Finite Group Databasebeta

← #41012659

Subgroups — #41012659

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#27213542C211yesmaximal, characteristic, complemented
3#27213428C211yesmaximal, characteristic, complemented
4#27212891C211yesmaximal, characteristic, complemented
5#27212876C211yesmaximal, characteristic, complemented
6D4⋊(C211⋊C8)C211yesmaximal, characteristic
7#461×C211C211yesmaximal, characteristic
8C22×Dic422—21no—
9#20269977C2211yescharacteristic
10#20269976—21no—
11#20269956—21no—
12#20269954—22no—
13C4×Dic422C2211yescharacteristic
14D4×Dic211—21no—
15#20269908—21no—
16#20269908—22no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18#20269900—21no—
19C4×C211⋊C8C2211yescharacteristic
20C2×SD16×C211—21no—
21C4⋊Q8×C211C2211yescharacteristic
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24#103×C211—21no—
25C2×Dic422—22no—
26C2×Dic422—21no—
27C2×Dic422—42no—
28C2×Dic422—42no—
29C2×Dic422D411yescharacteristic
30C22×Dic211—22no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D412yes—
36Dic211⋊C4—22no—
37Dic211⋊C4—21no—
38C22⋊Dic211—22no—
39C4×Dic211—21no—
40C2×Q8×C211—21no—
41C2×D4×C211D411yescharacteristic
42C22×C4×C211—21no—
43SD16×C211—42no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211C2311yescharacteristic
50Dic422—21no—
51Dic422—44no—
52Dic422—41no—
53Dic422—41no—
54Dic422SD1612yescomplemented
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—22no—
58C2×Dic211—44no—
59C211⋊C8—22no—
60C211⋊C8—41no—
61C23×C211—21no—
62Q8×C211—41no—
63Q8×C211—41no—
64D4×C211—21no—
65D4×C211SD1612yes—
66C2×C4×C211—21no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—41no—
74C22×C211—22no—
75C22×C211—41no—
76C22×C211—41no—
77C22×C211#12011yescharacteristic
78C4×C211—21no—
79C4×C211—41no—
80C4×C211—81no—
81C4×C211C2×SD1611yescharacteristic
82C4×C211C2×SD1611yescharacteristic
83C4×C211AGL(1,ℤ/8)11yescharacteristic
84C4×C211#13711yescharacteristic
85Dic211—22no—
86Dic211—42no—
87Dic211—41no—
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#461D21111yescharacteristic
98#332—2111no—
99Q8⋊C8—2111no—
100D4⋊C8—2111no—
101C22×Q8—4221no—
102C2×SD16—21no—
103C4⋊Q8—4221no—
104C4⋊Q8—2111no—
105C4⋊Q8C2×D21111yescharacteristic
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—21no—
116#103—4221no—
117C4×C8—2111no—
118C2×Q8—21no—
119C2×Q8—8442no—
120C2×Q8—8442no—
121C2×Q8—4222no—
122C2×Q8—4221no—
123C2×Q8—2111no—
124C2×D4C22⋊D21111yescharacteristic
125C22×C4—21no—
126C22×C4—4222no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—4221no—
130C2×C8—4221no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—4222no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C4⋊C4—2111no—
137C4⋊C4—2112no—
138C4⋊C4C22⋊D21111yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—4222no—
141C42—4221no—
142C42C22×D21111yescharacteristic
143C23—21no—
144Q8—41no—
145Q8—41no—
146Q8—8444no—
147Q8—8441no—
148Q8—8441no—
149Q8—4221no—
150Q8—2112no—
151D4—21no—
152D4#1680995012yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—8444no—
158C2×C4—4221no—
159C2×C4—4222no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4D4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C8—41no—
165C8—8441no—
166C8—4222no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2026994511yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—8441no—
175C4—8442no—
176C4—4222no—
177C4SD16×D21111yescharacteristic
178C4C2×#1680995011yescharacteristic
179C4#2027000411yescharacteristic
180C4#2027000711yescharacteristic
181C2—42no—
182C2#2721328411yescharacteristic
183C2#2721329511yescharacteristic
184C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#461×C211C2
socleC22×C211#120
prime radicalC2×D4×C211D4
hypercenter#461D211
hypocenterC211#1651
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #27213542C2semidirect
3. #27213428C2semidirect
4. #27212891C2semidirect
5. #27212876C2semidirect
54. 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
#272135422
#272134282
#272128912
#272128762
D4⋊(C211⋊C8)2
#461×C2112
#1651211