Finite Group Databasebeta

← C23×#20846 #19168472

Subgroups — C23×#20846 #19168472

Order 5888.

1630 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#117C211yesmaximal, characteristic
2C2×C23×#428C212yesmaximal, complemented
3C2×C23×#407C211yesmaximal, characteristic
4C2×C23×#380C211yesmaximal, characteristic
5C23×#1854C214yesmaximal, complemented
6C23×#1853C212yesmaximal, complemented
7C23×#1852C212yesmaximal, complemented
8C23×#1736C212yesmaximal, complemented
9C24×C4×C23C2211yescharacteristic
10C22×M16×C23C2212yes—
11C2×C23×#117—21no—
12C2×C23×#117—21no—
13C2×C23×#117—28no—
14C2×C23×#117C2212yes—
15C2×C23×#117C2212yes—
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C4⋊C4×C23C2211yescharacteristic
18C22×C4⋊C4×C23C2211yescharacteristic
19C22×C4⋊C4×C23C2211yescharacteristic
20C22×C22⋊C4×C23—21no—
21C22×C22⋊C4×C23—21no—
22C22×C42×C23—21no—
23C23×#428—28no—
24C2×C23×#107C2214yescomplemented
25C2×C23×#106C2214yescomplemented
26C23×#407C418yescomplemented
27C2×C23×C22⋊C8C2212yes—
28C2×C23×C22⋊C8C2212yes—
29C23×#380—21no—
30C23×#380—21no—
31C23×#380C2212yes—
32C23×#380C2212yes—
33C2×C23×#95—21no—
34C23×#340C2214yescomplemented
35C23×#340C2214yescomplemented
36C25×C23D411yescharacteristic
37C23×C4×C23—24no—
38C23×C4×C23—24no—
39C23×C4×C23—21no—
40C23×C4×C23—21no—
41C23×C4×C23—21no—
42C23×C4×C23—21no—
43C23×C4×C23C2×C414yes—
44C23×C4×C23D411yescharacteristic
45C23×C4×C23D411yescharacteristic
46C23×C4×C23D411yescharacteristic
47C23×C4×C23Q814yes—
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2311yescharacteristic
50C23×C4×C23C2311yescharacteristic
51C2×M16×C23—24no—
52C2×M16×C23—24no—
53C2×M16×C23C2×C418yescomplemented
54C22×C8×C23C2314yes—
55C23×#117—28no—
56C23×#117—28no—
57C23×#117—24no—
58C23×#117—24no—
59C23×#117—24no—
60C23×#117D418yes—
61C2×C4⋊C4×C23—21no—
62C2×C4⋊C4×C23—21no—
63C2×C4⋊C4×C23—24no—
64C2×C4⋊C4×C23—24no—
65C2×C4⋊C4×C23—24no—
66C2×C4⋊C4×C23—24no—
67C2×C4⋊C4×C23—21no—
68C2×C4⋊C4×C23—21no—
69C2×C4⋊C4×C23C2312yes—
70C2×C4⋊C4×C23C2312yes—
71C2×C4⋊C4×C23C2312yes—
72C2×C4⋊C4×C23C2312yes—
73C2×C22⋊C4×C23—28no—
74C2×C22⋊C4×C23—22no—
75C2×C22⋊C4×C23—22no—
76C2×C22⋊C4×C23—22no—
77C2×C22⋊C4×C23—22no—
78C2×C22⋊C4×C23—44no—
79C2×C42×C23—28no—
80C2×C42×C23—22no—
81C2×C42×C23—22no—
82C23×#107—28no—
83C23×#106—28no—
84C23×C22⋊C8C2×C418yescomplemented
85C23×C22⋊C8C2×C418yescomplemented
86C23×#95—22no—
87C23×#95—22no—
88C24×C23—22no—
89C24×C23—22no—
90C24×C23—22no—
91C24×C23—22no—
92C24×C23—41no—
93C24×C23—41no—
94C24×C23C4⋊C414yes—
95C24×C23C2×D411yescharacteristic
96C24×C23C2×D411yescharacteristic
97C24×C23C2×D411yescharacteristic
98C22×C4×C23—22no—
99C22×C4×C23—22no—
100C22×C4×C23—216no—
101C22×C4×C23—22no—
102C22×C4×C23—22no—
103C22×C4×C23—216no—
104C22×C4×C23—22no—
105C22×C4×C23—22no—
106C22×C4×C23—22no—
107C22×C4×C23—22no—
108C22×C4×C23—22no—
109C22×C4×C23—22no—
110C22×C4×C23—24no—
111C22×C4×C23—24no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—42no—
117C22×C4×C23—42no—
118C22×C4×C23—42no—
119C22×C4×C23—42no—
120C22×C4×C23—42no—
121C22×C4×C23—42no—
122C22×C4×C23—42no—
123C22×C4×C23—42no—
124C22×C4×C23—42no—
125C22×C4×C23—42no—
126C22×C4×C23C4⋊C414yes—
127C22×C4×C23C4⋊C414yes—
128C22×C4×C23C4⋊C414yes—
129C22×C4×C23C22×C412yes—
130C22×C4×C23C22×C412yes—
131C22×C4×C23C22×C412yes—
132C22×C4×C23C2×D411yescharacteristic
133C22×C4×C23C2×D411yescharacteristic
134C22×C4×C23C2×D411yescharacteristic
135C22×C4×C23C2×D414yes—
136C22×C4×C23C2×Q812yes—
137C22×C4×C23C2×Q812yes—
138C22×C4×C23C2×Q812yes—
139C22×C4×C23#4114yes—
140C22×C4×C23C2411yescharacteristic
141M16×C23—216no—
142C2×C8×C23—24no—
143C2×C8×C23—24no—
144C2×C8×C23C22×C418yes—
145C4⋊C4×C23—24no—
146C4⋊C4×C23—24no—
147C4⋊C4×C23—22no—
148C4⋊C4×C23—22no—
149C4⋊C4×C23—24no—
150C4⋊C4×C23—24no—
151C4⋊C4×C23—42no—
152C4⋊C4×C23—42no—
153C4⋊C4×C23C2×D414yes—
154C4⋊C4×C23C2×D414yes—
155C22⋊C4×C23—28no—
156C22⋊C4×C23—28no—
157C22⋊C4×C23—44no—
158C22⋊C4×C23—44no—
159C42×C23—28no—
160C42×C23—28no—
161#20846C2311yesmaximal, characteristic, direct product factor, Sylow
162C23×C23—21no—
163C23×C23—21no—
164C23×C23—21no—
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—21no—
169C23×C23—21no—
170C23×C23—21no—
171C23×C23—21no—
172C23×C23—21no—
173C23×C23—21no—
174C23×C23—21no—
175C23×C23—21no—
176C23×C23—28no—
177C23×C23—28no—
178C23×C23—44no—
179C23×C23—44no—
180C23×C23—44no—
181C23×C23—44no—
182C23×C23—44no—
183C23×C23C2×C4⋊C412yes—
184C23×C23C2×C4⋊C412yes—
185C23×C23C2×C4⋊C412yes—
186C23×C23#12012yes—
187C23×C23#12312yes—
188C23×C23AGL(1,ℤ/8)12yes—
189C23×C23#13712yes—
190C23×C23C22×D411yescharacteristic
191C2×C4×C23—21no—
192C2×C4×C23—21no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—28no—
196C2×C4×C23—24no—
197C2×C4×C23—24no—
198C2×C4×C23—24no—
199C2×C4×C23—24no—
200C2×C4×C23—28no—
201C2×C4×C23—28no—
202C2×C4×C23—24no—
203C2×C4×C23—24no—
204C2×C4×C23—48no—
205C2×C4×C23—48no—
206C2×C4×C23—42no—
207C2×C4×C23—42no—
208C2×C4×C23—42no—
209C2×C4×C23—42no—
210C2×C4×C23—42no—
211C2×C4×C23—42no—
212C2×C4×C23—42no—
213C2×C4×C23—42no—
214C2×C4×C23—42no—
215C2×C4×C23—42no—
216C2×C4×C23—84no—
217C2×C4×C23—84no—
218C2×C4×C23C2×C4⋊C412yes—
219C2×C4×C23C2×C4⋊C412yes—
220C2×C4×C23C2×C4⋊C412yes—
221C2×C4×C23C4×D418yes—
222C2×C4×C23#12012yes—
223C2×C4×C23C22⋊Q818yes—
224C2×C4×C23#12312yes—
225C2×C4×C23C23×C411yescharacteristic
226C2×C4×C23C22×D412yes—
227C2×C4×C23C22×Q811yescharacteristic
228C2×C4×C23C2×#4112yes—
229C8×C23—28no—
230C22×#117C2×C2311yescharacteristic
231C2×#428C2×C2312yescomplemented
232C2×#407C2×C2311yescharacteristic
233C2×#380C2×C2311yescharacteristic
234#1854C2×C2314yescomplemented
235#1853C2×C2312yescomplemented
236#1852C2×C2312yescomplemented
237#1736C2×C2312yescomplemented
238C22×C23—22no—
239C22×C23—22no—
240C22×C23—22no—
241C22×C23—22no—
242C22×C23—22no—
243C22×C23—22no—
244C22×C23—22no—
245C22×C23—22no—
246C22×C23—22no—
247C22×C23—22no—
248C22×C23—22no—
249C22×C23—22no—
250C22×C23—22no—
251C22×C23—22no—
252C22×C23—22no—
253C22×C23—416no—
254C22×C23—41no—
255C22×C23—41no—
256C22×C23—41no—
257C22×C23—41no—
258C22×C23#38514yes—
259C22×C23#42814yescomplemented
260C22×C23C22×C4⋊C411yescharacteristic
261C22×C23C2×#12011yescharacteristic
262C22×C23C2×#12311yescharacteristic
263C22×C23C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C23C2×#13711yescharacteristic
265C22×C23#57511yescharacteristic
266C22×C23#57511yescharacteristic
267C4×C23—22no—
268C4×C23—22no—
269C4×C23—44no—
270C4×C23—44no—
271C4×C23—82no—
272C4×C23—82no—
273C4×C23#38514yes—
274C4×C23C2×C4×D412yes—
275C4×C23C2×C22⋊Q812yes—
276C24×C4C22×C2311yescharacteristic
277C22×M16C22×C2312yes—
278C2×#117—21no—
279C2×#117—21no—
280C2×#117—28no—
281C2×#117C22×C2312yes—
282C2×#117C22×C2312yes—
283C22×C4⋊C4C22×C2311yescharacteristic
284C22×C4⋊C4C22×C2311yescharacteristic
285C22×C4⋊C4C22×C2311yescharacteristic
286C22×C4⋊C4C22×C2311yescharacteristic
287C22×C22⋊C4—21no—
288C22×C22⋊C4—21no—
289C22×C42—21no—
290#428—28no—
291C2×#107C22×C2314yescomplemented
292C2×#106C22×C2314yescomplemented
293#407C4×C2318yescomplemented
294C2×C22⋊C8C22×C2312yes—
295C2×C22⋊C8C22×C2312yes—
296#380—21no—
297#380—21no—
298#380C22×C2312yes—
299#380C22×C2312yes—
300C2×#95—21no—
301#340C22×C2314yescomplemented
302#340C22×C2314yescomplemented
303C2×C23—24no—
304C2×C23—24no—
305C2×C23—42no—
306C2×C23C2×#38511yescharacteristic
307C2×C23C2×#42811yescharacteristic
308C2×C23#291111yescharacteristic
309C2×C23#300611yescharacteristic
310C2×C23#300711yescharacteristic
311C2×C23#309411yescharacteristic
312C2×C23#309511yescharacteristic
313C25D4×C2311yescharacteristic
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—21no—
319C23×C4—21no—
320C23×C4C2×C4×C2314yes—
321C23×C4D4×C2311yescharacteristic
322C23×C4D4×C2311yescharacteristic
323C23×C4D4×C2311yescharacteristic
324C23×C4Q8×C2314yes—
325C23×C4C23×C2311yescharacteristic
326C23×C4C23×C2311yescharacteristic
327C23×C4C23×C2311yescharacteristic
328C2×M16—24no—
329C2×M16—24no—
330C2×M16C2×C4×C2318yescomplemented
331C22×C8C23×C2314yes—
332#117—28no—
333#117—28no—
334#117—24no—
335#117—24no—
336#117—24no—
337#117D4×C2318yes—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C23×C2312yes—
347C2×C4⋊C4C23×C2312yes—
348C2×C4⋊C4C23×C2312yes—
349C2×C4⋊C4C23×C2312yes—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—44no—
356C2×C42—28no—
357C2×C42—22no—
358C2×C42—22no—
359#107—28no—
360#106—28no—
361C22⋊C8C2×C4×C2318yescomplemented
362C22⋊C8C2×C4×C2318yescomplemented
363#95—22no—
364#95—22no—
365C23#2084611yescharacteristic, direct product factor, Sylow
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—41no—
371C24—41no—
372C24C4⋊C4×C2314yes—
373C24C2×D4×C2311yescharacteristic
374C24C2×D4×C2311yescharacteristic
375C24C2×D4×C2311yescharacteristic
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C4⋊C4×C2314yes—
405C22×C4C4⋊C4×C2314yes—
406C22×C4C4⋊C4×C2314yes—
407C22×C4C22×C4×C2312yes—
408C22×C4C22×C4×C2312yes—
409C22×C4C22×C4×C2312yes—
410C22×C4C2×D4×C2311yescharacteristic
411C22×C4C2×D4×C2311yescharacteristic
412C22×C4C2×D4×C2311yescharacteristic
413C22×C4C2×D4×C2314yes—
414C22×C4C2×Q8×C2312yes—
415C22×C4C2×Q8×C2312yes—
416C22×C4C2×Q8×C2312yes—
417C22×C4#41×C2314yes—
418C22×C4C24×C2311yescharacteristic
419M16—216no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8C22×C4×C2318yes—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4C2×D4×C2314yes—
432C4⋊C4C2×D4×C2314yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C42—28no—
438C42—28no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—28no—
452C23—28no—
453C23—21no—
454C23—21no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23C2×C4⋊C4×C2312yes—
461C23C2×C4⋊C4×C2312yes—
462C23C2×C4⋊C4×C2312yes—
463C23C23×#12012yes—
464C23C23×#12312yes—
465C23C23×AGL(1,ℤ/8)12yes—
466C23C23×#13712yes—
467C23C22×D4×C2311yescharacteristic
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4C2×C4⋊C4×C2312yes—
496C2×C4C2×C4⋊C4×C2312yes—
497C2×C4C2×C4⋊C4×C2312yes—
498C2×C4C4×D4×C2318yes—
499C2×C4C23×#12012yes—
500C2×C4C23×C22⋊Q818yes—
501C2×C4C23×#12312yes—
502C2×C4C23×C4×C2311yescharacteristic
503C2×C4C22×D4×C2312yes—
504C2×C4C22×Q8×C2311yescharacteristic
505C2×C4C2×#41×C2312yes—
506C8—28no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—416no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22C23×#38514yes—
528C22C23×#42814yescomplemented
529C22C22×C4⋊C4×C2311yescharacteristic
530C22C2×C23×#12011yescharacteristic
531C22C2×C23×#12311yescharacteristic
532C22C2×C23×AGL(1,ℤ/8)11yescharacteristic
533C22C2×C23×#13711yescharacteristic
534C22C23×#57511yescharacteristic
535C22C23×#57511yescharacteristic
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4C23×#38514yes—
543C4C2×C4×D4×C2312yes—
544C4C2×C23×C22⋊Q812yes—
545C2—24no—
546C2—24no—
547C2—42no—
548C2C2×C23×#38511yescharacteristic
549C2C2×C23×#42811yescharacteristic
550C2C23×#291111yescharacteristic
551C2C23×#300611yescharacteristic
552C2C23×#300711yescharacteristic
553C2C23×#309411yescharacteristic
554C2C23×#309511yescharacteristic
subgroupG/H
center, socleC23×C23C22×D4
commutatorC2×C4C23×C4×C23
Frattini, cosocleC22×C4C24×C23
prime radicalC25×C23D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#428C2semidirect
5. C23×#1854C2semidirect
6. C23×#1853C2semidirect
7. C23×#1852C2semidirect
8. C23×#1736C2semidirect
24. C2×C23×#107C22semidirect
25. C2×C23×#106C22semidirect
26. C23×#407C4semidirect
34. C23×#340C22semidirect
35. C23×#340C22semidirect
53. C2×M16×C23C2×C4semidirect
84. C23×C22⋊C8C2×C4semidirect
85. C23×C22⋊C8C2×C4semidirect
161. #20846C23direct (N × H)
231. C2×#428C2×C23semidirect
234. #1854C2×C23semidirect
235. #1853C2×C23semidirect
236. #1852C2×C23semidirect
237. #1736C2×C23semidirect
259. C22×C23#428semidirect
291. C2×#107C22×C23semidirect
292. C2×#106C22×C23semidirect
293. #407C4×C23semidirect
301. #340C22×C23semidirect
302. #340C22×C23semidirect
330. C2×M16C2×C4×C23semidirect
361. C22⋊C8C2×C4×C23semidirect
362. C22⋊C8C2×C4×C23semidirect
365. C23#20846direct (N × H)
528. C22C23×#428semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1172
C2×C23×#4282
C2×C23×#4072
C2×C23×#3802
C23×#18542
C23×#18532
C23×#18522
C23×#17362
#2084623