Finite Group Databasebeta

← #41012663

Subgroups — #41012663

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
1D4×Dic422C211yesmaximal, characteristic
2#27213544C211yesmaximal, characteristic, complemented
3#27213432C211yesmaximal, characteristic, complemented
4#27213351C211yesmaximal, characteristic
5#27212893C211yesmaximal, characteristic, complemented
6#27212878C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic
8C22×Dic422—21no—
9C2×#16809950—21no—
10#20269977C2211yescharacteristic
11#20269977C2211yescharacteristic
12#20269976—21no—
13#20269956—21no—
14#20269954—22no—
15C4×Dic422C2211yescharacteristic
16D4×Dic211—21no—
17#20269909—21no—
18#20269909—22no—
19C4⋊(C211⋊C8)C2211yescharacteristic
20#20269902—21no—
21#20269900—21no—
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—22no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—42no—
29C2×Dic422—42no—
30C2×Dic422D411yescharacteristic
31C22×Dic211—22no—
32#16809950—42no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D412yes—
39Dic211⋊C4—22no—
40Dic211⋊C4—21no—
41C22⋊Dic211—22no—
42C4×Dic211—21no—
43C2×D4×C211D411yescharacteristic
44C22×C4×C211—21no—
45C2×C8×C211—21no—
46C2×C8×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211C2311yescharacteristic
50Dic422—21no—
51Dic422—44no—
52Dic422—41no—
53Dic422—41no—
54Dic422—41no—
55Dic422—41no—
56Dic422SD1612yescomplemented
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—44no—
61C2×Dic211—41no—
62C211⋊C8—41no—
63C23×C211—21no—
64D4×C211—21no—
65D4×C211SD1612yes—
66C2×C4×C211—21no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C8×C211—22no—
73C8×C211—41no—
74C22×C211—22no—
75C22×C211—41no—
76C22×C211—41no—
77C22×C211#12011yescharacteristic
78C4×C211—21no—
79C4×C211—41no—
80C4×C211C2×SD1611yescharacteristic
81C4×C211C2×SD1611yescharacteristic
82C4×C211AGL(1,ℤ/8)11yescharacteristic
83C4×C211#13711yescharacteristic
84Dic211—22no—
85Dic211—42no—
86Dic211—41no—
87Dic211—81no—
88C2×C211—42no—
89C2×C211#44811yescharacteristic
90C2×C211#45011yescharacteristic
91C2×C211#45611yescharacteristic
92C211#165111yescharacteristic, complemented, Sylow
93#1651—2111nomaximal, Sylow
94D4×Q8—2111no—
95#498—2111no—
96#475—2111no—
97#461—2111no—
98#332—2111no—
99Q8⋊C8—2111no—
100D4⋊C8D21111yescharacteristic
101C22×Q8—4221no—
102C2×SD16—4221no—
103C4⋊Q8—4221no—
104C4⋊Q8—2111no—
105C4⋊Q8—2111no—
106C22⋊Q8—4222no—
107C22⋊Q8—4221no—
108C4×Q8—2111no—
109C4×D4—4221no—
110C4×D4C2×D21111yescharacteristic
111#106—4221no—
112#106—4222no—
113C4⋊C8—2111no—
114C4⋊C8C2×D21111yescharacteristic
115#103—4221no—
116#103—4221no—
117C4×C8C2×D21111yescharacteristic
118C2×Q8—8442no—
119C2×Q8—8442no—
120C2×Q8—4222no—
121C2×Q8—4221no—
122C2×Q8—4221no—
123C2×Q8—2111no—
124C2×D4D84411yescharacteristic
125C22×C4—21no—
126C22×C4—4222no—
127SD16—8442no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—4221no—
131C4⋊C4—4222no—
132C4⋊C4—4221no—
133C4⋊C4—4221no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C4⋊C4—2111no—
137C4⋊C4—2112no—
138C4⋊C4D84411yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—4222no—
141C42—4221no—
142C42C22×D21111yescharacteristic
143C23—21no—
144Q8—8444no—
145Q8—8441no—
146Q8—8441no—
147Q8—8441no—
148Q8—8441no—
149Q8—4221no—
150Q8—2112no—
151D4—21no—
152D4#1680995112yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—8444no—
157C2×C4—8441no—
158C2×C4—4221no—
159C2×C4—4222no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×D84411yescharacteristic
164C8—22no—
165C8—41no—
166C8—8441no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2026994711yescharacteristic
171C4—21no—
172C4—41no—
173C4—8441no—
174C4—8442no—
175C4—16881no—
176C4—4222no—
177C4SD16×D21111yescharacteristic
178C4C2×#1680995111yescharacteristic
179C4#2027000111yescharacteristic
180C4#2027001011yescharacteristic
181C2—42no—
182C2#2721328711yescharacteristic
183C2#2721329811yescharacteristic
184C2#2721333011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingD4⋊C8×C211C2
socleC22×C211#120
prime radicalC2×D4×C211D4
hypercenterD4⋊C8D211
hypocenterC211#1651
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213544C2semidirect
3. #27213432C2semidirect
5. #27212893C2semidirect
6. #27212878C2semidirect
56. Dic422SD16semidirect
92. C211#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic4222
#272135442
#272134322
#272133512
#272128932
#272128782
D4⋊C8×C2112
#1651211