Finite Group Databasebeta

← #41014690

Subgroups — #41014690

Order 27008.

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

normal subgroups only

This group has 282 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
1C2×#20270003C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213148C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic, complemented
5#27213114C211yesmaximal, characteristic, complemented
6#27212914C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270003—22no—
10#20270003—22no—
11C2×#16809950—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16#20269912—22no—
17#20269902—21no—
18#20269899—21no—
19#20269898—22no—
20(C2×D211)⋊C8—22no—
21C22×D4×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24#100×C211—22no—
25#16809962—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29C2×Dic422D411yescharacteristic
30C2×D844D411yescharacteristic
31C2×C22⋊D211—21no—
32C22×Dic211—21no—
33C2×C4×D211—21no—
34#16809950—24no—
35#16809947—24no—
36#16809946—21no—
37#16809946D412yes—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8D412yes—
42C4⋊Dic211—21no—
43C4⋊Dic211—21no—
44C24×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211D412yes—
51C22×C4×C211C2311yescharacteristic
52M16×C211—21no—
53M16×C211—21no—
54M16×C211—21no—
55M16×C211—21no—
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C22×D211—21no—
59Dic422—21no—
60Dic422—21no—
61D844—21no—
62D844—21no—
63C22⋊D211—41no—
64C22⋊D211—41no—
65C2×Dic211—21no—
66C2×Dic211—41no—
67C2×Dic211—81no—
68C4×D211—41no—
69C4×D211—41no—
70C211⋊C8—22no—
71C211⋊C8—22no—
72C23×C211—22no—
73C23×C211—41no—
74C23×C211—41no—
75C23×C211—81no—
76C23×C211—81no—
77C23×C211#4111yescharacteristic
78D4×C211—22no—
79D4×C211—22no—
80D4×C211—41no—
81D4×C211—41no—
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C8×C211—41no—
89C8×C211—41no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211—42no—
93C22×C211—82no—
94C22×C211—82no—
95C22×C211—81no—
96C22×C211—81no—
97C22×C211#12111yescharacteristic
98C22×C211#12311yescharacteristic
99C22×C211#12411yescharacteristic
100C2×D211—41no—
101C2×D211—41no—
102C2×D211—41no—
103C4×C211#12011yescharacteristic
104C4×C211#12111yescharacteristic
105C4×C211#12111yescharacteristic
106C4×C211#12311yescharacteristic
107Dic211—41no—
108Dic211—81no—
109C2×C211—21no—
110C2×C211—21no—
111C2×C211—82no—
112C2×C211#39411yescharacteristic
113C2×C211#47111yescharacteristic
114C2×C211#47211yescharacteristic
115D211—81no—
116C211#205111yescharacteristic, complemented, Sylow
117#2051—2111nomaximal, Sylow
118C2×AGL(1,ℤ/8)—2111no—
119C2×#108—2111no—
120#417—2111no—
121C2×#102—2111no—
122C2×#100D21111yescharacteristic, complemented
123#408—2111no—
124#343—2111no—
125C2×#41—2111no—
126C22×D4C2×D21111yescharacteristic
127AGL(1,ℤ/8)—4222no—
128AGL(1,ℤ/8)—4222no—
129C2×SD16—4221no—
130C2×D8—4221no—
131C2×M16—2111no—
132C2×M16C2×D21111yescharacteristic
133C2×M16C2×D21111yescharacteristic
134C22×C8—2111no—
135C2×C4⋊C4—2111no—
136#108—4222no—
137#103—4221no—
138#102—4221no—
139#102—4222no—
140#100—22no—
141C22⋊C8—4222no—
142C24—21no—
143#41—4221no—
144#41—4221no—
145#41—4221no—
146#41—4221no—
147C2×Q8—2111no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—4221no—
154C2×D4—2111no—
155C2×D4C22⋊D21112yes—
156C22×C4—4221no—
157C22×C4—4221no—
158C22×C4C22×D21111yescharacteristic
159SD16—4224no—
160D8—4224no—
161M16—21no—
162M16—21no—
163M16—21no—
164M16—21no—
165M16—4221no—
166M16—2112no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—4221no—
170C2×C8—4221no—
171C2×C8—4221no—
172C2×C8—2112no—
173C4⋊C4—4221no—
174C4⋊C4—4221no—
175C23—22no—
176C23—41no—
177C23—41no—
178C23—81no—
179C23—81no—
180C23—4221no—
181C23#1680996211yescharacteristic
182Q8—4221no—
183Q8—4221no—
184D4—22no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4—8441no—
189D4—8441no—
190D4—4221no—
191D4—4221no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—16881no—
196C2×C4—4221no—
197C2×C4D4×D21111yescharacteristic
198C2×C4D4×D21111yescharacteristic
199C2×C4D4×D21111yescharacteristic
200C2×C4C2×C22⋊D21111yescharacteristic
201C2×C4#1680996111yescharacteristic
202C2×C4#1680996211yescharacteristic
203C8—41no—
204C8—41no—
205C8—4222no—
206C8—4222no—
207C22—21no—
208C22—21no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—81no—
213C22—81no—
214C22—8441no—
215C22—8441no—
216C22—8441no—
217C22#2026995011yescharacteristic
218C22#2026996011yescharacteristic
219C22#2026996311yescharacteristic
220C4—8441no—
221C4—16881no—
222C4#2026994511yescharacteristic
223C4#2026994911yescharacteristic
224C4#2026995011yescharacteristic
225C4#2026995911yescharacteristic
226C2—21no—
227C2—21no—
228C2—82no—
229C2—16881no—
230C2#2721306111yescharacteristic
231C2#2721340811yescharacteristic
232C2#2721341511yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211#121
prime radicalC2×#20270003C2
hypercenterC2×#100D211
hypocenterC211#2051
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. C2×#20269912C2semidirect
3. #27213148C2semidirect
4. C2×#20269898C2semidirect
5. #27213114C2semidirect
6. #27212914C2semidirect
7. C2×#100×C211C2semidirect
116. C211#2051semidirect
122. C2×#100D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700032
C2×#202699122
#272131482
C2×#202698982
#272131142
#272129142
C2×#100×C2112
#2051211