Finite Group Databasebeta

← #1907×C211 #41009527

Subgroups — #1907×C211 #41009527

Order 27008.

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

normal subgroups only

This group has 332 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
1C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
2C2×#131×C211C211yesmaximal, characteristic, complemented
3#417×C211C211yesmaximal, characteristic, complemented
4C2×#102×C211C211yesmaximal, characteristic, complemented
5C2×#100×C211C211yesmaximal, characteristic, complemented
6#343×C211C211yesmaximal, characteristic, complemented
7#341×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C22×D4×C211C2211yescharacteristic
10AGL(1,ℤ/8)×C211—22no—
11AGL(1,ℤ/8)×C211C414yescomplemented
12C2×SD16×C211—21no—
13C2×D8×C211—21no—
14#131×C211—22no—
15#131×C211—22no—
16C2×M16×C211—21no—
17C2×M16×C211C2211yescharacteristic
18C2×M16×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20C22×C8×C211—21no—
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#103×C211—21no—
24#102×C211—21no—
25#102×C211—21no—
26#102×C211—21no—
27#100×C211—22no—
28C24×C211—21no—
29#41×C211—21no—
30#41×C211—21no—
31#41×C211C2×C412yes—
32#41×C211D412yes—
33C2×Q8×C211D411yescharacteristic
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36C2×D4×C211—21no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211C2×C412yes—
41C2×D4×C211D411yescharacteristic
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45SD16×C211—24no—
46D8×C211—24no—
47M16×C211—24no—
48M16×C211—21no—
49M16×C211—21no—
50M16×C211—21no—
51M16×C211—21no—
52M16×C211C2×C412yes—
53M16×C211D412yes—
54C2×C8×C211—24no—
55C2×C8×C211—21no—
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59C2×C8×C211—21no—
60C2×C8×C211D411yescharacteristic
61C2×C8×C211D411yescharacteristic
62C4⋊C4×C211—21no—
63C4⋊C4×C211—21no—
64C23×C211—21no—
65C23×C211—22no—
66C23×C211—41no—
67C23×C211—41no—
68C23×C211—81no—
69C23×C211—81no—
70C23×C211#4111yescharacteristic
71Q8×C211—21no—
72Q8×C211C22⋊C412yes—
73D4×C211—22no—
74D4×C211—22no—
75D4×C211—22no—
76D4×C211—22no—
77D4×C211—21no—
78D4×C211—41no—
79D4×C211—41no—
80D4×C211C22⋊C412yes—
81C2×C4×C211—21no—
82C2×C4×C211—22no—
83C2×C4×C211—22no—
84C2×C4×C211—41no—
85C2×C4×C211—81no—
86C2×C4×C211C22×C411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211#4111yescharacteristic
92C8×C211—22no—
93C8×C211—21no—
94C8×C211—22no—
95C8×C211—21no—
96C8×C211—41no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—22no—
100C22×C211—41no—
101C22×C211—41no—
102C22×C211—42no—
103C22×C211—82no—
104C22×C211—82no—
105C22×C211—81no—
106C22×C211—81no—
107C22×C211C4×D411yescharacteristic
108C22×C211#12111yescharacteristic
109C22×C211#12311yescharacteristic
110C4×C211—22no—
111C4×C211—81no—
112C4×C211C2×C22⋊C411yescharacteristic
113C4×C211C4×D411yescharacteristic
114C4×C211#12011yescharacteristic
115C4×C211#12111yescharacteristic
116C2×C211—21no—
117C2×C211—21no—
118C2×C211—42no—
119C2×C211—82no—
120C2×C211#38611yescharacteristic
121C2×C211#47111yescharacteristic
122C2×C211#47211yescharacteristic
123C211#190711yescharacteristic, direct product factor, Sylow
124#1907C21111yesmaximal, characteristic, direct product factor, Sylow
125C2×AGL(1,ℤ/8)C2×C21111yescharacteristic
126C2×#131C2×C21111yescharacteristic, complemented
127#417C2×C21111yescharacteristic, complemented
128C2×#102C2×C21111yescharacteristic, complemented
129C2×#100C2×C21111yescharacteristic, complemented
130#343C2×C21111yescharacteristic, complemented
131#341C2×C21111yescharacteristic, complemented
132C2×#41C22×C21111yescharacteristic
133C22×D4C22×C21111yescharacteristic
134AGL(1,ℤ/8)—22no—
135AGL(1,ℤ/8)C4×C21114yescomplemented
136C2×SD16—21no—
137C2×D8—21no—
138#131—22no—
139#131—22no—
140C2×M16—21no—
141C2×M16C22×C21111yescharacteristic
142C2×M16C22×C21111yescharacteristic
143C2×M16C22×C21111yescharacteristic
144C22×C8—21no—
145C22×C8C22×C21111yescharacteristic
146C2×C4⋊C4C22×C21111yescharacteristic
147#103—21no—
148#102—21no—
149#102—21no—
150#102—21no—
151#100—22no—
152C24—21no—
153#41—21no—
154#41—21no—
155#41C2×C4×C21112yes—
156#41D4×C21112yes—
157C2×Q8D4×C21111yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4C2×C4×C21112yes—
165C2×D4D4×C21111yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C23×C21111yescharacteristic
169SD16—24no—
170D8—24no—
171M16—24no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—21no—
176M16C2×C4×C21112yes—
177M16D4×C21112yes—
178C2×C8—24no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8D4×C21111yescharacteristic
185C2×C8D4×C21111yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C23—21no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—81no—
193C23—81no—
194C23#41×C21111yescharacteristic
195Q8—21no—
196Q8C22⋊C4×C21112yes—
197D4—22no—
198D4—22no—
199D4—22no—
200D4—22no—
201D4—21no—
202D4—41no—
203D4—41no—
204D4C22⋊C4×C21112yes—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4C22×C4×C21111yescharacteristic
211C2×C4C2×D4×C21111yescharacteristic
212C2×C4C2×D4×C21111yescharacteristic
213C2×C4C2×D4×C21111yescharacteristic
214C2×C4C2×D4×C21111yescharacteristic
215C2×C4#41×C21111yescharacteristic
216C8—22no—
217C8—21no—
218C8—22no—
219C8—21no—
220C8—41no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—82no—
228C22—82no—
229C22—81no—
230C22—81no—
231C22C4×D4×C21111yescharacteristic
232C22#121×C21111yescharacteristic
233C22#123×C21111yescharacteristic
234C4—22no—
235C4—81no—
236C4C2×C22⋊C4×C21111yescharacteristic
237C4C4×D4×C21111yescharacteristic
238C4#120×C21111yescharacteristic
239C4#121×C21111yescharacteristic
240C2—21no—
241C2—21no—
242C2—42no—
243C2—82no—
244C2#386×C21111yescharacteristic
245C2#471×C21111yescharacteristic
246C2#472×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#121
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×AGL(1,ℤ/8)×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×#131×C211C2semidirect
3. #417×C211C2semidirect
4. C2×#102×C211C2semidirect
5. C2×#100×C211C2semidirect
6. #343×C211C2semidirect
7. #341×C211C2semidirect
11. AGL(1,ℤ/8)×C211C4semidirect
123. C211#1907direct (N × H)
124. #1907C211direct (N × H)
126. C2×#131C2×C211semidirect
127. #417C2×C211semidirect
128. C2×#102C2×C211semidirect
129. C2×#100C2×C211semidirect
130. #343C2×C211semidirect
131. #341C2×C211semidirect
135. AGL(1,ℤ/8)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
C2×#131×C2112
#417×C2112
C2×#102×C2112
C2×#100×C2112
#343×C2112
#341×C2112
#1907211