Finite Group Databasebeta

← #41012628

Subgroups — #41012628

Order 27008.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#27213861C211yesmaximal, characteristic
2#27213497C211yesmaximal, characteristic, complemented
3#27213427C211yesmaximal, characteristic, complemented
4#27213351C211yesmaximal, characteristic, complemented
5#27212894C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#16809950—21no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269963—22no—
12#20269951—21no—
13#20269948—21no—
14C4×D844C2211yescharacteristic
15#20269936—21no—
16#20269935—21no—
17#20269911—21no—
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269902—22no—
20#20269900—21no—
21#20269899—21no—
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422D412yes—
27C2×D844D411yescharacteristic
28C2×C22⋊D211—22no—
29C2×C4×D211—21no—
30#16809950—42no—
31C2×C211⋊C8—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37C22⋊Dic211—22no—
38(C2×D211)⋊C4—22no—
39(C2×D211)⋊C4—21no—
40C4×Dic211—22no—
41C2×D4×C211D411yescharacteristic
42C22×C4×C211—21no—
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211D411yescharacteristic
46C22⋊C4×C211—21no—
47C42×C211C2311yescharacteristic
48C22×D211—21no—
49Dic422—42no—
50Dic422—41no—
51Dic422—41no—
52D844—21no—
53D844—21no—
54C22⋊D211—82no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C2×Dic211—41no—
58C4×D211—41no—
59C4×D211—41no—
60C211⋊C8—41no—
61C23×C211—21no—
62D4×C211—21no—
63D4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C8×C211—41no—
71C8×C211—41no—
72C22×C211—41no—
73C22×C211—41no—
74C22×C211—41no—
75C22×C211#12011yescharacteristic
76C2×D211—41no—
77C2×D211—41no—
78C2×D211—41no—
79C4×C211—21no—
80C4×C211—41no—
81C4×C211#13511yescharacteristic
82C4×C211#13511yescharacteristic
83C4×C211AGL(1,ℤ/8)11yescharacteristic
84C4×C211#13711yescharacteristic
85Dic211—41no—
86Dic211—42no—
87Dic211—81no—
88C2×C211—81no—
89C2×C211#44911yescharacteristic
90C2×C211#45211yescharacteristic
91C2×C211#45511yescharacteristic
92D211—81no—
93C211#164611yescharacteristic, complemented, Sylow
94#1646—2111nomaximal, Sylow
95#555—2111no—
96#487—2111no—
97#474—2111no—
98#461—2111no—
99#333—2111no—
100D4⋊C8—2111no—
101D4⋊C8D21111yescharacteristic, complemented
102C2×SD16—4221no—
103C4⋊Q8—2111no—
104C4⋊Q8—2111no—
105#124—4222no—
106#121—4221no—
107#121—4221no—
108C4×D4—2111no—
109C4×D4C2×D21111yescharacteristic
110#117—4221no—
111#117—4221no—
112#107—4221no—
113C4⋊C8—2111no—
114C4⋊C8C2×D21111yescharacteristic
115#103—4222no—
116#103—4221no—
117#102—4221no—
118C4×C8C2×D21111yescharacteristic
119C2×Q8—4221no—
120C2×Q8—2112no—
121C2×D4—4222no—
122C2×D4—2111no—
123C2×D4D84411yescharacteristic
124C22×C4—21no—
125C22×C4—4221no—
126SD16—8442no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—4221no—
130C4⋊C4—4221no—
131C4⋊C4—4221no—
132C4⋊C4—4221no—
133C4⋊C4—4221no—
134C4⋊C4—2111no—
135C4⋊C4D84411yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—4222no—
138C22⋊C4—4222no—
139C22⋊C4—4221no—
140C42—4222no—
141C42C22×D21111yescharacteristic
142C23—21no—
143C23—4221no—
144Q8—8442no—
145Q8—8441no—
146Q8—8441no—
147D4—21no—
148D4—21no—
149D4—16882no—
150D4—4221no—
151D4—4221no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—8441no—
156C2×C4—8441no—
157C2×C4—8441no—
158C2×C4—4221no—
159C2×C4—4222no—
160C2×C4D4×D21111yescharacteristic
161C2×C4D4×D21111yescharacteristic
162C2×C4C2×D84411yescharacteristic
163C8—41no—
164C8—41no—
165C8—8441no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—8441no—
170C22—8441no—
171C22—8441no—
172C22#2026994711yescharacteristic
173C4—21no—
174C4—41no—
175C4—8442no—
176C4—8441no—
177C4—16881no—
178C4#2026999611yescharacteristic
179C4#2026999911yescharacteristic
180C4#2027000511yescharacteristic
181C4#2027000611yescharacteristic
182C2—81no—
183C2—16881no—
184C2#2721329311yescharacteristic
185C2#2721330911yescharacteristic
186C2#2721332511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingD4⋊C8×C211C2
socleC22×C211#120
prime radical#27213861C2
hypercenterD4⋊C8D211
hypocenterC211#1646
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. #27213497C2semidirect
3. #27213427C2semidirect
4. #27213351C2semidirect
5. #27212894C2semidirect
6. #27212875C2semidirect
7. D4⋊C8×C211C2semidirect
93. C211#1646semidirect
101. D4⋊C8D211semidirect

Maximal subgroups

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

subgroupindex
#272138612
#272134972
#272134272
#272133512
#272128942
#272128752
D4⋊C8×C2112
#1646211