Finite Group Databasebeta

← #1883×C211 #41009503

Subgroups — #1883×C211 #41009503

Order 27008.

370 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 112 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×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
2#520×C211C211yesmaximal, characteristic, complemented
3C2×#104×C211C211yesmaximal, characteristic, complemented
4#418×C211C211yesmaximal, characteristic, complemented
5#390×C211C211yesmaximal, characteristic, complemented
6#343×C211C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C22×D4×C211C2211yescharacteristic
10AGL(1,ℤ/8)×C211—22no—
11AGL(1,ℤ/8)×C211C414yescomplemented
12C2×SD16×C211—21no—
13C2×D8×C211—21no—
14C2×M16×C211C2211yescharacteristic
15C2×M16×C211C2211yescharacteristic
16C4×Q8×C211—21no—
17C4×D4×C211—21no—
18C4×D4×C211—22no—
19#117×C211—21no—
20#117×C211C2211yescharacteristic
21C2×C4⋊C4×C211—21no—
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C22⋊C4×C211—22no—
24C2×C42×C211C2211yescharacteristic
25#104×C211—22no—
26#102×C211—21no—
27#102×C211—21no—
28C24×C211—21no—
29#41×C211—21no—
30#41×C211—21no—
31#41×C211C2×C412yes—
32#41×C211D412yes—
33C2×Q8×C211D411yescharacteristic
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36C2×D4×C211—21no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211C2×C412yescomplemented
41C2×D4×C211D411yescharacteristic
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211—22no—
45C22×C4×C211D412yes—
46C22×C4×C211C2311yescharacteristic
47SD16×C211—24no—
48D8×C211—24no—
49M16×C211—21no—
50M16×C211—21no—
51M16×C211—21no—
52M16×C211C2×C412yescomplemented
53C2×C8×C211—21no—
54C2×C8×C211—21no—
55C4⋊C4×C211—22no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211D411yescharacteristic
59C4⋊C4×C211D411yescharacteristic
60C22⋊C4×C211—22no—
61C22⋊C4×C211—21no—
62C22⋊C4×C211—44no—
63C42×C211—22no—
64C42×C211—21no—
65C42×C211—21no—
66C42×C211—21no—
67C23×C211—21no—
68C23×C211—22no—
69C23×C211—42no—
70C23×C211—42no—
71C23×C211—41no—
72C23×C211—41no—
73C23×C211C2×D411yescharacteristic
74Q8×C211—21no—
75Q8×C211C22⋊C412yescomplemented
76D4×C211—22no—
77D4×C211—22no—
78D4×C211—22no—
79D4×C211—22no—
80D4×C211—21no—
81D4×C211—41no—
82D4×C211—41no—
83D4×C211C22⋊C412yescomplemented
84C2×C4×C211—21no—
85C2×C4×C211—22no—
86C2×C4×C211—22no—
87C2×C4×C211—22no—
88C2×C4×C211—21no—
89C2×C4×C211—22no—
90C2×C4×C211—21no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211—42no—
94C2×C4×C211—82no—
95C2×C4×C211C22×C411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C8×C211—22no—
102C8×C211—41no—
103C22×C211—21no—
104C22×C211—21no—
105C22×C211—22no—
106C22×C211—44no—
107C22×C211—41no—
108C22×C211—41no—
109C22×C211—42no—
110C22×C211—82no—
111C22×C211—81no—
112C22×C211—81no—
113C22×C211C4×D411yescharacteristic
114C22×C211#12011yescharacteristic
115C22×C211#12111yescharacteristic
116C4×C211—22no—
117C4×C211—42no—
118C4×C211—42no—
119C4×C211—41no—
120C4×C211—41no—
121C4×C211C2×C22⋊C411yescharacteristic
122C4×C211C4×D411yescharacteristic
123C4×C211#12111yescharacteristic
124C4×C211#12311yescharacteristic
125C2×C211—21no—
126C2×C211—21no—
127C2×C211—42no—
128C2×C211—82no—
129C2×C211#38611yescharacteristic
130C2×C211#45311yescharacteristic
131C2×C211#45411yescharacteristic
132C211#188311yescharacteristic, direct product factor, Sylow
133#1883C21111yesmaximal, characteristic, direct product factor, Sylow
134C2×AGL(1,ℤ/8)C2×C21111yescharacteristic
135#520C2×C21111yescharacteristic, complemented
136C2×#104C2×C21111yescharacteristic, complemented
137#418C2×C21111yescharacteristic, complemented
138#390C2×C21111yescharacteristic, complemented
139#343C2×C21111yescharacteristic, complemented
140#339C2×C21111yescharacteristic, complemented
141C2×#41C22×C21111yescharacteristic
142C22×D4C22×C21111yescharacteristic
143AGL(1,ℤ/8)—22no—
144AGL(1,ℤ/8)C4×C21114yescomplemented
145C2×SD16—21no—
146C2×D8—21no—
147C2×M16C22×C21111yescharacteristic
148C2×M16C22×C21111yescharacteristic
149C4×Q8—21no—
150C4×D4—21no—
151C4×D4—22no—
152#117—21no—
153#117C22×C21111yescharacteristic
154C2×C4⋊C4—21no—
155C2×C4⋊C4C22×C21111yescharacteristic
156C2×C22⋊C4—22no—
157C2×C42C22×C21111yescharacteristic
158#104—22no—
159#102—21no—
160#102—21no—
161C24—21no—
162#41—21no—
163#41—21no—
164#41C2×C4×C21112yes—
165#41D4×C21112yes—
166C2×Q8D4×C21111yescharacteristic
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4C2×C4×C21112yescomplemented
174C2×D4D4×C21111yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4D4×C21112yes—
179C22×C4C23×C21111yescharacteristic
180SD16—24no—
181D8—24no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C2×C4×C21112yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4D4×C21111yescharacteristic
192C4⋊C4D4×C21111yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—21no—
195C22⋊C4—44no—
196C42—22no—
197C42—21no—
198C42—21no—
199C42—21no—
200C23—21no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—41no—
205C23—41no—
206C23C2×D4×C21111yescharacteristic
207Q8—21no—
208Q8C22⋊C4×C21112yescomplemented
209D4—22no—
210D4—22no—
211D4—22no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4C22⋊C4×C21112yescomplemented
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4C22×C4×C21111yescharacteristic
229C2×C4C2×D4×C21111yescharacteristic
230C2×C4C2×D4×C21111yescharacteristic
231C2×C4C2×D4×C21111yescharacteristic
232C2×C4#41×C21111yescharacteristic
233C2×C4#41×C21111yescharacteristic
234C8—22no—
235C8—41no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—44no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—81no—
245C22—81no—
246C22C4×D4×C21111yescharacteristic
247C22#120×C21111yescharacteristic
248C22#121×C21111yescharacteristic
249C4—22no—
250C4—42no—
251C4—42no—
252C4—41no—
253C4—41no—
254C4C2×C22⋊C4×C21111yescharacteristic
255C4C4×D4×C21111yescharacteristic
256C4#121×C21111yescharacteristic
257C4#123×C21111yescharacteristic
258C2—21no—
259C2—21no—
260C2—42no—
261C2—82no—
262C2#386×C21111yescharacteristic
263C2#453×C21111yescharacteristic
264C2#454×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×AGL(1,ℤ/8)×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #520×C211C2semidirect
3. C2×#104×C211C2semidirect
4. #418×C211C2semidirect
5. #390×C211C2semidirect
6. #343×C211C2semidirect
7. #339×C211C2semidirect
11. AGL(1,ℤ/8)×C211C4semidirect
40. C2×D4×C211C2×C4semidirect
52. M16×C211C2×C4semidirect
75. Q8×C211C22⋊C4semidirect
83. D4×C211C22⋊C4semidirect
132. C211#1883direct (N × H)
133. #1883C211direct (N × H)
135. #520C2×C211semidirect
136. C2×#104C2×C211semidirect
137. #418C2×C211semidirect
138. #390C2×C211semidirect
139. #343C2×C211semidirect
140. #339C2×C211semidirect
144. AGL(1,ℤ/8)C4×C211semidirect
173. C2×D4C2×C4×C211semidirect
185. M16C2×C4×C211semidirect
208. Q8C22⋊C4×C211semidirect
216. D4C22⋊C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)×C2112
#520×C2112
C2×#104×C2112
#418×C2112
#390×C2112
#343×C2112
#339×C2112
#1883211