Finite Group Databasebeta

← #41014413

Subgroups — #41014413

Order 27008.

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

normal subgroups only

This group has 478 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×#20270003C211yesmaximal, characteristic, complemented
2#27213713C211yesmaximal, characteristic, complemented
3C2×#20269905C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic, complemented
5C2×#20269894C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#127×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270003—22no—
10#20270003—22no—
11C2×#16809950—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15#20269973—21no—
16#20269964—21no—
17#20269948—22no—
18#20269946—22no—
19#20269936C2211yescharacteristic
20#20269905—22no—
21#20269898—22no—
22#20269894—22no—
23C22×D4×C211—22no—
24C22×D4×C211C2211yescharacteristic
25C22×D4×C211C2211yescharacteristic
26#127×C211—22no—
27#127×C211—22no—
28C2×C42×C211C2211yescharacteristic
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33C2×Dic422D411yescharacteristic, complemented
34C2×D844D411yescharacteristic, complemented
35C2×C22⋊D211—24no—
36C2×C22⋊D211—21no—
37C2×C4×D211—21no—
38#16809950—24no—
39#16809947—24no—
40#16809946—21no—
41#16809946—21no—
42#16809946—21no—
43#16809946D412yescomplemented
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C4⋊Dic211D412yescomplemented
47C22⋊Dic211—24no—
48C22⋊Dic211—21no—
49C4×Dic211—21no—
50C24×C211—21no—
51C24×C211D412yes—
52C2×D4×C211—28no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
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×C211—21no—
63C2×D4×C211—21no—
64C2×D4×C211—48no—
65C2×D4×C211D412yescomplemented
66C22×C4×C211—21no—
67C22×C4×C211C2311yescharacteristic
68C42×C211—21no—
69C42×C211D412yescomplemented
70C22×D211—21no—
71Dic422—21no—
72Dic422—21no—
73D844—21no—
74D844—21no—
75C22⋊D211—41no—
76C22⋊D211—41no—
77C22⋊D211—84no—
78C2×Dic211—22no—
79C2×Dic211—21no—
80C4×D211—41no—
81C4×D211—41no—
82C211⋊C8—22no—
83C211⋊C8—41no—
84C23×C211—22no—
85C23×C211—22no—
86C23×C211—22no—
87C23×C211—22no—
88C23×C211—44no—
89C23×C211—44no—
90C23×C211—41no—
91C23×C211—41no—
92C23×C211C2×D411yescharacteristic
93D4×C211—22no—
94D4×C211—22no—
95D4×C211—416no—
96D4×C211—41no—
97D4×C211—41no—
98D4×C211—41no—
99D4×C211—41no—
100D4×C211—41no—
101D4×C211—41no—
102C2×C4×C211—24no—
103C2×C4×C211—22no—
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211#4111yescharacteristic
110C22×C211—22no—
111C22×C211—44no—
112C22×C211—44no—
113C22×C211—42no—
114C22×C211—42no—
115C22×C211—44no—
116C22×C211—82no—
117C22×C211—82no—
118C22×C211—82no—
119C22×C211#12011yescharacteristic
120C22×C211#12011yescharacteristic
121C22×C211#12111yescharacteristic
122C2×D211—41no—
123C2×D211—42no—
124C4×C211—24no—
125C4×C211#12011yescharacteristic
126C4×C211#12111yescharacteristic
127C4×C211#12111yescharacteristic
128C4×C211#12711yescharacteristic
129Dic211—41no—
130Dic211—42no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—82no—
134C2×C211—82no—
135C2×C211#39211yescharacteristic
136C2×C211#45312yes—
137D211—81no—
138C211#200211yescharacteristic, complemented, Sylow
139#2002—2111nomaximal, Sylow
140C2×AGL(1,ℤ/8)—2111no—
141#535—2111no—
142C2×#127D21111yescharacteristic, complemented
143C2×#104—2111no—
144#418—2111no—
145C2×#100—2111no—
146#339—2111no—
147C2×#41—2111no—
148C22×D4—22no—
149C22×D4C2×D21111yescharacteristic
150C22×D4C2×D21111yescharacteristic
151AGL(1,ℤ/8)—4222no—
152AGL(1,ℤ/8)—4222no—
153C2×SD16—4221no—
154C2×D8—4221no—
155C2×M16—2111no—
156C2×M16—2111no—
157#127—22no—
158#127—22no—
159#127—4221no—
160#124—4221no—
161#121—4222no—
162#120—4222no—
163#117—2111no—
164C2×C42C2×D21111yescharacteristic
165#104—4222no—
166#102—4222no—
167#100—4222no—
168C24—21no—
169C24C22⋊D21112yes—
170#41—4221no—
171#41—4221no—
172#41—4221no—
173#41—4221no—
174C2×Q8—2111no—
175C2×D4—28no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—48no—
188C2×D4—4221no—
189C2×D4—4224no—
190C2×D4—2111no—
191C2×D4C22⋊D21112yescomplemented
192C22×C4—21no—
193C22×C4—4221no—
194C22×C4C22×D21111yescharacteristic
195SD16—4224no—
196D8—4224no—
197M16—4221no—
198M16—4221no—
199M16—4221no—
200M16—2112no—
201C2×C8—4221no—
202C2×C8—4221no—
203C4⋊C4—2112no—
204C22⋊C4—4224no—
205C22⋊C4—4221no—
206C42—21no—
207C42—4221no—
208C42C22⋊D21112yescomplemented
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—44no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23—4221no—
218C23C2×C22⋊D21111yescharacteristic
219Q8—4221no—
220Q8—4221no—
221D4—22no—
222D4—22no—
223D4—416no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—8441no—
231D4—8441no—
232D4—16884no—
233D4—4221no—
234D4—4221no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—8441no—
238C2×C4—8441no—
239C2×C4—4222no—
240C2×C4—4221no—
241C2×C4D4×D21111yescharacteristic
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4C2×C22⋊D21111yescharacteristic
245C2×C4C2×C22⋊D21111yescharacteristic
246C2×C4#1680996111yescharacteristic
247C8—8441no—
248C8—4222no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—44no—
253C22—42no—
254C22—42no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—8442no—
259C22—8441no—
260C22#2026994511yescharacteristic
261C22#2026994611yescharacteristic
262C22#2026994911yescharacteristic
263C4—24no—
264C4—8442no—
265C4—8441no—
266C4#2026994511yescharacteristic
267C4#2026994811yescharacteristic
268C4#2026994911yescharacteristic
269C4#2026997311yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—82no—
274C2—16881no—
275C2#2721305411yescharacteristic
276C2#2721331312yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#127×C211C2
socleC22×C211#120
hypercenterC2×#127D211
hypocenterC211#2002
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#20270003C2semidirect
2. #27213713C2semidirect
3. C2×#20269905C2semidirect
4. #27213151C2semidirect
5. C2×#20269894C2semidirect
6. #27212906C2semidirect
7. C2×#127×C211C2semidirect
33. C2×Dic422D4semidirect
34. C2×D844D4semidirect
43. #16809946D4semidirect
46. C4⋊Dic211D4semidirect
65. C2×D4×C211D4semidirect
69. C42×C211D4semidirect
138. C211#2002semidirect
142. C2×#127D211semidirect
191. C2×D4C22⋊D211semidirect
208. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700032
#272137132
C2×#202699052
#272131512
C2×#202698942
#272129062
C2×#127×C2112
#2002211