Finite Group Databasebeta

← C23×#9922 #19157548

Subgroups — C23×#9922 #19157548

Order 5888.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#450C211yesmaximal, characteristic, complemented
2C23×#2626C211yesmaximal, characteristic, complemented
3C2×C23×#392C211yesmaximal, characteristic
4C23×#1786C211yesmaximal, characteristic, complemented
5C2×C23×#328C211yesmaximal, characteristic, complemented
6C2×C23×#323C211yesmaximal, characteristic, complemented
7C23×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C23—21no—
9C23×D4×C23C2211yescharacteristic
10C22×SD16×C23—21no—
11C2×C23×C22⋊Q8C2211yescharacteristic
12C22×C22⋊C4×C23—21no—
13C22×C22⋊C4×C23C2211yescharacteristic
14C23×#450—22no—
15C23×#450—22no—
16C2×C23×#102—21no—
17C2×C23×#102—22no—
18C2×C23×C22⋊C8C2211yescharacteristic
19C2×C23×C22⋊C8C2211yescharacteristic
20C23×#399—21no—
21C23×#396—22no—
22C23×#393—21no—
23C23×#392—22no—
24C23×#392—22no—
25C23×#385—21no—
26C23×#380C2211yescharacteristic
27C23×#379—21no—
28C2×C23×#95C2211yescharacteristic
29C23×#328—22no—
30C23×#323—22no—
31C25×C23—21no—
32C25×C23D411yescharacteristic
33C22×Q8×C23D411yescharacteristic, complemented
34C22×D4×C23—24no—
35C22×D4×C23—21no—
36C22×D4×C23—21no—
37C22×D4×C23—21no—
38C22×D4×C23—21no—
39C22×D4×C23—21no—
40C22×D4×C23—44no—
41C22×D4×C23—44no—
42C22×D4×C23—44no—
43C22×D4×C23—42no—
44C22×D4×C23—42no—
45C22×D4×C23D412yes—
46C23×C4×C23—21no—
47C23×C4×C23D411yescharacteristic
48C23×C4×C23C2311yescharacteristic
49C2×SD16×C23—24no—
50C2×SD16×C23—44no—
51C22×C8×C23—21no—
52C22×C8×C23—21no—
53C23×C22⋊Q8—21no—
54C23×C22⋊Q8—21no—
55C23×C22⋊Q8—21no—
56C23×C22⋊Q8—21no—
57C2×C4⋊C4×C23—21no—
58C2×C4⋊C4×C23—22no—
59C2×C4⋊C4×C23D411yescharacteristic, complemented
60C2×C4⋊C4×C23D412yescomplemented
61C2×C22⋊C4×C23—24no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—21no—
65C2×C22⋊C4×C23—21no—
66C2×C22⋊C4×C23—21no—
67C2×C22⋊C4×C23—21no—
68C2×C22⋊C4×C23—21no—
69C2×C22⋊C4×C23—21no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—44no—
73C23×#102—24no—
74C23×#102—44no—
75C23×C22⋊C8—21no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×C22⋊C8D412yescomplemented
79C23×#95—22no—
80C23×#95—21no—
81C23×#95—21no—
82C23×#95D412yes—
83C24×C23—24no—
84C24×C23—21no—
85C24×C23—22no—
86C24×C23—21no—
87C24×C23—21no—
88C24×C23—21no—
89C24×C23—21no—
90C24×C23—21no—
91C24×C23—42no—
92C24×C23—42no—
93C24×C23—42no—
94C24×C23—42no—
95C24×C23—42no—
96C24×C23—41no—
97C24×C23—41no—
98C24×C23—41no—
99C24×C23—41no—
100C24×C23—41no—
101C24×C23—41no—
102C24×C23SD1612yescomplemented
103C24×C23C2×D411yescharacteristic
104C2×Q8×C23—21no—
105C2×Q8×C23—21no—
106C2×Q8×C23—42no—
107C2×D4×C23—24no—
108C2×D4×C23—24no—
109C2×D4×C23—22no—
110C2×D4×C23—22no—
111C2×D4×C23—42no—
112C2×D4×C23—42no—
113C2×D4×C23—48no—
114C2×D4×C23—44no—
115C2×D4×C23—44no—
116C2×D4×C23—44no—
117C2×D4×C23—44no—
118C2×D4×C23—48no—
119C2×D4×C23—48no—
120C2×D4×C23—416no—
121C2×D4×C23—416no—
122C2×D4×C23—41no—
123C2×D4×C23—41no—
124C22×C4×C23—24no—
125C22×C4×C23—22no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—22no—
129C22×C4×C23—21no—
130C22×C4×C23—21no—
131C22×C4×C23—21no—
132C22×C4×C23—42no—
133C22×C4×C23—42no—
134C22×C4×C23—41no—
135C22×C4×C23—41no—
136C22×C4×C23—42no—
137C22×C4×C23C2×D411yescharacteristic
138C22×C4×C23C2×D411yescharacteristic
139C22×C4×C23C2×D411yescharacteristic
140C22×C4×C23C2×D411yescharacteristic
141C22×C4×C23C2×D411yescharacteristic
142C22×C4×C23#4111yescharacteristic
143SD16×C23—48no—
144C2×C8×C23—22no—
145C2×C8×C23—42no—
146C2×C8×C23—42no—
147C2×C8×C23—41no—
148C4⋊C4×C23—21no—
149C4⋊C4×C23—21no—
150C4⋊C4×C23—42no—
151C4⋊C4×C23—41no—
152C4⋊C4×C23—41no—
153C4⋊C4×C23—82no—
154C22⋊C4×C23—48no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23—41no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23—41no—
159C22⋊C4×C23—41no—
160C22⋊C4×C23—41no—
161C22⋊C4×C23—41no—
162C22⋊C4×C23—41no—
163C22⋊C4×C23—82no—
164C22⋊C4×C23—82no—
165#9922C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—22no—
167C23×C23—22no—
168C23×C23—24no—
169C23×C23—22no—
170C23×C23—22no—
171C23×C23—22no—
172C23×C23—44no—
173C23×C23—42no—
174C23×C23—44no—
175C23×C23—44no—
176C23×C23—44no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—44no—
182C23×C23—44no—
183C23×C23—44no—
184C23×C23—41no—
185C23×C23—42no—
186C23×C23—41no—
187C23×C23—42no—
188C23×C23—41no—
189C23×C23—42no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—44no—
193C23×C23—41no—
194C23×C23—82no—
195C23×C23—82no—
196C23×C23—81no—
197C23×C23—81no—
198C23×C23—82no—
199C23×C23—82no—
200C23×C23—82no—
201C23×C23—82no—
202C23×C23—82no—
203C23×C23—81no—
204C23×C23—81no—
205C23×C23—82no—
206C23×C23—81no—
207C23×C23—81no—
208C23×C23#12011yescharacteristic
209C23×C23#12011yescharacteristic
210C23×C23#12111yescharacteristic
211C23×C23C2×SD1611yescharacteristic
212C23×C23#13511yescharacteristic
213C23×C23AGL(1,ℤ/8)11yescharacteristic
214C23×C23#13711yescharacteristic
215Q8×C23—42no—
216Q8×C23—81no—
217D4×C23—24no—
218D4×C23—24no—
219D4×C23—42no—
220D4×C23—44no—
221D4×C23—42no—
222D4×C23—44no—
223D4×C23—432no—
224C2×C4×C23—22no—
225C2×C4×C23—22no—
226C2×C4×C23—24no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—44no—
232C2×C4×C23—44no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23—81no—
246C2×C4×C23#12011yescharacteristic
247C2×C4×C23#12111yescharacteristic
248C2×C4×C23#12111yescharacteristic
249C2×C4×C23#12711yescharacteristic
250C8×C23—42no—
251C8×C23—81no—
252C2×#450C2×C2311yescharacteristic, complemented
253#2626C2×C2311yescharacteristic, complemented
254C2×#392C2×C2311yescharacteristic
255#1786C2×C2311yescharacteristic, complemented
256C2×#328C2×C2311yescharacteristic, complemented
257C2×#323C2×C2311yescharacteristic, complemented
258#1294C2×C2311yescharacteristic, complemented
259C22×C23—22no—
260C22×C23—22no—
261C22×C23—22no—
262C22×C23—22no—
263C22×C23—21no—
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—21no—
267C22×C23—44no—
268C22×C23—44no—
269C22×C23—44no—
270C22×C23—44no—
271C22×C23—42no—
272C22×C23—44no—
273C22×C23—82no—
274C22×C23—82no—
275C22×C23—82no—
276C22×C23—82no—
277C22×C23—84no—
278C22×C23—82no—
279C22×C23—84no—
280C22×C23—84no—
281C22×C23—82no—
282C22×C23—84no—
283C22×C23—82no—
284C22×C23—81no—
285C22×C23—82no—
286C22×C23—81no—
287C22×C23#39211yescharacteristic
288C22×C23#44811yescharacteristic
289C22×C23#44911yescharacteristic
290C22×C23#45011yescharacteristic
291C22×C23#45211yescharacteristic
292C22×C23#45312yes—
293C22×C23B(4,𝔽2)12yes—
294C22×C23#46511yescharacteristic
295C22×C23#46811yescharacteristic
296C4×C23—21no—
297C4×C23—21no—
298C4×C23—21no—
299C4×C23—21no—
300C4×C23—44no—
301C4×C23—82no—
302C4×C23—81no—
303C4×C23—81no—
304C4×C23—81no—
305C23×D4—21no—
306C23×D4C22×C2311yescharacteristic
307C22×SD16—21no—
308C2×C22⋊Q8C22×C2311yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4C22×C2311yescharacteristic
311#450—22no—
312#450—22no—
313C2×#102—21no—
314C2×#102—22no—
315C2×C22⋊C8C22×C2311yescharacteristic
316C2×C22⋊C8C22×C2311yescharacteristic
317#399—21no—
318#396—22no—
319#393—21no—
320#392—22no—
321#392—22no—
322#385—21no—
323#380C22×C2311yescharacteristic
324#379—21no—
325C2×#95C22×C2311yescharacteristic
326#328—22no—
327#323—22no—
328C2×C23—21no—
329C2×C23—21no—
330C2×C23—21no—
331C2×C23—21no—
332C2×C23—82no—
333C2×C23—84no—
334C2×C23#159612yes—
335C2×C23#160012yes—
336C2×C23#200011yescharacteristic
337C2×C23#200211yescharacteristic
338C2×C23#202111yescharacteristic
339C25—21no—
340C25D4×C2311yescharacteristic
341C22×Q8D4×C2311yescharacteristic, complemented
342C22×D4—24no—
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—44no—
349C22×D4—44no—
350C22×D4—44no—
351C22×D4—42no—
352C22×D4—42no—
353C22×D4D4×C2312yes—
354C23×C4—21no—
355C23×C4D4×C2311yescharacteristic
356C23×C4C23×C2311yescharacteristic
357C2×SD16—24no—
358C2×SD16—44no—
359C22×C8—21no—
360C22×C8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4D4×C2311yescharacteristic, complemented
368C2×C4⋊C4D4×C2312yescomplemented
369C2×C22⋊C4—24no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—44no—
381#102—24no—
382#102—44no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8D4×C2312yescomplemented
387#95—22no—
388#95—21no—
389#95—21no—
390#95D4×C2312yes—
391C23#992211yescharacteristic, direct product factor, Sylow
392C24—24no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24SD16×C2312yescomplemented
412C24C2×D4×C2311yescharacteristic
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—41no—
432C2×D4—41no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4C2×D4×C2311yescharacteristic
447C22×C4C2×D4×C2311yescharacteristic
448C22×C4C2×D4×C2311yescharacteristic
449C22×C4C2×D4×C2311yescharacteristic
450C22×C4C2×D4×C2311yescharacteristic
451C22×C4#41×C2311yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—41no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—82no—
463C22⋊C4—48no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—41no—
502C23—82no—
503C23—81no—
504C23—81no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23C23×#12011yescharacteristic
517C23C23×#12011yescharacteristic
518C23C23×#12111yescharacteristic
519C23C2×SD16×C2311yescharacteristic
520C23C23×#13511yescharacteristic
521C23C23×AGL(1,ℤ/8)11yescharacteristic
522C23C23×#13711yescharacteristic
523Q8—42no—
524Q8—81no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—44no—
531D4—432no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4C23×#12011yescharacteristic
555C2×C4C23×#12111yescharacteristic
556C2×C4C23×#12111yescharacteristic
557C2×C4C23×#12711yescharacteristic
558C8—42no—
559C8—81no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22—84no—
581C22—82no—
582C22—81no—
583C22—82no—
584C22—81no—
585C22—84no—
586C22—82no—
587C22—84no—
588C22C23×#39211yescharacteristic
589C22C23×#44811yescharacteristic
590C22C23×#44911yescharacteristic
591C22C23×#45011yescharacteristic
592C22C23×#45211yescharacteristic
593C22C23×#45312yes—
594C22C23×B(4,𝔽2)12yes—
595C22C23×#46511yescharacteristic
596C22C23×#46811yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—82no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—84no—
612C2C23×#159612yes—
613C2C23×#160012yes—
614C2C23×#200011yescharacteristic
615C2C23×#200211yescharacteristic
616C2C23×#202111yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#392C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#450C2semidirect
2. C23×#2626C2semidirect
4. C23×#1786C2semidirect
5. C2×C23×#328C2semidirect
6. C2×C23×#323C2semidirect
7. C23×#1294C2semidirect
33. C22×Q8×C23D4semidirect
59. C2×C4⋊C4×C23D4semidirect
60. C2×C4⋊C4×C23D4semidirect
78. C23×C22⋊C8D4semidirect
102. C24×C23SD16semidirect
165. #9922C23direct (N × H)
252. C2×#450C2×C23semidirect
253. #2626C2×C23semidirect
255. #1786C2×C23semidirect
256. C2×#328C2×C23semidirect
257. C2×#323C2×C23semidirect
258. #1294C2×C23semidirect
341. C22×Q8D4×C23semidirect
367. C2×C4⋊C4D4×C23semidirect
368. C2×C4⋊C4D4×C23semidirect
386. C22⋊C8D4×C23semidirect
391. C23#9922direct (N × H)
411. C24SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#4502
C23×#26262
C2×C23×#3922
C23×#17862
C2×C23×#3282
C2×C23×#3232
C23×#12942
#992223