Finite Group Databasebeta

← #1882×C211 #41009502

Subgroups — #1882×C211 #41009502

Order 27008.

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

normal subgroups only

This group has 334 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
2#520×C211C211yesmaximal, characteristic
3C2×#104×C211C211yesmaximal, characteristic
4#419×C211C211yesmaximal, characteristic, complemented
5#391×C211C211yesmaximal, characteristic, complemented
6#343×C211C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C22×Q8×C211C2211yescharacteristic
10#137×C211—22no—
11#137×C211C414yescomplemented
12C2×Q16×C211—21no—
13C2×SD16×C211—21no—
14C2×M16×C211C2211yescharacteristic
15C2×M16×C211C2211yescharacteristic
16C4×Q8×C211—21no—
17C4×D4×C211—21no—
18C4×D4×C211—22no—
19#117×C211—21no—
20#117×C211C2211yescharacteristic
21C2×C4⋊C4×C211—21no—
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C42×C211C2211yescharacteristic
24#104×C211—22no—
25#103×C211—21no—
26#103×C211—21no—
27#95×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×C412yescomplemented
38C2×Q8×C211D411yescharacteristic
39C2×D4×C211—21no—
40C2×D4×C211D411yescharacteristic
41C22×C4×C211—21no—
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211—22no—
45C22×C4×C211D412yes—
46C22×C4×C211C2311yescharacteristic
47Q16×C211—24no—
48SD16×C211—24no—
49M16×C211—21no—
50M16×C211—21no—
51M16×C211—21no—
52M16×C211C2×C412yescomplemented
53C2×C8×C211—21no—
54C2×C8×C211—21no—
55C4⋊C4×C211—22no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211D411yescharacteristic
59C4⋊C4×C211D411yescharacteristic
60C22⋊C4×C211—22no—
61C22⋊C4×C211—21no—
62C42×C211—22no—
63C42×C211—21no—
64C42×C211—21no—
65C42×C211—21no—
66C23×C211—21no—
67C23×C211C2×D411yescharacteristic
68Q8×C211—21no—
69Q8×C211—22no—
70Q8×C211—22no—
71Q8×C211—41no—
72Q8×C211—41no—
73Q8×C211C22⋊C412yes—
74D4×C211—22no—
75D4×C211—22no—
76D4×C211—21no—
77D4×C211C22⋊C412yes—
78C2×C4×C211—21no—
79C2×C4×C211—22no—
80C2×C4×C211—22no—
81C2×C4×C211—22no—
82C2×C4×C211—22no—
83C2×C4×C211—21no—
84C2×C4×C211—22no—
85C2×C4×C211—21no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—42no—
91C2×C4×C211—82no—
92C2×C4×C211C22×C411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211#4111yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C8×C211—22no—
99C8×C211—41no—
100C22×C211—21no—
101C22×C211—21no—
102C22×C211—22no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211C4×D411yescharacteristic
106C22×C211#12011yescharacteristic
107C22×C211#12111yescharacteristic
108C4×C211—22no—
109C4×C211—42no—
110C4×C211—42no—
111C4×C211—42no—
112C4×C211—41no—
113C4×C211—41no—
114C4×C211C2×C22⋊C411yescharacteristic
115C4×C211C4×D411yescharacteristic
116C4×C211#12111yescharacteristic
117C4×C211#12311yescharacteristic
118C2×C211—21no—
119C2×C211—21no—
120C2×C211—42no—
121C2×C211#38611yescharacteristic
122C2×C211#45511yescharacteristic
123C2×C211#45611yescharacteristic
124C211#188211yescharacteristic, direct product factor, Sylow
125#1882C21111yesmaximal, characteristic, direct product factor, Sylow
126C2×#137C2×C21111yescharacteristic
127#520C2×C21111yescharacteristic
128C2×#104C2×C21111yescharacteristic
129#419C2×C21111yescharacteristic, complemented
130#391C2×C21111yescharacteristic, complemented
131#343C2×C21111yescharacteristic, complemented
132#339C2×C21111yescharacteristic, complemented
133C2×#41C22×C21111yescharacteristic
134C22×Q8C22×C21111yescharacteristic
135#137—22no—
136#137C4×C21114yescomplemented
137C2×Q16—21no—
138C2×SD16—21no—
139C2×M16C22×C21111yescharacteristic
140C2×M16C22×C21111yescharacteristic
141C4×Q8—21no—
142C4×D4—21no—
143C4×D4—22no—
144#117—21no—
145#117C22×C21111yescharacteristic
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22×C21111yescharacteristic
148C2×C42C22×C21111yescharacteristic
149#104—22no—
150#103—21no—
151#103—21no—
152#95—22no—
153#41—21no—
154#41—21no—
155#41C2×C4×C21112yes—
156#41D4×C21112yes—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8C2×C4×C21112yescomplemented
163C2×Q8D4×C21111yescharacteristic
164C2×D4—21no—
165C2×D4D4×C21111yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4D4×C21112yes—
171C22×C4C23×C21111yescharacteristic
172Q16—24no—
173SD16—24no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16C2×C4×C21112yescomplemented
178C2×C8—21no—
179C2×C8—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4D4×C21111yescharacteristic
184C4⋊C4D4×C21111yescharacteristic
185C22⋊C4—22no—
186C22⋊C4—21no—
187C42—22no—
188C42—21no—
189C42—21no—
190C42—21no—
191C23—21no—
192C23C2×D4×C21111yescharacteristic
193Q8—21no—
194Q8—22no—
195Q8—22no—
196Q8—41no—
197Q8—41no—
198Q8C22⋊C4×C21112yes—
199D4—22no—
200D4—22no—
201D4—21no—
202D4C22⋊C4×C21112yes—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4C22×C4×C21111yescharacteristic
218C2×C4C2×D4×C21111yescharacteristic
219C2×C4C2×D4×C21111yescharacteristic
220C2×C4C2×D4×C21111yescharacteristic
221C2×C4#41×C21111yescharacteristic
222C2×C4#41×C21111yescharacteristic
223C8—22no—
224C8—41no—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22C4×D4×C21111yescharacteristic
231C22#120×C21111yescharacteristic
232C22#121×C21111yescharacteristic
233C4—22no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4C2×C22⋊C4×C21111yescharacteristic
240C4C4×D4×C21111yescharacteristic
241C4#121×C21111yescharacteristic
242C4#123×C21111yescharacteristic
243C2—21no—
244C2—21no—
245C2—42no—
246C2#386×C21111yescharacteristic
247C2#455×C21111yescharacteristic
248C2#456×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
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. #419×C211C2semidirect
5. #391×C211C2semidirect
6. #343×C211C2semidirect
7. #339×C211C2semidirect
11. #137×C211C4semidirect
37. C2×Q8×C211C2×C4semidirect
52. M16×C211C2×C4semidirect
124. C211#1882direct (N × H)
125. #1882C211direct (N × H)
129. #419C2×C211semidirect
130. #391C2×C211semidirect
131. #343C2×C211semidirect
132. #339C2×C211semidirect
136. #137C4×C211semidirect
162. C2×Q8C2×C4×C211semidirect
177. M16C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#137×C2112
#520×C2112
C2×#104×C2112
#419×C2112
#391×C2112
#343×C2112
#339×C2112
#1882211