Finite Group Databasebeta

← #41014412

Subgroups — #41014412

Order 27008.

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

normal subgroups only

This group has 478 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×#20270002C211yesmaximal, characteristic, complemented
2#27213710C211yesmaximal, characteristic, complemented
3C2×#20269974C211yesmaximal, characteristic, complemented
4#27213152C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C22×D844—22no—
9C22×D844C2211yescharacteristic
10C22×D844C2211yescharacteristic
11#20270002—22no—
12#20270002—22no—
13C2×#16809949—21no—
14C2×#16809947—21no—
15C2×#16809946C2211yescharacteristic
16#20269974—22no—
17#20269974—22no—
18#20269973—21no—
19#20269966—21no—
20#20269951—22no—
21#20269945—22no—
22#20269936C2211yescharacteristic
23#20269899—22no—
24#20269893—22no—
25C2×#41×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×C42×C211C2211yescharacteristic
28#104×C211—22no—
29C23×D211—21no—
30C23×D211D412yes—
31C2×D844—28no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844—21no—
40C2×D844—21no—
41C2×D844—21no—
42C2×D844—21no—
43C2×D844—48no—
44C2×D844D412yescomplemented
45C2×C22⋊D211—24no—
46#16809949—24no—
47#16809947—24no—
48#16809946—21no—
49#16809946D412yescomplemented
50C2×C211⋊C8—21no—
51C4⋊Dic211D412yescomplemented
52C22⋊Dic211—21no—
53(C2×D211)⋊C4—24no—
54C4×Dic211—21no—
55#41×C211—21no—
56#41×C211—21no—
57#41×C211—21no—
58#41×C211—21no—
59C2×Q8×C211D411yescharacteristic, complemented
60C2×D4×C211—21no—
61C2×D4×C211D411yescharacteristic, complemented
62C22×C4×C211—21no—
63C22×C4×C211—21no—
64C22×C4×C211C2311yescharacteristic
65M16×C211—21no—
66M16×C211—21no—
67C2×C8×C211—21no—
68C42×C211—21no—
69C42×C211D412yescomplemented
70C22×D211—22no—
71C22×D211—22no—
72C22×D211—22no—
73C22×D211—22no—
74C22×D211—44no—
75C22×D211—44no—
76C22×D211—41no—
77C22×D211—41no—
78D844—22no—
79D844—22no—
80D844—416no—
81D844—41no—
82D844—41no—
83D844—41no—
84D844—41no—
85D844—41no—
86D844—41no—
87C22⋊D211—84no—
88C2×Dic211—22no—
89C211⋊C8—22no—
90C23×C211—21no—
91C23×C211C2×D411yescharacteristic
92Q8×C211—21no—
93Q8×C211—21no—
94D4×C211—21no—
95D4×C211—21no—
96D4×C211—41no—
97D4×C211—41no—
98C2×C4×C211—24no—
99C2×C4×C211—22no—
100C2×C4×C211—21no—
101C2×C4×C211—41no—
102C2×C4×C211—41no—
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211#4111yescharacteristic
109C8×C211—41no—
110C22×C211—22no—
111C22×C211—41no—
112C22×C211—42no—
113C22×C211#12011yescharacteristic
114C22×C211#12011yescharacteristic
115C22×C211#12111yescharacteristic
116C2×D211—44no—
117C2×D211—44no—
118C2×D211—42no—
119C2×D211—42no—
120C2×D211—44no—
121C2×D211—82no—
122C2×D211—82no—
123C2×D211—82no—
124C4×C211—24no—
125C4×C211—41no—
126C4×C211#12011yescharacteristic
127C4×C211#12111yescharacteristic
128C4×C211#12111yescharacteristic
129C4×C211#12711yescharacteristic
130Dic211—42no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—81no—
134C2×C211#39211yescharacteristic
135C2×C211#45312yes—
136D211—82no—
137D211—82no—
138C211#200211yescharacteristic, complemented, Sylow
139#2002—2111nomaximal, Sylow
140C2×AGL(1,ℤ/8)—2111no—
141#535—2111no—
142C2×#127—2111no—
143C2×#104D21111yescharacteristic, complemented
144#418—2111no—
145C2×#100—2111no—
146#339—2111no—
147C2×#41C2×D21111yescharacteristic
148C22×D4—4222no—
149C22×D4—2111no—
150C22×D4—2111no—
151AGL(1,ℤ/8)—4222no—
152AGL(1,ℤ/8)—4222no—
153C2×SD16—4221no—
154C2×D8—4221no—
155C2×M16—2111no—
156C2×M16C2×D21111yescharacteristic
157#127—4222no—
158#127—4221no—
159#127—4222no—
160#124—4221no—
161#121—4222no—
162#120—4222no—
163#117—2111no—
164C2×C42C2×D21111yescharacteristic
165#104—22no—
166#102—4222no—
167#100—4222no—
168C24—4221no—
169C24—2112no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8D84411yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—8448no—
177C2×D4—4228no—
178C2×D4—4224no—
179C2×D4—4221no—
180C2×D4—4221no—
181C2×D4—4221no—
182C2×D4—4221no—
183C2×D4—4221no—
184C2×D4—4221no—
185C2×D4—4221no—
186C2×D4—4221no—
187C2×D4—4221no—
188C2×D4—4221no—
189C2×D4—4221no—
190C2×D4—2112no—
191C2×D4D84411yescharacteristic, complemented
192C22×C4—21no—
193C22×C4—21no—
194C22×C4C22×D21111yescharacteristic
195SD16—4224no—
196D8—4224no—
197M16—21no—
198M16—21no—
199M16—4221no—
200M16—2112no—
201C2×C8—21no—
202C2×C8—4221no—
203C4⋊C4—2112no—
204C22⋊C4—4224no—
205C22⋊C4—4221no—
206C42—21no—
207C42—4221no—
208C42C22⋊D21112yescomplemented
209C23—21no—
210C23—8444no—
211C23—8444no—
212C23—8441no—
213C23—8441no—
214C23—4222no—
215C23—4222no—
216C23—4222no—
217C23—4222no—
218C23D4×D21111yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—21no—
223D4—41no—
224D4—41no—
225D4—84416no—
226D4—8441no—
227D4—8441no—
228D4—8441no—
229D4—8441no—
230D4—8441no—
231D4—8441no—
232D4—16884no—
233D4—4222no—
234D4—4222no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—4222no—
241C2×C4D4×D21111yescharacteristic
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4C2×C22⋊D21111yescharacteristic
245C2×C4C2×D84411yescharacteristic
246C2×C4#1680996211yescharacteristic
247C8—41no—
248C8—4222no—
249C22—22no—
250C22—42no—
251C22—41no—
252C22—8444no—
253C22—8444no—
254C22—8444no—
255C22—8442no—
256C22—8442no—
257C22—16882no—
258C22—16882no—
259C22—16882no—
260C22#2026994511yescharacteristic
261C22#2026994711yescharacteristic
262C22#2026995011yescharacteristic
263C4—24no—
264C4—41no—
265C4—8442no—
266C4#2026994711yescharacteristic
267C4#2026995011yescharacteristic
268C4#2026995111yescharacteristic
269C4#2026997311yescharacteristic
270C2—21no—
271C2—21no—
272C2—81no—
273C2—16882no—
274C2—16882no—
275C2#2721305511yescharacteristic
276C2#2721331512yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211#120
hypercenterC2×#104D211
hypocenterC211#2002
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#20270002C2semidirect
2. #27213710C2semidirect
3. C2×#20269974C2semidirect
4. #27213152C2semidirect
5. C2×#20269893C2semidirect
6. #27212906C2semidirect
7. C2×#104×C211C2semidirect
44. C2×D844D4semidirect
49. #16809946D4semidirect
51. C4⋊Dic211D4semidirect
59. C2×Q8×C211D4semidirect
61. C2×D4×C211D4semidirect
69. C42×C211D4semidirect
138. C211#2002semidirect
143. C2×#104D211semidirect
174. C2×Q8D844semidirect
191. C2×D4D844semidirect
208. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700022
#272137102
C2×#202699742
#272131522
C2×#202698932
#272129062
C2×#104×C2112
#2002211