Finite Group Databasebeta

← #41014451

Subgroups — #41014451

Order 27008.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213944C211yesmaximal, characteristic, complemented
2C2×#20270003C211yesmaximal, characteristic, complemented
3#27213163C211yesmaximal, characteristic, complemented
4#27213151C211yesmaximal, characteristic, complemented
5#27213119C211yesmaximal, characteristic, complemented
6#27212916C211yesmaximal, characteristic, complemented
7#535×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007—21no—
11#20270003—22no—
12#20270003—22no—
13#20270003—21no—
14#20270002—21no—
15#20269997—22no—
16C2×#16809950—21no—
17C2×#16809947—21no—
18C2×#16809946C2211yescharacteristic
19C2×#16809946C2211yescharacteristic
20#20269936C2211yescharacteristic
21#20269904—22no—
22#20269898—22no—
23#20269891—22no—
24C2×#41×C211C2211yescharacteristic
25C22×D4×C211C2211yescharacteristic
26#127×C211—21no—
27#124×C211—21no—
28#121×C211—22no—
29#120×C211—22no—
30#117×C211C2211yescharacteristic
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35C2×Dic422D411yescharacteristic, complemented
36C2×D844D411yescharacteristic, complemented
37C2×C22⋊D211—21no—
38C2×C4×D211—21no—
39#16809952—22no—
40#16809950—24no—
41#16809950—22no—
42#16809949—22no—
43#16809947—24no—
44#16809947—22no—
45#16809946—21no—
46#16809946—21no—
47#16809946—21no—
48#16809946D412yescomplemented
49C2×C211⋊C8—21no—
50C2×C211⋊C8D412yescomplemented
51C4⋊Dic211—21no—
52C22⋊Dic211—21no—
53C4×Dic211—21no—
54C24×C211—21no—
55#41×C211—21no—
56#41×C211—21no—
57#41×C211—21no—
58#41×C211—21no—
59C2×Q8×C211D411yescharacteristic
60C2×D4×C211—21no—
61C2×D4×C211—24no—
62C2×D4×C211—21no—
63C2×D4×C211—21no—
64C2×D4×C211—21no—
65C2×D4×C211—21no—
66C2×D4×C211—21no—
67C2×D4×C211D412yes—
68C2×D4×C211D411yescharacteristic
69C22×C4×C211—21no—
70C22×C4×C211C2311yescharacteristic
71C4⋊C4×C211—21no—
72C22⋊C4×C211—24no—
73C22⋊C4×C211D412yes—
74C42×C211—21no—
75C22×D211—21no—
76Dic422—21no—
77Dic422—21no—
78D844—21no—
79D844—21no—
80C22⋊D211—41no—
81C22⋊D211—41no—
82C2×Dic211—21no—
83C2×Dic211—41no—
84C4×D211—41no—
85C4×D211—41no—
86C211⋊C8—22no—
87C211⋊C8—22no—
88C23×C211—22no—
89C23×C211—21no—
90C23×C211—41no—
91C23×C211—41no—
92C23×C211—82no—
93C23×C211C2×D411yescharacteristic
94Q8×C211—21no—
95Q8×C211—21no—
96D4×C211—22no—
97D4×C211—21no—
98D4×C211—22no—
99D4×C211—21no—
100D4×C211—41no—
101D4×C211—41no—
102D4×C211—41no—
103D4×C211—41no—
104D4×C211—84no—
105C2×C4×C211—22no—
106C2×C4×C211—21no—
107C2×C4×C211—41no—
108C2×C4×C211—41no—
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211C2×D411yescharacteristic
111C2×C4×C211C2×D411yescharacteristic
112C2×C4×C211C2×D411yescharacteristic
113C2×C4×C211C2×D411yescharacteristic
114C2×C4×C211#4111yescharacteristic
115C22×C211—41no—
116C22×C211—42no—
117C22×C211—44no—
118C22×C211—41no—
119C22×C211—42no—
120C22×C211—82no—
121C22×C211—82no—
122C22×C211#12011yescharacteristic
123C22×C211#12111yescharacteristic
124C22×C211#12711yescharacteristic
125C2×D211—41no—
126C2×D211—81no—
127C4×C211—42no—
128C4×C211—41no—
129C4×C211#12011yescharacteristic
130C4×C211#12011yescharacteristic
131C4×C211#12111yescharacteristic
132C4×C211#12111yescharacteristic
133Dic211—41no—
134Dic211—81no—
135C2×C211—21no—
136C2×C211—21no—
137C2×C211—21no—
138C2×C211—81no—
139C2×C211—82no—
140C2×C211#39211yescharacteristic
141D211—81no—
142C211#200811yescharacteristic, complemented, Sylow
143#2008—2111nomaximal, Sylow
144#575—2111no—
145C2×AGL(1,ℤ/8)—2111no—
146#535D21111yescharacteristic, complemented
147#421—2111no—
148#418—2111no—
149#410—2111no—
150#344—2111no—
151C2×#41—2111no—
152C2×#41C2×D21111yescharacteristic
153C22×D4C2×D21111yescharacteristic
154#137—4221no—
155#137—4221no—
156AGL(1,ℤ/8)—4221no—
157AGL(1,ℤ/8)—4222no—
158AGL(1,ℤ/8)—4221no—
159AGL(1,ℤ/8)—4222no—
160#135—4222no—
161C2×SD16—4221no—
162C2×D8—4221no—
163C2×M16—2111no—
164C2×M16—2111no—
165#127—21no—
166#124—21no—
167#121—22no—
168#120—22no—
169#117—2111no—
170#117C2×D21111yescharacteristic
171#104—4222no—
172#102—4222no—
173#99—4222no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—4221no—
180#41—4221no—
181#41—4221no—
182#41—4221no—
183C2×Q8—2111no—
184C2×Q8C22⋊D21111yescharacteristic
185C2×D4—21no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—4221no—
193C2×D4—2111no—
194C2×D4C22⋊D21112yes—
195C2×D4C22⋊D21111yescharacteristic
196C22×C4—21no—
197C22×C4—4221no—
198C22×C4C22×D21111yescharacteristic
199Q16—4222no—
200SD16—4224no—
201SD16—4222no—
202SD16—4222no—
203D8—4224no—
204D8—4222no—
205M16—4221no—
206M16—4221no—
207M16—4221no—
208M16—2112no—
209C2×C8—4221no—
210C2×C8—2112no—
211C4⋊C4—21no—
212C4⋊C4—4221no—
213C22⋊C4—24no—
214C22⋊C4—4221no—
215C22⋊C4C22⋊D21112yes—
216C42—21no—
217C42—4221no—
218C23—22no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23—4221no—
224C23C2×C22⋊D21111yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—4221no—
228Q8—4221no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—84no—
238D4—8441no—
239D4—8441no—
240D4—4221no—
241D4—4221no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—8441no—
247C2×C4—8441no—
248C2×C4—8441no—
249C2×C4—4221no—
250C2×C4D4×D21111yescharacteristic
251C2×C4D4×D21111yescharacteristic
252C2×C4D4×D21111yescharacteristic
253C2×C4C2×C22⋊D21111yescharacteristic
254C2×C4C2×C22⋊D21111yescharacteristic
255C2×C4#1680996111yescharacteristic
256C8—4222no—
257C8—4222no—
258C22—41no—
259C22—42no—
260C22—44no—
261C22—41no—
262C22—42no—
263C22—82no—
264C22—82no—
265C22—8441no—
266C22—16881no—
267C22#2026994611yescharacteristic
268C22#2026994811yescharacteristic
269C22#2026997311yescharacteristic
270C4—42no—
271C4—41no—
272C4—8441no—
273C4—16881no—
274C4#2026994511yescharacteristic
275C4#2026994511yescharacteristic
276C4#2026994911yescharacteristic
277C4#2026994911yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—81no—
283C2—16881no—
284C2#2721305411yescharacteristic
subgroupG/H
centerC2#27213054
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#535×C211C2
socleC2×C211#392
hypercenter#535D211
hypocenterC211#2008
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #27213944C2semidirect
2. C2×#20270003C2semidirect
3. #27213163C2semidirect
4. #27213151C2semidirect
5. #27213119C2semidirect
6. #27212916C2semidirect
7. #535×C211C2semidirect
35. C2×Dic422D4semidirect
36. C2×D844D4semidirect
48. #16809946D4semidirect
50. C2×C211⋊C8D4semidirect
142. C211#2008semidirect
146. #535D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
C2×#202700032
#272131632
#272131512
#272131192
#272129162
#535×C2112
#2008211