Finite Group Databasebeta

← #41014502

Subgroups — #41014502

Order 27008.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213946C211yesmaximal, characteristic, complemented
2#27213944C212yesmaximal, complemented
3#27213112C212yesmaximal, complemented
4C211⋊#338C211yesmaximal, characteristic, complemented
5#408×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7#20270010—22no—
8#20270008—22no—
9#20270007—22no—
10#20270005—22no—
11#20270003—22no—
12#20270002—22no—
13#20269999—22no—
14#20269997—24no—
15C2×#16809946C2212yes—
16(C2×D211)⋊C8—24no—
17C2×#41×C211C2211yescharacteristic
18C2×M16×C211C2211yescharacteristic
19C22×C8×C211C2211yescharacteristic
20C22⋊C8×C211—22no—
21#16809962—22no—
22#16809962—22no—
23#16809962—22no—
24#16809962—22no—
25C2×Dic422D412yes—
26C2×D844D412yes—
27C2×C22⋊D211—22no—
28C2×C4×D211—22no—
29Dic844—22no—
30#16809952—24no—
31#16809951—24no—
32#16809950—24no—
33#16809949—24no—
34D1688—22no—
35#16809947—24no—
36#16809946—22no—
37#16809946—22no—
38C2×C211⋊C8D414yes—
39#41×C211—22no—
40#41×C211—22no—
41C2×Q8×C211D411yescharacteristic
42C2×D4×C211—21no—
43C2×D4×C211D411yescharacteristic
44C22×C4×C211—21no—
45C22×C4×C211C2311yescharacteristic
46M16×C211—22no—
47C2×C8×C211—21no—
48C2×C8×C211—22no—
49C2×C8×C211D412yes—
50C22×D211—22no—
51Dic422—22no—
52Dic422—22no—
53D844—22no—
54D844—22no—
55C22⋊D211—42no—
56C22⋊D211—42no—
57C2×Dic211—22no—
58C4×D211—42no—
59C4×D211—42no—
60C211⋊C8—24no—
61C23×C211—21no—
62C23×C211#4111yescharacteristic
63Q8×C211—22no—
64D4×C211—22no—
65D4×C211—42no—
66C2×C4×C211—21no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211C2×D412yes—
70C2×C4×C211C2×D412yes—
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—22no—
74C8×C211—41no—
75C22×C211—41no—
76C22×C211—42no—
77C22×C211—41no—
78C22×C211#12112yes—
79C22×C211#12111yescharacteristic
80C2×D211—44no—
81C2×D211—42no—
82C4×C211—41no—
83C4×C211#12011yescharacteristic
84C4×C211#12012yes—
85C4×C211#12711yescharacteristic
86Dic211—42no—
87C2×C211—22no—
88C2×C211—21no—
89C2×C211—81no—
90C2×C211#39211yescharacteristic
91D211—82no—
92C211#201711yescharacteristic, complemented, Sylow
93#2017—2111nomaximal, Sylow
94#575—2112no—
95#575—2111no—
96#408—2112no—
97#408D21111yescharacteristic, complemented
98#338—2111no—
99C2×#41—2112no—
100C2×#41C2×D21111yescharacteristic
101#137—4222no—
102#137—4222no—
103#137—4222no—
104AGL(1,ℤ/8)—4222no—
105AGL(1,ℤ/8)—4222no—
106AGL(1,ℤ/8)—4222no—
107#135—4224no—
108#135—4222no—
109C2×M16—2112no—
110C2×M16C2×D21111yescharacteristic
111C22×C8C2×D21111yescharacteristic
112C22⋊C8—22no—
113C22⋊C8—4224no—
114#41—22no—
115#41—22no—
116#41—4222no—
117#41—4222no—
118#41—4222no—
119#41—4222no—
120C2×Q8—2112no—
121C2×Q8D84411yescharacteristic
122C2×D4—21no—
123C2×D4—4222no—
124C2×D4—2112no—
125C2×D4D84411yescharacteristic
126C22×C4—21no—
127C22×C4—4222no—
128C22×C4C22×D21111yescharacteristic
129Q16—4222no—
130Q16—4224no—
131SD16—4224no—
132SD16—4224no—
133SD16—4224no—
134D8—4222no—
135D8—4224no—
136M16—22no—
137M16—4222no—
138M16—4222no—
139C2×C8—22no—
140C2×C8—21no—
141C2×C8—2114no—
142C2×C8C22⋊D21112yes—
143C23—21no—
144C23—4222no—
145C23#1680996211yescharacteristic
146Q8—22no—
147Q8—4222no—
148Q8—4222no—
149D4—22no—
150D4—42no—
151D4—8442no—
152D4—8442no—
153D4—4222no—
154D4—4222no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—8442no—
159C2×C4—8442no—
160C2×C4—4222no—
161C2×C4D4×D21112yes—
162C2×C4D4×D21112yes—
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4C2×D84411yescharacteristic
165C8—22no—
166C8—41no—
167C8—4224no—
168C22—41no—
169C22—42no—
170C22—41no—
171C22—8444no—
172C22—8442no—
173C22#2026995012yes—
174C22#2026995111yescharacteristic
175C4—41no—
176C4—8442no—
177C4#2026994511yescharacteristic
178C4#2026994712yes—
179C4#2026997311yescharacteristic
180C2—21no—
181C2—22no—
182C2—81no—
183C2—16882no—
184C2#2721305511yescharacteristic
subgroupG/H
centerC4#20269973
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#408×C211C2
socleC2×C211#392
hypercenter#408D211
hypocenterC211#2017
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #27213946C2semidirect
2. #27213944C2semidirect
3. #27213112C2semidirect
4. C211⋊#338C2semidirect
5. #408×C211C2semidirect
92. C211#2017semidirect
97. #408D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272139442
#272131122
C211⋊#3382
#408×C2112
#2017211