Finite Group Databasebeta

← C23×#18173 #19165799

Subgroups — C23×#18173 #19165799

Order 5888.

1242 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3496C211yesmaximal, characteristic
2C23×#3094C212yesmaximal, complemented
3C23×#2395C211yesmaximal, characteristic, complemented
4C23×#1616C212yesmaximal, complemented
5C23×#1613C212yesmaximal, complemented
6C23×#1610C212yesmaximal, complemented
7C23×#1607C212yesmaximal, complemented
8C23×#1507C212yesmaximal, complemented
9C23×#1465C211yesmaximal, characteristic, complemented
10C2×C23×H(5,𝔽2)—21no—
11C23×#547—22no—
12C23×#546—24no—
13D42×C23—22no—
14C23×#537C2212yes—
15C23×#534—21no—
16C23×#534C2212yes—
17C23×#532—21no—
18C2×C23×#121—21no—
19C2×C23×#121C2212yes—
20C2×C4×D4×C23C2211yescharacteristic
21C2×C23×#117C2211yescharacteristic, complemented
22C23×#482—24no—
23C23×#481C2214yescomplemented
24C23×#480C2214yescomplemented
25C23×#428—22no—
26C23×#418—22no—
27C23×#407C2212yescomplemented
28C23×#400C2211yescharacteristic, complemented
29C23×#399C2211yescharacteristic, complemented
30C23×#398C2211yescharacteristic, complemented
31C23×#397—21no—
32C23×#397C2211yescharacteristic, complemented
33C23×#394—21no—
34C4×C4⋊C4×C23—21no—
35C4×C22⋊C4×C23—21no—
36C4×C22⋊C4×C23C2211yescharacteristic, complemented
37C23×#327C2214yescomplemented
38C23×#327C2214yescomplemented
39C23×#324C2212yescomplemented
40C23×#323C2212yescomplemented
41C23×H(5,𝔽2)—44no—
42C23×H(5,𝔽2)—44no—
43C2×#41×C23—24no—
44C2×#41×C23—22no—
45C22×D4×C23—24no—
46C22×D4×C23—22no—
47C22×D4×C23—22no—
48C22×D4×C23—21no—
49C22×D4×C23D411yescharacteristic
50C22×D4×C23D412yes—
51C23×C4×C23D411yescharacteristic
52C23×C4×C23C2311yescharacteristic
53C2×M16×C23—22no—
54C23×C4⋊Q8—21no—
55C23×#127—21no—
56C23×#124—22no—
57C23×#123—24no—
58C23×#123—24no—
59C23×C22⋊Q8—24no—
60C23×#121—22no—
61C23×#121—22no—
62C23×#121—24no—
63C23×#121—22no—
64C23×#121—22no—
65C23×#121—22no—
66C23×#121—22no—
67C23×#121D414yes—
68C23×#121C2314yes—
69C23×#120—24no—
70C23×#120—24no—
71C4×D4×C23—22no—
72C4×D4×C23—22no—
73C4×D4×C23—22no—
74C4×D4×C23—22no—
75C23×#117—22no—
76C23×#117—22no—
77C2×C4⋊C4×C23—21no—
78C2×C4⋊C4×C23—21no—
79C2×C4⋊C4×C23—21no—
80C2×C4⋊C4×C23D412yes—
81C2×C4⋊C4×C23C2311yescharacteristic
82C2×C4⋊C4×C23C2311yescharacteristic
83C2×C4⋊C4×C23C2311yescharacteristic
84C2×C22⋊C4×C23—22no—
85C2×C22⋊C4×C23—22no—
86C2×C22⋊C4×C23—21no—
87C2×C22⋊C4×C23—21no—
88C2×C22⋊C4×C23—21no—
89C2×C22⋊C4×C23—21no—
90C2×C22⋊C4×C23D411yescharacteristic
91C2×C22⋊C4×C23C2311yescharacteristic
92C2×C42×C23—21no—
93C2×C42×C23—21no—
94C2×C42×C23D411yescharacteristic
95C23×#107—24no—
96C23×#106—24no—
97C23×#102—24no—
98C23×#102—24no—
99C23×C22⋊C8—22no—
100C23×C22⋊C8C2314yes—
101C23×#95—21no—
102C23×#95—21no—
103C23×#95—21no—
104C23×#95—21no—
105C23×#95C2311yescharacteristic
106C23×#95C2311yescharacteristic
107C24×C23—22no—
108C24×C23—22no—
109C24×C23C2×D411yescharacteristic
110C24×C23C2×D411yescharacteristic
111#41×C23—44no—
112#41×C23—416no—
113#41×C23—44no—
114C2×Q8×C23—22no—
115C2×D4×C23—22no—
116C2×D4×C23—22no—
117C2×D4×C23—24no—
118C2×D4×C23—22no—
119C2×D4×C23—24no—
120C2×D4×C23—24no—
121C2×D4×C23—22no—
122C2×D4×C23—22no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—44no—
126C2×D4×C23—416no—
127C2×D4×C23—44no—
128C2×D4×C23—44no—
129C2×D4×C23—44no—
130C2×D4×C23—44no—
131C2×D4×C23—82no—
132C2×D4×C23C2×D414yes—
133C22×C4×C23—22no—
134C22×C4×C23—24no—
135C22×C4×C23—22no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—21no—
140C22×C4×C23—21no—
141C22×C4×C23—21no—
142C22×C4×C23—42no—
143C22×C4×C23—42no—
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D412yes—
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2411yescharacteristic
151M16×C23—44no—
152C2×C8×C23—24no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—22no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—42no—
160C4⋊C4×C23—42no—
161C4⋊C4×C23C2×D414yes—
162C22⋊C4×C23—24no—
163C22⋊C4×C23—24no—
164C22⋊C4×C23—22no—
165C22⋊C4×C23—24no—
166C22⋊C4×C23—24no—
167C22⋊C4×C23—22no—
168C22⋊C4×C23—42no—
169C22⋊C4×C23—42no—
170C22⋊C4×C23—42no—
171C22⋊C4×C23—42no—
172C42×C23—21no—
173C42×C23—21no—
174C42×C23—42no—
175C42×C23—41no—
176C42×C23—41no—
177#18173C2311yesmaximal, characteristic, direct product factor, Sylow
178C23×C23—21no—
179C23×C23—21no—
180C23×C23—21no—
181C23×C23—21no—
182C23×C23—24no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—41no—
187C23×C23—44no—
188C23×C23—44no—
189C23×C23—82no—
190C23×C23—82no—
191C23×C23—82no—
192C23×C23—82no—
193C23×C23#12012yes—
194C23×C23AGL(1,ℤ/8)12yes—
195C23×C23C22×D411yescharacteristic
196Q8×C23—44no—
197D4×C23—28no—
198D4×C23—44no—
199D4×C23—44no—
200D4×C23—48no—
201D4×C23—48no—
202D4×C23—44no—
203D4×C23—44no—
204D4×C23—44no—
205D4×C23—82no—
206D4×C23—82no—
207C2×C4×C23—21no—
208C2×C4×C23—21no—
209C2×C4×C23—22no—
210C2×C4×C23—24no—
211C2×C4×C23—22no—
212C2×C4×C23—21no—
213C2×C4×C23—21no—
214C2×C4×C23—24no—
215C2×C4×C23—42no—
216C2×C4×C23—41no—
217C2×C4×C23—48no—
218C2×C4×C23—41no—
219C2×C4×C23—44no—
220C2×C4×C23—44no—
221C2×C4×C23—41no—
222C2×C4×C23—42no—
223C2×C4×C23—41no—
224C2×C4×C23—41no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—41no—
232C2×C4×C23—41no—
233C2×C4×C23—41no—
234C2×C4×C23—81no—
235C2×C4×C23#12011yescharacteristic
236C2×C4×C23#12011yescharacteristic
237C2×C4×C23AGL(1,ℤ/8)12yes—
238C2×C4×C23C22×D412yes—
239C8×C23—44no—
240#3496C2×C2311yescharacteristic
241#3094C2×C2312yescomplemented
242#2395C2×C2311yescharacteristic, complemented
243#1616C2×C2312yescomplemented
244#1613C2×C2312yescomplemented
245#1610C2×C2312yescomplemented
246#1607C2×C2312yescomplemented
247#1507C2×C2312yescomplemented
248#1465C2×C2311yescharacteristic, complemented
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—22no—
252C22×C23—22no—
253C22×C23—22no—
254C22×C23—22no—
255C22×C23—42no—
256C22×C23—41no—
257C22×C23—41no—
258C22×C23—44no—
259C22×C23—44no—
260C22×C23—44no—
261C22×C23—81no—
262C22×C23—81no—
263C22×C23—81no—
264C22×C23—164no—
265C22×C23—164no—
266C22×C23C2×#12011yescharacteristic
267C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
269C4×C23—22no—
270C4×C23—22no—
271C4×C23—21no—
272C4×C23—42no—
273C4×C23—44no—
274C4×C23—41no—
275C4×C23—41no—
276C4×C23—81no—
277C4×C23—81no—
278C4×C23—81no—
279C4×C23#45812yes—
280C2×H(5,𝔽2)—21no—
281#547—22no—
282#546—24no—
283D42—22no—
284#537C22×C2312yes—
285#534—21no—
286#534C22×C2312yes—
287#532—21no—
288C2×#121—21no—
289C2×#121C22×C2312yes—
290C2×C4×D4C22×C2311yescharacteristic
291C2×#117C22×C2311yescharacteristic, complemented
292#482—24no—
293#481C22×C2314yescomplemented
294#480C22×C2314yescomplemented
295#428—22no—
296#418—22no—
297#407C22×C2312yescomplemented
298#400C22×C2311yescharacteristic, complemented
299#399C22×C2311yescharacteristic, complemented
300#398C22×C2311yescharacteristic, complemented
301#397—21no—
302#397C22×C2311yescharacteristic, complemented
303#394—21no—
304C4×C4⋊C4—21no—
305C4×C22⋊C4—21no—
306C4×C22⋊C4C22×C2311yescharacteristic, complemented
307#327C22×C2314yescomplemented
308#327C22×C2314yescomplemented
309#324C22×C2312yescomplemented
310#323C22×C2312yescomplemented
311C2×C23—21no—
312C2×C23—21no—
313C2×C23—42no—
314C2×C23—42no—
315C2×C23—84no—
316C2×C23#300311yescharacteristic
317C2×C23#301811yescharacteristic
318C2×C23C2×#45811yescharacteristic
319H(5,𝔽2)—44no—
320H(5,𝔽2)—44no—
321C2×#41—24no—
322C2×#41—22no—
323C22×D4—24no—
324C22×D4—22no—
325C22×D4—22no—
326C22×D4—21no—
327C22×D4D4×C2311yescharacteristic
328C22×D4D4×C2312yes—
329C23×C4D4×C2311yescharacteristic
330C23×C4C23×C2311yescharacteristic
331C2×M16—22no—
332C4⋊Q8—21no—
333#127—21no—
334#124—22no—
335#123—24no—
336#123—24no—
337C22⋊Q8—24no—
338#121—22no—
339#121—22no—
340#121—24no—
341#121—22no—
342#121—22no—
343#121—22no—
344#121—22no—
345#121D4×C2314yes—
346#121C23×C2314yes—
347#120—24no—
348#120—24no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353#117—22no—
354#117—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4D4×C2312yes—
359C2×C4⋊C4C23×C2311yescharacteristic
360C2×C4⋊C4C23×C2311yescharacteristic
361C2×C4⋊C4C23×C2311yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4D4×C2311yescharacteristic
369C2×C22⋊C4C23×C2311yescharacteristic
370C2×C42—21no—
371C2×C42—21no—
372C2×C42D4×C2311yescharacteristic
373#107—24no—
374#106—24no—
375#102—24no—
376#102—24no—
377C22⋊C8—22no—
378C22⋊C8C23×C2314yes—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95C23×C2311yescharacteristic
384#95C23×C2311yescharacteristic
385C23#1817311yescharacteristic, direct product factor, Sylow
386C24—22no—
387C24—22no—
388C24C2×D4×C2311yescharacteristic
389C24C2×D4×C2311yescharacteristic
390#41—44no—
391#41—416no—
392#41—44no—
393C2×Q8—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—416no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—82no—
411C2×D4C2×D4×C2314yes—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4C2×D4×C2311yescharacteristic
424C22×C4C2×D4×C2312yes—
425C22×C4C2×D4×C2311yescharacteristic
426C22×C4C2×D4×C2311yescharacteristic
427C22×C4C2×D4×C2311yescharacteristic
428C22×C4C2×D4×C2312yes—
429C22×C4C24×C2311yescharacteristic
430M16—44no—
431C2×C8—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4C2×D4×C2314yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—41no—
455C42—41no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—44no—
466C23—44no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23C23×#12012yes—
472C23C23×AGL(1,ℤ/8)12yes—
473C23C22×D4×C2311yescharacteristic
474Q8—44no—
475D4—28no—
476D4—44no—
477D4—44no—
478D4—48no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—82no—
484D4—82no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—48no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—81no—
513C2×C4C23×#12011yescharacteristic
514C2×C4C23×#12011yescharacteristic
515C2×C4C23×AGL(1,ℤ/8)12yes—
516C2×C4C22×D4×C2312yes—
517C8—44no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—164no—
534C22—164no—
535C22C2×C23×#12011yescharacteristic
536C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
537C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
538C4—22no—
539C4—22no—
540C4—21no—
541C4—42no—
542C4—44no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4C23×#45812yes—
549C2—21no—
550C2—21no—
551C2—42no—
552C2—42no—
553C2—84no—
554C2C23×#300311yescharacteristic
555C2C23×#301811yescharacteristic
556C2C2×C23×#45811yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3496C2
LCSG ▷ C22×C4 ▷ 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×#3094C2semidirect
3. C23×#2395C2semidirect
4. C23×#1616C2semidirect
5. C23×#1613C2semidirect
6. C23×#1610C2semidirect
7. C23×#1607C2semidirect
8. C23×#1507C2semidirect
9. C23×#1465C2semidirect
21. C2×C23×#117C22semidirect
23. C23×#481C22semidirect
24. C23×#480C22semidirect
27. C23×#407C22semidirect
28. C23×#400C22semidirect
29. C23×#399C22semidirect
30. C23×#398C22semidirect
32. C23×#397C22semidirect
36. C4×C22⋊C4×C23C22semidirect
37. C23×#327C22semidirect
38. C23×#327C22semidirect
39. C23×#324C22semidirect
40. C23×#323C22semidirect
177. #18173C23direct (N × H)
241. #3094C2×C23semidirect
242. #2395C2×C23semidirect
243. #1616C2×C23semidirect
244. #1613C2×C23semidirect
245. #1610C2×C23semidirect
246. #1607C2×C23semidirect
247. #1507C2×C23semidirect
248. #1465C2×C23semidirect
291. C2×#117C22×C23semidirect
293. #481C22×C23semidirect
294. #480C22×C23semidirect
297. #407C22×C23semidirect
298. #400C22×C23semidirect
299. #399C22×C23semidirect
300. #398C22×C23semidirect
302. #397C22×C23semidirect
306. C4×C22⋊C4C22×C23semidirect
307. #327C22×C23semidirect
308. #327C22×C23semidirect
309. #324C22×C23semidirect
310. #323C22×C23semidirect
385. C23#18173direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34962
C23×#30942
C23×#23952
C23×#16162
C23×#16132
C23×#16102
C23×#16072
C23×#15072
C23×#14652
#1817323