Finite Group Databasebeta

← #1904×C211 #41009524

Subgroups — #1904×C211 #41009524

Order 27008.

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

normal subgroups only

This group has 346 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
1C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
2#518×C211C211yesmaximal, characteristic, complemented
3#421×C211C211yesmaximal, characteristic, complemented
4#418×C211C211yesmaximal, characteristic, complemented
5#410×C211C211yesmaximal, characteristic, complemented
6#344×C211C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C22×D4×C211C2211yescharacteristic
10AGL(1,ℤ/8)×C211—22no—
11AGL(1,ℤ/8)×C211—22no—
12C2×SD16×C211C412yescomplemented
13C2×D8×C211C412yescomplemented
14C2×M16×C211C2211yescharacteristic
15C2×M16×C211C2211yescharacteristic
16C4×D4×C211—22no—
17C4×D4×C211—22no—
18#117×C211C2211yescharacteristic
19#117×C211C2211yescharacteristic
20#117×C211C2211yescharacteristic
21C2×C22⋊C4×C211—22no—
22#104×C211—22no—
23#102×C211—22no—
24#99×C211—22no—
25C24×C211—21no—
26#41×C211—21no—
27#41×C211—21no—
28#41×C211—21no—
29#41×C211—21no—
30C2×Q8×C211C2×C411yescharacteristic, complemented
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34C2×D4×C211—21no—
35C2×D4×C211C2×C412yes—
36C2×D4×C211C2×C411yescharacteristic, complemented
37C2×D4×C211D412yes—
38C2×D4×C211D412yes—
39C22×C4×C211—24no—
40C22×C4×C211—21no—
41C22×C4×C211C2311yescharacteristic
42SD16×C211—24no—
43D8×C211—24no—
44M16×C211—21no—
45M16×C211—21no—
46M16×C211—21no—
47M16×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211C2×C412yescomplemented
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C22⋊C4×C211—24no—
54C22⋊C4×C211—21no—
55C22⋊C4×C211—21no—
56C22⋊C4×C211D412yes—
57C42×C211—21no—
58C42×C211—21no—
59C42×C211D412yes—
60C23×C211—22no—
61C23×C211—22no—
62C23×C211—22no—
63C23×C211—21no—
64C23×C211—82no—
65C23×C211C2×D411yescharacteristic
66Q8×C211—21no—
67Q8×C211—21no—
68D4×C211—21no—
69D4×C211—22no—
70D4×C211—21no—
71D4×C211—22no—
72D4×C211—22no—
73D4×C211—41no—
74D4×C211—41no—
75D4×C211C22⋊C414yes—
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—84no—
83C2×C4×C211C22×C411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C8×C211—22no—
90C8×C211—41no—
91C22×C211—24no—
92C22×C211—41no—
93C22×C211—44no—
94C22×C211—41no—
95C22×C211—82no—
96C22×C211—82no—
97C22×C211—81no—
98C22×C211C4×D411yescharacteristic
99C22×C211#12011yescharacteristic
100C22×C211#12111yescharacteristic
101C4×C211—42no—
102C4×C211—42no—
103C4×C211—41no—
104C4×C211—81no—
105C4×C211C2×C22⋊C411yescharacteristic
106C4×C211C4×D411yescharacteristic
107C4×C211#12111yescharacteristic
108C4×C211#12311yescharacteristic
109C2×C211—21no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211—44no—
113C2×C211—81no—
114C2×C211#38611yescharacteristic
115C211#190411yescharacteristic, direct product factor, Sylow
116#1904C21111yesmaximal, characteristic, direct product factor, Sylow
117C2×AGL(1,ℤ/8)C2×C21111yescharacteristic
118#518C2×C21111yescharacteristic, complemented
119#421C2×C21111yescharacteristic, complemented
120#418C2×C21111yescharacteristic, complemented
121#410C2×C21111yescharacteristic, complemented
122#344C2×C21111yescharacteristic, complemented
123H(3,ℤ/4)C2×C21111yescharacteristic, complemented
124C2×#41C22×C21111yescharacteristic
125C22×D4C22×C21111yescharacteristic
126AGL(1,ℤ/8)—22no—
127AGL(1,ℤ/8)—22no—
128C2×SD16C4×C21112yescomplemented
129C2×D8C4×C21112yescomplemented
130C2×M16C22×C21111yescharacteristic
131C2×M16C22×C21111yescharacteristic
132C4×D4—22no—
133C4×D4—22no—
134#117C22×C21111yescharacteristic
135#117C22×C21111yescharacteristic
136#117C22×C21111yescharacteristic
137C2×C22⋊C4—22no—
138#104—22no—
139#102—22no—
140#99—22no—
141C24—21no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146C2×Q8C2×C4×C21111yescharacteristic, complemented
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4C2×C4×C21112yes—
152C2×D4C2×C4×C21111yescharacteristic, complemented
153C2×D4D4×C21112yes—
154C2×D4D4×C21112yes—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4C23×C21111yescharacteristic
158SD16—24no—
159D8—24no—
160M16—21no—
161M16—21no—
162M16—21no—
163M16—21no—
164C2×C8—21no—
165C2×C8C2×C4×C21112yescomplemented
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C22⋊C4—24no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4D4×C21112yes—
173C42—21no—
174C42—21no—
175C42D4×C21112yes—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—21no—
180C23—82no—
181C23C2×D4×C21111yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—22no—
186D4—21no—
187D4—22no—
188D4—22no—
189D4—41no—
190D4—41no—
191D4C22⋊C4×C21114yes—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—84no—
199C2×C4C22×C4×C21111yescharacteristic
200C2×C4C2×D4×C21111yescharacteristic
201C2×C4C2×D4×C21111yescharacteristic
202C2×C4C2×D4×C21111yescharacteristic
203C2×C4#41×C21111yescharacteristic
204C2×C4#41×C21111yescharacteristic
205C8—22no—
206C8—41no—
207C22—24no—
208C22—41no—
209C22—44no—
210C22—41no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22C4×D4×C21111yescharacteristic
215C22#120×C21111yescharacteristic
216C22#121×C21111yescharacteristic
217C4—42no—
218C4—42no—
219C4—41no—
220C4—81no—
221C4C2×C22⋊C4×C21111yescharacteristic
222C4C4×D4×C21111yescharacteristic
223C4#121×C21111yescharacteristic
224C4#123×C21111yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—44no—
229C2—81no—
230C2#386×C21111yescharacteristic
subgroupG/H
center, socleC2×C211#386
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×AGL(1,ℤ/8)×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #518×C211C2semidirect
3. #421×C211C2semidirect
4. #418×C211C2semidirect
5. #410×C211C2semidirect
6. #344×C211C2semidirect
7. H(3,ℤ/4)×C211C2semidirect
12. C2×SD16×C211C4semidirect
13. C2×D8×C211C4semidirect
30. C2×Q8×C211C2×C4semidirect
36. C2×D4×C211C2×C4semidirect
49. C2×C8×C211C2×C4semidirect
115. C211#1904direct (N × H)
116. #1904C211direct (N × H)
118. #518C2×C211semidirect
119. #421C2×C211semidirect
120. #418C2×C211semidirect
121. #410C2×C211semidirect
122. #344C2×C211semidirect
123. H(3,ℤ/4)C2×C211semidirect
128. C2×SD16C4×C211semidirect
129. C2×D8C4×C211semidirect
146. C2×Q8C2×C4×C211semidirect
152. C2×D4C2×C4×C211semidirect
165. C2×C8C2×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
#518×C2112
#421×C2112
#418×C2112
#410×C2112
#344×C2112
H(3,ℤ/4)×C2112
#1904211