Finite Group Databasebeta

← #1903×C211 #41009523

Subgroups — #1903×C211 #41009523

Order 27008.

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

normal subgroups only

This group has 324 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#568×C211C211yesmaximal, characteristic, complemented
3#421×C211C211yesmaximal, characteristic, complemented
4#418×C211C211yesmaximal, characteristic, complemented
5C2×#100×C211C211yesmaximal, characteristic, complemented
6#344×C211C211yesmaximal, characteristic, complemented
7#338×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
14#131×C211—22no—
15#131×C211—22no—
16C2×M16×C211—22no—
17C2×M16×C211C2211yescharacteristic
18C2×M16×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#104×C211—22no—
23#102×C211—22no—
24#100×C211—22no—
25C24×C211—21no—
26#41×C211—21no—
27#41×C211—21no—
28#41×C211D412yes—
29#41×C211D412yes—
30C2×Q8×C211C2×C411yescharacteristic
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36C2×D4×C211—21no—
37C2×D4×C211C2×C412yes—
38C2×D4×C211C2×C411yescharacteristic
39C22×C4×C211—21no—
40C22×C4×C211C2311yescharacteristic
41SD16×C211—24no—
42D8×C211—24no—
43M16×C211—24no—
44M16×C211—21no—
45M16×C211—21no—
46M16×C211—21no—
47M16×C211—21no—
48M16×C211—21no—
49M16×C211—21no—
50M16×C211—21no—
51M16×C211D412yes—
52C2×C8×C211—24no—
53C2×C8×C211—21no—
54C2×C8×C211—21no—
55C2×C8×C211C2×C412yes—
56C2×C8×C211D412yes—
57C4⋊C4×C211—21no—
58C22⋊C4×C211—21no—
59C42×C211—21no—
60C23×C211—21no—
61C23×C211—22no—
62C23×C211—41no—
63C23×C211—41no—
64C23×C211—82no—
65C23×C211#4111yescharacteristic
66Q8×C211—21no—
67Q8×C211C22⋊C412yes—
68D4×C211—22no—
69D4×C211—22no—
70D4×C211—22no—
71D4×C211—21no—
72D4×C211—22no—
73D4×C211—41no—
74D4×C211—41no—
75D4×C211C22⋊C412yes—
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—41no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
87C8×C211—22no—
88C8×C211—22no—
89C8×C211—41no—
90C22×C211—22no—
91C22×C211—41no—
92C22×C211—42no—
93C22×C211—81no—
94C22×C211—82no—
95C22×C211—82no—
96C22×C211—82no—
97C22×C211C4×D411yescharacteristic
98C22×C211#12111yescharacteristic
99C22×C211#12311yescharacteristic
100C4×C211—22no—
101C4×C211—81no—
102C4×C211C2×C22⋊C411yescharacteristic
103C4×C211C4×D411yescharacteristic
104C4×C211#12011yescharacteristic
105C4×C211#12111yescharacteristic
106C2×C211—21no—
107C2×C211—21no—
108C2×C211—21no—
109C2×C211—42no—
110C2×C211—82no—
111C2×C211#38611yescharacteristic
112C211#190311yescharacteristic, direct product factor, Sylow
113#1903C21111yesmaximal, characteristic, direct product factor, Sylow
114C2×AGL(1,ℤ/8)C2×C21111yescharacteristic
115#568C2×C21111yescharacteristic, complemented
116#421C2×C21111yescharacteristic, complemented
117#418C2×C21111yescharacteristic, complemented
118C2×#100C2×C21111yescharacteristic, complemented
119#344C2×C21111yescharacteristic, complemented
120#338C2×C21111yescharacteristic, complemented
121C2×#41C22×C21111yescharacteristic
122C22×D4C22×C21111yescharacteristic
123AGL(1,ℤ/8)—22no—
124AGL(1,ℤ/8)—22no—
125C2×SD16C4×C21112yescomplemented
126C2×D8C4×C21112yescomplemented
127#131—22no—
128#131—22no—
129C2×M16—22no—
130C2×M16C22×C21111yescharacteristic
131C2×M16C22×C21111yescharacteristic
132C2×M16C22×C21111yescharacteristic
133C2×M16C22×C21111yescharacteristic
134#117C22×C21111yescharacteristic
135#104—22no—
136#102—22no—
137#100—22no—
138C24—21no—
139#41—21no—
140#41—21no—
141#41D4×C21112yes—
142#41D4×C21112yes—
143C2×Q8C2×C4×C21111yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×C4×C21112yes—
151C2×D4C2×C4×C21111yescharacteristic
152C22×C4—21no—
153C22×C4C23×C21111yescharacteristic
154SD16—24no—
155D8—24no—
156M16—24no—
157M16—21no—
158M16—21no—
159M16—21no—
160M16—21no—
161M16—21no—
162M16—21no—
163M16—21no—
164M16D4×C21112yes—
165C2×C8—24no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8C2×C4×C21112yes—
169C2×C8D4×C21112yes—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—82no—
178C23#41×C21111yescharacteristic
179Q8—21no—
180Q8C22⋊C4×C21112yes—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—21no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊C4×C21112yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4C22×C4×C21111yescharacteristic
194C2×C4C2×D4×C21111yescharacteristic
195C2×C4C2×D4×C21111yescharacteristic
196C2×C4C2×D4×C21111yescharacteristic
197C2×C4C2×D4×C21111yescharacteristic
198C2×C4#41×C21111yescharacteristic
199C8—22no—
200C8—22no—
201C8—22no—
202C8—41no—
203C22—22no—
204C22—41no—
205C22—42no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22C4×D4×C21111yescharacteristic
211C22#121×C21111yescharacteristic
212C22#123×C21111yescharacteristic
213C4—22no—
214C4—81no—
215C4C2×C22⋊C4×C21111yescharacteristic
216C4C4×D4×C21111yescharacteristic
217C4#120×C21111yescharacteristic
218C4#121×C21111yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—82no—
224C2#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. #568×C211C2semidirect
3. #421×C211C2semidirect
4. #418×C211C2semidirect
5. C2×#100×C211C2semidirect
6. #344×C211C2semidirect
7. #338×C211C2semidirect
12. C2×SD16×C211C4semidirect
13. C2×D8×C211C4semidirect
112. C211#1903direct (N × H)
113. #1903C211direct (N × H)
115. #568C2×C211semidirect
116. #421C2×C211semidirect
117. #418C2×C211semidirect
118. C2×#100C2×C211semidirect
119. #344C2×C211semidirect
120. #338C2×C211semidirect
125. C2×SD16C4×C211semidirect
126. C2×D8C4×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
#568×C2112
#421×C2112
#418×C2112
C2×#100×C2112
#344×C2112
#338×C2112
#1903211