Finite Group Databasebeta

← C23×#36676 #19184302

Subgroups — C23×#36676 #19184302

Order 5888.

2006 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2842C211yesmaximal, characteristic, complemented
2C23×#2834C211yesmaximal, characteristic, complemented
3C23×#2827C216yesmaximal, complemented
4C23×#2826C213yesmaximal, complemented
5C23×#2825C212yesmaximal, complemented
6C23×#2812C213yesmaximal, complemented
7C23×#2401C212yesmaximal, complemented
8C23×#2397C211yesmaximal, characteristic, complemented
9C23×#2395C213yesmaximal, complemented
10C23×#2392C216yesmaximal, complemented
11C4×C23×#120C213yesmaximal, complemented
12C24×C4×C23C2211yescharacteristic
13C2×C23×#120C2213yes—
14C2×C4×D4×C23C2213yes—
15C2×C4×D4×C23C2216yes—
16C2×C4×D4×C23C2213yescomplemented
17C2×C23×#117C2213yescomplemented
18C23×#401C2212yescomplemented
19C23×#400C2216yescomplemented
20C23×#400C2213yescomplemented
21C23×#399C2211yescharacteristic
22C23×#399C2213yes—
23C23×#399C2213yescomplemented
24C23×#398C2213yescomplemented
25C23×#398C2216yescomplemented
26C23×#397C2216yes—
27C23×#397C2213yes—
28C23×#397C2216yescomplemented
29C23×#397C2216yescomplemented
30C23×#396C2212yes—
31C23×#396C2216yes—
32C23×#396C2216yescomplemented
33C23×#394C2216yes—
34C23×#394C2216yes—
35C23×#394C2216yes—
36C23×#394C2216yescomplemented
37C23×#392C2213yes—
38C23×#392C2216yes—
39C23×#392C2211yescharacteristic, complemented
40C23×#386C2216yes—
41C23×#386C2213yes—
42C23×#385C2213yes—
43C23×#385C2216yes—
44C23×#379C2213yes—
45C4×C4⋊C4×C23C2216yescomplemented
46C4×C22⋊C4×C23C2216yes—
47C4×C22⋊C4×C23C2213yes—
48C4×C22⋊C4×C23C2213yescomplemented
49C23×#376C2211yescharacteristic, complemented
50C25×C23C2311yescharacteristic
51C22×D4×C23C2316yes—
52C22×D4×C23C2313yes—
53C22×D4×C23C2313yes—
54C23×C4×C23—46no—
55C23×C4×C23C2312yes—
56C23×C4×C23C2311yescharacteristic
57C23×C4×C23C2313yes—
58C23×C4×C23C2311yescharacteristic
59C23×C4×C23C2312yes—
60C23×#120—212no—
61C4×D4×C23—412no—
62C4×D4×C23—46no—
63C4×D4×C23—46no—
64C23×#117—46no—
65C2×C4⋊C4×C23C2316yes—
66C2×C4⋊C4×C23C2316yes—
67C2×C4⋊C4×C23C2316yes—
68C2×C4⋊C4×C23C2313yes—
69C2×C4⋊C4×C23C2313yes—
70C2×C4⋊C4×C23C2313yes—
71C2×C4⋊C4×C23C2313yes—
72C2×C22⋊C4×C23C2316yes—
73C2×C22⋊C4×C23C2313yes—
74C2×C22⋊C4×C23C2316yes—
75C2×C22⋊C4×C23C2313yes—
76C2×C22⋊C4×C23C2316yes—
77C2×C22⋊C4×C23C2313yes—
78C2×C22⋊C4×C23C2316yes—
79C2×C22⋊C4×C23C2313yes—
80C2×C22⋊C4×C23C2316yes—
81C2×C22⋊C4×C23C2316yes—
82C2×C22⋊C4×C23C2316yes—
83C2×C42×C23C2313yes—
84C2×C42×C23C2316yes—
85C23×#95C2313yes—
86C23×#95C2311yescharacteristic
87C23×#95C2316yes—
88C23×#95C2316yes—
89C23×#95C2316yes—
90C23×#95C2313yes—
91C23×#95C2313yes—
92C23×#95C2316yes—
93C23×#95C2316yes—
94C24×C23—43no—
95C24×C23—43no—
96C24×C23#4114yes—
97C24×C23C2413yes—
98C24×C23C2412yes—
99C24×C23C2411yescharacteristic
100C2×D4×C23—212no—
101C2×D4×C23—26no—
102C2×D4×C23—43no—
103C2×D4×C23—43no—
104C2×D4×C23—46no—
105C2×D4×C23—46no—
106C2×D4×C23—46no—
107C2×D4×C23—43no—
108C22×C4×C23—26no—
109C22×C4×C23—212no—
110C22×C4×C23—212no—
111C22×C4×C23—26no—
112C22×C4×C23—43no—
113C22×C4×C23—46no—
114C22×C4×C23—416no—
115C22×C4×C23C2416yes—
116C22×C4×C23C2413yes—
117C22×C4×C23C2416yes—
118C22×C4×C23C2413yes—
119C22×C4×C23C2413yes—
120C22×C4×C23C2411yescharacteristic
121C22×C4×C23C2412yes—
122C22×C4×C23C2411yescharacteristic
123C4⋊C4×C23—46no—
124C4⋊C4×C23—46no—
125C4⋊C4×C23—43no—
126C4⋊C4×C23—43no—
127C4⋊C4×C23—43no—
128C4⋊C4×C23—43no—
129C4⋊C4×C23—46no—
130C22⋊C4×C23—212no—
131C22⋊C4×C23—26no—
132C22⋊C4×C23—43no—
133C22⋊C4×C23—46no—
134C22⋊C4×C23—46no—
135C22⋊C4×C23—46no—
136C22⋊C4×C23—46no—
137C22⋊C4×C23—46no—
138C22⋊C4×C23—46no—
139C22⋊C4×C23—43no—
140C22⋊C4×C23—43no—
141C42×C23—212no—
142C42×C23—26no—
143#36676C2311yesmaximal, characteristic, direct product factor, Sylow
144C23×C23—26no—
145C23×C23—23no—
146C23×C23—23no—
147C23×C23—26no—
148C23×C23—23no—
149C23×C23—23no—
150C23×C23—26no—
151C23×C23—26no—
152C23×C23—23no—
153C23×C23—412no—
154C23×C23—44no—
155C23×C23—412no—
156C23×C23C2×#4114yes—
157C23×C23C2×#4112yes—
158C23×C23C2511yescharacteristic
159D4×C23—412no—
160D4×C23—424no—
161D4×C23—412no—
162C2×C4×C23—23no—
163C2×C4×C23—26no—
164C2×C4×C23—26no—
165C2×C4×C23—23no—
166C2×C4×C23—23no—
167C2×C4×C23—26no—
168C2×C4×C23—26no—
169C2×C4×C23—23no—
170C2×C4×C23—26no—
171C2×C4×C23—23no—
172C2×C4×C23—26no—
173C2×C4×C23—26no—
174C2×C4×C23—26no—
175C2×C4×C23—26no—
176C2×C4×C23—23no—
177C2×C4×C23—44no—
178C2×C4×C23—48no—
179C2×C4×C23—412no—
180C2×C4×C23—48no—
181C2×C4×C23—44no—
182C2×C4×C23H(5,𝔽2)16yes—
183#2842C2×C2311yescharacteristic, complemented
184#2834C2×C2311yescharacteristic, complemented
185#2827C2×C2316yescomplemented
186#2826C2×C2313yescomplemented
187#2825C2×C2312yescomplemented
188#2812C2×C2313yescomplemented
189#2401C2×C2312yescomplemented
190#2397C2×C2311yescharacteristic, complemented
191#2395C2×C2313yescomplemented
192#2392C2×C2316yescomplemented
193C4×#120C2×C2313yescomplemented
194C22×C23—26no—
195C22×C23—212no—
196C22×C23—416no—
197C22×C23—42no—
198C22×C23—41no—
199C22×C23—46no—
200C22×C23—43no—
201C22×C23—43no—
202C22×C23—46no—
203C22×C23—43no—
204C22×C23—46no—
205C22×C23—43no—
206C22×C23C22×#4111yescharacteristic
207C22×C23C2×H(5,𝔽2)13yes—
208C22×C23#58513yes—
209C4×C23—43no—
210C4×C23—46no—
211C4×C23—43no—
212C4×C23—42no—
213C4×C23—41no—
214C4×C23—46no—
215C4×C23—43no—
216C4×C23#56014yes—
217C24×C4C22×C2311yescharacteristic
218C2×#120C22×C2313yes—
219C2×C4×D4C22×C2316yes—
220C2×C4×D4C22×C2313yes—
221C2×C4×D4C22×C2313yescomplemented
222C2×#117C22×C2313yescomplemented
223#401C22×C2312yescomplemented
224#400C22×C2316yescomplemented
225#400C22×C2313yescomplemented
226#399C22×C2313yes—
227#399C22×C2311yescharacteristic
228#399C22×C2313yescomplemented
229#398C22×C2313yescomplemented
230#398C22×C2316yescomplemented
231#397C22×C2316yes—
232#397C22×C2313yes—
233#397C22×C2316yescomplemented
234#397C22×C2316yescomplemented
235#396C22×C2312yes—
236#396C22×C2316yes—
237#396C22×C2316yescomplemented
238#394C22×C2316yes—
239#394C22×C2316yes—
240#394C22×C2316yes—
241#394C22×C2316yescomplemented
242#392C22×C2316yes—
243#392C22×C2313yes—
244#392C22×C2311yescharacteristic, complemented
245#386C22×C2316yes—
246#386C22×C2313yes—
247#385C22×C2313yes—
248#385C22×C2316yes—
249#379C22×C2313yes—
250C4×C4⋊C4C22×C2316yescomplemented
251C4×C22⋊C4C22×C2316yes—
252C4×C22⋊C4C22×C2313yes—
253C4×C22⋊C4C22×C2313yescomplemented
254#376C22×C2311yescharacteristic, complemented
255C2×C23—42no—
256C2×C23—44no—
257C2×C23—83no—
258C2×C23#345913yes—
259C2×C23C2×#56011yescharacteristic
260C2×C23#352913yes—
261C25C23×C2311yescharacteristic
262C22×D4C23×C2316yes—
263C22×D4C23×C2313yes—
264C22×D4C23×C2313yes—
265C23×C4—46no—
266C23×C4C23×C2313yes—
267C23×C4C23×C2312yes—
268C23×C4C23×C2311yescharacteristic
269C23×C4C23×C2311yescharacteristic
270C23×C4C23×C2312yes—
271#120—212no—
272C4×D4—412no—
273C4×D4—46no—
274C4×D4—46no—
275#117—46no—
276C2×C4⋊C4C23×C2313yes—
277C2×C4⋊C4C23×C2313yes—
278C2×C4⋊C4C23×C2316yes—
279C2×C4⋊C4C23×C2313yes—
280C2×C4⋊C4C23×C2313yes—
281C2×C4⋊C4C23×C2316yes—
282C2×C4⋊C4C23×C2316yes—
283C2×C22⋊C4C23×C2316yes—
284C2×C22⋊C4C23×C2313yes—
285C2×C22⋊C4C23×C2316yes—
286C2×C22⋊C4C23×C2313yes—
287C2×C22⋊C4C23×C2316yes—
288C2×C22⋊C4C23×C2313yes—
289C2×C22⋊C4C23×C2316yes—
290C2×C22⋊C4C23×C2316yes—
291C2×C22⋊C4C23×C2316yes—
292C2×C22⋊C4C23×C2313yes—
293C2×C22⋊C4C23×C2316yes—
294C2×C42C23×C2316yes—
295C2×C42C23×C2313yes—
296#95C23×C2313yes—
297#95C23×C2311yescharacteristic
298#95C23×C2316yes—
299#95C23×C2316yes—
300#95C23×C2316yes—
301#95C23×C2313yes—
302#95C23×C2313yes—
303#95C23×C2316yes—
304#95C23×C2316yes—
305C23#3667611yescharacteristic, direct product factor, Sylow
306C24—43no—
307C24—43no—
308C24#41×C2314yes—
309C24C24×C2313yes—
310C24C24×C2311yescharacteristic
311C24C24×C2312yes—
312C2×D4—212no—
313C2×D4—26no—
314C2×D4—43no—
315C2×D4—43no—
316C2×D4—46no—
317C2×D4—46no—
318C2×D4—46no—
319C2×D4—43no—
320C22×C4—26no—
321C22×C4—212no—
322C22×C4—212no—
323C22×C4—26no—
324C22×C4—43no—
325C22×C4—46no—
326C22×C4—416no—
327C22×C4C24×C2313yes—
328C22×C4C24×C2316yes—
329C22×C4C24×C2313yes—
330C22×C4C24×C2312yes—
331C22×C4C24×C2311yescharacteristic
332C22×C4C24×C2316yes—
333C22×C4C24×C2313yes—
334C22×C4C24×C2311yescharacteristic
335C4⋊C4—43no—
336C4⋊C4—43no—
337C4⋊C4—43no—
338C4⋊C4—43no—
339C4⋊C4—46no—
340C4⋊C4—46no—
341C4⋊C4—46no—
342C22⋊C4—212no—
343C22⋊C4—26no—
344C22⋊C4—46no—
345C22⋊C4—46no—
346C22⋊C4—46no—
347C22⋊C4—46no—
348C22⋊C4—43no—
349C22⋊C4—43no—
350C22⋊C4—43no—
351C22⋊C4—46no—
352C22⋊C4—46no—
353C42—212no—
354C42—26no—
355C23—26no—
356C23—23no—
357C23—23no—
358C23—26no—
359C23—23no—
360C23—23no—
361C23—26no—
362C23—26no—
363C23—23no—
364C23—412no—
365C23—44no—
366C23—412no—
367C23C2×#41×C2312yes—
368C23C2×#41×C2314yes—
369C23C25×C2311yescharacteristic
370D4—412no—
371D4—424no—
372D4—412no—
373C2×C4—23no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—23no—
377C2×C4—23no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—23no—
381C2×C4—26no—
382C2×C4—23no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—26no—
387C2×C4—23no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—412no—
393C2×C4C23×H(5,𝔽2)16yes—
394C22—26no—
395C22—212no—
396C22—416no—
397C22—42no—
398C22—41no—
399C22—46no—
400C22—43no—
401C22—43no—
402C22—46no—
403C22—43no—
404C22—46no—
405C22—43no—
406C22C22×#41×C2311yescharacteristic
407C22C2×C23×H(5,𝔽2)13yes—
408C22C23×#58513yes—
409C4—43no—
410C4—46no—
411C4—42no—
412C4—41no—
413C4—43no—
414C4—43no—
415C4—46no—
416C4C23×#56014yes—
417C2—42no—
418C2—44no—
419C2—83no—
420C2C23×#345913yes—
421C2C2×C23×#56011yescharacteristic
422C2C23×#352913yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2842C2semidirect
2. C23×#2834C2semidirect
3. C23×#2827C2semidirect
4. C23×#2826C2semidirect
5. C23×#2825C2semidirect
6. C23×#2812C2semidirect
7. C23×#2401C2semidirect
8. C23×#2397C2semidirect
9. C23×#2395C2semidirect
10. C23×#2392C2semidirect
11. C4×C23×#120C2semidirect
16. C2×C4×D4×C23C22semidirect
17. C2×C23×#117C22semidirect
18. C23×#401C22semidirect
19. C23×#400C22semidirect
20. C23×#400C22semidirect
23. C23×#399C22semidirect
24. C23×#398C22semidirect
25. C23×#398C22semidirect
28. C23×#397C22semidirect
29. C23×#397C22semidirect
32. C23×#396C22semidirect
36. C23×#394C22semidirect
39. C23×#392C22semidirect
45. C4×C4⋊C4×C23C22semidirect
48. C4×C22⋊C4×C23C22semidirect
49. C23×#376C22semidirect
143. #36676C23direct (N × H)
183. #2842C2×C23semidirect
184. #2834C2×C23semidirect
185. #2827C2×C23semidirect
186. #2826C2×C23semidirect
187. #2825C2×C23semidirect
188. #2812C2×C23semidirect
189. #2401C2×C23semidirect
190. #2397C2×C23semidirect
191. #2395C2×C23semidirect
192. #2392C2×C23semidirect
193. C4×#120C2×C23semidirect
221. C2×C4×D4C22×C23semidirect
222. C2×#117C22×C23semidirect
223. #401C22×C23semidirect
224. #400C22×C23semidirect
225. #400C22×C23semidirect
228. #399C22×C23semidirect
229. #398C22×C23semidirect
230. #398C22×C23semidirect
233. #397C22×C23semidirect
234. #397C22×C23semidirect
237. #396C22×C23semidirect
241. #394C22×C23semidirect
244. #392C22×C23semidirect
250. C4×C4⋊C4C22×C23semidirect
253. C4×C22⋊C4C22×C23semidirect
254. #376C22×C23semidirect
305. C23#36676direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28422
C23×#28342
C23×#28272
C23×#28262
C23×#28252
C23×#28122
C23×#24012
C23×#23972
C23×#23952
C23×#23922
C4×C23×#1202
#3667623