Finite Group Databasebeta

← #41014417

Subgroups — #41014417

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
1#27213717C211yesmaximal, characteristic, complemented
2C2×#20269964C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic, complemented
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269975—21no—
11#20269964—22no—
12#20269964—22no—
13#20269964—21no—
14#20269958—22no—
15#20269956—22no—
16#20269936C2211yescharacteristic
17C2×C22⋊Dic211—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269904—22no—
20#20269898—22no—
21#20269895—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×Dic422—21no—
35C22×Dic211—21no—
36C22×Dic211D412yes—
37#16809946—21no—
38#16809946—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211D412yes—
41Dic211⋊C4—24no—
42C22⋊Dic211—28no—
43C22⋊Dic211—24no—
44C22⋊Dic211—21no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C4×Dic211D412yes—
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—41no—
70Dic422—41no—
71Dic422—41no—
72Dic422—41no—
73C2×Dic211—22no—
74C2×Dic211—22no—
75C2×Dic211—22no—
76C2×Dic211—22no—
77C2×Dic211—22no—
78C2×Dic211—42no—
79C211⋊C8—41no—
80C23×C211—21no—
81C23×C211—22no—
82C23×C211—44no—
83C23×C211—41no—
84C23×C211—41no—
85C23×C211C2×D411yescharacteristic
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×C211#4111yescharacteristic
105C8×C211—22no—
106C22×C211—22no—
107C22×C211—44no—
108C22×C211—41no—
109C22×C211—42no—
110C22×C211—42no—
111C22×C211—82no—
112C22×C211—82no—
113C22×C211#12011yescharacteristic
114C22×C211#12011yescharacteristic
115C22×C211#12111yescharacteristic
116C4×C211—41no—
117C4×C211#12011yescharacteristic
118C4×C211#12111yescharacteristic
119C4×C211#12111yescharacteristic
120C4×C211#12711yescharacteristic
121Dic211—42no—
122Dic211—42no—
123Dic211—42no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211—81no—
127C2×C211—82no—
128C2×C211#39211yescharacteristic
129C2×C211#45412yes—
130C211#200311yescharacteristic, complemented, Sylow
131#2003—2111nomaximal, Sylow
132C2×AGL(1,ℤ/8)D21111yescharacteristic
133#536—2111no—
134C2×#124—2111no—
135C2×#104—2111no—
136#418—2111no—
137C2×#101—2111no—
138#339—2111no—
139C2×#41C2×D21111yescharacteristic
140C22×Q8—2111no—
141C22×D4C2×D21111yescharacteristic
142AGL(1,ℤ/8)—22no—
143AGL(1,ℤ/8)—22no—
144C2×SD16—21no—
145C2×D8—21no—
146C2×M16—2111no—
147C2×M16C2×D21111yescharacteristic
148C4⋊Q8—4221no—
149#124—4222no—
150#124—4221no—
151#124—4222no—
152#123—4222no—
153C22⋊Q8—4222no—
154#117—2111no—
155C2×C22⋊C4—4222no—
156C2×C42—2111no—
157#104—4222no—
158#102—4222no—
159#101—4222no—
160C24—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8—4221no—
171C2×Q8C22⋊D21111yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4C22⋊D21111yescharacteristic
179C2×D4C22⋊D21112yes—
180C22×C4—21no—
181C22×C4—4221no—
182C22×C4—2112no—
183C22×C4C22×D21111yescharacteristic
184SD16—24no—
185D8—24no—
186M16—21no—
187M16—4221no—
188M16—4221no—
189M16C22⋊D21112yes—
190C2×C8—21no—
191C2×C8—4221no—
192C4⋊C4—4224no—
193C4⋊C4—2112no—
194C22⋊C4—4228no—
195C22⋊C4—4224no—
196C22⋊C4—4221no—
197C42—4221no—
198C42—4221no—
199C42—2112no—
200C23—22no—
201C23—21no—
202C23—44no—
203C23—41no—
204C23—41no—
205C23D4×D21111yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—8441no—
209Q8—8441no—
210Q8—8441no—
211Q8—8441no—
212D4—21no—
213D4—22no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—8442no—
224C2×C4—4222no—
225C2×C4—4222no—
226C2×C4—4222no—
227C2×C4—4222no—
228C2×C4—4222no—
229C2×C4D4×D21111yescharacteristic
230C2×C4D4×D21111yescharacteristic
231C2×C4C2×C22⋊D21111yescharacteristic
232C2×C4C2×C22⋊D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4#1680996111yescharacteristic
235C8—22no—
236C8—8441no—
237C22—22no—
238C22—44no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—82no—
243C22—82no—
244C22#2026994511yescharacteristic
245C22#2026994511yescharacteristic
246C22#2026994811yescharacteristic
247C4—41no—
248C4—8442no—
249C4—8442no—
250C4—8442no—
251C4#2026994611yescharacteristic
252C4#2026994911yescharacteristic
253C4#2026994911yescharacteristic
254C4#2026997311yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—81no—
259C2#2721305411yescharacteristic
260C2#2721331712yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×AGL(1,ℤ/8)×C211C2
socleC22×C211#120
hypercenterC2×AGL(1,ℤ/8)D211
hypocenterC211#2003
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. #27213717C2semidirect
2. C2×#20269964C2semidirect
3. C2×#20269904C2semidirect
4. #27213151C2semidirect
5. C2×#20269895C2semidirect
6. #27212907C2semidirect
130. C211#2003semidirect

Maximal subgroups

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

subgroupindex
#272137172
C2×#202699642
C2×#202699042
#272131512
C2×#202698952
#272129072
C2×AGL(1,ℤ/8)×C2112
#2003211