Finite Group Databasebeta

← #1908×C211 #41009528

Subgroups — #1908×C211 #41009528

Order 27008.

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

normal subgroups only

This group has 312 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×#137×C211C211yesmaximal, characteristic
2C2×#131×C211C211yesmaximal, characteristic
3#417×C211C211yesmaximal, characteristic
4C2×#103×C211C211yesmaximal, characteristic, complemented
5C2×#101×C211C211yesmaximal, characteristic, complemented
6#343×C211C211yesmaximal, characteristic, complemented
7#341×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C22×Q8×C211C2211yescharacteristic
10#137×C211—22no—
11#137×C211C414yescomplemented
12C2×Q16×C211—21no—
13C2×SD16×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#103×C211—21no—
25#103×C211—21no—
26#102×C211—21no—
27#101×C211—22no—
28#41×C211—21no—
29#41×C211—21no—
30#41×C211C2×C412yes—
31#41×C211D412yes—
32C2×Q8×C211—21no—
33C2×Q8×C211—21no—
34C2×Q8×C211—21no—
35C2×Q8×C211—21no—
36C2×Q8×C211—21no—
37C2×Q8×C211C2×C412yes—
38C2×Q8×C211D411yescharacteristic
39C2×D4×C211—21no—
40C2×D4×C211D411yescharacteristic
41C22×C4×C211—21no—
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45Q16×C211—24no—
46SD16×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#4111yescharacteristic
66Q8×C211—21no—
67Q8×C211—22no—
68Q8×C211—22no—
69Q8×C211—41no—
70Q8×C211—41no—
71Q8×C211C22⋊C412yes—
72D4×C211—22no—
73D4×C211—22no—
74D4×C211—21no—
75D4×C211C22⋊C412yes—
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—22no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211—81no—
84C2×C4×C211C22×C411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211#4111yescharacteristic
90C8×C211—22no—
91C8×C211—21no—
92C8×C211—22no—
93C8×C211—21no—
94C8×C211—41no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211—22no—
98C22×C211—41no—
99C22×C211—41no—
100C22×C211C4×D411yescharacteristic
101C22×C211#12111yescharacteristic
102C22×C211#12311yescharacteristic
103C4×C211—22no—
104C4×C211—42no—
105C4×C211—81no—
106C4×C211C2×C22⋊C411yescharacteristic
107C4×C211C4×D411yescharacteristic
108C4×C211#12011yescharacteristic
109C4×C211#12111yescharacteristic
110C2×C211—21no—
111C2×C211—21no—
112C2×C211—42no—
113C2×C211#38611yescharacteristic
114C2×C211#47111yescharacteristic
115C2×C211#47311yescharacteristic
116C211#190811yescharacteristic, direct product factor, Sylow
117#1908C21111yesmaximal, characteristic, direct product factor, Sylow
118C2×#137C2×C21111yescharacteristic
119C2×#131C2×C21111yescharacteristic
120#417C2×C21111yescharacteristic
121C2×#103C2×C21111yescharacteristic, complemented
122C2×#101C2×C21111yescharacteristic, complemented
123#343C2×C21111yescharacteristic, complemented
124#341C2×C21111yescharacteristic, complemented
125C2×#41C22×C21111yescharacteristic
126C22×Q8C22×C21111yescharacteristic
127#137—22no—
128#137C4×C21114yescomplemented
129C2×Q16—21no—
130C2×SD16—21no—
131#131—22no—
132#131—22no—
133C2×M16—21no—
134C2×M16C22×C21111yescharacteristic
135C2×M16C22×C21111yescharacteristic
136C2×M16C22×C21111yescharacteristic
137C22×C8—21no—
138C22×C8C22×C21111yescharacteristic
139C2×C4⋊C4C22×C21111yescharacteristic
140#103—21no—
141#103—21no—
142#103—21no—
143#102—21no—
144#101—22no—
145#41—21no—
146#41—21no—
147#41C2×C4×C21112yes—
148#41D4×C21112yes—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×C4×C21112yes—
155C2×Q8D4×C21111yescharacteristic
156C2×D4—21no—
157C2×D4D4×C21111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C23×C21111yescharacteristic
162Q16—24no—
163SD16—24no—
164M16—24no—
165M16—21no—
166M16—21no—
167M16—21no—
168M16—21no—
169M16C2×C4×C21112yes—
170M16D4×C21112yes—
171C2×C8—24no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8D4×C21111yescharacteristic
178C2×C8D4×C21111yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C23—21no—
182C23#41×C21111yescharacteristic
183Q8—21no—
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8C22⋊C4×C21112yes—
189D4—22no—
190D4—22no—
191D4—21no—
192D4C22⋊C4×C21112yes—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4C22×C4×C21111yescharacteristic
202C2×C4C2×D4×C21111yescharacteristic
203C2×C4C2×D4×C21111yescharacteristic
204C2×C4C2×D4×C21111yescharacteristic
205C2×C4C2×D4×C21111yescharacteristic
206C2×C4#41×C21111yescharacteristic
207C8—22no—
208C8—21no—
209C8—22no—
210C8—21no—
211C8—41no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22C4×D4×C21111yescharacteristic
218C22#121×C21111yescharacteristic
219C22#123×C21111yescharacteristic
220C4—22no—
221C4—42no—
222C4—81no—
223C4C2×C22⋊C4×C21111yescharacteristic
224C4C4×D4×C21111yescharacteristic
225C4#120×C21111yescharacteristic
226C4#121×C21111yescharacteristic
227C2—21no—
228C2—21no—
229C2—42no—
230C2#386×C21111yescharacteristic
231C2#471×C21111yescharacteristic
232C2#473×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#121
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
4. C2×#103×C211C2semidirect
5. C2×#101×C211C2semidirect
6. #343×C211C2semidirect
7. #341×C211C2semidirect
11. #137×C211C4semidirect
116. C211#1908direct (N × H)
117. #1908C211direct (N × H)
121. C2×#103C2×C211semidirect
122. C2×#101C2×C211semidirect
123. #343C2×C211semidirect
124. #341C2×C211semidirect
128. #137C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#137×C2112
C2×#131×C2112
#417×C2112
C2×#103×C2112
C2×#101×C2112
#343×C2112
#341×C2112
#1908211