Finite Group Databasebeta

← #41014510

Subgroups — #41014510

Order 27008.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20270008C211yesmaximal, characteristic, complemented
2C2×#20270003C211yesmaximal, characteristic
3C2×#20269896C211yesmaximal, characteristic, complemented
4C2×#20269894C211yesmaximal, characteristic, complemented
5#27213112C211yesmaximal, characteristic, complemented
6#27212911C211yesmaximal, characteristic, complemented
7C22×SD16×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—22no—
10#20270008—22no—
11#20270003—22no—
12#20270003—22no—
13C2×#16809952—21no—
14C2×#16809950—21no—
15C2×#16809949—21no—
16C2×#16809947—21no—
17C2×#16809946C2211yescharacteristic
18C2×#16809946C2211yescharacteristic
19C2×#16809946C2211yescharacteristic
20#20269896—22no—
21#20269894—22no—
22(C2×D211)⋊C8—22no—
23C22×Q8×C211C2211yescharacteristic
24C22×D4×C211C2211yescharacteristic
25C2×SD16×C211—22no—
26C2×SD16×C211—22no—
27C2×SD16×C211—22no—
28C22×C8×C211C2211yescharacteristic
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33C2×Dic422D411yescharacteristic
34C2×D844D411yescharacteristic
35C2×C22⋊D211—21no—
36C2×C4×D211—21no—
37#16809952—24no—
38#16809950—24no—
39#16809949—24no—
40#16809947—24no—
41#16809946—21no—
42#16809946—21no—
43#16809946—21no—
44#16809946—21no—
45#16809946D412yes—
46#16809946D412yes—
47C2×C211⋊C8—21no—
48C2×C211⋊C8—21no—
49C2×C211⋊C8—21no—
50C24×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211D412yes—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211D412yes—
63C22×C4×C211—21no—
64C22×C4×C211C2311yescharacteristic
65SD16×C211—28no—
66C2×C8×C211—21no—
67C2×C8×C211—21no—
68C2×C8×C211D412yes—
69C22×D211—21no—
70Dic422—21no—
71Dic422—21no—
72D844—21no—
73D844—21no—
74C22⋊D211—41no—
75C22⋊D211—41no—
76C2×Dic211—21no—
77C4×D211—41no—
78C4×D211—41no—
79C211⋊C8—22no—
80C211⋊C8—22no—
81C211⋊C8—41no—
82C23×C211—22no—
83C23×C211—41no—
84C23×C211—41no—
85C23×C211—82no—
86C23×C211#4111yescharacteristic
87Q8×C211—22no—
88Q8×C211—22no—
89Q8×C211—41no—
90Q8×C211—41no—
91D4×C211—22no—
92D4×C211—22no—
93D4×C211—41no—
94D4×C211—41no—
95C2×C4×C211—22no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C8×C211—22no—
105C22×C211—22no—
106C22×C211—42no—
107C22×C211—82no—
108C22×C211—82no—
109C22×C211—82no—
110C22×C211#12111yescharacteristic
111C22×C211#12111yescharacteristic
112C22×C211#12111yescharacteristic
113C2×D211—41no—
114C2×D211—42no—
115C4×C211—42no—
116C4×C211#12011yescharacteristic
117C4×C211#12011yescharacteristic
118C4×C211#12011yescharacteristic
119C4×C211#12711yescharacteristic
120Dic211—41no—
121C2×C211—21no—
122C2×C211—21no—
123C2×C211—82no—
124C2×C211#39211yescharacteristic
125C2×C211#47112yes—
126D211—81no—
127C211#201911yescharacteristic, complemented, Sylow
128#2019—2111nomaximal, Sylow
129C2×#137—2111no—
130C2×AGL(1,ℤ/8)—2111no—
131C22×SD16D21111yescharacteristic, complemented
132C2×#101—2111no—
133C2×#100—2111no—
134#408—2111no—
135#341—2111no—
136C2×#41—2111no—
137C22×Q8C2×D21111yescharacteristic
138C22×D4C2×D21111yescharacteristic
139#137—4222no—
140#137—4222no—
141AGL(1,ℤ/8)—4222no—
142AGL(1,ℤ/8)—4222no—
143C2×Q16—4221no—
144C2×SD16—22no—
145C2×SD16—22no—
146C2×SD16—22no—
147C2×SD16—4221no—
148C2×SD16—4221no—
149C2×D8—4221no—
150C2×M16—2111no—
151C2×M16—2111no—
152C2×M16—2111no—
153C22×C8C2×D21111yescharacteristic
154#101—4222no—
155#100—4222no—
156C22⋊C8—4222no—
157C24—21no—
158#41—4221no—
159#41—4221no—
160#41—4221no—
161#41—4221no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—2111no—
168C2×Q8C22⋊D21112yes—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—4221no—
175C2×D4—2111no—
176C2×D4C22⋊D21112yes—
177C22×C4—21no—
178C22×C4—4221no—
179C22×C4C22×D21111yescharacteristic
180Q16—4224no—
181SD16—28no—
182SD16—4224no—
183SD16—4224no—
184D8—4224no—
185M16—4221no—
186M16—4221no—
187M16—4221no—
188M16—4221no—
189M16—2112no—
190M16—2112no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—4221no—
194C2×C8—4221no—
195C2×C8—4221no—
196C2×C8C22⋊D21112yes—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—82no—
201C23—4221no—
202C23#1680996111yescharacteristic
203Q8—22no—
204Q8—22no—
205Q8—41no—
206Q8—41no—
207Q8—4221no—
208Q8—4221no—
209D4—22no—
210D4—22no—
211D4—41no—
212D4—41no—
213D4—8441no—
214D4—8441no—
215D4—4221no—
216D4—4221no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—8441no—
221C2×C4—8441no—
222C2×C4—4221no—
223C2×C4D4×D21111yescharacteristic
224C2×C4D4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4C2×C22⋊D21111yescharacteristic
227C2×C4C2×C22⋊D21111yescharacteristic
228C2×C4C2×C22⋊D21111yescharacteristic
229C8—22no—
230C8—8441no—
231C8—4222no—
232C8—4222no—
233C22—22no—
234C22—42no—
235C22—82no—
236C22—82no—
237C22—82no—
238C22—8442no—
239C22—8441no—
240C22#2026994811yescharacteristic
241C22#2026994911yescharacteristic
242C22#2026994911yescharacteristic
243C4—42no—
244C4—8441no—
245C4#2026994511yescharacteristic
246C4#2026994511yescharacteristic
247C4#2026994611yescharacteristic
248C4#2026997311yescharacteristic
249C2—21no—
250C2—21no—
251C2—82no—
252C2—16881no—
253C2#2721305411yescharacteristic
254C2#2721340612yes—
subgroupG/H
centerC22#20269948
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×SD16×C211C2
socleC22×C211#121
prime radicalC2×#20270003C2
hypercenterC22×SD16D211
hypocenterC211#2019
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20270008C2semidirect
3. C2×#20269896C2semidirect
4. C2×#20269894C2semidirect
5. #27213112C2semidirect
6. #27212911C2semidirect
7. C22×SD16×C211C2semidirect
127. C211#2019semidirect
131. C22×SD16D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
C2×#202700032
C2×#202698962
C2×#202698942
#272131122
#272129112
C22×SD16×C2112
#2019211