Finite Group Databasebeta

← #1799×C211 #41009419

Subgroups — #1799×C211 #41009419

Order 27008.

380 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 124 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×C4×D4×C211C211yesmaximal, characteristic
2C2×#99×C211C212yesmaximal
3#380×C211C212yesmaximal, complemented
4#342×C211C212yesmaximal, complemented
5C22×D4×C211C2211yescharacteristic
6C23×C4×C211C411yescharacteristic, complemented
7C23×C4×C211C2211yescharacteristic
8C4×D4×C211—22no—
9C4×D4×C211—22no—
10C2×C4⋊C4×C211—22no—
11C2×C4⋊C4×C211C411yescharacteristic, complemented
12C2×C22⋊C4×C211C411yescharacteristic, complemented
13C2×C22⋊C4×C211C2214yes—
14C2×C22⋊C4×C211C2211yescharacteristic
15C2×C42×C211C411yescharacteristic, complemented
16#99×C211—24no—
17#95×C211—22no—
18C24×C211C2×C411yescharacteristic
19C24×C211C2311yescharacteristic
20C2×D4×C211—22no—
21C2×D4×C211—22no—
22C2×D4×C211D412yes—
23C2×D4×C211Q812yes—
24C22×C4×C211—21no—
25C22×C4×C211—21no—
26C22×C4×C211—24no—
27C22×C4×C211—42no—
28C22×C4×C211—42no—
29C22×C4×C211C2×C411yescharacteristic
30C22×C4×C211C2×C411yescharacteristic
31C22×C4×C211C2×C411yescharacteristic
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211C2×C411yescharacteristic
34C22×C4×C211D412yes—
35C22×C4×C211D412yes—
36C4⋊C4×C211—22no—
37C4⋊C4×C211—44no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C22⋊C4×C211—24no—
41C22⋊C4×C211—24no—
42C42×C211—21no—
43C42×C211—21no—
44C23×C211—21no—
45C23×C211—21no—
46C23×C211—41no—
47C23×C211—41no—
48C23×C211—41no—
49C23×C211—41no—
50C23×C211C4⋊C412yescomplemented
51C23×C211C22×C411yescharacteristic
52C23×C211C2×D412yes—
53C23×C211C2×D411yescharacteristic
54C23×C211C2×Q811yescharacteristic
55C23×C211#4112yes—
56D4×C211—24no—
57D4×C211—24no—
58C2×C4×C211—21no—
59C2×C4×C211—21no—
60C2×C4×C211—24no—
61C2×C4×C211—22no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—41no—
65C2×C4×C211—42no—
66C2×C4×C211—42no—
67C2×C4×C211—42no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211—44no—
71C2×C4×C211—44no—
72C2×C4×C211—44no—
73C2×C4×C211C22⋊C411yescharacteristic
74C2×C4×C211C22⋊C412yes—
75C2×C4×C211C22⋊C411yescharacteristic
76C2×C4×C211C4⋊C412yescomplemented
77C22×C211—22no—
78C22×C211—21no—
79C22×C211—22no—
80C22×C211—21no—
81C22×C211—22no—
82C22×C211—41no—
83C22×C211—41no—
84C22×C211—42no—
85C22×C211—42no—
86C22×C211—42no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211C2×C22⋊C411yescharacteristic
90C22×C211C2×C4⋊C411yescharacteristic
91C22×C211#11711yescharacteristic
92C22×C211#12112yes—
93C22×C211C22⋊Q812yes—
94C4×C211—22no—
95C4×C211—21no—
96C4×C211—22no—
97C4×C211—41no—
98C4×C211—41no—
99C4×C211—84no—
100C4×C211#9912yescomplemented
101C2×C211—22no—
102C2×C211—21no—
103C2×C211—21no—
104C2×C211—22no—
105C2×C211—42no—
106C2×C211#38011yescharacteristic
107C2×C211C2×#9911yescharacteristic
108C2×C211#41011yescharacteristic
109C211#179911yescharacteristic, direct product factor, Sylow
110#1799C21111yesmaximal, characteristic, direct product factor, Sylow
111C2×C4×D4C2×C21111yescharacteristic
112C2×#99C2×C21112yes—
113#380C2×C21112yescomplemented
114#342C2×C21112yescomplemented
115C22×D4C22×C21111yescharacteristic
116C23×C4C4×C21111yescharacteristic, complemented
117C23×C4C22×C21111yescharacteristic
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4C4×C21111yescharacteristic, complemented
122C2×C22⋊C4C4×C21111yescharacteristic, complemented
123C2×C22⋊C4C22×C21114yes—
124C2×C22⋊C4C22×C21111yescharacteristic
125C2×C42C4×C21111yescharacteristic, complemented
126#99—24no—
127#95—22no—
128C24C2×C4×C21111yescharacteristic
129C24C23×C21111yescharacteristic
130C2×D4—22no—
131C2×D4—22no—
132C2×D4D4×C21112yes—
133C2×D4Q8×C21112yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×C4×C21111yescharacteristic
140C22×C4C2×C4×C21111yescharacteristic
141C22×C4C2×C4×C21111yescharacteristic
142C22×C4C2×C4×C21111yescharacteristic
143C22×C4C2×C4×C21111yescharacteristic
144C22×C4D4×C21112yes—
145C22×C4D4×C21112yes—
146C4⋊C4—22no—
147C4⋊C4—44no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C42—21no—
153C42—21no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23C4⋊C4×C21112yescomplemented
161C23C22×C4×C21111yescharacteristic
162C23C2×D4×C21112yes—
163C23C2×D4×C21111yescharacteristic
164C23C2×Q8×C21111yescharacteristic
165C23#41×C21112yes—
166D4—24no—
167D4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4C22⋊C4×C21111yescharacteristic
184C2×C4C22⋊C4×C21112yes—
185C2×C4C22⋊C4×C21111yescharacteristic
186C2×C4C4⋊C4×C21112yescomplemented
187C22—22no—
188C22—21no—
189C22—22no—
190C22—21no—
191C22—22no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—41no—
199C22C2×C22⋊C4×C21111yescharacteristic
200C22C2×C4⋊C4×C21111yescharacteristic
201C22#117×C21111yescharacteristic
202C22#121×C21112yes—
203C22C22⋊Q8×C21112yes—
204C4—22no—
205C4—21no—
206C4—22no—
207C4—41no—
208C4—41no—
209C4—84no—
210C4#99×C21112yescomplemented
211C2—22no—
212C2—21no—
213C2—21no—
214C2—22no—
215C2—42no—
216C2#380×C21111yescharacteristic
217C2C2×#99×C21111yescharacteristic
218C2#410×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC23C22×C4×C211
Frattini, cosocleC24C23×C211
prime radicalC22×D4×C211C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #380×C211C2semidirect
4. #342×C211C2semidirect
6. C23×C4×C211C4semidirect
11. C2×C4⋊C4×C211C4semidirect
12. C2×C22⋊C4×C211C4semidirect
15. C2×C42×C211C4semidirect
50. C23×C211C4⋊C4semidirect
76. C2×C4×C211C4⋊C4semidirect
100. C4×C211#99semidirect
109. C211#1799direct (N × H)
110. #1799C211direct (N × H)
113. #380C2×C211semidirect
114. #342C2×C211semidirect
116. C23×C4C4×C211semidirect
121. C2×C4⋊C4C4×C211semidirect
122. C2×C22⋊C4C4×C211semidirect
125. C2×C42C4×C211semidirect
160. C23C4⋊C4×C211semidirect
186. C2×C4C4⋊C4×C211semidirect
210. C4#99×C211semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×C2112
C2×#99×C2112
#380×C2112
#342×C2112
#1799211