Finite Group Databasebeta

← #41014418

Subgroups — #41014418

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
2C2×#20269966C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213150C211yesmaximal, characteristic, complemented
5C2×#20269896C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#536×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9#20270002—22no—
10#20270002—22no—
11C2×#16809949—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15#20269966—22no—
16#20269966—22no—
17C2×(C2×D211)⋊C4—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269904—22no—
20#20269897—22no—
21#20269896—22no—
22C2×#41×C211C2211yescharacteristic
23C22×Q8×C211C2211yescharacteristic
24C4⋊Q8×C211—21no—
25#124×C211—21no—
26#123×C211—22no—
27C22⋊Q8×C211—22no—
28#117×C211C2211yescharacteristic
29C23×D211—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844D412yes—
36C22×Dic211—21no—
37#16809949—24no—
38#16809947—24no—
39#16809946—21no—
40#16809946—21no—
41#16809946—21no—
42#16809946D412yes—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45(C2×D211)⋊C4—28no—
46C4×Dic211—21no—
47C4×Dic211D412yes—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211—21no—
58C2×Q8×C211D411yescharacteristic
59C2×D4×C211—21no—
60C2×D4×C211D411yescharacteristic
61C22×C4×C211—21no—
62C22×C4×C211D412yes—
63C22×C4×C211C2311yescharacteristic
64C4⋊C4×C211—24no—
65C4⋊C4×C211D412yes—
66C22⋊C4×C211—24no—
67C22⋊C4×C211—21no—
68C42×C211—21no—
69C22×D211—22no—
70C22×D211—44no—
71C22×D211—41no—
72C22×D211—41no—
73D844—22no—
74D844—22no—
75D844—41no—
76D844—41no—
77C2×Dic211—22no—
78C2×Dic211—42no—
79C211⋊C8—22no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211C2×D411yescharacteristic
83Q8×C211—21no—
84Q8×C211—21no—
85Q8×C211—41no—
86Q8×C211—41no—
87Q8×C211—41no—
88Q8×C211—41no—
89D4×C211—21no—
90D4×C211—21no—
91D4×C211—41no—
92D4×C211—41no—
93C2×C4×C211—22no—
94C2×C4×C211—22no—
95C2×C4×C211—21no—
96C2×C4×C211—22no—
97C2×C4×C211—22no—
98C2×C4×C211—41no—
99C2×C4×C211—41no—
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211#4111yescharacteristic
106C22×C211—22no—
107C22×C211—41no—
108C22×C211—42no—
109C22×C211#12011yescharacteristic
110C22×C211#12011yescharacteristic
111C22×C211#12111yescharacteristic
112C2×D211—44no—
113C2×D211—42no—
114C2×D211—82no—
115C2×D211—82no—
116C4×C211—42no—
117C4×C211—41no—
118C4×C211—42no—
119C4×C211#12011yescharacteristic
120C4×C211#12111yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12711yescharacteristic
123Dic211—42no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211—81no—
127C2×C211#39211yescharacteristic
128C2×C211#45412yes—
129D211—82no—
130C211#200311yescharacteristic, complemented, Sylow
131#2003—2111nomaximal, Sylow
132C2×AGL(1,ℤ/8)—2111no—
133#536D21111yescharacteristic, complemented
134C2×#124—2111no—
135C2×#104—2111no—
136#418—2111no—
137C2×#101—2111no—
138#339—2111no—
139C2×#41C2×D21111yescharacteristic
140C22×Q8C2×D21111yescharacteristic
141C22×D4—2111no—
142AGL(1,ℤ/8)—4222no—
143AGL(1,ℤ/8)—4222no—
144C2×SD16—4221no—
145C2×D8—4221no—
146C2×M16—2111no—
147C2×M16—2111no—
148C4⋊Q8—21no—
149#124—21no—
150#124—4222no—
151#124—4222no—
152#123—22no—
153C22⋊Q8—22no—
154#117C2×D21111yescharacteristic
155C2×C22⋊C4—4222no—
156C2×C42—2111no—
157#104—4222no—
158#102—4222no—
159#101—4222no—
160C24—4221no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D21111yescharacteristic
172C2×D4—21no—
173C2×D4—4221no—
174C2×D4—4221no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—2112no—
179C2×D4C22⋊D21111yescharacteristic
180C22×C4—21no—
181C22×C4—4221no—
182C22×C4C22⋊D21112yes—
183C22×C4C22×D21111yescharacteristic
184SD16—4224no—
185D8—4224no—
186M16—4221no—
187M16—4221no—
188M16—4221no—
189M16—2112no—
190C2×C8—4221no—
191C2×C8—4221no—
192C4⋊C4—24no—
193C4⋊C4C22⋊D21112yes—
194C22⋊C4—24no—
195C22⋊C4—21no—
196C22⋊C4—4228no—
197C42—21no—
198C42—4221no—
199C42—2112no—
200C23—21no—
201C23—8444no—
202C23—8441no—
203C23—8441no—
204C23—4222no—
205C23C2×C22⋊D21111yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—41no—
209Q8—41no—
210Q8—41no—
211Q8—41no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—8441no—
217D4—8441no—
218D4—4222no—
219D4—4222no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—8442no—
228C2×C4—4222no—
229C2×C4D4×D21111yescharacteristic
230C2×C4D4×D21111yescharacteristic
231C2×C4D4×D21111yescharacteristic
232C2×C4C2×C22⋊D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4#1680996111yescharacteristic
235C8—8441no—
236C8—4222no—
237C22—22no—
238C22—42no—
239C22—41no—
240C22—8444no—
241C22—8442no—
242C22—16882no—
243C22—16882no—
244C22#2026994511yescharacteristic
245C22#2026994611yescharacteristic
246C22#2026994911yescharacteristic
247C4—42no—
248C4—42no—
249C4—41no—
250C4—8442no—
251C4#2026994511yescharacteristic
252C4#2026994811yescharacteristic
253C4#2026994911yescharacteristic
254C4#2026997311yescharacteristic
255C2—21no—
256C2—21no—
257C2—81no—
258C2—16882no—
259C2#2721305411yescharacteristic
260C2#2721331912yes—
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#536×C211C2
socleC22×C211#120
prime radicalC2×#20270002C2
hypercenter#536D211
hypocenterC211#2003
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#20269966C2semidirect
3. C2×#20269904C2semidirect
4. #27213150C2semidirect
5. C2×#20269896C2semidirect
6. #27212905C2semidirect
7. #536×C211C2semidirect
130. C211#2003semidirect
133. #536D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700022
C2×#202699662
C2×#202699042
#272131502
C2×#202698962
#272129052
#536×C2112
#2003211