Finite Group Databasebeta

← #41012347

Subgroups — #41012347

Order 27008.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#27213736C211yesmaximal, characteristic, complemented
2#27213296C211yesmaximal, characteristic, complemented
3#27213282C211yesmaximal, characteristic, complemented
4#27213056C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#328×C211C211yesmaximal, characteristic, complemented
8C22×D844—21no—
9C22×D844C2211yescharacteristic
10C2×#16809949—21no—
11C2×D1688—21no—
12#20269972—21no—
13#20269966—21no—
14#20269962—21no—
15#20269952—21no—
16#20269951C2211yescharacteristic
17#20269947—21no—
18(C22⋊D211)⋊C4—21no—
19C2×(C2×D211)⋊C4—21no—
20C2×(C2×D211)⋊C4C2211yescharacteristic
21#20269899—21no—
22#20269897—21no—
23C22⋊(C211⋊C8)C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211—21no—
28C23×D211D411yescharacteristic
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—42no—
34C2×D844—42no—
35C2×D844—42no—
36C2×D844—42no—
37C2×D844D411yescharacteristic, complemented
38C2×C22⋊D211—21no—
39C22×Dic211D411yescharacteristic
40C2×C4×D211—21no—
41#16809949—42no—
42D1688—42no—
43C2×C211⋊C8—21no—
44C4⋊Dic211D411yescharacteristic, complemented
45Dic211⋊C4—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—21no—
51(C2×D211)⋊C4—21no—
52(C2×D211)⋊C4—42no—
53C4×Dic211—21no—
54C2×Q8×C211D411yescharacteristic, complemented
55C22×C4×C211—21no—
56C22×C4×C211C2311yescharacteristic
57C2×C8×C211—21no—
58C4⋊C4×C211—21no—
59C4⋊C4×C211D411yescharacteristic, complemented
60C22⋊C4×C211—21no—
61C22×D211—22no—
62C22×D211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—21no—
66C22×D211—41no—
67C22×D211—41no—
68C22×D211—41no—
69C22×D211—41no—
70C22×D211—41no—
71C22×D211—41no—
72C22×D211—41no—
73C22×D211—41no—
74D844—22no—
75D844—22no—
76D844—41no—
77D844—41no—
78D844—41no—
79D844—48no—
80D844—81no—
81D844—81no—
82C22⋊D211—41no—
83C22⋊D211—41no—
84C2×Dic211—21no—
85C2×Dic211—21no—
86C2×Dic211—41no—
87C4×D211—81no—
88C211⋊C8—41no—
89C23×C211C2×D411yescharacteristic
90Q8×C211—41no—
91C2×C4×C211—22no—
92C2×C4×C211—21no—
93C2×C4×C211—21no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C8×C211—41no—
100C22×C211—21no—
101C22×C211—21no—
102C22×C211#12011yescharacteristic
103C22×C211#13511yescharacteristic
104C22×C211AGL(1,ℤ/8)11yescharacteristic
105C2×D211—42no—
106C2×D211—42no—
107C2×D211—42no—
108C2×D211—41no—
109C2×D211—41no—
110C2×D211—41no—
111C2×D211—41no—
112C2×D211—42no—
113C2×D211—41no—
114C2×D211—41no—
115C2×D211—81no—
116C2×D211—81no—
117C2×D211—81no—
118C2×D211—81no—
119C2×D211—81no—
120C4×C211—21no—
121C4×C211—21no—
122C4×C211—42no—
123C4×C211—41no—
124C4×C211—41no—
125Dic211—41no—
126Dic211—41no—
127C2×C211—21no—
128C2×C211—21no—
129C2×C211#44911yescharacteristic
130C2×C211#45311yescharacteristic
131C2×C211B(4,𝔽2)11yescharacteristic
132D211—81no—
133D211—81no—
134D211—82no—
135C211#160011yescharacteristic, complemented, Sylow
136#1600—2111nomaximal, Sylow
137#538—2111no—
138#450—2111no—
139#447—2111no—
140#392—2111no—
141#328D21111yescharacteristic, complemented
142#327—2111no—
143#323—2111no—
144C22×D4—4221no—
145C22×D4—2111no—
146C2×SD16—4221no—
147C2×D8—4221no—
148#126—4221no—
149#124—4221no—
150#123—4221no—
151C22⋊Q8C2×D21111yescharacteristic
152#121—4221no—
153#121—2111no—
154#120—4221no—
155C4×D4—4221no—
156C2×C22⋊C4—4221no—
157C2×C22⋊C4—2111no—
158#102—4221no—
159#102—4221no—
160C22⋊C8—2111no—
161C22⋊C8C2×D21111yescharacteristic
162#95C2×D21111yescharacteristic
163C24—4221no—
164C24—2111no—
165C2×Q8D84411yescharacteristic, complemented
166C2×D4—8442no—
167C2×D4—8442no—
168C2×D4—8442no—
169C2×D4—8442no—
170C2×D4—4221no—
171C2×D4—4221no—
172C2×D4—4221no—
173C2×D4—4221no—
174C2×D4—4221no—
175C2×D4—2111no—
176C22×C4—21no—
177C22×C4—4221no—
178C22×C4—2111no—
179C22×C4C22×D21111yescharacteristic
180SD16—8442no—
181D8—8442no—
182C2×C8—21no—
183C2×C8—4221no—
184C4⋊C4—21no—
185C4⋊C4—4221no—
186C4⋊C4—2111no—
187C4⋊C4D84411yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—8442no—
190C22⋊C4—4221no—
191C22⋊C4—4221no—
192C22⋊C4—4221no—
193C22⋊C4—4221no—
194C22⋊C4—4221no—
195C22⋊C4—4221no—
196C42—4221no—
197C23—8441no—
198C23—8441no—
199C23—8441no—
200C23—8441no—
201C23—8441no—
202C23—8441no—
203C23—8441no—
204C23—8441no—
205C23—4222no—
206C23—4221no—
207C23—4221no—
208C23—4221no—
209C23—4221no—
210C23D4×D21111yescharacteristic
211Q8—41no—
212D4—8441no—
213D4—8441no—
214D4—8441no—
215D4—8448no—
216D4—8441no—
217D4—8441no—
218D4—16881no—
219D4—16881no—
220D4—4222no—
221D4—4222no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—8441no—
229C2×C4—16881no—
230C2×C4—4221no—
231C2×C4—4221no—
232C2×C4D4×D21111yescharacteristic
233C2×C4C2×D84411yescharacteristic
234C8—41no—
235C8—8441no—
236C22—21no—
237C22—21no—
238C22—8442no—
239C22—8442no—
240C22—8442no—
241C22—8441no—
242C22—8441no—
243C22—8442no—
244C22—8441no—
245C22—8441no—
246C22—8441no—
247C22—8441no—
248C22—16881no—
249C22—16881no—
250C22—16881no—
251C22—16881no—
252C22—16881no—
253C22#2026994711yescharacteristic
254C22#2027000011yescharacteristic
255C22#2027000411yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—8441no—
262C4—8441no—
263C2—21no—
264C2—21no—
265C2—16881no—
266C2—16882no—
267C2—16881no—
268C2#2721329411yescharacteristic
269C2#2721331511yescharacteristic
270C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211#120
hypercenter#328D211
hypocenterC211#1600
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27213736C2semidirect
2. #27213296C2semidirect
3. #27213282C2semidirect
4. #27213056C2semidirect
5. #27212879C2semidirect
6. #27212869C2semidirect
7. #328×C211C2semidirect
37. C2×D844D4semidirect
44. C4⋊Dic211D4semidirect
54. C2×Q8×C211D4semidirect
59. C4⋊C4×C211D4semidirect
135. C211#1600semidirect
141. #328D211semidirect
165. C2×Q8D844semidirect
187. C4⋊C4D844semidirect

Maximal subgroups

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

subgroupindex
#272137362
#272132962
#272132822
#272130562
#272128792
#272128692
#328×C2112
#1600211