Finite Group Databasebeta

← #41013906

Subgroups — #41013906

Order 27008.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20270002C211yesmaximal, characteristic
2#27213147C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic, complemented
4C2×#20269893C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212911C211yesmaximal, characteristic, complemented
7C2×#131×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9#20270002—22no—
10#20270002C414yescomplemented
11C2×#16809949—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16#20269901—21no—
17#20269899—21no—
18#20269899—21no—
19#20269898—21no—
20#20269893—22no—
21C2×#41×C211C2211yescharacteristic
22#131×C211—22no—
23#131×C211—22no—
24C2×M16×C211—21no—
25C2×M16×C211C2211yescharacteristic
26C22×C8×C211—21no—
27C22×C8×C211C2211yescharacteristic
28C23×D211—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844C2×C412yes—
35C22×Dic211—21no—
36#16809949—24no—
37#16809947—24no—
38#16809946—21no—
39#16809946—21no—
40#16809946—21no—
41#16809946C2×C412yes—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C4⋊Dic211—21no—
45C4⋊Dic211—21no—
46#41×C211—21no—
47#41×C211—21no—
48#41×C211C2×C412yes—
49#41×C211D412yes—
50C2×Q8×C211D411yescharacteristic
51C2×D4×C211—21no—
52C2×D4×C211D411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55M16×C211—24no—
56M16×C211—21no—
57M16×C211D412yes—
58C2×C8×C211—24no—
59C2×C8×C211—21no—
60C2×C8×C211—21no—
61C2×C8×C211—21no—
62C2×C8×C211D411yescharacteristic
63C2×C8×C211D411yescharacteristic
64C22×D211—22no—
65C22×D211—41no—
66C22×D211—41no—
67C22×D211—81no—
68C22×D211—81no—
69D844—22no—
70D844—22no—
71D844—41no—
72D844—41no—
73C2×Dic211—41no—
74C2×Dic211—81no—
75C211⋊C8—22no—
76C211⋊C8—41no—
77C23×C211—21no—
78C23×C211#4111yescharacteristic
79Q8×C211—21no—
80Q8×C211C22⋊C412yes—
81D4×C211—22no—
82D4×C211—22no—
83D4×C211—21no—
84D4×C211C22⋊C412yes—
85C2×C4×C211—21no—
86C2×C4×C211—22no—
87C2×C4×C211—22no—
88C2×C4×C211C22×C411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211#4111yescharacteristic
94C8×C211—21no—
95C8×C211—22no—
96C8×C211—21no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—22no—
100C22×C211—41no—
101C22×C211—41no—
102C22×C211C4×D411yescharacteristic
103C22×C211#12111yescharacteristic
104C22×C211#12311yescharacteristic
105C2×D211—42no—
106C2×D211—82no—
107C2×D211—81no—
108C2×D211—81no—
109C2×D211—82no—
110C4×C211—22no—
111C4×C211C2×C22⋊C411yescharacteristic
112C4×C211C4×D411yescharacteristic
113C4×C211#12011yescharacteristic
114C4×C211#12111yescharacteristic
115Dic211—81no—
116C2×C211—21no—
117C2×C211—21no—
118C2×C211—42no—
119C2×C211#38611yescharacteristic
120C2×C211#47111yescharacteristic
121C2×C211#47211yescharacteristic
122D211—82no—
123C211#190711yescharacteristic, complemented, Sylow
124#1907—2111nomaximal, Sylow
125C2×AGL(1,ℤ/8)—2111no—
126C2×#131D21111yescharacteristic, complemented
127#417—2111no—
128C2×#102—2111no—
129C2×#100—2111no—
130#343—2111no—
131#341—2111no—
132C2×#41C2×D21111yescharacteristic
133C22×D4—2111no—
134AGL(1,ℤ/8)—4222no—
135AGL(1,ℤ/8)—2114no—
136C2×SD16—4221no—
137C2×D8—4221no—
138#131—22no—
139#131—22no—
140C2×M16—21no—
141C2×M16—2111no—
142C2×M16—2111no—
143C2×M16C2×D21111yescharacteristic
144C22×C8—21no—
145C22×C8C2×D21111yescharacteristic
146C2×C4⋊C4—2111no—
147#103—4221no—
148#102—4221no—
149#102—4221no—
150#102—4221no—
151#100—4222no—
152C24—4221no—
153#41—21no—
154#41—21no—
155#41C4×D21112yes—
156#41C22⋊D21112yes—
157C2×Q8D84411yescharacteristic
158C2×D4—21no—
159C2×D4—4221no—
160C2×D4—4221no—
161C2×D4—4221no—
162C2×D4—4221no—
163C2×D4—4221no—
164C2×D4—2112no—
165C2×D4D84411yescharacteristic
166C22×C4—21no—
167C22×C4—4221no—
168C22×C4C22×D21111yescharacteristic
169SD16—4224no—
170D8—4224no—
171M16—24no—
172M16—21no—
173M16—4221no—
174M16—4221no—
175M16—4221no—
176M16—2112no—
177M16C22⋊D21112yes—
178C2×C8—24no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—4221no—
183C2×C8—4221no—
184C2×C8C22⋊D21111yescharacteristic
185C2×C8C22⋊D21111yescharacteristic
186C4⋊C4—4221no—
187C4⋊C4—4221no—
188C23—21no—
189C23—8441no—
190C23—8441no—
191C23—16881no—
192C23—16881no—
193C23—4222no—
194C23#1680996211yescharacteristic
195Q8—21no—
196Q8(C2×D211)⋊C412yes—
197D4—22no—
198D4—22no—
199D4—21no—
200D4—8441no—
201D4—8441no—
202D4—4222no—
203D4—4222no—
204D4(C2×D211)⋊C412yes—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—8441no—
209C2×C4—16881no—
210C2×C4C2×C4×D21111yescharacteristic
211C2×C4C2×C22⋊D21111yescharacteristic
212C2×C4C2×C22⋊D21111yescharacteristic
213C2×C4C2×C22⋊D21111yescharacteristic
214C2×C4C2×D84411yescharacteristic
215C2×C4#1680996211yescharacteristic
216C8—21no—
217C8—22no—
218C8—21no—
219C8—8441no—
220C8—4222no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—8442no—
227C22—16882no—
228C22—16882no—
229C22—16881no—
230C22—16881no—
231C22C4×C22⋊D21111yescharacteristic
232C22#2026995111yescharacteristic
233C22#2026996011yescharacteristic
234C4—22no—
235C4—16881no—
236C4C2×(C2×D211)⋊C411yescharacteristic
237C4C4×C22⋊D21111yescharacteristic
238C4#2026994611yescharacteristic
239C4#2026995111yescharacteristic
240C2—21no—
241C2—21no—
242C2—42no—
243C2—16882no—
244C2#2721302511yescharacteristic
245C2#2721340911yescharacteristic
246C2#2721341611yescharacteristic
subgroupG/H
centerC22#20269951
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#131×C211C2
socleC22×C211#121
prime radicalC2×#20270002C2
cosocleC22×C4×C211C23
hypercenterC2×#131D211
hypocenterC211#1907
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
2. #27213147C2semidirect
3. C2×#20269899C2semidirect
4. C2×#20269893C2semidirect
5. #27212914C2semidirect
6. #27212911C2semidirect
7. C2×#131×C211C2semidirect
10. #20270002C4semidirect
123. C211#1907semidirect
126. C2×#131D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700022
#272131472
C2×#202698992
C2×#202698932
#272129142
#272129112
C2×#131×C2112
#1907211