Finite Group Databasebeta

← #1906×C211 #41009526

Subgroups — #1906×C211 #41009526

Order 27008.

332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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
2C4×#41×C211C211yesmaximal, characteristic, complemented
3#421×C211C211yesmaximal, characteristic, complemented
4C2×#104×C211C211yesmaximal, characteristic, complemented
5#410×C211C211yesmaximal, characteristic, complemented
6#344×C211C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C2×#41×C211C2211yescharacteristic
10#137×C211—21no—
11#137×C211C412yescomplemented
12AGL(1,ℤ/8)×C211—21no—
13AGL(1,ℤ/8)×C211C412yescomplemented
14#135×C211—22no—
15C2×M16×C211C2211yescharacteristic
16C2×M16×C211C2211yescharacteristic
17C4×Q8×C211—21no—
18C4×D4×C211—21no—
19C4×D4×C211—22no—
20#117×C211—21no—
21#117×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23C2×C42×C211—21no—
24C2×C42×C211C2211yescharacteristic
25#104×C211—22no—
26#104×C211—21no—
27#104×C211—21no—
28#99×C211—22no—
29#41×C211—21no—
30#41×C211—21no—
31#41×C211—21no—
32#41×C211—21no—
33#41×C211—21no—
34#41×C211—21no—
35#41×C211C2×C412yes—
36#41×C211D412yes—
37C2×Q8×C211C2×C411yescharacteristic, complemented
38C2×Q8×C211D411yescharacteristic
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211C2×C411yescharacteristic, complemented
42C2×D4×C211D411yescharacteristic
43C22×C4×C211—21no—
44C22×C4×C211—22no—
45C22×C4×C211—21no—
46C22×C4×C211—21no—
47C22×C4×C211C2311yescharacteristic
48Q16×C211—22no—
49SD16×C211—22no—
50SD16×C211—22no—
51D8×C211—22no—
52M16×C211—21no—
53M16×C211—21no—
54M16×C211—21no—
55M16×C211C2×C412yescomplemented
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C4⋊C4×C211—22no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211—21no—
61C22⋊C4×C211—22no—
62C22⋊C4×C211—21no—
63C22⋊C4×C211D412yes—
64C42×C211—22no—
65C42×C211—21no—
66C42×C211—21no—
67C42×C211—21no—
68C42×C211D411yescharacteristic
69C42×C211D411yescharacteristic
70C23×C211—21no—
71C23×C211—21no—
72C23×C211C2×D411yescharacteristic
73Q8×C211—21no—
74Q8×C211—21no—
75Q8×C211—21no—
76Q8×C211C22⋊C412yes—
77D4×C211—22no—
78D4×C211—22no—
79D4×C211—21no—
80D4×C211—21no—
81D4×C211—21no—
82D4×C211—41no—
83D4×C211—41no—
84D4×C211C22⋊C412yes—
85C2×C4×C211—21no—
86C2×C4×C211—21no—
87C2×C4×C211—22no—
88C2×C4×C211—22no—
89C2×C4×C211—21no—
90C2×C4×C211—22no—
91C2×C4×C211—21no—
92C2×C4×C211—42no—
93C2×C4×C211—44no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211C22×C411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211#4111yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C8×C211—22no—
104C8×C211—41no—
105C22×C211—22no—
106C22×C211—41no—
107C22×C211—42no—
108C22×C211—41no—
109C22×C211—81no—
110C22×C211C4×D411yescharacteristic
111C22×C211#12011yescharacteristic
112C22×C211#12111yescharacteristic
113C4×C211—22no—
114C4×C211—22no—
115C4×C211—42no—
116C4×C211—41no—
117C4×C211—41no—
118C4×C211—81no—
119C4×C211C2×C22⋊C411yescharacteristic
120C4×C211C4×D411yescharacteristic
121C4×C211#12111yescharacteristic
122C4×C211#12311yescharacteristic
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—21no—
126C2×C211—42no—
127C2×C211—81no—
128C2×C211#38611yescharacteristic
129C211#190611yescharacteristic, direct product factor, Sylow
130#1906C21111yesmaximal, characteristic, direct product factor, Sylow
131#575C2×C21111yescharacteristic
132C4×#41C2×C21111yescharacteristic, complemented
133#421C2×C21111yescharacteristic, complemented
134C2×#104C2×C21111yescharacteristic, complemented
135#410C2×C21111yescharacteristic, complemented
136#344C2×C21111yescharacteristic, complemented
137#339C2×C21111yescharacteristic, complemented
138C2×#41C22×C21111yescharacteristic
139C2×#41C22×C21111yescharacteristic
140#137—21no—
141#137C4×C21112yescomplemented
142AGL(1,ℤ/8)—21no—
143AGL(1,ℤ/8)C4×C21112yescomplemented
144#135—22no—
145C2×M16C22×C21111yescharacteristic
146C2×M16C22×C21111yescharacteristic
147C4×Q8—21no—
148C4×D4—21no—
149C4×D4—22no—
150#117—21no—
151#117C22×C21111yescharacteristic
152#117C22×C21111yescharacteristic
153C2×C42—21no—
154C2×C42C22×C21111yescharacteristic
155#104—22no—
156#104—21no—
157#104—21no—
158#99—22no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41C2×C4×C21112yes—
166#41D4×C21112yes—
167C2×Q8C2×C4×C21111yescharacteristic, complemented
168C2×Q8D4×C21111yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×C4×C21111yescharacteristic, complemented
172C2×D4D4×C21111yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C23×C21111yescharacteristic
178Q16—22no—
179SD16—22no—
180SD16—22no—
181D8—22no—
182M16—21no—
183M16—21no—
184M16—21no—
185M16C2×C4×C21112yescomplemented
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—21no—
193C22⋊C4D4×C21112yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42D4×C21111yescharacteristic
199C42D4×C21111yescharacteristic
200C23—21no—
201C23—21no—
202C23C2×D4×C21111yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—21no—
206Q8C22⋊C4×C21112yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—21no—
211D4—21no—
212D4—41no—
213D4—41no—
214D4C22⋊C4×C21112yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4C22×C4×C21111yescharacteristic
228C2×C4C2×D4×C21111yescharacteristic
229C2×C4C2×D4×C21111yescharacteristic
230C2×C4C2×D4×C21111yescharacteristic
231C2×C4#41×C21111yescharacteristic
232C2×C4#41×C21111yescharacteristic
233C8—22no—
234C8—41no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—81no—
240C22C4×D4×C21111yescharacteristic
241C22#120×C21111yescharacteristic
242C22#121×C21111yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—41no—
248C4—81no—
249C4C2×C22⋊C4×C21111yescharacteristic
250C4C4×D4×C21111yescharacteristic
251C4#121×C21111yescharacteristic
252C4#123×C21111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—81no—
258C2#386×C21111yescharacteristic
subgroupG/H
centerC4×C211#121
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC2×C211#386
prime radical#575×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#41×C211C2semidirect
3. #421×C211C2semidirect
4. C2×#104×C211C2semidirect
5. #410×C211C2semidirect
6. #344×C211C2semidirect
7. #339×C211C2semidirect
11. #137×C211C4semidirect
13. AGL(1,ℤ/8)×C211C4semidirect
37. C2×Q8×C211C2×C4semidirect
41. C2×D4×C211C2×C4semidirect
55. M16×C211C2×C4semidirect
129. C211#1906direct (N × H)
130. #1906C211direct (N × H)
132. C4×#41C2×C211semidirect
133. #421C2×C211semidirect
134. C2×#104C2×C211semidirect
135. #410C2×C211semidirect
136. #344C2×C211semidirect
137. #339C2×C211semidirect
141. #137C4×C211semidirect
143. AGL(1,ℤ/8)C4×C211semidirect
167. C2×Q8C2×C4×C211semidirect
171. C2×D4C2×C4×C211semidirect
185. M16C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
#575×C2112
C4×#41×C2112
#421×C2112
C2×#104×C2112
#410×C2112
#344×C2112
#339×C2112
#1906211