×C211C2211yescharacteristic
22#106×C211—21no—
23C4⋊C8×C211C2211yescharacteristic
24#103×C211—21no—
25C2×Dic422—22no—
26C2×Dic422—21no—
27Q8×D211—42no—
28Q8×D211—42no—
29C2×D844D411yescharacteristic
30C2×C4×D211—22no—
31C2×C4×D211—21no—
32#16809949—42no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—22no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D412yes—
39(C2×D211)⋊C4—22no—
40(C2×D211)⋊C4—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211D411yescharacteristic
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—44no—
52Dic422—41no—
53Dic422—41no—
54D844—21no—
55D844SD1612yes—
56C2×Dic211—22no—
57C2×Dic211—21no—
58C4×D211—44no—
59C4×D211—41no—
60C4×D211—41no—
61C211⋊C8—22no—
62C211⋊C8—41no—
63Q8×C211—21no—
64Q8×C211—41no—
65Q8×C211—41no—
66Q8×C211SD1612yescomplemented
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—41no—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—41no—
74C22×C211#12011yescharacteristic
75C2×D211—22no—
76C2×D211—41no—
77C2×D211—41no—
78C4×C211—22no—
79C4×C211—21no—
80C4×C211—41no—
81C4×C211—81no—
82C4×C211C2×SD1611yescharacteristic
83C4×C211C2×SD1611yescharacteristic
84C4×C211AGL(1,ℤ/8)11yescharacteristic
85C4×C211#13711yescharacteristic
86Dic211—42no—
87Dic211—41no—
88C2×C211#44811yescharacteristic
89C2×C211#45011yescharacteristic
90C2×C211#45611yescharacteristic
91D211—42no—
92C211#165111yescharacteristic, complemented, Sylow
93#1651—2111nomaximal, Sylow
94D4×Q8—2111no—
95#498—2111no—
96#475D21111yescharacteristic, complemented
97#461—2111no—
98#332—2111no—
99Q8⋊C8—2111no—
100D4⋊C8—2111no—
101C22×Q8—4221no—
102C2×SD16—4221no—
103C4⋊Q8—4221no—
104C4⋊Q8—2111no—
105C4⋊Q8C2×D21111yescharacteristic
106C22⋊Q8—4222no—
107C22⋊Q8—4221no—
108C4×Q8C2×D21111yescharacteristic
109C4×D4—4221no—
110C4×D4—2111no—
111#106—21no—
112#106—4222no—
113C4⋊C8—2111no—
114C4⋊C8C2×D21111yescharacteristic
115#103—21no—
116#103—4221no—
117C4×C8—2111no—
118C2×Q8—21no—
119C2×Q8—8442no—
120C2×Q8—8442no—
121C2×Q8—4222no—
122C2×Q8—4221no—
123C2×Q8C22⋊D21111yescharacteristic
124C2×D4—2111no—
125C22×C4—4222no—
126C22×C4—4221no—
127SD16—8442no—
128C2×C8—21no—
129C2×C8—4221no—
130C2×C8—4221no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—4222no—
135C4⋊C4—4221no—
136C4⋊C4—2111no—
137C4⋊C4—2112no—
138C4⋊C4C22⋊D21111yescharacteristic
139C22⋊C4—4222no—
140C22⋊C4—4221no—
141C42—21no—
142C42C22×D21111yescharacteristic
143C23—4221no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—8444no—
148Q8—8441no—
149Q8—8441no—
150Q8#1680994912yescomplemented
151D4—4221no—
152D4—2112no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—8444no—
157C2×C4—8441no—
158C2×C4—8441no—
159C2×C4—4222no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C8—41no—
165C8—8441no—
166C8—4222no—
167C22—8441no—
168C22—8441no—
169C22—4222no—
170C22#2026994511yescharacteristic
171C4—21no—
172C4—22no—
173C4—41no—
174C4—81no—
175C4—8442no—
176C4—8441no—
177C4SD16×D21111yescharacteristic
178C4C2×#1680994911yescharacteristic
179C4#2027000211yescharacteristic
180C4#2027000611yescharacteristic
181C2—8442no—
182C2#2721328311yescharacteristic
183C2#2721329611yescharacteristic
184C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#475×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#475D211
hypocenterC211#1651
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #27213542C2semidirect
4. #27212893C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
7. #475×C211C2semidirect
66. Q8×C211SD16semidirect
92. C211#1651semidirect
96. #475D211semidirect
150. Q8#16809949semidirect

Maximal subgroups

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

subgroupindex
Q8×D8442
#272135422
#272133482
#272128932
Q8⋊(C211⋊C8)2
#272128732
#475×C2112
#1651211