Finite Group Databasebeta

← #41014693

Subgroups — #41014693

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×#20270005C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213145C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27213114C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×D844C2211yescharacteristic
10#20270005—22no—
11#20270005—22no—
12C2×#16809951—21no—
13C2×D1688—21no—
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8C2211yescharacteristic
16#20269912—22no—
17#20269900—21no—
18#20269897—21no—
19#20269897—22no—
20#20269893—22no—
21(C2×D211)⋊C8—22no—
22C2×M16×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C23×D211—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30C2×Dic422D411yescharacteristic
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844D412yes—
37C2×D844D411yescharacteristic
38C2×C22⋊D211—21no—
39C2×C4×D211—21no—
40#16809951—24no—
41D1688—24no—
42#16809946—21no—
43#16809946—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50M16×C211—21no—
51M16×C211—21no—
52M16×C211—21no—
53M16×C211D412yes—
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C22×D211—21no—
59C22×D211—22no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—81no—
63C22×D211—81no—
64Dic422—21no—
65Dic422—21no—
66D844—21no—
67D844—22no—
68D844—22no—
69D844—21no—
70D844—41no—
71D844—41no—
72C22⋊D211—41no—
73C22⋊D211—41no—
74C2×Dic211—21no—
75C4×D211—41no—
76C4×D211—41no—
77C211⋊C8—22no—
78C211⋊C8—41no—
79C23×C211#4111yescharacteristic
80C2×C4×C211—41no—
81C2×C4×C211—81no—
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—22no—
89C8×C211—41no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211#12111yescharacteristic
93C22×C211#12311yescharacteristic
94C22×C211#12411yescharacteristic
95C2×D211—41no—
96C2×D211—42no—
97C2×D211—41no—
98C2×D211—41no—
99C2×D211—82no—
100C2×D211—81no—
101C2×D211—81no—
102C2×D211—82no—
103C4×C211—81no—
104C4×C211#12011yescharacteristic
105C4×C211#12111yescharacteristic
106C4×C211#12111yescharacteristic
107C4×C211#12311yescharacteristic
108Dic211—41no—
109C2×C211—21no—
110C2×C211—21no—
111C2×C211#39411yescharacteristic
112C2×C211#47111yescharacteristic
113C2×C211#47211yescharacteristic
114D211—81no—
115D211—82no—
116C211#205111yescharacteristic, complemented, Sylow
117#2051—2111nomaximal, Sylow
118C2×AGL(1,ℤ/8)—2111no—
119C2×#108—2111no—
120#417—2111no—
121C2×#102—2111no—
122C2×#100—2111no—
123#408—2111no—
124#343D21111yescharacteristic, complemented
125C2×#41—2111no—
126C22×D4—2111no—
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⋊C4C2×D21111yescharacteristic
136#108—4222no—
137#103—4221no—
138#102—4221no—
139#102—4222no—
140#100—4222no—
141C22⋊C8—4222no—
142C24—4221no—
143#41—4221no—
144#41—4221no—
145#41—4221no—
146#41—4221no—
147C2×Q8—2111no—
148C2×D4—4221no—
149C2×D4—4221no—
150C2×D4—4221no—
151C2×D4—4221no—
152C2×D4—4221no—
153C2×D4—4221no—
154C2×D4—2112no—
155C2×D4—2111no—
156C22×C4—21no—
157C22×C4—4221no—
158C22×C4C22×D21111yescharacteristic
159SD16—4224no—
160D8—4224no—
161M16—21no—
162M16—21no—
163M16—21no—
164M16—4221no—
165M16—4221no—
166M16D84412yes—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—4221no—
170C2×C8—4221no—
171C2×C8—4221no—
172C2×C8—2112no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C23—8441no—
176C23—8441no—
177C23—16881no—
178C23—16881no—
179C23—4221no—
180C23—4222no—
181C23#1680996211yescharacteristic
182Q8—4221no—
183Q8—4221no—
184D4—8441no—
185D4—8441no—
186D4—8441no—
187D4—8441no—
188D4—4221no—
189D4—4222no—
190D4—4222no—
191D4—4221no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—8441no—
195C2×C4—8441no—
196C2×C4—4221no—
197C2×C4D4×D21111yescharacteristic
198C2×C4D4×D21111yescharacteristic
199C2×C4D4×D21111yescharacteristic
200C2×C4C2×D84411yescharacteristic
201C2×C4#1680996211yescharacteristic
202C2×C4#1680996311yescharacteristic
203C8—22no—
204C8—41no—
205C8—8441no—
206C8—4222no—
207C22—21no—
208C22—21no—
209C22—8441no—
210C22—8442no—
211C22—8441no—
212C22—8441no—
213C22—16882no—
214C22—16882no—
215C22—16881no—
216C22—16881no—
217C22#2026995011yescharacteristic
218C22#2026996211yescharacteristic
219C22#2026996511yescharacteristic
220C4—81no—
221C4—8441no—
222C4#2026994711yescharacteristic
223C4#2026995011yescharacteristic
224C4#2026995211yescharacteristic
225C4#2026996211yescharacteristic
226C2—21no—
227C2—21no—
228C2—16881no—
229C2—16882no—
230C2#2721306411yescharacteristic
231C2#2721341211yescharacteristic
232C2#2721341511yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#121
prime radicalC2×#20270005C2
hypercenter#343D211
hypocenterC211#2051
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
2. C2×#20269912C2semidirect
3. #27213145C2semidirect
4. C2×#20269897C2semidirect
5. C2×#20269893C2semidirect
6. #27213114C2semidirect
7. #343×C211C2semidirect
116. C211#2051semidirect
124. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700052
C2×#202699122
#272131452
C2×#202698972
C2×#202698932
#272131142
#343×C2112
#2051211