Finite Group Databasebeta

← #41014659

Subgroups — #41014659

Order 27008.

276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C22×#16809950C211yesmaximal, characteristic
2#27213174C211yesmaximal, characteristic, complemented
3#27213156C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—22no—
10C2×#16809950—22no—
11C2×#16809950—22no—
12C2×#16809946C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14#20269936C2211yescharacteristic
15Dic211⋊C8—22no—
16#20269902—22no—
17#20269898—22no—
18#20269895—22no—
19C22×D4×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#100×C211—22no—
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422D412yes—
29C22×Dic211—21no—
30#16809950—28no—
31#16809946—21no—
32#16809946—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8D412yes—
37C4⋊Dic211D412yes—
38C22⋊Dic211—21no—
39C4×Dic211—21no—
40C24×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211D412yes—
47C22×C4×C211C2311yescharacteristic
48M16×C211—21no—
49M16×C211—21no—
50M16×C211—21no—
51M16×C211—21no—
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54Dic422—22no—
55Dic422—22no—
56Dic422—41no—
57Dic422—41no—
58C2×Dic211—22no—
59C2×Dic211—22no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C211⋊C8—22no—
63C211⋊C8—41no—
64C23×C211—22no—
65C23×C211—41no—
66C23×C211—41no—
67C23×C211—82no—
68C23×C211#4111yescharacteristic
69D4×C211—22no—
70D4×C211—22no—
71D4×C211—41no—
72D4×C211—41no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C8×C211—41no—
80C8×C211—41no—
81C22×C211—22no—
82C22×C211—42no—
83C22×C211—82no—
84C22×C211—82no—
85C22×C211—82no—
86C22×C211#12111yescharacteristic
87C22×C211#12311yescharacteristic
88C22×C211#12311yescharacteristic
89C4×C211#12011yescharacteristic
90C4×C211#12111yescharacteristic
91C4×C211#12111yescharacteristic
92C4×C211#12411yescharacteristic
93Dic211—42no—
94Dic211—42no—
95C2×C211—21no—
96C2×C211—21no—
97C2×C211—82no—
98C2×C211#39411yescharacteristic
99C2×C211#47112yes—
100C211#204611yescharacteristic, complemented, Sylow
101#2046—2111nomaximal, Sylow
102C22×SD16—2111no—
103#424—2111no—
104#419—2111no—
105#418—2111no—
106C2×#101—2111no—
107C2×#100D21111yescharacteristic
108#341—2111no—
109C22×Q8—2111no—
110C22×D4C2×D21111yescharacteristic
111C2×SD16—4222no—
112C2×SD16—4222no—
113C2×SD16—4222no—
114C2×M16—2111no—
115C2×M16C2×D21111yescharacteristic
116C2×M16C2×D21111yescharacteristic
117C22×C8—2111no—
118#117—2111no—
119C4⋊C8—4222no—
120#103—4222no—
121#102—4222no—
122#101—4222no—
123#100—22no—
124C24—21no—
125C2×Q8—4221no—
126C2×Q8—4221no—
127C2×Q8—4221no—
128C2×Q8—4221no—
129C2×Q8—4221no—
130C2×Q8—2112no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C22⋊D21112yes—
137C22×C4—4221no—
138C22×C4C22×D21111yescharacteristic
139SD16—4228no—
140M16—21no—
141M16—21no—
142M16—21no—
143M16—21no—
144M16—4221no—
145M16—4221no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—4221no—
149C2×C8—4221no—
150C2×C8—4221no—
151C2×C8—2112no—
152C4⋊C4—2112no—
153C22⋊C4—4221no—
154C42—4221no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—82no—
159C23#1680996211yescharacteristic
160Q8—8441no—
161Q8—8441no—
162Q8—4222no—
163Q8—4222no—
164D4—22no—
165D4—22no—
166D4—41no—
167D4—41no—
168C2×C4—8441no—
169C2×C4—8441no—
170C2×C4—4222no—
171C2×C4—4222no—
172C2×C4D4×D21111yescharacteristic
173C2×C4D4×D21111yescharacteristic
174C2×C4D4×D21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C2×C4#1680996111yescharacteristic
177C2×C4#1680996211yescharacteristic
178C8—41no—
179C8—41no—
180C8—8441no—
181C8—4222no—
182C22—22no—
183C22—42no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22#2026995011yescharacteristic
188C22#2026995911yescharacteristic
189C22#2026996011yescharacteristic
190C4—8442no—
191C4—8442no—
192C4#2026994511yescharacteristic
193C4#2026994911yescharacteristic
194C4#2026995011yescharacteristic
195C4#2026996311yescharacteristic
196C2—21no—
197C2—21no—
198C2—82no—
199C2#2721306111yescharacteristic
200C2#2721340812yes—
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211#121
prime radicalC22×D4×C211C22
hypercenterC2×#100D211
hypocenterC211#2046
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #27213174C2semidirect
3. #27213156C2semidirect
5. C2×#20269895C2semidirect
6. #27212910C2semidirect
100. C211#2046semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
#272131742
#272131562
#272131512
C2×#202698952
#272129102
C2×#100×C2112
#2046211