Finite Group Databasebeta

← C23×#8540 #19156166

Subgroups — C23×#8540 #19156166

Order 5888.

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C23×#3491C211yesmaximal, characteristic
2C23×#1822C211yesmaximal, characteristic, complemented
3C23×#1818C211yesmaximal, characteristic, complemented
4C23×#1468C214yesmaximal, complemented
5C23×#552—24no—
6C23×#551—24no—
7C23×#543C411yescharacteristic
8C23×#539—22no—
9C23×#534C411yescharacteristic
10C23×#529—22no—
11C2×C23×#123C2212yes—
12C23×#518C412yes—
13C2×C23×#117C2211yescharacteristic
14C23×#407C2212yes—
15C2×C23×C22⋊C8C2212yes—
16C23×#343—22no—
17C23×#336—22no—
18C23×#324—24no—
19C23×#323—24no—
20C22×D4×C23C2×C412yes—
21C23×C4×C23C2311yescharacteristic
22C2×M16×C23—22no—
23C22×C8×C23—22no—
24C23×#126—24no—
25C23×#126—24no—
26C23×#125—22no—
27C23×#125—24no—
28C23×#124—22no—
29C23×#123—24no—
30C23×#123—22no—
31C23×#123—24no—
32C23×#123—22no—
33C23×#123—22no—
34C23×C22⋊Q8—24no—
35C23×C22⋊Q8—24no—
36C23×C22⋊Q8—22no—
37C23×#121—22no—
38C23×#120—22no—
39C4×Q8×C23—22no—
40C4×D4×C23—22no—
41C4×D4×C23—24no—
42C4×D4×C23—24no—
43C23×#117—24no—
44C23×#117—22no—
45C23×#117—24no—
46C2×C4⋊C4×C23D414yes—
47C2×C4⋊C4×C23D412yes—
48C2×C22⋊C4×C23—22no—
49C2×C22⋊C4×C23C2×C411yescharacteristic
50C2×C22⋊C4×C23C2×C411yescharacteristic
51C2×C22⋊C4×C23C2×C412yes—
52C2×C22⋊C4×C23D414yes—
53C2×C42×C23D412yes—
54C23×C22⋊C8—24no—
55C23×C22⋊C8—24no—
56C24×C23C22⋊C412yes—
57C24×C23C22×C411yescharacteristic
58C2×Q8×C23—22no—
59C2×D4×C23—22no—
60C2×D4×C23—22no—
61C2×D4×C23—22no—
62C2×D4×C23—44no—
63C22×C4×C23—24no—
64C22×C4×C23—42no—
65C22×C4×C23C22⋊C412yes—
66C22×C4×C23C22⋊C412yes—
67C22×C4×C23C22⋊C414yes—
68C22×C4×C23C22⋊C412yes—
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23C2×D411yescharacteristic
71C22×C4×C23C2×D414yes—
72M16×C23—44no—
73C2×C8×C23—44no—
74C2×C8×C23—42no—
75C2×C8×C23—42no—
76C4⋊C4×C23—24no—
77C4⋊C4×C23—24no—
78C4⋊C4×C23—22no—
79C4⋊C4×C23—22no—
80C4⋊C4×C23—24no—
81C4⋊C4×C23—24no—
82C4⋊C4×C23—24no—
83C22⋊C4×C23—24no—
84C22⋊C4×C23—24no—
85C22⋊C4×C23—22no—
86C22⋊C4×C23—22no—
87C22⋊C4×C23—22no—
88C22⋊C4×C23—22no—
89C22⋊C4×C23—24no—
90C22⋊C4×C23—24no—
91C42×C23—24no—
92C42×C23—22no—
93C42×C23—22no—
94#8540C2311yesmaximal, characteristic, direct product factor, Sylow
95C23×C23—22no—
96C23×C23—22no—
97C23×C23—42no—
98C23×C23—42no—
99C23×C23—44no—
100C23×C23C2×C22⋊C412yes—
101C23×C23C2×C22⋊C411yescharacteristic
102Q8×C23—82no—
103D4×C23—44no—
104D4×C23—44no—
105D4×C23—82no—
106C2×C4×C23—22no—
107C2×C4×C23—22no—
108C2×C4×C23—22no—
109C2×C4×C23—22no—
110C2×C4×C23—22no—
111C2×C4×C23—24no—
112C2×C4×C23—44no—
113C2×C4×C23—42no—
114C2×C4×C23—44no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—84no—
118C2×C4×C23#12012yes—
119C2×C4×C23#12012yes—
120C8×C23—82no—
121C8×C23—82no—
122#3491C2×C2311yescharacteristic
123#1822C2×C2311yescharacteristic, complemented
124#1818C2×C2311yescharacteristic, complemented
125#1468C2×C2314yescomplemented
126C22×C23—22no—
127C22×C23—22no—
128C22×C23—24no—
129C22×C23—42no—
130C22×C23—44no—
131C22×C23—82no—
132C22×C23—162no—
133C22×C23#37911yescharacteristic
134C22×C23#41012yes—
135C4×C23—22no—
136C4×C23—22no—
137C4×C23—42no—
138C4×C23—42no—
139C4×C23—44no—
140C4×C23—42no—
141#552—24no—
142#551—24no—
143#543C4×C2311yescharacteristic
144#539—22no—
145#534C4×C2311yescharacteristic
146#529—22no—
147C2×#123C22×C2312yes—
148#518C4×C2312yes—
149C2×#117C22×C2311yescharacteristic
150#407C22×C2312yes—
151C2×C22⋊C8C22×C2312yes—
152#343—22no—
153#336—22no—
154#324—24no—
155#323—24no—
156C2×C23—22no—
157C2×C23—42no—
158C2×C23—82no—
159C2×C23#178211yescharacteristic
160C2×C23#179112yes—
161C22×D4C2×C4×C2312yes—
162C23×C4C23×C2311yescharacteristic
163C2×M16—22no—
164C22×C8—22no—
165#126—24no—
166#126—24no—
167#125—22no—
168#125—24no—
169#124—22no—
170#123—24no—
171#123—22no—
172#123—22no—
173#123—22no—
174#123—24no—
175C22⋊Q8—24no—
176C22⋊Q8—22no—
177C22⋊Q8—24no—
178#121—22no—
179#120—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—24no—
183C4×D4—24no—
184#117—24no—
185#117—22no—
186#117—24no—
187C2×C4⋊C4D4×C2312yes—
188C2×C4⋊C4D4×C2314yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C2×C4×C2311yescharacteristic
191C2×C22⋊C4C2×C4×C2311yescharacteristic
192C2×C22⋊C4C2×C4×C2312yes—
193C2×C22⋊C4D4×C2314yes—
194C2×C42D4×C2312yes—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C23#854011yescharacteristic, direct product factor, Sylow
198C24C22⋊C4×C2312yes—
199C24C22×C4×C2311yescharacteristic
200C2×Q8—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—44no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4C22⋊C4×C2312yes—
208C22×C4C22⋊C4×C2312yes—
209C22×C4C22⋊C4×C2314yes—
210C22×C4C22⋊C4×C2312yes—
211C22×C4C2×D4×C2311yescharacteristic
212C22×C4C2×D4×C2311yescharacteristic
213C22×C4C2×D4×C2314yes—
214M16—44no—
215C2×C8—44no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—22no—
235C42—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—44no—
241C23C2×C22⋊C4×C2312yes—
242C23C2×C22⋊C4×C2311yescharacteristic
243Q8—82no—
244D4—44no—
245D4—44no—
246D4—82no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4C23×#12012yes—
260C2×C4C23×#12012yes—
261C8—82no—
262C8—82no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—42no—
267C22—44no—
268C22—82no—
269C22—162no—
270C22C23×#37911yescharacteristic
271C22C23×#41012yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—42no—
278C2—22no—
279C2—42no—
280C2—82no—
281C2C23×#178211yescharacteristic
282C2C23×#179112yes—
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#534C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1822C2semidirect
3. C23×#1818C2semidirect
4. C23×#1468C2semidirect
94. #8540C23direct (N × H)
123. #1822C2×C23semidirect
124. #1818C2×C23semidirect
125. #1468C2×C23semidirect
197. C23#8540direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34912
C23×#18222
C23×#18182
C23×#14682
#854023