Finite Group Databasebeta

← #1859×C211 #41009479

Subgroups — #1859×C211 #41009479

Order 27008.

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

normal subgroups only

This group has 302 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
1#520×C211C211yesmaximal, characteristic
2#428×C211C211yesmaximal, characteristic, complemented
3C2×#104×C211C211yesmaximal, characteristic
4#417×C211C211yesmaximal, characteristic
5#384×C211C211yesmaximal, characteristic, complemented
6#343×C211C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C2×M16×C211C2211yescharacteristic
10C2×M16×C211C2211yescharacteristic
11C4×Q8×C211—21no—
12C4×D4×C211—21no—
13C4×D4×C211—22no—
14#117×C211—21no—
15#117×C211C2211yescharacteristic
16C2×C4⋊C4×C211—21no—
17C2×C4⋊C4×C211—22no—
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C42×C211C2211yescharacteristic
21#107×C211—21no—
22#106×C211—21no—
23#104×C211C414yescomplemented
24#103×C211—21no—
25#102×C211—21no—
26#41×C211—21no—
27#41×C211—21no—
28#41×C211C2×C412yes—
29#41×C211Q812yes—
30C2×Q8×C211D411yescharacteristic
31C2×D4×C211—21no—
32C2×D4×C211D411yescharacteristic
33C22×C4×C211—21no—
34C22×C4×C211—21no—
35C22×C4×C211—22no—
36C22×C4×C211—21no—
37C22×C4×C211D412yes—
38C22×C4×C211C2311yescharacteristic
39M16×C211—21no—
40M16×C211—21no—
41M16×C211—21no—
42M16×C211C2×C412yescomplemented
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—22no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211—44no—
51C4⋊C4×C211D411yescharacteristic
52C4⋊C4×C211D411yescharacteristic
53C22⋊C4×C211—22no—
54C22⋊C4×C211—21no—
55C42×C211—22no—
56C42×C211—21no—
57C42×C211—21no—
58C42×C211C2×C412yescomplemented
59C23×C211—21no—
60C23×C211C2×D411yescharacteristic
61Q8×C211—21no—
62Q8×C211C4⋊C412yescomplemented
63D4×C211—22no—
64D4×C211—22no—
65D4×C211—21no—
66D4×C211C4⋊C412yescomplemented
67C2×C4×C211—22no—
68C2×C4×C211—22no—
69C2×C4×C211—22no—
70C2×C4×C211—21no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—21no—
74C2×C4×C211—22no—
75C2×C4×C211—21no—
76C2×C4×C211—42no—
77C2×C4×C211—42no—
78C2×C4×C211—41no—
79C2×C4×C211—82no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×Q811yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
87C8×C211—41no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—22no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211C4×D411yescharacteristic
94C22×C211#12011yescharacteristic
95C22×C211C22⋊Q811yescharacteristic
96C4×C211—24no—
97C4×C211—22no—
98C4×C211—42no—
99C4×C211—41no—
100C4×C211—41no—
101C4×C211—81no—
102C4×C211C2×C4⋊C411yescharacteristic
103C4×C211C4×D411yescharacteristic
104C4×C211C22⋊Q811yescharacteristic
105C4×C211#12311yescharacteristic
106C2×C211—21no—
107C2×C211—21no—
108C2×C211—42no—
109C2×C211#38511yescharacteristic
110C2×C211#45311yescharacteristic
111C2×C211#45611yescharacteristic
112C211#185911yescharacteristic, direct product factor, Sylow
113#1859C21111yesmaximal, characteristic, direct product factor, Sylow
114#520C2×C21111yescharacteristic
115#428C2×C21111yescharacteristic, complemented
116C2×#104C2×C21111yescharacteristic
117#417C2×C21111yescharacteristic
118#384C2×C21111yescharacteristic, complemented
119#343C2×C21111yescharacteristic, complemented
120#339C2×C21111yescharacteristic, complemented
121C2×#41C22×C21111yescharacteristic
122C2×M16C22×C21111yescharacteristic
123C2×M16C22×C21111yescharacteristic
124C4×Q8—21no—
125C4×D4—21no—
126C4×D4—22no—
127#117—21no—
128#117C22×C21111yescharacteristic
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4C22×C21111yescharacteristic
132C2×C4⋊C4C22×C21111yescharacteristic
133C2×C42C22×C21111yescharacteristic
134#107—21no—
135#106—21no—
136#104C4×C21114yescomplemented
137#103—21no—
138#102—21no—
139#41—21no—
140#41—21no—
141#41C2×C4×C21112yes—
142#41Q8×C21112yes—
143C2×Q8D4×C21111yescharacteristic
144C2×D4—21no—
145C2×D4D4×C21111yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4D4×C21112yes—
151C22×C4C23×C21111yescharacteristic
152M16—21no—
153M16—21no—
154M16—21no—
155M16C2×C4×C21112yescomplemented
156C2×C8—21no—
157C2×C8—21no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—44no—
164C4⋊C4D4×C21111yescharacteristic
165C4⋊C4D4×C21111yescharacteristic
166C22⋊C4—22no—
167C22⋊C4—21no—
168C42—22no—
169C42—21no—
170C42—21no—
171C42C2×C4×C21112yescomplemented
172C23—21no—
173C23C2×D4×C21111yescharacteristic
174Q8—21no—
175Q8C4⋊C4×C21112yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4C4⋊C4×C21112yescomplemented
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4C22×C4×C21111yescharacteristic
194C2×C4C2×D4×C21111yescharacteristic
195C2×C4C2×D4×C21111yescharacteristic
196C2×C4C2×Q8×C21111yescharacteristic
197C2×C4#41×C21111yescharacteristic
198C2×C4#41×C21111yescharacteristic
199C8—22no—
200C8—41no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22C4×D4×C21111yescharacteristic
207C22#120×C21111yescharacteristic
208C22C22⋊Q8×C21111yescharacteristic
209C4—24no—
210C4—22no—
211C4—42no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4C2×C4⋊C4×C21111yescharacteristic
216C4C4×D4×C21111yescharacteristic
217C4C22⋊Q8×C21111yescharacteristic
218C4#123×C21111yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#385×C21111yescharacteristic
223C2#453×C21111yescharacteristic
224C2#456×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#41×C211C22
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. #428×C211C2semidirect
5. #384×C211C2semidirect
6. #343×C211C2semidirect
7. #339×C211C2semidirect
23. #104×C211C4semidirect
42. M16×C211C2×C4semidirect
58. C42×C211C2×C4semidirect
62. Q8×C211C4⋊C4semidirect
66. D4×C211C4⋊C4semidirect
112. C211#1859direct (N × H)
113. #1859C211direct (N × H)
115. #428C2×C211semidirect
118. #384C2×C211semidirect
119. #343C2×C211semidirect
120. #339C2×C211semidirect
136. #104C4×C211semidirect
155. M16C2×C4×C211semidirect
171. C42C2×C4×C211semidirect
175. Q8C4⋊C4×C211semidirect
179. D4C4⋊C4×C211semidirect

Maximal subgroups

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

subgroupindex
#520×C2112
#428×C2112
C2×#104×C2112
#417×C2112
#384×C2112
#343×C2112
#339×C2112
#1859211