Finite Group Databasebeta

← #41013907

Subgroups — #41013907

Order 27008.

332 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269985C211yesmaximal, characteristic, complemented
3#27213145C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#343×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×D844C2211yescharacteristic
10#20270005—22no—
11#20270005C414yescomplemented
12C2×#16809951—21no—
13C2×D1688—21no—
14#20269985—22no—
15#20269985—22no—
16C2×#16809946C2211yescharacteristic
17C2×#16809945—21no—
18C22×C211⋊C8C2211yescharacteristic
19C2×C8×D211—21no—
20#20269900—21no—
21#20269897—21no—
22#20269897—21no—
23#20269897—21no—
24#20269893—22no—
25C2×M16×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28C23×D211—21no—
29#16809962—21no—
30#16809962—21no—
31#16809962C2×C412yes—
32#16809962D412yes—
33C2×Dic422D411yescharacteristic
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844C2×C412yes—
40C2×D844D411yescharacteristic
41C2×C22⋊D211—21no—
42C2×C4×D211—21no—
43#16809951—24no—
44D1688—24no—
45#16809946—21no—
46#16809946—21no—
47#16809945—24no—
48C2×C211⋊C8—21no—
49C2×C211⋊C8—21no—
50C2×C211⋊C8—21no—
51C2×C211⋊C8D411yescharacteristic
52C2×C211⋊C8D411yescharacteristic
53C8×D211—24no—
54C22×C4×C211—21no—
55C22×C4×C211C2311yescharacteristic
56M16×C211—21no—
57M16×C211—21no—
58M16×C211C2×C412yes—
59M16×C211D412yes—
60C2×C8×C211—21no—
61C2×C8×C211—21no—
62C4⋊C4×C211—21no—
63C4⋊C4×C211—21no—
64C22×D211—21no—
65C22×D211—22no—
66C22×D211—41no—
67C22×D211—41no—
68C22×D211—81no—
69C22×D211—81no—
70Dic422—21no—
71Dic422C22⋊C412yes—
72D844—22no—
73D844—22no—
74D844—21no—
75D844—41no—
76D844—41no—
77D844C22⋊C412yes—
78C22⋊D211—22no—
79C22⋊D211—22no—
80C2×Dic211—21no—
81C4×D211—22no—
82C4×D211—22no—
83C211⋊C8—21no—
84C211⋊C8—21no—
85C211⋊C8—41no—
86C23×C211#4111yescharacteristic
87C2×C4×C211—41no—
88C2×C4×C211—81no—
89C2×C4×C211C22×C411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C8×C211—22no—
96C8×C211—22no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211C4×D411yescharacteristic
100C22×C211#12111yescharacteristic
101C22×C211#12311yescharacteristic
102C2×D211—22no—
103C2×D211—41no—
104C2×D211—41no—
105C2×D211—42no—
106C2×D211—82no—
107C2×D211—81no—
108C2×D211—81no—
109C2×D211—82no—
110C4×C211—81no—
111C4×C211C2×C22⋊C411yescharacteristic
112C4×C211C4×D411yescharacteristic
113C4×C211#12011yescharacteristic
114C4×C211#12111yescharacteristic
115Dic211—22no—
116C2×C211—21no—
117C2×C211—21no—
118C2×C211#38611yescharacteristic
119C2×C211#47111yescharacteristic
120C2×C211#47211yescharacteristic
121D211—42no—
122D211—82no—
123C211#190711yescharacteristic, complemented, Sylow
124#1907—2111nomaximal, Sylow
125C2×AGL(1,ℤ/8)—2111no—
126C2×#131—2111no—
127#417—2111no—
128C2×#102—2111no—
129C2×#100—2111no—
130#343D21111yescharacteristic, complemented
131#341—2111no—
132C2×#41—2111no—
133C22×D4—2111no—
134AGL(1,ℤ/8)—4222no—
135AGL(1,ℤ/8)—2114no—
136C2×SD16—4221no—
137C2×D8—4221no—
138#131—4222no—
139#131—4222no—
140C2×M16—4221no—
141C2×M16—2111no—
142C2×M16C2×D21111yescharacteristic
143C2×M16C2×D21111yescharacteristic
144C22×C8—4221no—
145C22×C8—2111no—
146C2×C4⋊C4C2×D21111yescharacteristic
147#103—4221no—
148#102—4221no—
149#102—4221no—
150#102—4221no—
151#100—4222no—
152C24—4221no—
153#41—4221no—
154#41—4221no—
155#41—2112no—
156#41—2112no—
157C2×Q8—2111no—
158C2×D4—4221no—
159C2×D4—4221no—
160C2×D4—4221no—
161C2×D4—4221no—
162C2×D4—4221no—
163C2×D4—4221no—
164C2×D4—2112no—
165C2×D4—2111no—
166C22×C4—21no—
167C22×C4—4221no—
168C22×C4C22×D21111yescharacteristic
169SD16—4224no—
170D8—4224no—
171M16—21no—
172M16—21no—
173M16—4224no—
174M16—4221no—
175M16—4221no—
176M16C4×D21112yes—
177M16D84412yes—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—4224no—
181C2×C8—4221no—
182C2×C8—4221no—
183C2×C8—4221no—
184C2×C8—2111no—
185C2×C8—2111no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C23—8441no—
189C23—8441no—
190C23—16881no—
191C23—16881no—
192C23—4221no—
193C23—4222no—
194C23#1680996211yescharacteristic
195Q8—4221no—
196Q8—2112no—
197D4—8441no—
198D4—8441no—
199D4—4222no—
200D4—4222no—
201D4—4222no—
202D4—4222no—
203D4—4221no—
204D4—2112no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—4221no—
208C2×C4—4222no—
209C2×C4—4222no—
210C2×C4C2×C4×D21111yescharacteristic
211C2×C4D4×D21111yescharacteristic
212C2×C4D4×D21111yescharacteristic
213C2×C4D4×D21111yescharacteristic
214C2×C4C2×D84411yescharacteristic
215C2×C4#1680996311yescharacteristic
216C8—22no—
217C8—22no—
218C8—8441no—
219C8—4221no—
220C8—4221no—
221C22—21no—
222C22—21no—
223C22—8441no—
224C22—8441no—
225C22—8442no—
226C22—16882no—
227C22—16882no—
228C22—16881no—
229C22—16881no—
230C22—4222no—
231C22C4×D84411yescharacteristic
232C22#2026995011yescharacteristic
233C22#2026996211yescharacteristic
234C4—81no—
235C4—4222no—
236C4C22⋊C4×D21111yescharacteristic
237C4#2026993911yescharacteristic
238C4#2026994711yescharacteristic
239C4#2026995211yescharacteristic
240C2—21no—
241C2—21no—
242C2—8442no—
243C2—16882no—
244C2#2721302411yescharacteristic
245C2#2721341211yescharacteristic
246C2#2721341511yescharacteristic
subgroupG/H
centerC22#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#343×C211C2
socleC22×C211#121
prime radicalC2×#20270005C2
cosocleC22×C4×C211C23
hypercenter#343D211
hypocenterC211#1907
LCSG ▷ C2×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×#20269985C2semidirect
3. #27213145C2semidirect
4. C2×#20269897C2semidirect
5. C2×#20269893C2semidirect
6. #27212910C2semidirect
7. #343×C211C2semidirect
11. #20270005C4semidirect
123. C211#1907semidirect
130. #343D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700052
C2×#202699852
#272131452
C2×#202698972
C2×#202698932
#272129102
#343×C2112
#1907211