Finite Group Databasebeta

← #41014424

Subgroups — #41014424

Order 27008.

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

normal subgroups only

This group has 394 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#27213710C211yesmaximal, characteristic
2C2×#20269966C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213155C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269973—21no—
11#20269966—22no—
12#20269966—21no—
13#20269966—22no—
14#20269951—22no—
15#20269945—22no—
16#20269936C2211yescharacteristic
17C2×(C2×D211)⋊C4—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269904—22no—
20#20269901—22no—
21#20269893—22no—
22C2×#41×C211C2211yescharacteristic
23C22×Q8×C211C2211yescharacteristic
24#137×C211—22no—
25#137×C211—22no—
26C2×Q16×C211—21no—
27C2×SD16×C211—21no—
28C2×M16×C211C2211yescharacteristic
29C23×D211D412yes—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×C22⋊D211—24no—
37C22×Dic211—21no—
38#16809946—21no—
39#16809946—21no—
40C2×C211⋊C8—21no—
41C4⋊Dic211D412yes—
42C22⋊Dic211—21no—
43(C2×D211)⋊C4—28no—
44(C2×D211)⋊C4—24no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C4×Dic211D412yescomplemented
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×C211D411yescharacteristic
58C2×Q8×C211D412yescomplemented
59C2×D4×C211—21no—
60C2×D4×C211D411yescharacteristic
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64Q16×C211—24no—
65SD16×C211—24no—
66M16×C211—21no—
67M16×C211D412yescomplemented
68C2×C8×C211—21no—
69C22×D211—22no—
70C22×D211—22no—
71C22×D211—22no—
72C22×D211—44no—
73D844—41no—
74D844—41no—
75D844—41no—
76D844—41no—
77C22⋊D211—84no—
78C2×Dic211—22no—
79C2×Dic211—22no—
80C2×Dic211—42no—
81C211⋊C8—41no—
82C23×C211—21no—
83C23×C211C2×D411yescharacteristic
84Q8×C211—22no—
85Q8×C211—21no—
86Q8×C211—22no—
87Q8×C211—21no—
88Q8×C211—41no—
89Q8×C211—41no—
90D4×C211—21no—
91D4×C211—21no—
92D4×C211—41no—
93D4×C211—41no—
94C2×C4×C211—21no—
95C2×C4×C211—22no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
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—22no—
107C22×C211—22no—
108C22×C211—41no—
109C22×C211—42no—
110C22×C211#12011yescharacteristic
111C22×C211#12011yescharacteristic
112C22×C211#12111yescharacteristic
113C2×D211—44no—
114C2×D211—42no—
115C2×D211—44no—
116C2×D211—82no—
117C4×C211—41no—
118C4×C211—42no—
119C4×C211#12011yescharacteristic
120C4×C211#12111yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12711yescharacteristic
123Dic211—42no—
124Dic211—42no—
125C2×C211—21no—
126C2×C211—21no—
127C2×C211—81no—
128C2×C211#39211yescharacteristic
129C2×C211#45512yes—
130D211—82no—
131C211#200411yescharacteristic, complemented, Sylow
132#2004—2111nomaximal, Sylow
133C2×#137D21111yescharacteristic, complemented
134#535—2111no—
135C2×#124—2111no—
136C2×#104—2111no—
137#419—2111no—
138C2×#100—2111no—
139#339—2111no—
140C2×#41C2×D21111yescharacteristic
141C22×Q8C2×D21111yescharacteristic
142C22×D4—2111no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
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#117—2111no—
156C2×C22⋊C4—4222no—
157C2×C42—2111no—
158#104—4222no—
159#103—4222no—
160#100—4222no—
161C24—2112no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D21111yescharacteristic
172C2×Q8C22⋊D21112yescomplemented
173C2×D4—21no—
174C2×D4—4224no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—4221no—
180C2×D4—4221no—
181C2×D4C22⋊D21111yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—4221no—
185C22×C4C22×D21111yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—4221no—
190M16—4221no—
191M16C22⋊D21112yescomplemented
192C2×C8—21no—
193C2×C8—4221no—
194C4⋊C4—2112no—
195C22⋊C4—4228no—
196C22⋊C4—4224no—
197C22⋊C4—4221no—
198C42—4221no—
199C42—4221no—
200C42—2112no—
201C23—21no—
202C23—8444no—
203C23—4222no—
204C23—4222no—
205C23—4222no—
206C23D4×D21111yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—8441no—
218D4—8441no—
219D4—8441no—
220D4—8441no—
221D4—16884no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—8442no—
229C2×C4—4222no—
230C2×C4—4222no—
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4C2×C22⋊D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C2×C4C2×C22⋊D21111yescharacteristic
236C2×C4#1680996111yescharacteristic
237C8—22no—
238C8—8441no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—8444no—
243C22—8444no—
244C22—8442no—
245C22—16882no—
246C22#2026994511yescharacteristic
247C22#2026994511yescharacteristic
248C22#2026994811yescharacteristic
249C4—41no—
250C4—42no—
251C4—8442no—
252C4—8442no—
253C4#2026994611yescharacteristic
254C4#2026994911yescharacteristic
255C4#2026994911yescharacteristic
256C4#2026997311yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—16882no—
261C2#2721305411yescharacteristic
262C2#2721332212yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC22×C211#120
prime radical#27213710C2
hypercenterC2×#137D211
hypocenterC211#2004
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#20269966C2semidirect
3. C2×#20269904C2semidirect
4. #27213155C2semidirect
5. C2×#20269893C2semidirect
6. #27212907C2semidirect
7. C2×#137×C211C2semidirect
47. C4×Dic211D4semidirect
58. C2×Q8×C211D4semidirect
67. M16×C211D4semidirect
131. C211#2004semidirect
133. C2×#137D211semidirect
172. C2×Q8C22⋊D211semidirect
191. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272137102
C2×#202699662
C2×#202699042
#272131552
C2×#202698932
#272129072
C2×#137×C2112
#2004211