Finite Group Databasebeta

← #41014425

Subgroups — #41014425

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
1C2×#20270007C211yesmaximal, characteristic, complemented
2C2×#20269964C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4#27213154C211yesmaximal, characteristic, complemented
5C2×#20269894C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#535×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007—22no—
11C2×#16809952—21no—
12C2×#16809950—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15#20269964—22no—
16#20269964—22no—
17C2×C22⋊Dic211—22no—
18C2×C4×Dic211C2211yescharacteristic
19#20269904—22no—
20#20269900—22no—
21#20269894—22no—
22C2×#41×C211C2211yescharacteristic
23C22×D4×C211C2211yescharacteristic
24#127×C211—21no—
25#124×C211—21no—
26#121×C211—22no—
27#120×C211—22no—
28#117×C211C2211yescharacteristic
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422D412yescomplemented
35C22×Dic211—21no—
36C22×Dic211—21no—
37#16809952—24no—
38#16809950—24no—
39#16809946—21no—
40#16809946—21no—
41#16809946—21no—
42#16809946D412yescomplemented
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C22⋊Dic211—28no—
46C4×Dic211—21no—
47C4×Dic211D412yescomplemented
48C24×C211D412yes—
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—24no—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211D411yescharacteristic
63C22×C4×C211—21no—
64C22×C4×C211C2311yescharacteristic
65C4⋊C4×C211D412yes—
66C22⋊C4×C211—24no—
67C22⋊C4×C211—21no—
68C42×C211—21no—
69Dic422—22no—
70Dic422—22no—
71Dic422—41no—
72Dic422—41no—
73C2×Dic211—22no—
74C2×Dic211—22no—
75C2×Dic211—42no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C211⋊C8—22no—
79C211⋊C8—41no—
80C23×C211—22no—
81C23×C211—21no—
82C23×C211—22no—
83C23×C211—22no—
84C23×C211—44no—
85C23×C211C2×D411yescharacteristic
86Q8×C211—21no—
87Q8×C211—21no—
88D4×C211—21no—
89D4×C211—21no—
90D4×C211—41no—
91D4×C211—41no—
92D4×C211—41no—
93D4×C211—41no—
94D4×C211—41no—
95D4×C211—41no—
96D4×C211—84no—
97C2×C4×C211—22no—
98C2×C4×C211—21no—
99C2×C4×C211—41no—
100C2×C4×C211—41no—
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211#4111yescharacteristic
107C22×C211—22no—
108C22×C211—44no—
109C22×C211—41no—
110C22×C211—42no—
111C22×C211—42no—
112C22×C211—44no—
113C22×C211—82no—
114C22×C211#12011yescharacteristic
115C22×C211#12011yescharacteristic
116C22×C211#12111yescharacteristic
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—82no—
129C2×C211#39211yescharacteristic
130C2×C211#45512yes—
131C211#200411yescharacteristic, complemented, Sylow
132#2004—2111nomaximal, Sylow
133C2×#137—2111no—
134#535D21111yescharacteristic
135C2×#124—2111no—
136C2×#104—2111no—
137#419—2111no—
138C2×#100—2111no—
139#339—2111no—
140C2×#41C2×D21111yescharacteristic
141C22×Q8—2111no—
142C22×D4C2×D21111yescharacteristic
143#137—4222no—
144#137—4222no—
145C2×Q16—4221no—
146C2×SD16—4221no—
147C2×M16—2111no—
148C2×M16—2111no—
149#127—21no—
150#124—21no—
151#124—4222no—
152#124—4222no—
153#121—22no—
154#120—22no—
155#117C2×D21111yescharacteristic
156C2×C22⋊C4—4222no—
157C2×C42—2111no—
158#104—4222no—
159#103—4222no—
160#100—4222no—
161C24C22⋊D21112yes—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8—4221no—
171C2×Q8—2112no—
172C2×Q8C22⋊D21111yescharacteristic
173C2×D4—21no—
174C2×D4—24no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4C22⋊D21111yescharacteristic
182C22×C4—21no—
183C22×C4—4221no—
184C22×C4—4221no—
185C22×C4C22×D21111yescharacteristic
186Q16—4224no—
187SD16—4224no—
188M16—4221no—
189M16—4221no—
190M16—4221no—
191M16—2112no—
192C2×C8—4221no—
193C2×C8—4221no—
194C4⋊C4C22⋊D21112yes—
195C22⋊C4—24no—
196C22⋊C4—21no—
197C22⋊C4—4228no—
198C42—21no—
199C42—4221no—
200C42—2112no—
201C23—22no—
202C23—22no—
203C23—21no—
204C23—22no—
205C23—44no—
206C23C2×C22⋊D21111yescharacteristic
207Q8—21no—
208Q8—21no—
209Q8—8441no—
210Q8—8441no—
211Q8—4222no—
212Q8—4222no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221D4—84no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—8442no—
227C2×C4—8441no—
228C2×C4—8441no—
229C2×C4—4222no—
230C2×C4—4222no—
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4D4×D21111yescharacteristic
234C2×C4C2×C22⋊D21111yescharacteristic
235C2×C4C2×C22⋊D21111yescharacteristic
236C2×C4#1680996111yescharacteristic
237C8—8441no—
238C8—4222no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—41no—
244C22—42no—
245C22—82no—
246C22#2026994511yescharacteristic
247C22#2026994611yescharacteristic
248C22#2026994911yescharacteristic
249C4—42no—
250C4—41no—
251C4—8442no—
252C4—8442no—
253C4#2026994511yescharacteristic
254C4#2026994811yescharacteristic
255C4#2026994911yescharacteristic
256C4#2026997311yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—81no—
261C2#2721305411yescharacteristic
262C2#2721332412yes—
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#535×C211C2
socleC22×C211#120
hypercenter#535D211
hypocenterC211#2004
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20270007C2semidirect
2. C2×#20269964C2semidirect
3. C2×#20269904C2semidirect
4. #27213154C2semidirect
5. C2×#20269894C2semidirect
6. #27212905C2semidirect
34. C2×Dic422D4semidirect
42. #16809946D4semidirect
47. C4×Dic211D4semidirect
131. C211#2004semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
C2×#202699642
C2×#202699042
#272131542
C2×#202698942
#272129052
#535×C2112
#2004211