Finite Group Databasebeta

← #41012668

Subgroups — #41012668

Order 27008.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213542C211yesmaximal, characteristic, complemented
2#27213426C211yesmaximal, characteristic
3#27213356C211yesmaximal, characteristic, complemented
4#27212892C211yesmaximal, characteristic, complemented
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C211⋊C8)C211yesmaximal, characteristic
7#556×C211C211yesmaximal, characteristic
8C2×#16809952—21no—
9#20269977C2211yescharacteristic
10#20269910—21no—
11#20269908—22no—
12C4⋊(C211⋊C8)C2211yescharacteristic
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269900—21no—
15#20269898—21no—
16C4×C211⋊C8C2211yescharacteristic
17C4⋊Q8×C211C2211yescharacteristic
18#126×C211—22no—
19#124×C211—21no—
20C22⋊Q8×C211—21no—
21C4×Q8×C211—21no—
22C4×Q8×C211C2211yescharacteristic
23C4×D4×C211C2211yescharacteristic
24#117×C211—21no—
25C2×Dic422—21no—
26#16809952—42no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C2×Q8×C211—21no—
33C2×Q8×C211D411yescharacteristic
34C2×D4×C211D411yescharacteristic
35C22×C4×C211—21no—
36C4⋊C4×C211—22no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211D411yescharacteristic
40C4⋊C4×C211D411yescharacteristic
41C4⋊C4×C211D412yes—
42C22⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—21no—
45C42×C211—22no—
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48Dic422—41no—
49Dic422—41no—
50C2×Dic211—41no—
51C211⋊C8—22no—
52C211⋊C8—41no—
53C211⋊C8—41no—
54C23×C211—21no—
55Q8×C211—21no—
56Q8×C211—21no—
57Q8×C211—41no—
58Q8×C211—41no—
59D4×C211—21no—
60D4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—22no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C22×C211—41no—
71C22×C211—41no—
72C22×C211—41no—
73C22×C211#12011yescharacteristic
74C4×C211—21no—
75C4×C211—41no—
76C4×C211—41no—
77C4×C211—42no—
78C4×C211—41no—
79C4×C211#13511yescharacteristic
80C4×C211#13511yescharacteristic
81C4×C211AGL(1,ℤ/8)11yescharacteristic
82C4×C211#13711yescharacteristic
83Dic211—81no—
84C2×C211—81no—
85C2×C211#44911yescharacteristic
86C2×C211#45211yescharacteristic
87C2×C211#45611yescharacteristic
88C211#165211yescharacteristic, complemented, Sylow
89#1652—2111nomaximal, Sylow
90#556D21111yescharacteristic
91#498—2111no—
92#474—2111no—
93#462—2111no—
94#332—2111no—
95Q8⋊C8—2111no—
96D4⋊C8—2111no—
97C2×Q16—4221no—
98C4⋊Q8—2111no—
99C4⋊Q8C2×D21111yescharacteristic
100#126—22no—
101#124—21no—
102C22⋊Q8—21no—
103C4×Q8—21no—
104C4×Q8C2×D21111yescharacteristic
105C4×D4C2×D21111yescharacteristic
106#117—21no—
107#107—4221no—
108#106—4222no—
109C4⋊C8—2111no—
110C4⋊C8—2111no—
111#103—4221no—
112#102—4221no—
113C4×C8—2111no—
114C2×Q8—21no—
115C2×Q8—4221no—
116C2×Q8C22⋊D21111yescharacteristic
117C2×D4C22⋊D21111yescharacteristic
118C22×C4—21no—
119Q16—8442no—
120C2×C8—4221no—
121C2×C8—4221no—
122C2×C8—4221no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4C22⋊D21111yescharacteristic
129C4⋊C4C22⋊D21112yes—
130C4⋊C4C22⋊D21111yescharacteristic
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—22no—
135C42—21no—
136C42C22×D21111yescharacteristic
137C23—21no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—8441no—
143Q8—8441no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—8441no—
153C2×C4C2×C22⋊D21111yescharacteristic
154C2×C4C2×C22⋊D21111yescharacteristic
155C2×C4C2×C22⋊D21111yescharacteristic
156C8—8441no—
157C8—8441no—
158C8—4222no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2026994611yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—41no—
167C4—42no—
168C4—16881no—
169C4#2026999711yescharacteristic
170C4#2026999711yescharacteristic
171C4#2027000211yescharacteristic
172C4#2027000711yescharacteristic
173C2—81no—
174C2#2721329111yescharacteristic
175C2#2721330811yescharacteristic
176C2#2721332911yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#556×C211C2
socleC22×C211#120
prime radicalC2×D4×C211D4
hypercenter#556D211
hypocenterC211#1652
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #27213542C2semidirect
3. #27213356C2semidirect
4. #27212892C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
88. C211#1652semidirect

Maximal subgroups

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

subgroupindex
#272135422
#272134262
#272133562
#272128922
Q8⋊(C211⋊C8)2
D4⋊(C211⋊C8)2
#556×C2112
#1652211