Finite Group Databasebeta

← #41014428

Subgroups — #41014428

Order 27008.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20270008C211yesmaximal, characteristic, complemented
2#27213715C211yesmaximal, characteristic
3C2×#20269966C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#419×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×D844C2211yescharacteristic
10#20270008—22no—
11#20270008—22no—
12C2×#16809952—21no—
13C2×#16809949—21no—
14C2×#16809946C2211yescharacteristic
15#20269974—21no—
16#20269966—22no—
17#20269966—22no—
18#20269965—21no—
19#20269952—22no—
20#20269947—22no—
21C2×(C2×D211)⋊C4—22no—
22C2×C4×Dic211C2211yescharacteristic
23#20269903—22no—
24#20269893—22no—
25C22×Q8×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#117×C211C2211yescharacteristic
28#103×C211—22no—
29C23×D211D412yes—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34C2×Dic422D411yescharacteristic
35C2×D844—24no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844—21no—
40C2×D844—21no—
41C2×D844—21no—
42C2×D844D411yescharacteristic
43C2×C22⋊D211—21no—
44C22×Dic211—21no—
45C2×C4×D211—21no—
46#16809952—24no—
47#16809949—24no—
48#16809946—21no—
49#16809946D412yescomplemented
50C2×C211⋊C8—21no—
51(C2×D211)⋊C4—28no—
52(C2×D211)⋊C4—24no—
53C4×Dic211—21no—
54C4×Dic211D412yescomplemented
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211—21no—
58C2×Q8×C211—21no—
59C2×Q8×C211—21no—
60C2×Q8×C211D412yescomplemented
61C22×C4×C211—21no—
62C22×C4×C211C2311yescharacteristic
63M16×C211—21no—
64M16×C211—21no—
65C2×C8×C211—21no—
66C4⋊C4×C211D412yes—
67C22⋊C4×C211—21no—
68C42×C211—21no—
69C22×D211—22no—
70C22×D211—22no—
71C22×D211—21no—
72C22×D211—22no—
73C22×D211—44no—
74Dic422—21no—
75Dic422—21no—
76D844—21no—
77D844—21no—
78D844—41no—
79D844—41no—
80D844—41no—
81D844—41no—
82D844—84no—
83C22⋊D211—41no—
84C22⋊D211—41no—
85C2×Dic211—22no—
86C2×Dic211—21no—
87C2×Dic211—42no—
88C4×D211—41no—
89C4×D211—41no—
90C211⋊C8—22no—
91C23×C211C2×D411yescharacteristic
92Q8×C211—22no—
93Q8×C211—22no—
94Q8×C211—41no—
95Q8×C211—41no—
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
106C8×C211—41no—
107C22×C211—22no—
108C22×C211#12011yescharacteristic
109C22×C211#12011yescharacteristic
110C22×C211#12111yescharacteristic
111C2×D211—44no—
112C2×D211—41no—
113C2×D211—42no—
114C2×D211—42no—
115C2×D211—44no—
116C2×D211—82no—
117C4×C211—42no—
118C4×C211—42no—
119C4×C211#12011yescharacteristic
120C4×C211#12111yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12711yescharacteristic
123Dic211—42no—
124Dic211—41no—
125C2×C211—21no—
126C2×C211—21no—
127C2×C211#39211yescharacteristic
128C2×C211#45512yes—
129D211—81no—
130D211—82no—
131C211#200411yescharacteristic, complemented, Sylow
132#2004—2111nomaximal, Sylow
133C2×#137—2111no—
134#535—2111no—
135C2×#124—2111no—
136C2×#104—2111no—
137#419D21111yescharacteristic, complemented
138C2×#100—2111no—
139#339—2111no—
140C2×#41—2111no—
141C22×Q8C2×D21111yescharacteristic
142C22×D4—2111no—
143#137—4222no—
144#137—4222no—
145C2×Q16—4221no—
146C2×SD16—4221no—
147C2×M16—2111no—
148C2×M16C2×D21111yescharacteristic
149#127—4221no—
150#124—4221no—
151#124—4222no—
152#124—4222no—
153#121—4222no—
154#120—4222no—
155#117C2×D21111yescharacteristic
156C2×C22⋊C4—4222no—
157C2×C42—2111no—
158#104—4222no—
159#103—22no—
160#100—4222no—
161C24—2112no—
162#41—4221no—
163#41—4221no—
164#41—4221no—
165#41—4221no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—2111no—
172C2×Q8D84412yescomplemented
173C2×D4—4224no—
174C2×D4—4221no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—4221no—
180C2×D4—4221no—
181C2×D4—2111no—
182C22×C4—21no—
183C22×C4—4221no—
184C22×C4—4221no—
185C22×C4C22×D21111yescharacteristic
186Q16—4224no—
187SD16—4224no—
188M16—21no—
189M16—21no—
190M16—4221no—
191M16—2112no—
192C2×C8—21no—
193C2×C8—4221no—
194C4⋊C4C22⋊D21112yes—
195C22⋊C4—21no—
196C22⋊C4—4228no—
197C22⋊C4—4224no—
198C42—21no—
199C42—4221no—
200C42—2112no—
201C23—8444no—
202C23—4222no—
203C23—4221no—
204C23—4222no—
205C23—4222no—
206C23D4×D21111yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—4221no—
212Q8—4221no—
213D4—8441no—
214D4—8441no—
215D4—8441no—
216D4—8441no—
217D4—8441no—
218D4—8441no—
219D4—16884no—
220D4—4221no—
221D4—4221no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—8442no—
227C2×C4—8441no—
228C2×C4—8441no—
229C2×C4—4221no—
230C2×C4—4222no—
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4D4×D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C2×C4C2×D84411yescharacteristic
236C2×C4#1680996211yescharacteristic
237C8—41no—
238C8—4222no—
239C22—22no—
240C22—8444no—
241C22—8442no—
242C22—8444no—
243C22—8441no—
244C22—8442no—
245C22—16882no—
246C22#2026994511yescharacteristic
247C22#2026994711yescharacteristic
248C22#2026995011yescharacteristic
249C4—42no—
250C4—42no—
251C4—8442no—
252C4—8441no—
253C4#2026994711yescharacteristic
254C4#2026995011yescharacteristic
255C4#2026995111yescharacteristic
256C4#2026997311yescharacteristic
257C2—21no—
258C2—21no—
259C2—16882no—
260C2—16881no—
261C2#2721305511yescharacteristic
262C2#2721332212yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#120
prime radical#27213715C2
hypercenter#419D211
hypocenterC211#2004
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#20270008C2semidirect
3. C2×#20269966C2semidirect
4. C2×#20269903C2semidirect
5. C2×#20269893C2semidirect
6. #27212905C2semidirect
7. #419×C211C2semidirect
49. #16809946D4semidirect
54. C4×Dic211D4semidirect
60. C2×Q8×C211D4semidirect
131. C211#2004semidirect
137. #419D211semidirect
172. C2×Q8D844semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
#272137152
C2×#202699662
C2×#202699032
C2×#202698932
#272129052
#419×C2112
#2004211