Finite Group Databasebeta

← #41014847

Subgroups — #41014847

Order 27008.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×#20269908C211yesmaximal, characteristic, complemented
2#27213148C211yesmaximal, characteristic
3#27213145C211yesmaximal, characteristic
4#27213114C211yesmaximal, characteristic
5#27212914C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269908—22no—
13#20269902—21no—
14#20269900—21no—
15#20269899—21no—
16#20269897—21no—
17(C2×D211)⋊C8—22no—
18C2×M16×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21#16809962—21no—
22#16809962—21no—
23#16809962—21no—
24#16809962—21no—
25C2×Dic422D411yescharacteristic
26C2×D844D411yescharacteristic
27C2×C22⋊D211—21no—
28C22×Dic211—21no—
29C2×C4×D211—21no—
30#16809946—21no—
31#16809946—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8D412yes—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38C22×C4×C211—21no—
39C22×C4×C211C2311yescharacteristic
40M16×C211—21no—
41M16×C211—21no—
42M16×C211—21no—
43M16×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C22×D211—21no—
49Dic422—21no—
50Dic422—21no—
51D844—21no—
52D844—21no—
53C22⋊D211—41no—
54C22⋊D211—41no—
55C2×Dic211—21no—
56C2×Dic211—41no—
57C2×Dic211—81no—
58C4×D211—41no—
59C4×D211—41no—
60C211⋊C8—22no—
61C211⋊C8—41no—
62C23×C211#4111yescharacteristic
63C2×C4×C211—41no—
64C2×C4×C211—81no—
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211#4111yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C8×C211—41no—
72C8×C211—41no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211#12111yescharacteristic
76C22×C211#12611yescharacteristic
77C22×C211#12611yescharacteristic
78C2×D211—41no—
79C2×D211—41no—
80C2×D211—41no—
81C4×C211—81no—
82C4×C211#12311yescharacteristic
83C4×C211#12311yescharacteristic
84C4×C211#12311yescharacteristic
85C4×C211#12411yescharacteristic
86Dic211—41no—
87Dic211—81no—
88C2×C211—21no—
89C2×C211—21no—
90C2×C211#39711yescharacteristic
91C2×C211#47111yescharacteristic
92C2×C211#47111yescharacteristic
93D211—81no—
94C211#207911yescharacteristic, complemented, Sylow
95#2079—2111nomaximal, Sylow
96C2×#106—2111no—
97#417—2111no—
98#417—2111no—
99#408—2111no—
100#343—2111no—
101#343D21111yescharacteristic, complemented
102#341—2111no—
103C2×#41—2111no—
104C2×M16—2111no—
105C2×M16C2×D21111yescharacteristic
106C2×M16C2×D21111yescharacteristic
107C22×C8—2111no—
108C2×C4⋊C4—2111no—
109C2×C4⋊C4C2×D21111yescharacteristic
110#106—4222no—
111#103—4221no—
112#103—4221no—
113#102—4221no—
114#102—4221no—
115C22⋊C8—4222no—
116#41—4221no—
117#41—4221no—
118#41—4221no—
119#41—4221no—
120C2×Q8—2111no—
121C2×D4—4221no—
122C2×D4—2111no—
123C22×C4—21no—
124C22×C4—4221no—
125C22×C4—4221no—
126C22×C4C22×D21111yescharacteristic
127M16—21no—
128M16—21no—
129M16—21no—
130M16—21no—
131M16—4221no—
132M16—4221no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—4221no—
136C2×C8—4221no—
137C2×C8—4221no—
138C2×C8—2112no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C23—4221no—
144C23#1680996211yescharacteristic
145Q8—4221no—
146Q8—4221no—
147D4—8441no—
148D4—8441no—
149D4—4221no—
150D4—4221no—
151C2×C4—41no—
152C2×C4—81no—
153C2×C4—8441no—
154C2×C4—8441no—
155C2×C4—8441no—
156C2×C4—16881no—
157C2×C4—4221no—
158C2×C4D4×D21111yescharacteristic
159C2×C4D4×D21111yescharacteristic
160C2×C4#1680996111yescharacteristic
161C2×C4#1680996211yescharacteristic
162C2×C4#1680996211yescharacteristic
163C2×C4#1680996311yescharacteristic
164C8—41no—
165C8—41no—
166C8—8441no—
167C8—4222no—
168C22—21no—
169C22—21no—
170C22—8441no—
171C22—8441no—
172C22—8441no—
173C22#2026995011yescharacteristic
174C22#2026997111yescharacteristic
175C22#2026997211yescharacteristic
176C4—81no—
177C4—8441no—
178C4—16881no—
179C4#2026995911yescharacteristic
180C4#2026996211yescharacteristic
181C4#2026996211yescharacteristic
182C4#2026996311yescharacteristic
183C2—21no—
184C2—21no—
185C2—16881no—
186C2#2721307511yescharacteristic
187C2#2721340811yescharacteristic
188C2#2721341211yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
hypercenter#343D211
hypocenterC211#2079
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#20269908C2semidirect
5. #27212914C2semidirect
6. #27212910C2semidirect
7. #343×C211C2semidirect
94. C211#2079semidirect
101. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
#272131482
#272131452
#272131142
#272129142
#272129102
#343×C2112
#2079211