Finite Group Databasebeta

← #1639×C211 #41009259

Subgroups — #1639×C211 #41009259

Order 27008.

308 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 76 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
1D42×C211C211yesmaximal, characteristic
2#486×C211C211yesmaximal, characteristic, complemented
3#474×C211C211yesmaximal, characteristic, complemented
4#461×C211C211yesmaximal, characteristic, complemented
5#331×C211C211yesmaximal, characteristic, complemented
6D4⋊C8×C211C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C22×D4×C211—21no—
9C22×D4×C211—21no—
10C2×SD16×C211—21no—
11C4⋊Q8×C211C2211yescharacteristic, complemented
12#127×C211C2211yescharacteristic
13#121×C211—21no—
14#121×C211—21no—
15#120×C211—22no—
16C4×D4×C211C2211yescharacteristic
17C4×D4×C211C2211yescharacteristic
18#107×C211—21no—
19C4⋊C8×C211C2211yescharacteristic, complemented
20C4⋊C8×C211C2211yescharacteristic, complemented
21#103×C211—21no—
22#102×C211—21no—
23#102×C211—22no—
24C4×C8×C211C2211yescharacteristic, complemented
25C24×C211—22no—
26C2×Q8×C211—21no—
27C2×D4×C211—22no—
28C2×D4×C211—22no—
29C2×D4×C211—21no—
30C2×D4×C211—21no—
31C2×D4×C211—42no—
32C2×D4×C211—42no—
33C2×D4×C211—42no—
34C2×D4×C211—42no—
35C2×D4×C211D411yescharacteristic
36C2×D4×C211D411yescharacteristic
37C2×D4×C211D412yes—
38C22×C4×C211—21no—
39C22×C4×C211—21no—
40SD16×C211—42no—
41C2×C8×C211—21no—
42C2×C8×C211—21no—
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D411yescharacteristic
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—22no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C42×C211C2311yescharacteristic
52C23×C211—21no—
53C23×C211—21no—
54C23×C211—22no—
55C23×C211—44no—
56C23×C211—44no—
57C23×C211—82no—
58Q8×C211—41no—
59Q8×C211—41no—
60D4×C211—21no—
61D4×C211—21no—
62D4×C211—42no—
63D4×C211—44no—
64D4×C211—44no—
65D4×C211—41no—
66D4×C211—41no—
67D4×C211—41no—
68D4×C211—41no—
69D4×C211D812yes—
70D4×C211SD1612yescomplemented
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C8×C211—41no—
82C8×C211—41no—
83C8×C211—41no—
84C22×C211—22no—
85C22×C211—22no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211—42no—
91C22×C211—42no—
92C22×C211—42no—
93C22×C211—88no—
94C22×C211#12011yescharacteristic
95C4×C211—21no—
96C4×C211—41no—
97C4×C211—41no—
98C4×C211—81no—
99C4×C211C2×D811yescharacteristic
100C4×C211C2×SD1611yescharacteristic
101C4×C211AGL(1,ℤ/8)11yescharacteristic
102C4×C211AGL(1,ℤ/8)11yescharacteristic
103C2×C211—42no—
104C2×C211—42no—
105C2×C211—82no—
106C2×C211#44711yescharacteristic
107C2×C211#45011yescharacteristic
108C2×C211#45511yescharacteristic
109C211#163911yescharacteristic, direct product factor, Sylow
110#1639C21111yesmaximal, characteristic, direct product factor, Sylow
111D42C2×C21111yescharacteristic
112#486C2×C21111yescharacteristic, complemented
113#474C2×C21111yescharacteristic, complemented
114#461C2×C21111yescharacteristic, complemented
115#331C2×C21111yescharacteristic, complemented
116D4⋊C8C2×C21111yescharacteristic, complemented
117D4⋊C8C2×C21111yescharacteristic, complemented
118C22×D4—21no—
119C22×D4—21no—
120C2×SD16—21no—
121C4⋊Q8C22×C21111yescharacteristic, complemented
122#127C22×C21111yescharacteristic
123#121—21no—
124#121—21no—
125#120—22no—
126C4×D4C22×C21111yescharacteristic
127C4×D4C22×C21111yescharacteristic
128#107—21no—
129C4⋊C8C22×C21111yescharacteristic, complemented
130C4⋊C8C22×C21111yescharacteristic, complemented
131#103—21no—
132#102—21no—
133#102—22no—
134C4×C8C22×C21111yescharacteristic, complemented
135C24—22no—
136C2×Q8—21no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4D4×C21111yescharacteristic
146C2×D4D4×C21111yescharacteristic
147C2×D4D4×C21112yes—
148C22×C4—21no—
149C22×C4—21no—
150SD16—42no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4D4×C21111yescharacteristic
157C4⋊C4D4×C21111yescharacteristic
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C42C23×C21111yescharacteristic
162C23—21no—
163C23—21no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—82no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—21no—
172D4—42no—
173D4—44no—
174D4—44no—
175D4—41no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4D8×C21112yes—
180D4SD16×C21112yescomplemented
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4C2×D4×C21111yescharacteristic
189C2×C4C2×D4×C21111yescharacteristic
190C2×C4C2×D4×C21111yescharacteristic
191C8—41no—
192C8—41no—
193C8—41no—
194C22—22no—
195C22—22no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—88no—
204C22#120×C21111yescharacteristic
205C4—21no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4C2×D8×C21111yescharacteristic
210C4C2×SD16×C21111yescharacteristic
211C4AGL(1,ℤ/8)×C21111yescharacteristic
212C4AGL(1,ℤ/8)×C21111yescharacteristic
213C2—42no—
214C2—42no—
215C2—82no—
216C2#447×C21111yescharacteristic
217C2#450×C21111yescharacteristic
218C2#455×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
prime radicalD42×C211C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #486×C211C2semidirect
3. #474×C211C2semidirect
4. #461×C211C2semidirect
5. #331×C211C2semidirect
6. D4⋊C8×C211C2semidirect
7. D4⋊C8×C211C2semidirect
11. C4⋊Q8×C211C22semidirect
19. C4⋊C8×C211C22semidirect
20. C4⋊C8×C211C22semidirect
24. C4×C8×C211C22semidirect
70. D4×C211SD16semidirect
109. C211#1639direct (N × H)
110. #1639C211direct (N × H)
112. #486C2×C211semidirect
113. #474C2×C211semidirect
114. #461C2×C211semidirect
115. #331C2×C211semidirect
116. D4⋊C8C2×C211semidirect
117. D4⋊C8C2×C211semidirect
121. C4⋊Q8C22×C211semidirect
129. C4⋊C8C22×C211semidirect
130. C4⋊C8C22×C211semidirect
134. C4×C8C22×C211semidirect
180. D4SD16×C211semidirect

Maximal subgroups

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

subgroupindex
D42×C2112
#486×C2112
#474×C2112
#461×C2112
#331×C2112
D4⋊C8×C2112
#1639211