Finite Group Databasebeta

← #2008×C211 #41009628

Subgroups — #2008×C211 #41009628

Order 27008.

380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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#575×C211C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
3#535×C211C211yesmaximal, characteristic, complemented
4#421×C211C211yesmaximal, characteristic, complemented
5#418×C211C211yesmaximal, characteristic, complemented
6#410×C211C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C2×#41×C211C2211yescharacteristic
10C22×D4×C211C2211yescharacteristic
11#137×C211—21no—
12#137×C211—21no—
13AGL(1,ℤ/8)×C211—22no—
14AGL(1,ℤ/8)×C211—21no—
15AGL(1,ℤ/8)×C211—21no—
16AGL(1,ℤ/8)×C211—22no—
17#135×C211—22no—
18C2×SD16×C211—21no—
19C2×D8×C211—21no—
20C2×M16×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#127×C211—21no—
23#124×C211—21no—
24#121×C211—22no—
25#120×C211—22no—
26#117×C211C2211yescharacteristic
27#117×C211C2211yescharacteristic
28#104×C211—22no—
29#102×C211—22no—
30#99×C211—22no—
31C24×C211—21no—
32#41×C211—21no—
33#41×C211—21no—
34#41×C211—21no—
35#41×C211—21no—
36#41×C211—21no—
37#41×C211—21no—
38#41×C211—21no—
39#41×C211—21no—
40C2×Q8×C211D411yescharacteristic
41C2×Q8×C211D411yescharacteristic, complemented
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—24no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211D412yes—
51C2×D4×C211D411yescharacteristic
52C2×D4×C211D411yescharacteristic, complemented
53C22×C4×C211—21no—
54C22×C4×C211—21no—
55C22×C4×C211C2311yescharacteristic
56Q16×C211—22no—
57SD16×C211—24no—
58SD16×C211—22no—
59SD16×C211—22no—
60D8×C211—24no—
61D8×C211—22no—
62M16×C211—21no—
63M16×C211—21no—
64M16×C211—21no—
65M16×C211D412yescomplemented
66C2×C8×C211—21no—
67C2×C8×C211D412yescomplemented
68C4⋊C4×C211—21no—
69C4⋊C4×C211—21no—
70C22⋊C4×C211—24no—
71C22⋊C4×C211—21no—
72C22⋊C4×C211D412yes—
73C42×C211—21no—
74C42×C211—21no—
75C23×C211—22no—
76C23×C211—21no—
77C23×C211—21no—
78C23×C211—41no—
79C23×C211—41no—
80C23×C211—82no—
81C23×C211C2×D411yescharacteristic
82Q8×C211—21no—
83Q8×C211—21no—
84Q8×C211—21no—
85Q8×C211—21no—
86D4×C211—21no—
87D4×C211—22no—
88D4×C211—21no—
89D4×C211—21no—
90D4×C211—22no—
91D4×C211—21no—
92D4×C211—41no—
93D4×C211—41no—
94D4×C211—41no—
95D4×C211—41no—
96D4×C211—41no—
97D4×C211—41no—
98D4×C211—84no—
99C2×C4×C211—22no—
100C2×C4×C211—21no—
101C2×C4×C211—21no—
102C2×C4×C211—41no—
103C2×C4×C211—41no—
104C2×C4×C211—41no—
105C2×C4×C211—41no—
106C2×C4×C211—41no—
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211C2×D411yescharacteristic
111C2×C4×C211C2×D411yescharacteristic
112C2×C4×C211#4111yescharacteristic
113C8×C211—22no—
114C8×C211—22no—
115C22×C211—41no—
116C22×C211—42no—
117C22×C211—44no—
118C22×C211—41no—
119C22×C211—41no—
120C22×C211—42no—
121C22×C211—81no—
122C22×C211—82no—
123C22×C211—82no—
124C22×C211#12011yescharacteristic
125C22×C211#12111yescharacteristic
126C22×C211#12711yescharacteristic
127C4×C211—42no—
128C4×C211—41no—
129C4×C211—41no—
130C4×C211—81no—
131C4×C211#12011yescharacteristic
132C4×C211#12011yescharacteristic
133C4×C211#12111yescharacteristic
134C4×C211#12111yescharacteristic
135C2×C211—21no—
136C2×C211—21no—
137C2×C211—21no—
138C2×C211—81no—
139C2×C211—82no—
140C2×C211—81no—
141C2×C211#39211yescharacteristic
142C211#200811yescharacteristic, direct product factor, Sylow
143#2008C21111yesmaximal, characteristic, direct product factor, Sylow
144#575C2×C21111yescharacteristic, complemented
145C2×AGL(1,ℤ/8)C2×C21111yescharacteristic, complemented
146#535C2×C21111yescharacteristic, complemented
147#421C2×C21111yescharacteristic, complemented
148#418C2×C21111yescharacteristic, complemented
149#410C2×C21111yescharacteristic, complemented
150#344C2×C21111yescharacteristic, complemented
151C2×#41C22×C21111yescharacteristic
152C2×#41C22×C21111yescharacteristic
153C22×D4C22×C21111yescharacteristic
154#137—21no—
155#137—21no—
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—21no—
158AGL(1,ℤ/8)—21no—
159AGL(1,ℤ/8)—22no—
160#135—22no—
161C2×SD16—21no—
162C2×D8—21no—
163C2×M16C22×C21111yescharacteristic
164C2×M16C22×C21111yescharacteristic
165#127—21no—
166#124—21no—
167#121—22no—
168#120—22no—
169#117C22×C21111yescharacteristic
170#117C22×C21111yescharacteristic
171#104—22no—
172#102—22no—
173#99—22no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183C2×Q8D4×C21111yescharacteristic
184C2×Q8D4×C21111yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—24no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4D4×C21112yes—
194C2×D4D4×C21111yescharacteristic
195C2×D4D4×C21111yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—21no—
198C22×C4C23×C21111yescharacteristic
199Q16—22no—
200SD16—24no—
201SD16—22no—
202SD16—22no—
203D8—24no—
204D8—22no—
205M16—21no—
206M16—21no—
207M16—21no—
208M16D4×C21112yescomplemented
209C2×C8—21no—
210C2×C8D4×C21112yescomplemented
211C4⋊C4—21no—
212C4⋊C4—21no—
213C22⋊C4—24no—
214C22⋊C4—21no—
215C22⋊C4D4×C21112yes—
216C42—21no—
217C42—21no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23—82no—
224C23C2×D4×C21111yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—21no—
228Q8—21no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—21no—
233D4—22no—
234D4—21no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—84no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4C2×D4×C21111yescharacteristic
251C2×C4C2×D4×C21111yescharacteristic
252C2×C4C2×D4×C21111yescharacteristic
253C2×C4C2×D4×C21111yescharacteristic
254C2×C4C2×D4×C21111yescharacteristic
255C2×C4#41×C21111yescharacteristic
256C8—22no—
257C8—22no—
258C22—41no—
259C22—42no—
260C22—44no—
261C22—41no—
262C22—41no—
263C22—42no—
264C22—81no—
265C22—82no—
266C22—82no—
267C22#120×C21111yescharacteristic
268C22#121×C21111yescharacteristic
269C22#127×C21111yescharacteristic
270C4—42no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4#120×C21111yescharacteristic
275C4#120×C21111yescharacteristic
276C4#121×C21111yescharacteristic
277C4#121×C21111yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—81no—
283C2—81no—
284C2#392×C21111yescharacteristic
subgroupG/H
center, socleC2×C211#392
commutator, Frattini, cosocleC22×C4C23×C211
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. #575×C211C2semidirect
2. C2×AGL(1,ℤ/8)×C211C2semidirect
3. #535×C211C2semidirect
4. #421×C211C2semidirect
5. #418×C211C2semidirect
6. #410×C211C2semidirect
7. #344×C211C2semidirect
41. C2×Q8×C211D4semidirect
52. C2×D4×C211D4semidirect
65. M16×C211D4semidirect
67. C2×C8×C211D4semidirect
142. C211#2008direct (N × H)
143. #2008C211direct (N × H)
144. #575C2×C211semidirect
145. C2×AGL(1,ℤ/8)C2×C211semidirect
146. #535C2×C211semidirect
147. #421C2×C211semidirect
148. #418C2×C211semidirect
149. #410C2×C211semidirect
150. #344C2×C211semidirect
184. C2×Q8D4×C211semidirect
195. C2×D4D4×C211semidirect
208. M16D4×C211semidirect
210. C2×C8D4×C211semidirect

Maximal subgroups

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

subgroupindex
#575×C2112
C2×AGL(1,ℤ/8)×C2112
#535×C2112
#421×C2112
#418×C2112
#410×C2112
#344×C2112
#2008211