Finite Group Databasebeta

← #41014415

Subgroups — #41014415

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×#20270005C211yesmaximal, characteristic, complemented
2#27213713C211yesmaximal, characteristic, complemented
3C2×#20269973C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27213152C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C22×C22⋊D211—22no—
11#20270005—22no—
12#20270005—22no—
13C2×#16809951—21no—
14C2×D1688—21no—
15#20269973—22no—
16#20269973—21no—
17#20269973—22no—
18#20269964—21no—
19#20269948—22no—
20#20269946—22no—
21#20269936C2211yescharacteristic
22C2×C4×Dic211C2211yescharacteristic
23#20269903—22no—
24#20269899—22no—
25C22×D4×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28#100×C211—22no—
29C23×D211—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34C2×Dic422D411yescharacteristic, complemented
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844—21no—
40C2×D844D411yescharacteristic, complemented
41C2×D844D412yescomplemented
42C2×C22⋊D211—28no—
43C2×C22⋊D211—24no—
44C2×C22⋊D211—21no—
45C2×C22⋊D211—48no—
46C22×Dic211—21no—
47C2×C4×D211—21no—
48#16809951—24no—
49D1688—24no—
50C4⋊Dic211D412yescomplemented
51C22⋊Dic211—24no—
52C22⋊Dic211—21no—
53C4×Dic211—21no—
54C4×Dic211—21no—
55C4×Dic211D412yescomplemented
56C24×C211D412yes—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64M16×C211—21no—
65M16×C211—21no—
66M16×C211—21no—
67M16×C211D412yescomplemented
68C2×C8×C211—21no—
69C2×C8×C211—21no—
70C22×D211—21no—
71C22×D211—22no—
72C22×D211—44no—
73C22×D211—41no—
74C22×D211—41no—
75Dic422—21no—
76Dic422—21no—
77D844—21no—
78D844—22no—
79D844—22no—
80D844—21no—
81D844—41no—
82D844—41no—
83C22⋊D211—416no—
84C22⋊D211—41no—
85C22⋊D211—41no—
86C22⋊D211—84no—
87C2×Dic211—24no—
88C2×Dic211—22no—
89C2×Dic211—22no—
90C2×Dic211—21no—
91C4×D211—41no—
92C4×D211—41no—
93C23×C211—22no—
94C23×C211—22no—
95C23×C211—22no—
96C23×C211—44no—
97C23×C211C2×D411yescharacteristic
98D4×C211—41no—
99D4×C211—41no—
100D4×C211—41no—
101D4×C211—41no—
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211#4111yescharacteristic
108C8×C211—22no—
109C8×C211—41no—
110C22×C211—22no—
111C22×C211—44no—
112C22×C211—42no—
113C22×C211—44no—
114C22×C211—82no—
115C22×C211#12011yescharacteristic
116C22×C211#12011yescharacteristic
117C22×C211#12111yescharacteristic
118C2×D211—44no—
119C2×D211—41no—
120C2×D211—42no—
121C2×D211—42no—
122C2×D211—82no—
123C2×D211—82no—
124C4×C211#12011yescharacteristic
125C4×C211#12111yescharacteristic
126C4×C211#12111yescharacteristic
127C4×C211#12711yescharacteristic
128Dic211—24no—
129Dic211—41no—
130Dic211—42no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—82no—
134C2×C211#39211yescharacteristic
135C2×C211#45312yes—
136D211—81no—
137D211—82no—
138C211#200211yescharacteristic, complemented, Sylow
139#2002—2111nomaximal, Sylow
140C2×AGL(1,ℤ/8)—2111no—
141#535—2111no—
142C2×#127—2111no—
143C2×#104—2111no—
144#418—2111no—
145C2×#100D21111yescharacteristic, complemented
146#339—2111no—
147C2×#41—2111no—
148C22×D4—4222no—
149C22×D4—2111no—
150C22×D4C2×D21111yescharacteristic
151AGL(1,ℤ/8)—4222no—
152AGL(1,ℤ/8)—4222no—
153C2×SD16—4221no—
154C2×D8—4221no—
155C2×M16C2×D21111yescharacteristic
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×C42—2111no—
165#104—4222no—
166#102—4222no—
167#100—22no—
168C24—4221no—
169C24D84412yes—
170#41—4221no—
171#41—4221no—
172#41—4221no—
173#41—4221no—
174C2×Q8—2111no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—8448no—
182C2×D4—4228no—
183C2×D4—4221no—
184C2×D4—4224no—
185C2×D4—4221no—
186C2×D4—4221no—
187C2×D4—4221no—
188C2×D4—4221no—
189C2×D4—4221no—
190C2×D4—2111no—
191C2×D4—2112no—
192C22×C4—4221no—
193C22×C4—4221no—
194C22×C4C22×D21111yescharacteristic
195SD16—4224no—
196D8—4224no—
197M16—21no—
198M16—21no—
199M16—21no—
200M16C22⋊D21112yescomplemented
201C2×C8—21no—
202C2×C8—21no—
203C4⋊C4—2112no—
204C22⋊C4—4224no—
205C22⋊C4—4221no—
206C42—4221no—
207C42—4221no—
208C42—2112no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—44no—
213C23—8444no—
214C23—8441no—
215C23—8441no—
216C23—4222no—
217C23—4221no—
218C23C2×D84411yescharacteristic
219Q8—4221no—
220Q8—4221no—
221D4—41no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—8441no—
226D4—8441no—
227D4—84416no—
228D4—8441no—
229D4—8441no—
230D4—16884no—
231D4—4221no—
232D4—4222no—
233D4—4222no—
234D4—4221no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—4222no—
238C2×C4—4221no—
239C2×C4—4224no—
240C2×C4—4222no—
241C2×C4D4×D21111yescharacteristic
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4D4×D21111yescharacteristic
245C2×C4C2×C22⋊D21111yescharacteristic
246C2×C4#1680996211yescharacteristic
247C8—22no—
248C8—41no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—82no—
254C22—8444no—
255C22—8442no—
256C22—8441no—
257C22—8442no—
258C22—16882no—
259C22—16882no—
260C22#2026994711yescharacteristic
261C22#2026994711yescharacteristic
262C22#2026995111yescharacteristic
263C4—8442no—
264C4—8441no—
265C4—4224no—
266C4#2026994511yescharacteristic
267C4#2026995011yescharacteristic
268C4#2026995011yescharacteristic
269C4#2026997311yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—16882no—
274C2—16881no—
275C2#2721305511yescharacteristic
276C2#2721331612yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211#120
hypercenterC2×#100D211
hypocenterC211#2002
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#20270005C2semidirect
2. #27213713C2semidirect
3. C2×#20269973C2semidirect
4. C2×#20269903C2semidirect
5. #27213152C2semidirect
6. #27212907C2semidirect
7. C2×#100×C211C2semidirect
34. C2×Dic422D4semidirect
40. C2×D844D4semidirect
41. C2×D844D4semidirect
50. C4⋊Dic211D4semidirect
55. C4×Dic211D4semidirect
67. M16×C211D4semidirect
138. C211#2002semidirect
145. C2×#100D211semidirect
200. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700052
#272137132
C2×#202699732
C2×#202699032
#272131522
#272129072
C2×#100×C2112
#2002211