Finite Group Databasebeta

← #1792×C211 #41009412

Subgroups — #1792×C211 #41009412

Order 27008.

620 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 136 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×H(5,𝔽2)×C211C211yesmaximal, characteristic
2C2×#104×C211C212yesmaximal, complemented
3#417×C211C212yesmaximal, complemented
4C2×#100×C211C211yesmaximal, characteristic, complemented
5#390×C211C211yesmaximal, characteristic, complemented
6H(5,𝔽2)×C211—24no—
7H(5,𝔽2)×C211—22no—
8H(5,𝔽2)×C211C414yescomplemented
9C2×#41×C211—22no—
10C2×#41×C211C2212yes—
11C22×D4×C211—22no—
12C22×D4×C211—22no—
13C22×D4×C211C2211yescharacteristic
14C2×M16×C211C2212yescomplemented
15C2×C4⋊C4×C211C2211yescharacteristic, complemented
16C2×C22⋊C4×C211—22no—
17C2×C42×C211C2211yescharacteristic, complemented
18#104×C211—24no—
19#103×C211—22no—
20#102×C211—22no—
21#100×C211—22no—
22C24×C211—22no—
23C24×C211D412yes—
24#41×C211—28no—
25#41×C211—28no—
26#41×C211—22no—
27#41×C211—22no—
28#41×C211C2×C414yes—
29#41×C211D414yes—
30C2×Q8×C211D412yes—
31C2×D4×C211—24no—
32C2×D4×C211—28no—
33C2×D4×C211—28no—
34C2×D4×C211—28no—
35C2×D4×C211—22no—
36C2×D4×C211—28no—
37C2×D4×C211—22no—
38C2×D4×C211—22no—
39C2×D4×C211C2×C412yes—
40C2×D4×C211D412yes—
41C2×D4×C211D412yes—
42C22×C4×C211—22no—
43C22×C4×C211—22no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22×C4×C211C2311yescharacteristic
47M16×C211—22no—
48M16×C211—22no—
49C2×C8×C211—22no—
50C4⋊C4×C211—22no—
51C22⋊C4×C211—44no—
52C42×C211—22no—
53C23×C211—28no—
54C23×C211—22no—
55C23×C211—22no—
56C23×C211—22no—
57C23×C211—42no—
58C23×C211—42no—
59C23×C211—82no—
60C23×C211—82no—
61C23×C211C22⋊C414yes—
62C23×C211C2×D411yescharacteristic
63Q8×C211—22no—
64Q8×C211C22⋊C414yescomplemented
65D4×C211—216no—
66D4×C211—24no—
67D4×C211—24no—
68D4×C211—24no—
69D4×C211—24no—
70D4×C211—22no—
71D4×C211C22⋊C414yescomplemented
72C2×C4×C211—28no—
73C2×C4×C211—22no—
74C2×C4×C211—24no—
75C2×C4×C211—24no—
76C2×C4×C211—22no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—42no—
80C2×C4×C211—41no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D412yes—
83C2×C4×C211C2×D412yes—
84C2×C4×C211C2×D411yescharacteristic
85C8×C211—42no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—24no—
89C22×C211—24no—
90C22×C211—44no—
91C22×C211—42no—
92C22×C211—42no—
93C22×C211—42no—
94C22×C211—42no—
95C22×C211—82no—
96C22×C211—88no—
97C22×C211C2×C22⋊C411yescharacteristic
98C22×C211#12011yescharacteristic
99C22×C211#12011yescharacteristic
100C4×C211—24no—
101C4×C211—42no—
102C4×C211—81no—
103C4×C211C2×C22⋊C412yes—
104C4×C211#12012yes—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—44no—
108C2×C211—44no—
109C2×C211#37911yescharacteristic
110C2×C211#45311yescharacteristic
111C2×C211#45511yescharacteristic
112C211#179211yescharacteristic, direct product factor, Sylow
113#1792C21111yesmaximal, characteristic, direct product factor, Sylow
114C2×H(5,𝔽2)C2×C21111yescharacteristic
115C2×#104C2×C21112yescomplemented
116#417C2×C21112yescomplemented
117C2×#100C2×C21111yescharacteristic, complemented
118#390C2×C21111yescharacteristic, complemented
119H(5,𝔽2)—24no—
120H(5,𝔽2)—22no—
121H(5,𝔽2)C4×C21114yescomplemented
122C2×#41—22no—
123C2×#41C22×C21112yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C22×C21111yescharacteristic
127C2×M16C22×C21112yescomplemented
128C2×C4⋊C4C22×C21111yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C22×C21111yescharacteristic, complemented
131#104—24no—
132#103—22no—
133#102—22no—
134#100—22no—
135C24—22no—
136C24D4×C21112yes—
137#41—28no—
138#41—28no—
139#41—22no—
140#41—22no—
141#41C2×C4×C21114yes—
142#41D4×C21114yes—
143C2×Q8D4×C21112yes—
144C2×D4—24no—
145C2×D4—28no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—22no—
149C2×D4—28no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4C2×C4×C21112yes—
153C2×D4D4×C21112yes—
154C2×D4D4×C21112yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4C23×C21111yescharacteristic
160M16—22no—
161M16—22no—
162C2×C8—22no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—28no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23C22⋊C4×C21114yes—
175C23C2×D4×C21111yescharacteristic
176Q8—22no—
177Q8C22⋊C4×C21114yescomplemented
178D4—216no—
179D4—24no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C22⋊C4×C21114yescomplemented
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4C22×C4×C21111yescharacteristic
195C2×C4C2×D4×C21112yes—
196C2×C4C2×D4×C21112yes—
197C2×C4C2×D4×C21111yescharacteristic
198C8—42no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—88no—
210C22C2×C22⋊C4×C21111yescharacteristic
211C22#120×C21111yescharacteristic
212C22#120×C21111yescharacteristic
213C4—24no—
214C4—42no—
215C4—81no—
216C4C2×C22⋊C4×C21112yes—
217C4#120×C21112yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2#379×C21111yescharacteristic
223C2#453×C21111yescharacteristic
224C2#455×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×H(5,𝔽2)×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. C2×#104×C211C2semidirect
3. #417×C211C2semidirect
4. C2×#100×C211C2semidirect
5. #390×C211C2semidirect
8. H(5,𝔽2)×C211C4semidirect
14. C2×M16×C211C22semidirect
15. C2×C4⋊C4×C211C22semidirect
17. C2×C42×C211C22semidirect
64. Q8×C211C22⋊C4semidirect
71. D4×C211C22⋊C4semidirect
112. C211#1792direct (N × H)
113. #1792C211direct (N × H)
115. C2×#104C2×C211semidirect
116. #417C2×C211semidirect
117. C2×#100C2×C211semidirect
118. #390C2×C211semidirect
121. H(5,𝔽2)C4×C211semidirect
127. C2×M16C22×C211semidirect
128. C2×C4⋊C4C22×C211semidirect
130. C2×C42C22×C211semidirect
177. Q8C22⋊C4×C211semidirect
184. D4C22⋊C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C2112
C2×#104×C2112
#417×C2112
C2×#100×C2112
#390×C2112
#1792211