Finite Group Databasebeta

← #41014423

Subgroups — #41014423

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×#20270005C211yesmaximal, characteristic
2#27213722C211yesmaximal, characteristic, complemented
3C2×#20269965C211yesmaximal, characteristic, complemented
4C2×#20269905C211yesmaximal, characteristic, complemented
5#27213150C211yesmaximal, characteristic, complemented
6C2×#20269895C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10C22×D844C2211yescharacteristic
11#20270005—22no—
12#20270005—22no—
13C2×#16809951—21no—
14C2×D1688—21no—
15C2×#16809946C2211yescharacteristic
16#20269977—21no—
17#20269965—22no—
18#20269965—21no—
19#20269965—22no—
20#20269961—22no—
21#20269957—22no—
22C2×(C2×D211)⋊C4—22no—
23#20269905—22no—
24#20269897—22no—
25#20269895—22no—
26C2×M16×C211C2211yescharacteristic
27#117×C211C2211yescharacteristic
28C2×C42×C211C2211yescharacteristic
29C23×D211—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34C2×Dic422—21no—
35C2×Dic422—21no—
36C2×Dic422—21no—
37C2×Dic422—21no—
38C2×Dic422—21no—
39C2×Dic422—21no—
40C2×Dic422D411yescharacteristic
41C2×D844—21no—
42C2×D844—21no—
43C2×D844—21no—
44C2×D844—21no—
45C2×D844—21no—
46C2×D844D411yescharacteristic
47C2×D844D412yes—
48C2×C22⋊D211—21no—
49C22×Dic211D412yes—
50C2×C4×D211—21no—
51#16809951—24no—
52D1688—24no—
53#16809946—21no—
54#16809946—21no—
55C2×C211⋊C8—21no—
56C4⋊Dic211—24no—
57(C2×D211)⋊C4—28no—
58(C2×D211)⋊C4—24no—
59C22×C4×C211—21no—
60C22×C4×C211C2311yescharacteristic
61M16×C211—21no—
62M16×C211D412yes—
63C2×C8×C211—21no—
64C4⋊C4×C211D412yes—
65C22⋊C4×C211—21no—
66C42×C211—21no—
67C42×C211—21no—
68C42×C211D412yes—
69C22×D211—21no—
70C22×D211—22no—
71C22×D211—44no—
72C22×D211—41no—
73C22×D211—41no—
74Dic422—21no—
75Dic422—21no—
76Dic422—41no—
77Dic422—41no—
78Dic422—41no—
79Dic422—41no—
80D844—21no—
81D844—22no—
82D844—22no—
83D844—21no—
84D844—41no—
85D844—41no—
86C22⋊D211—41no—
87C22⋊D211—41no—
88C2×Dic211—22no—
89C2×Dic211—22no—
90C2×Dic211—21no—
91C2×Dic211—22no—
92C4×D211—41no—
93C4×D211—41no—
94C211⋊C8—41no—
95C23×C211C2×D411yescharacteristic
96C2×C4×C211—22no—
97C2×C4×C211—22no—
98C2×C4×C211—42no—
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#12011yescharacteristic
108C22×C211#12011yescharacteristic
109C22×C211#12111yescharacteristic
110C2×D211—44no—
111C2×D211—41no—
112C2×D211—42no—
113C2×D211—42no—
114C2×D211—82no—
115C2×D211—82no—
116C4×C211—42no—
117C4×C211—42no—
118C4×C211#12011yescharacteristic
119C4×C211#12111yescharacteristic
120C4×C211#12111yescharacteristic
121C4×C211#12711yescharacteristic
122Dic211—41no—
123Dic211—42no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211#39211yescharacteristic
127C2×C211#45412yes—
128D211—81no—
129D211—82no—
130C211#200311yescharacteristic, complemented, Sylow
131#2003—2111nomaximal, Sylow
132C2×AGL(1,ℤ/8)—2111no—
133#536—2111no—
134C2×#124—2111no—
135C2×#104—2111no—
136#418—2111no—
137C2×#101—2111no—
138#339D21111yescharacteristic, complemented
139C2×#41—2111no—
140C22×Q8—2111no—
141C22×D4—2111no—
142AGL(1,ℤ/8)—4222no—
143AGL(1,ℤ/8)—4222no—
144C2×SD16—4221no—
145C2×D8—4221no—
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#117C2×D21111yescharacteristic
155C2×C22⋊C4—4222no—
156C2×C42C2×D21111yescharacteristic
157#104—4222no—
158#102—4222no—
159#101—4222no—
160C24—4221no—
161#41—4221no—
162#41—4221no—
163#41—4221no—
164#41—4221no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8—4221no—
171C2×Q8—2111no—
172C2×D4—4221no—
173C2×D4—4221no—
174C2×D4—4221no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—2111no—
179C2×D4—2112no—
180C22×C4—21no—
181C22×C4—4221no—
182C22×C4—2112no—
183C22×C4C22×D21111yescharacteristic
184SD16—4224no—
185D8—4224no—
186M16—21no—
187M16—4221no—
188M16—4221no—
189M16D84412yes—
190C2×C8—21no—
191C2×C8—4221no—
192C4⋊C4—4224no—
193C4⋊C4D84412yes—
194C22⋊C4—21no—
195C22⋊C4—4228no—
196C22⋊C4—4224no—
197C42—21no—
198C42—21no—
199C42D84412yes—
200C23—8444no—
201C23—8441no—
202C23—8441no—
203C23—4222no—
204C23—4221no—
205C23D4×D21111yescharacteristic
206Q8—8441no—
207Q8—8441no—
208Q8—8441no—
209Q8—8441no—
210Q8—4221no—
211Q8—4221no—
212D4—8441no—
213D4—8441no—
214D4—8441no—
215D4—8441no—
216D4—4221no—
217D4—4222no—
218D4—4222no—
219D4—4221no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—8441no—
224C2×C4—8441no—
225C2×C4—4222no—
226C2×C4—4222no—
227C2×C4—4221no—
228C2×C4—4222no—
229C2×C4D4×D21111yescharacteristic
230C2×C4D4×D21111yescharacteristic
231C2×C4C2×D84411yescharacteristic
232C2×C4C2×D84411yescharacteristic
233C2×C4C2×D84411yescharacteristic
234C2×C4#1680996311yescharacteristic
235C8—22no—
236C8—8441no—
237C22—22no—
238C22—8444no—
239C22—8441no—
240C22—8442no—
241C22—8442no—
242C22—16882no—
243C22—16882no—
244C22#2026994711yescharacteristic
245C22#2026994711yescharacteristic
246C22#2026995211yescharacteristic
247C4—42no—
248C4—42no—
249C4—8441no—
250C4—8442no—
251C4#2026994711yescharacteristic
252C4#2026995211yescharacteristic
253C4#2026995211yescharacteristic
254C4#2026997411yescharacteristic
255C2—21no—
256C2—21no—
257C2—16882no—
258C2—16881no—
259C2#2721305611yescharacteristic
260C2#2721332012yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#339×C211C2
socleC22×C211#120
prime radicalC2×#20270005C2
hypercenter#339D211
hypocenterC211#2003
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #27213722C2semidirect
3. C2×#20269965C2semidirect
4. C2×#20269905C2semidirect
5. #27213150C2semidirect
6. C2×#20269895C2semidirect
7. #339×C211C2semidirect
130. C211#2003semidirect
138. #339D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700052
#272137222
C2×#202699652
C2×#202699052
#272131502
C2×#202698952
#339×C2112
#2003211