Finite Group Databasebeta

← #41014512

Subgroups — #41014512

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×#20270007C211yesmaximal, characteristic, complemented
2C22×#16809950C211yesmaximal, characteristic, complemented
3C2×#20269895C211yesmaximal, characteristic, complemented
4C2×#20269894C211yesmaximal, characteristic, complemented
5#27213113C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007—22no—
11C2×#16809952—21no—
12C2×#16809950—22no—
13C2×#16809950—22no—
14C2×#16809950—22no—
15C2×#16809950—21no—
16C2×#16809946C2211yescharacteristic
17C2×#16809946C2211yescharacteristic
18C22×C211⋊C8C2211yescharacteristic
19#20269895—22no—
20#20269894—22no—
21C22⋊(C211⋊C8)—22no—
22C2×#41×C211C2211yescharacteristic
23C22×D4×C211C2211yescharacteristic
24AGL(1,ℤ/8)×C211—22no—
25AGL(1,ℤ/8)×C211—22no—
26C2×SD16×C211—21no—
27C2×D8×C211—21no—
28C2×M16×C211C2211yescharacteristic
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422D412yes—
35C22×Dic211—21no—
36#16809952—24no—
37#16809950—28no—
38#16809950—24no—
39#16809946—21no—
40#16809946—21no—
41#16809946—21no—
42#16809946D412yes—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48C24×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53C2×Q8×C211D411yescharacteristic
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211D411yescharacteristic
61C2×D4×C211D412yes—
62C22×C4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64SD16×C211—24no—
65D8×C211—24no—
66M16×C211—21no—
67M16×C211D412yes—
68C2×C8×C211—21no—
69Dic422—22no—
70Dic422—22no—
71Dic422—41no—
72Dic422—41no—
73C2×Dic211—22no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C211⋊C8—22no—
77C211⋊C8—22no—
78C211⋊C8—41no—
79C23×C211—21no—
80C23×C211—22no—
81C23×C211—41no—
82C23×C211—41no—
83C23×C211—82no—
84C23×C211#4111yescharacteristic
85Q8×C211—21no—
86Q8×C211—21no—
87D4×C211—21no—
88D4×C211—22no—
89D4×C211—22no—
90D4×C211—21no—
91D4×C211—41no—
92D4×C211—41no—
93D4×C211—41no—
94D4×C211—41no—
95C2×C4×C211—21no—
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—41no—
107C22×C211—42no—
108C22×C211—42no—
109C22×C211—82no—
110C22×C211—82no—
111C22×C211—82no—
112C22×C211#12111yescharacteristic
113C22×C211#12111yescharacteristic
114C22×C211#12111yescharacteristic
115C4×C211—41no—
116C4×C211#12011yescharacteristic
117C4×C211#12011yescharacteristic
118C4×C211#12011yescharacteristic
119C4×C211#12711yescharacteristic
120Dic211—42no—
121C2×C211—21no—
122C2×C211—21no—
123C2×C211—81no—
124C2×C211—82no—
125C2×C211#39211yescharacteristic
126C2×C211#47112yes—
127C211#201911yescharacteristic, complemented, Sylow
128#2019—2111nomaximal, Sylow
129C2×#137—2111no—
130C2×AGL(1,ℤ/8)D21111yescharacteristic
131C22×SD16—2111no—
132C2×#101—2111no—
133C2×#100—2111no—
134#408—2111no—
135#341—2111no—
136C2×#41C2×D21111yescharacteristic
137C22×Q8—2111no—
138C22×D4C2×D21111yescharacteristic
139#137—4222no—
140#137—4222no—
141AGL(1,ℤ/8)—22no—
142AGL(1,ℤ/8)—22no—
143C2×Q16—4221no—
144C2×SD16—21no—
145C2×SD16—4222no—
146C2×SD16—4222no—
147C2×SD16—4221no—
148C2×SD16—4222no—
149C2×D8—21no—
150C2×M16—2111no—
151C2×M16—2111no—
152C2×M16C2×D21111yescharacteristic
153C22×C8—2111no—
154#101—4222no—
155#100—4222no—
156C22⋊C8—4222no—
157C24—21no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8—4221no—
163C2×Q8—4221no—
164C2×Q8—4221no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—2112no—
168C2×Q8C22⋊D21111yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4C22⋊D21111yescharacteristic
176C2×D4C22⋊D21112yes—
177C22×C4—21no—
178C22×C4—4221no—
179C22×C4C22×D21111yescharacteristic
180Q16—4224no—
181SD16—24no—
182SD16—4228no—
183SD16—4224no—
184D8—24no—
185M16—21no—
186M16—4221no—
187M16—4221no—
188M16—4221no—
189M16—2112no—
190M16C22⋊D21112yes—
191C2×C8—21no—
192C2×C8—4221no—
193C2×C8—4221no—
194C2×C8—4221no—
195C2×C8—4221no—
196C2×C8—2112no—
197C23—21no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—82no—
202C23#1680996111yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—8441no—
206Q8—8441no—
207Q8—4222no—
208Q8—4222no—
209D4—22no—
210D4—21no—
211D4—21no—
212D4—22no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—8441no—
221C2×C4—8441no—
222C2×C4—4222no—
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—41no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22—82no—
240C22#2026994811yescharacteristic
241C22#2026994911yescharacteristic
242C22#2026994911yescharacteristic
243C4—41no—
244C4—8442no—
245C4#2026994511yescharacteristic
246C4#2026994511yescharacteristic
247C4#2026994611yescharacteristic
248C4#2026997311yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—82no—
253C2#2721305411yescharacteristic
254C2#2721340712yes—
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×AGL(1,ℤ/8)×C211C2
socleC22×C211#121
hypercenterC2×AGL(1,ℤ/8)D211
hypocenterC211#2019
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20270007C2semidirect
2. C22×#16809950C2semidirect
3. C2×#20269895C2semidirect
4. C2×#20269894C2semidirect
5. #27213113C2semidirect
6. #27212910C2semidirect
127. C211#2019semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
C22×#168099502
C2×#202698952
C2×#202698942
#272131132
#272129102
C2×AGL(1,ℤ/8)×C2112
#2019211