Finite Group Databasebeta

← #41012510

Subgroups — #41012510

Order 27008.

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

normal subgroups only

This group has 216 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#27213866C211yesmaximal, characteristic, complemented
2#27213493C211yesmaximal, characteristic, complemented
3#27213347C211yesmaximal, characteristic, complemented
4#27213341C211yesmaximal, characteristic
5#27212891C211yesmaximal, characteristic, complemented
6#27212878C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—21no—
9C2×#16809947—21no—
10#20269977C2211yescharacteristic
11#20269974C2211yescharacteristic
12#20269970—22no—
13#20269964—21no—
14#20269956—21no—
15C4×Dic422C2211yescharacteristic
16#20269942—21no—
17#20269936—21no—
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269899—21no—
20#20269899—22no—
21#20269897—21no—
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30#16809951—42no—
31#16809947—42no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D412yes—
36Dic211⋊C4—22no—
37Dic211⋊C4—21no—
38C22⋊Dic211—22no—
39C22⋊Dic211—22no—
40C4×Dic211—22no—
41C4×Dic211—21no—
42C2×D4×C211D411yescharacteristic
43C22×C4×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211D411yescharacteristic
47C22⋊C4×C211—21no—
48C42×C211C2311yescharacteristic
49C22×D211—41no—
50Dic422—21no—
51Dic422—21no—
52Dic422—41no—
53Dic422—41no—
54D844—42no—
55D844—41no—
56D844—41no—
57D844—41no—
58D844—41no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—22no—
62C211⋊C8—41no—
63C23×C211—21no—
64D4×C211—21no—
65D4×C211—21no—
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—41no—
73C8×C211—41no—
74C22×C211—41no—
75C22×C211—41no—
76C22×C211—41no—
77C22×C211#12011yescharacteristic
78C2×D211—81no—
79C2×D211—81no—
80C2×D211—81no—
81C4×C211—21no—
82C4×C211—41no—
83C4×C211#13511yescharacteristic
84C4×C211#13511yescharacteristic
85C4×C211AGL(1,ℤ/8)11yescharacteristic
86C4×C211AGL(1,ℤ/8)11yescharacteristic
87Dic211—41no—
88Dic211—42no—
89Dic211—41no—
90C2×C211—81no—
91C2×C211#44911yescharacteristic
92C2×C211#44911yescharacteristic
93C2×C211#45411yescharacteristic
94D211—161no—
95C211#162811yescharacteristic, complemented, Sylow
96#1628—2111nomaximal, Sylow
97#556—2111no—
98#486—2111no—
99#460—2111no—
100#459—2111no—
101#332—2111no—
102Q8⋊C8—2111no—
103D4⋊C8D21111yescharacteristic, complemented
104C2×SD16—4221no—
105C2×D8—4221no—
106C4⋊Q8—2111no—
107#127—2111no—
108#126—4222no—
109#124—4221no—
110C22⋊Q8—4221no—
111C4×Q8—4221no—
112C4×Q8—2111no—
113C4×D4C2×D21111yescharacteristic
114#117—4221no—
115C4⋊C8—2111no—
116C4⋊C8C2×D21111yescharacteristic
117#102—4221no—
118#102—4222no—
119#102—4221no—
120C4×C8C2×D21111yescharacteristic
121C2×Q8—4221no—
122C2×Q8—2111no—
123C2×D4—4221no—
124C2×D4—4221no—
125C2×D4—4221no—
126C2×D4D84411yescharacteristic
127C22×C4—21no—
128SD16—8442no—
129D8—8442no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—4221no—
133C4⋊C4—4222no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C4⋊C4—2111no—
137C4⋊C4—2112no—
138C4⋊C4D84411yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—4222no—
141C22⋊C4—4222no—
142C42—4222no—
143C42—4221no—
144C42C22×D21111yescharacteristic
145C23—21no—
146C23—8441no—
147Q8—8441no—
148Q8—8441no—
149Q8—4221no—
150Q8—4221no—
151D4—21no—
152D4—21no—
153D4—8442no—
154D4—8441no—
155D4—8441no—
156D4—8441no—
157D4—8441no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—4221no—
162C2×C4—4221no—
163C2×C4—4222no—
164C2×C4D4×D21111yescharacteristic
165C2×C4D4×D21111yescharacteristic
166C2×C4C2×D84411yescharacteristic
167C8—41no—
168C8—41no—
169C8—8441no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—16881no—
174C22—16881no—
175C22—16881no—
176C22#2026994711yescharacteristic
177C4—21no—
178C4—41no—
179C4—8441no—
180C4—8441no—
181C4—8442no—
182C4#2026999811yescharacteristic
183C4#2026999911yescharacteristic
184C4#2027000111yescharacteristic
185C4#2027000511yescharacteristic
186C2—81no—
187C2—33761no—
188C2#2721329211yescharacteristic
189C2#2721329311yescharacteristic
190C2#2721332011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingD4⋊C8×C211C2
socleC22×C211#120
prime radical#27213341C2
hypercenterD4⋊C8D211
hypocenterC211#1628
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
1. #27213866C2semidirect
2. #27213493C2semidirect
3. #27213347C2semidirect
5. #27212891C2semidirect
6. #27212878C2semidirect
7. D4⋊C8×C211C2semidirect
95. C211#1628semidirect
103. D4⋊C8D211semidirect

Maximal subgroups

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

subgroupindex
#272138662
#272134932
#272133472
#272133412
#272128912
#272128782
D4⋊C8×C2112
#1628211