Finite Group Databasebeta

← #41014504

Subgroups — #41014504

Order 27008.

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

normal subgroups only

This group has 382 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×#20270002C211yesmaximal, characteristic, complemented
2C22×#16809947C211yesmaximal, characteristic, complemented
3C2×#20269894C211yesmaximal, characteristic, complemented
4C2×#20269893C211yesmaximal, characteristic, complemented
5#27213113C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9#20270002—22no—
10#20270002—22no—
11C2×#16809949—21no—
12C2×#16809947—22no—
13C2×#16809947—22no—
14C2×#16809947—22no—
15C2×#16809947—21no—
16C2×#16809946C2211yescharacteristic
17C2×#16809946C2211yescharacteristic
18C22×C211⋊C8C2211yescharacteristic
19#20269894—22no—
20#20269893—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
29C23×D211—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844D412yes—
36#16809949—24no—
37#16809947—28no—
38#16809947—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—
69C22×D211—22no—
70C22×D211—41no—
71C22×D211—41no—
72C22×D211—82no—
73D844—22no—
74D844—22no—
75D844—41no—
76D844—41no—
77C211⋊C8—22no—
78C211⋊C8—22no—
79C211⋊C8—41no—
80C23×C211—21no—
81C23×C211—22no—
82C23×C211—41no—
83C23×C211—41no—
84C23×C211—82no—
85C23×C211#4111yescharacteristic
86Q8×C211—21no—
87Q8×C211—21no—
88D4×C211—21no—
89D4×C211—22no—
90D4×C211—22no—
91D4×C211—21no—
92D4×C211—41no—
93D4×C211—41no—
94D4×C211—41no—
95D4×C211—41no—
96C2×C4×C211—21no—
97C2×C4×C211—41no—
98C2×C4×C211—41no—
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C8×C211—22no—
106C22×C211—22no—
107C22×C211—41no—
108C22×C211—42no—
109C22×C211—42no—
110C22×C211—82no—
111C22×C211—82no—
112C22×C211—82no—
113C22×C211#12111yescharacteristic
114C22×C211#12111yescharacteristic
115C22×C211#12111yescharacteristic
116C2×D211—42no—
117C2×D211—82no—
118C2×D211—82no—
119C2×D211—82no—
120C4×C211—41no—
121C4×C211#12011yescharacteristic
122C4×C211#12011yescharacteristic
123C4×C211#12011yescharacteristic
124C4×C211#12711yescharacteristic
125C2×C211—21no—
126C2×C211—21no—
127C2×C211—81no—
128C2×C211—82no—
129C2×C211#39211yescharacteristic
130C2×C211#47212yes—
131D211—82no—
132C211#201811yescharacteristic, complemented, Sylow
133#2018—2111nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2111no—
135C2×AGL(1,ℤ/8)D21111yescharacteristic, complemented
136C22×D8—2111no—
137C2×#100—2111no—
138C2×#100—2111no—
139#408—2111no—
140#341—2111no—
141C2×#41C2×D21111yescharacteristic
142C22×D4—2111no—
143C22×D4C2×D21111yescharacteristic
144AGL(1,ℤ/8)—22no—
145AGL(1,ℤ/8)—22no—
146AGL(1,ℤ/8)—4222no—
147AGL(1,ℤ/8)—4222no—
148C2×SD16—21no—
149C2×SD16—4221no—
150C2×D8—21no—
151C2×D8—4222no—
152C2×D8—4222no—
153C2×D8—4221no—
154C2×D8—4222no—
155C2×M16—2111no—
156C2×M16—2111no—
157C2×M16C2×D21111yescharacteristic
158C22×C8—2111no—
159#100—4222no—
160#100—4222no—
161C22⋊C8—4222no—
162C24—21no—
163C24—4221no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168C2×Q8C22⋊D21111yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—4221no—
180C2×D4—2112no—
181C2×D4C22⋊D21111yescharacteristic
182C2×D4C22⋊D21112yes—
183C22×C4—21no—
184C22×C4C22×D21111yescharacteristic
185SD16—24no—
186SD16—4224no—
187D8—24no—
188D8—4228no—
189D8—4224no—
190M16—21no—
191M16—4221no—
192M16—4221no—
193M16—4221no—
194M16—2112no—
195M16C22⋊D21112yes—
196C2×C8—21no—
197C2×C8—4221no—
198C2×C8—4221no—
199C2×C8—4221no—
200C2×C8—4221no—
201C2×C8—2112no—
202C23—21no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—82no—
207C23—8441no—
208C23—8441no—
209C23—16882no—
210C23—4222no—
211C23#1680996111yescharacteristic
212Q8—21no—
213Q8—21no—
214D4—22no—
215D4—21no—
216D4—21no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—8441no—
223D4—8441no—
224D4—4222no—
225D4—4222no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4D4×D21111yescharacteristic
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4C2×C22⋊D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C8—22no—
236C8—8441no—
237C8—4222no—
238C8—4222no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22—8442no—
247C22—16882no—
248C22—16882no—
249C22—16882no—
250C22#2026994811yescharacteristic
251C22#2026994911yescharacteristic
252C22#2026994911yescharacteristic
253C4—41no—
254C4#2026994511yescharacteristic
255C4#2026994511yescharacteristic
256C4#2026994611yescharacteristic
257C4#2026997311yescharacteristic
258C2—21no—
259C2—21no—
260C2—81no—
261C2—82no—
262C2—16882no—
263C2#2721305411yescharacteristic
264C2#2721341412yes—
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×AGL(1,ℤ/8)×C211C2
socleC22×C211#121
hypercenterC2×AGL(1,ℤ/8)D211
hypocenterC211#2018
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×#20270002C2semidirect
2. C22×#16809947C2semidirect
3. C2×#20269894C2semidirect
4. C2×#20269893C2semidirect
5. #27213113C2semidirect
6. #27212910C2semidirect
7. C2×AGL(1,ℤ/8)×C211C2semidirect
132. C211#2018semidirect
135. C2×AGL(1,ℤ/8)D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700022
C22×#168099472
C2×#202698942
C2×#202698932
#272131132
#272129102
C2×AGL(1,ℤ/8)×C2112
#2018211