Finite Group Databasebeta

← C23×#9012 #19156638

Subgroups — C23×#9012 #19156638

Order 5888.

558 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2975C211yesmaximal, characteristic
2C23×#2395C211yesmaximal, characteristic
3C23×#1826C212yesmaximal, complemented
4C4×C23×#99C211yesmaximal, characteristic
5C23×#1297C212yesmaximal, complemented
6C2×C4×D4×C23C2211yescharacteristic
7C2×C23×#117C2211yescharacteristic
8C23×#436—24no—
9C23×#431—21no—
10C2×C23×#99—21no—
11C23×#407—21no—
12C23×#407C2212yes—
13C23×#400C411yescharacteristic, complemented
14C23×#399C411yescharacteristic, complemented
15C23×#398C411yescharacteristic, complemented
16C23×#397—21no—
17C23×#397C411yescharacteristic, complemented
18C23×#394—21no—
19C4×C4⋊C4×C23—21no—
20C4×C22⋊C4×C23—21no—
21C4×C22⋊C4×C23C2212yes—
22C4×C22⋊C4×C23C2211yescharacteristic
23C23×#342—21no—
24C23×#339—24no—
25C22×D4×C23D411yescharacteristic
26C23×C4×C23D411yescharacteristic
27C23×C4×C23C2311yescharacteristic
28C2×M16×C23—22no—
29C2×M16×C23D414yes—
30C4×D4×C23—22no—
31C4×D4×C23—22no—
32C23×#117—22no—
33C23×#117—22no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C4⋊C4×C23C2×C411yescharacteristic
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23C2×C411yescharacteristic
45C2×C22⋊C4×C23D411yescharacteristic
46C2×C42×C23—21no—
47C2×C42×C23—21no—
48C2×C42×C23—22no—
49C2×C42×C23D411yescharacteristic
50C23×C4⋊C8—22no—
51C23×#99—24no—
52C23×C22⋊C8—22no—
53C23×C22⋊C8—22no—
54C23×C22⋊C8—22no—
55C4×C8×C23—22no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—22no—
61C23×#95C2×C411yescharacteristic
62C23×#95C2×C411yescharacteristic
63C24×C23C22⋊C411yescharacteristic
64C24×C23C2×D411yescharacteristic
65C2×D4×C23—22no—
66C2×D4×C23—22no—
67C2×D4×C23—41no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—42no—
75C22×C4×C23—42no—
76C22×C4×C23—42no—
77C22×C4×C23C22⋊C411yescharacteristic
78C22×C4×C23C22⋊C411yescharacteristic
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22×C411yescharacteristic
81C22×C4×C23C2×D412yes—
82C22×C4×C23C2×D411yescharacteristic
83C22×C4×C23#4112yes—
84M16×C23—44no—
85M16×C23—82no—
86C2×C8×C23—24no—
87C4⋊C4×C23—22no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—22no—
90C4⋊C4×C23—42no—
91C4⋊C4×C23—42no—
92C22⋊C4×C23—22no—
93C22⋊C4×C23—22no—
94C22⋊C4×C23—24no—
95C22⋊C4×C23—42no—
96C22⋊C4×C23—42no—
97C22⋊C4×C23—42no—
98C22⋊C4×C23—42no—
99C22⋊C4×C23—42no—
100C42×C23—21no—
101C42×C23—21no—
102C42×C23—42no—
103C42×C23—44no—
104C42×C23—41no—
105C42×C23—41no—
106#9012C2311yesmaximal, characteristic, direct product factor, Sylow
107C23×C23—21no—
108C23×C23—21no—
109C23×C23—21no—
110C23×C23—21no—
111C23×C23—21no—
112C23×C23—41no—
113C23×C23—41no—
114C23×C23—41no—
115C23×C23—41no—
116C23×C23C2×C22⋊C411yescharacteristic
117C23×C23#12112yes—
118D4×C23—44no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—24no—
122C2×C4×C23—22no—
123C2×C4×C23—22no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—41no—
127C2×C4×C23—42no—
128C2×C4×C23—41no—
129C2×C4×C23—41no—
130C2×C4×C23—42no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—44no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—41no—
138C2×C4×C23—41no—
139C2×C4×C23—41no—
140C2×C4×C23—41no—
141C2×C4×C23—41no—
142C2×C4×C23—41no—
143C2×C4×C23—81no—
144C2×C4×C23—82no—
145C2×C4×C23—82no—
146C2×C4×C23#10412yes—
147C2×C4×C23C4×D412yes—
148C2×C4×C23#12011yescharacteristic
149C2×C4×C23#12311yescharacteristic
150C8×C23—44no—
151#2975C2×C2311yescharacteristic
152#2395C2×C2311yescharacteristic
153#1826C2×C2312yescomplemented
154C4×#99C2×C2311yescharacteristic
155#1297C2×C2312yescomplemented
156C22×C23—21no—
157C22×C23—21no—
158C22×C23—22no—
159C22×C23—22no—
160C22×C23—22no—
161C22×C23—42no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23—41no—
165C22×C23—81no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#38611yescharacteristic
169C22×C23C2×#10411yescharacteristic
170C22×C23#42111yescharacteristic
171C4×C23—22no—
172C4×C23—21no—
173C4×C23—22no—
174C4×C23—42no—
175C4×C23—41no—
176C4×C23—41no—
177C4×C23—81no—
178C4×C23—84no—
179C4×C23—81no—
180C4×C23—81no—
181C4×C23#45812yes—
182C2×C4×D4C22×C2311yescharacteristic
183C2×#117C22×C2311yescharacteristic
184#436—24no—
185#431—21no—
186C2×#99—21no—
187#407—21no—
188#407C22×C2312yes—
189#400C4×C2311yescharacteristic, complemented
190#399C4×C2311yescharacteristic, complemented
191#398C4×C2311yescharacteristic, complemented
192#397—21no—
193#397C4×C2311yescharacteristic, complemented
194#394—21no—
195C4×C4⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4C22×C2312yes—
198C4×C22⋊C4C22×C2311yescharacteristic
199#342—21no—
200#339—24no—
201C2×C23—21no—
202C2×C23—21no—
203C2×C23—42no—
204C2×C23—81no—
205C2×C23#187111yescharacteristic
206C2×C23#188411yescharacteristic
207C2×C23#189011yescharacteristic
208C22×D4D4×C2311yescharacteristic
209C23×C4D4×C2311yescharacteristic
210C23×C4C23×C2311yescharacteristic
211C2×M16—22no—
212C2×M16D4×C2314yes—
213C4×D4—22no—
214C4×D4—22no—
215#117—22no—
216#117—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C2×C4×C2311yescharacteristic
220C2×C4⋊C4C2×C4×C2311yescharacteristic
221C2×C4⋊C4C2×C4×C2311yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4C2×C4×C2311yescharacteristic
228C2×C22⋊C4D4×C2311yescharacteristic
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42D4×C2311yescharacteristic
233C4⋊C8—22no—
234#99—24no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C4×C8—22no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95C2×C4×C2311yescharacteristic
245#95C2×C4×C2311yescharacteristic
246C23#901211yescharacteristic, direct product factor, Sylow
247C24C22⋊C4×C2311yescharacteristic
248C24C2×D4×C2311yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C22⋊C4×C2311yescharacteristic
262C22×C4C22⋊C4×C2311yescharacteristic
263C22×C4C22⋊C4×C2311yescharacteristic
264C22×C4C22×C4×C2311yescharacteristic
265C22×C4C2×D4×C2312yes—
266C22×C4C2×D4×C2311yescharacteristic
267C22×C4#41×C2312yes—
268M16—44no—
269M16—82no—
270C2×C8—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—24no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C42—21no—
285C42—21no—
286C42—42no—
287C42—44no—
288C42—41no—
289C42—41no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C2311yescharacteristic
300C23C23×#12112yes—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C23×#10412yes—
330C2×C4C4×D4×C2312yes—
331C2×C4C23×#12011yescharacteristic
332C2×C4C23×#12311yescharacteristic
333C8—44no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C23×#38611yescharacteristic
347C22C2×C23×#10411yescharacteristic
348C22C23×#42111yescharacteristic
349C4—21no—
350C4—22no—
351C4—22no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—84no—
357C4—81no—
358C4—81no—
359C4C23×#45812yes—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2C23×#187111yescharacteristic
365C2C23×#188411yescharacteristic
366C2C23×#189011yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
socleC22×C23#386
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1826C2semidirect
5. C23×#1297C2semidirect
13. C23×#400C4semidirect
14. C23×#399C4semidirect
15. C23×#398C4semidirect
17. C23×#397C4semidirect
106. #9012C23direct (N × H)
153. #1826C2×C23semidirect
155. #1297C2×C23semidirect
189. #400C4×C23semidirect
190. #399C4×C23semidirect
191. #398C4×C23semidirect
193. #397C4×C23semidirect
246. C23#9012direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29752
C23×#23952
C23×#18262
C4×C23×#992
C23×#12972
#901223