Finite Group Databasebeta

← C23×#10991 #19158617

Subgroups — C23×#10991 #19158617

Order 5888.

3122 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#120C211yesmaximal, characteristic
2C22×C23×#99C211yesmaximal, characteristic, complemented
3C23×#2308C211yesmaximal, characteristic, complemented
4C2×C23×#379C212yesmaximal, complemented
5C2×C23×#342C212yesmaximal, complemented
6C26×C23C2211yescharacteristic
7C23×D4×C23—21no—
8C23×D4×C23C2211yescharacteristic
9C2×C23×#120—24no—
10C2×C23×#120—22no—
11C2×C23×#120—24no—
12C2×C23×#120C414yescomplemented
13C22×C22⋊C4×C23—21no—
14C22×C22⋊C4×C23—24no—
15C22×C22⋊C4×C23C2212yes—
16C22×C22⋊C4×C23C2212yes—
17C22×C22⋊C4×C23C2211yescharacteristic
18C2×C23×#99—22no—
19C2×C23×#99—24no—
20C23×#386—24no—
21C23×#381—21no—
22C23×#379—24no—
23C23×#379—21no—
24C23×#379—24no—
25C23×#342—24no—
26C25×C23—21no—
27C25×C23—22no—
28C25×C23—22no—
29C25×C23—21no—
30C25×C23—22no—
31C25×C23—22no—
32C25×C23—22no—
33C25×C23—48no—
34C25×C23C2×C412yes—
35C25×C23D411yescharacteristic
36C25×C23D414yes—
37C25×C23C2311yescharacteristic
38C22×D4×C23—28no—
39C22×D4×C23—24no—
40C22×D4×C23—22no—
41C22×D4×C23—22no—
42C22×D4×C23—22no—
43C22×D4×C23—22no—
44C22×D4×C23—22no—
45C22×D4×C23—22no—
46C22×D4×C23—48no—
47C22×D4×C23C2×C412yes—
48C22×D4×C23D412yes—
49C23×C4×C23—21no—
50C23×C4×C23—22no—
51C23×C4×C23D411yescharacteristic
52C23×C4×C23D414yescomplemented
53C23×#120—216no—
54C23×#120—28no—
55C23×#120—28no—
56C2×C22⋊C4×C23—28no—
57C2×C22⋊C4×C23—22no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—24no—
60C2×C22⋊C4×C23—22no—
61C2×C22⋊C4×C23—22no—
62C2×C22⋊C4×C23—22no—
63C2×C22⋊C4×C23—22no—
64C2×C22⋊C4×C23—216no—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—22no—
67C2×C22⋊C4×C23—22no—
68C2×C22⋊C4×C23—22no—
69C2×C22⋊C4×C23—22no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—24no—
73C2×C22⋊C4×C23—416no—
74C2×C22⋊C4×C23C2×C412yes—
75C2×C22⋊C4×C23D414yes—
76C2×C22⋊C4×C23D414yescomplemented
77C23×#99—28no—
78C23×#95—24no—
79C24×C23—216no—
80C24×C23—28no—
81C24×C23—22no—
82C24×C23—24no—
83C24×C23—24no—
84C24×C23—24no—
85C24×C23—24no—
86C24×C23—22no—
87C24×C23—24no—
88C24×C23—24no—
89C24×C23—22no—
90C24×C23—21no—
91C24×C23—24no—
92C24×C23—24no—
93C24×C23—22no—
94C24×C23—21no—
95C24×C23—48no—
96C24×C23—416no—
97C24×C23—44no—
98C24×C23—48no—
99C24×C23—44no—
100C24×C23—416no—
101C24×C23—48no—
102C24×C23—416no—
103C24×C23—416no—
104C24×C23—416no—
105C24×C23—44no—
106C24×C23—41no—
107C24×C23—41no—
108C24×C23—42no—
109C24×C23—42no—
110C24×C23—42no—
111C24×C23—42no—
112C24×C23—41no—
113C24×C23—41no—
114C24×C23—42no—
115C24×C23—42no—
116C24×C23C22⋊C414yescomplemented
117C24×C23C22⋊C412yescomplemented
118C24×C23C22×C411yescharacteristic
119C24×C23C2×D411yescharacteristic
120C24×C23C2×D412yes—
121C24×C23C2×D412yes—
122C24×C23C2×D411yescharacteristic
123C24×C23C2×D412yes—
124C24×C23C2×D412yes—
125C24×C23#4112yes—
126C24×C23#4112yes—
127C2×D4×C23—216no—
128C2×D4×C23—22no—
129C2×D4×C23—22no—
130C2×D4×C23—24no—
131C2×D4×C23—24no—
132C2×D4×C23—22no—
133C2×D4×C23—24no—
134C2×D4×C23—24no—
135C2×D4×C23—22no—
136C2×D4×C23—416no—
137C2×D4×C23—432no—
138C2×D4×C23—42no—
139C2×D4×C23—42no—
140C2×D4×C23—42no—
141C2×D4×C23—42no—
142C2×D4×C23—42no—
143C2×D4×C23—42no—
144C2×D4×C23—42no—
145C2×D4×C23—42no—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—24no—
149C22×C4×C23—21no—
150C22×C4×C23—22no—
151C22×C4×C23—21no—
152C22×C4×C23—24no—
153C22×C4×C23—24no—
154C22×C4×C23—24no—
155C22×C4×C23—24no—
156C22×C4×C23—44no—
157C22×C4×C23—48no—
158C22×C4×C23—48no—
159C22×C4×C23—42no—
160C22×C4×C23—42no—
161C22×C4×C23C22⋊C412yescomplemented
162C22⋊C4×C23—216no—
163C22⋊C4×C23—22no—
164C22⋊C4×C23—24no—
165C22⋊C4×C23—24no—
166C22⋊C4×C23—24no—
167C22⋊C4×C23—432no—
168C22⋊C4×C23—42no—
169C22⋊C4×C23—42no—
170C22⋊C4×C23—42no—
171C22⋊C4×C23—42no—
172C22⋊C4×C23—42no—
173C22⋊C4×C23—42no—
174C22⋊C4×C23—84no—
175C22⋊C4×C23C22⋊C414yescomplemented
176#10991C2311yesmaximal, characteristic, direct product factor, Sylow
177C23×C23—28no—
178C23×C23—24no—
179C23×C23—216no—
180C23×C23—28no—
181C23×C23—216no—
182C23×C23—216no—
183C23×C23—216no—
184C23×C23—24no—
185C23×C23—28no—
186C23×C23—22no—
187C23×C23—22no—
188C23×C23—22no—
189C23×C23—48no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—44no—
193C23×C23—44no—
194C23×C23—44no—
195C23×C23—44no—
196C23×C23—44no—
197C23×C23—44no—
198C23×C23—416no—
199C23×C23—48no—
200C23×C23—48no—
201C23×C23—48no—
202C23×C23—48no—
203C23×C23—416no—
204C23×C23—416no—
205C23×C23—48no—
206C23×C23—416no—
207C23×C23—48no—
208C23×C23—416no—
209C23×C23—416no—
210C23×C23—416no—
211C23×C23—416no—
212C23×C23—44no—
213C23×C23—48no—
214C23×C23—48no—
215C23×C23—416no—
216C23×C23—48no—
217C23×C23—416no—
218C23×C23—42no—
219C23×C23—42no—
220C23×C23—44no—
221C23×C23—44no—
222C23×C23—44no—
223C23×C23—44no—
224C23×C23—44no—
225C23×C23—48no—
226C23×C23—42no—
227C23×C23—42no—
228C23×C23—44no—
229C23×C23—82no—
230C23×C23—84no—
231C23×C23C2×C22⋊C411yescharacteristic
232C23×C23C2×C22⋊C411yescharacteristic
233C23×C23C2×C22⋊C411yescharacteristic
234C23×C23C4×D412yes—
235C23×C23C4×D412yes—
236C23×C23#12012yes—
237C23×C23#12012yes—
238C23×C23#12012yes—
239C23×C23#12011yescharacteristic
240C23×C23#12011yescharacteristic
241C23×C23#12112yes—
242C23×C23#12112yes—
243C23×C23#12112yes—
244C23×C23#12112yes—
245C23×C23#12112yes—
246C23×C23#12112yes—
247C23×C23#12311yescharacteristic
248C23×C23#12311yescharacteristic
249C23×C23#12312yes—
250C23×C23#12412yes—
251C23×C23#12712yes—
252D4×C23—432no—
253D4×C23—44no—
254D4×C23—44no—
255D4×C23—44no—
256D4×C23—44no—
257C2×C4×C23—22no—
258C2×C4×C23—24no—
259C2×C4×C23—22no—
260C2×C4×C23—22no—
261C2×C4×C23—44no—
262C2×C4×C23—48no—
263C2×C4×C23—42no—
264C2×C4×C23—42no—
265C2×C4×C23—48no—
266C2×C4×C23—48no—
267C2×C4×C23—44no—
268C2×C4×C23—44no—
269C2×C4×C23—48no—
270C2×C4×C23—84no—
271C2×C4×C23—84no—
272C2×C4×C23—84no—
273C22×#120C2×C2311yescharacteristic
274C22×#99C2×C2311yescharacteristic, complemented
275#2308C2×C2311yescharacteristic, complemented
276C2×#379C2×C2312yescomplemented
277C2×#342C2×C2312yescomplemented
278C22×C23—22no—
279C22×C23—24no—
280C22×C23—24no—
281C22×C23—22no—
282C22×C23—24no—
283C22×C23—24no—
284C22×C23—24no—
285C22×C23—24no—
286C22×C23—24no—
287C22×C23—24no—
288C22×C23—28no—
289C22×C23—216no—
290C22×C23—41no—
291C22×C23—41no—
292C22×C23—42no—
293C22×C23—42no—
294C22×C23—41no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—42no—
298C22×C23—41no—
299C22×C23—42no—
300C22×C23—44no—
301C22×C23—416no—
302C22×C23—48no—
303C22×C23—48no—
304C22×C23—416no—
305C22×C23—416no—
306C22×C23—416no—
307C22×C23—44no—
308C22×C23—48no—
309C22×C23—44no—
310C22×C23—416no—
311C22×C23—44no—
312C22×C23—42no—
313C22×C23—82no—
314C22×C23—84no—
315C22×C23—84no—
316C22×C23#37911yescharacteristic
317C22×C23#38611yescharacteristic
318C22×C23#38612yes—
319C22×C23#38611yescharacteristic
320C22×C23#39012yes—
321C22×C23#39212yes—
322C22×C23#39212yes—
323C22×C23#39412yes—
324C22×C23#39412yes—
325C22×C23B(4,𝔽2)14yes—
326C4×C23—44no—
327C4×C23—42no—
328C4×C23—84no—
329C4×C23—84no—
330C26C22×C2311yescharacteristic
331C23×D4—21no—
332C23×D4C22×C2311yescharacteristic
333C2×#120—24no—
334C2×#120—22no—
335C2×#120—24no—
336C2×#120C4×C2314yescomplemented
337C22×C22⋊C4—21no—
338C22×C22⋊C4—24no—
339C22×C22⋊C4C22×C2312yes—
340C22×C22⋊C4C22×C2312yes—
341C22×C22⋊C4C22×C2311yescharacteristic
342C2×#99—22no—
343C2×#99—24no—
344#386—24no—
345#381—21no—
346#379—21no—
347#379—24no—
348#379—24no—
349#342—24no—
350C2×C23—21no—
351C2×C23—22no—
352C2×C23—22no—
353C2×C23—21no—
354C2×C23—22no—
355C2×C23—22no—
356C2×C23—22no—
357C2×C23—48no—
358C2×C23—82no—
359C2×C23#143811yescharacteristic
360C2×C23#188912yes—
361C2×C23#202114yes—
362C25—22no—
363C25—21no—
364C25—21no—
365C25—22no—
366C25—22no—
367C25—22no—
368C25—22no—
369C25—48no—
370C25C2×C4×C2312yes—
371C25D4×C2314yes—
372C25D4×C2311yescharacteristic
373C25C23×C2311yescharacteristic
374C22×D4—28no—
375C22×D4—24no—
376C22×D4—22no—
377C22×D4—22no—
378C22×D4—22no—
379C22×D4—22no—
380C22×D4—22no—
381C22×D4—22no—
382C22×D4—48no—
383C22×D4C2×C4×C2312yes—
384C22×D4D4×C2312yes—
385C23×C4—21no—
386C23×C4—22no—
387C23×C4D4×C2311yescharacteristic
388C23×C4D4×C2314yescomplemented
389#120—216no—
390#120—28no—
391#120—28no—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—216no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—416no—
410C2×C22⋊C4C2×C4×C2312yes—
411C2×C22⋊C4D4×C2314yes—
412C2×C22⋊C4D4×C2314yescomplemented
413#99—28no—
414#95—24no—
415C23#1099111yescharacteristic, direct product factor, Sylow
416C24—216no—
417C24—28no—
418C24—22no—
419C24—24no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—22no—
424C24—24no—
425C24—21no—
426C24—24no—
427C24—22no—
428C24—24no—
429C24—24no—
430C24—22no—
431C24—21no—
432C24—48no—
433C24—416no—
434C24—44no—
435C24—48no—
436C24—44no—
437C24—416no—
438C24—48no—
439C24—416no—
440C24—416no—
441C24—416no—
442C24—44no—
443C24—41no—
444C24—41no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—41no—
450C24—41no—
451C24—42no—
452C24—42no—
453C24C22⋊C4×C2314yescomplemented
454C24C22⋊C4×C2312yescomplemented
455C24C22×C4×C2311yescharacteristic
456C24C2×D4×C2311yescharacteristic
457C24C2×D4×C2312yes—
458C24C2×D4×C2312yes—
459C24C2×D4×C2311yescharacteristic
460C24C2×D4×C2312yes—
461C24C2×D4×C2312yes—
462C24#41×C2312yes—
463C24#41×C2312yes—
464C2×D4—216no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—22no—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—22no—
473C2×D4—416no—
474C2×D4—432no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4C22⋊C4×C2312yescomplemented
499C22⋊C4—216no—
500C22⋊C4—22no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—432no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—84no—
512C22⋊C4C22⋊C4×C2314yescomplemented
513C23—28no—
514C23—24no—
515C23—216no—
516C23—28no—
517C23—216no—
518C23—216no—
519C23—216no—
520C23—24no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—416no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—416no—
540C23—416no—
541C23—48no—
542C23—416no—
543C23—48no—
544C23—416no—
545C23—416no—
546C23—416no—
547C23—416no—
548C23—44no—
549C23—48no—
550C23—48no—
551C23—416no—
552C23—48no—
553C23—416no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—48no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—82no—
566C23—84no—
567C23C2×C22⋊C4×C2311yescharacteristic
568C23C2×C22⋊C4×C2311yescharacteristic
569C23C2×C22⋊C4×C2311yescharacteristic
570C23C4×D4×C2312yes—
571C23C4×D4×C2312yes—
572C23C23×#12012yes—
573C23C23×#12012yes—
574C23C23×#12012yes—
575C23C23×#12011yescharacteristic
576C23C23×#12011yescharacteristic
577C23C23×#12112yes—
578C23C23×#12112yes—
579C23C23×#12112yes—
580C23C23×#12112yes—
581C23C23×#12112yes—
582C23C23×#12112yes—
583C23C23×#12311yescharacteristic
584C23C23×#12311yescharacteristic
585C23C23×#12312yes—
586C23C23×#12412yes—
587C23C23×#12712yes—
588D4—432no—
589D4—44no—
590D4—44no—
591D4—44no—
592D4—44no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C22—22no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—28no—
620C22—216no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—42no—
631C22—44no—
632C22—416no—
633C22—48no—
634C22—48no—
635C22—416no—
636C22—416no—
637C22—416no—
638C22—44no—
639C22—48no—
640C22—44no—
641C22—416no—
642C22—44no—
643C22—42no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22C23×#37911yescharacteristic
648C22C23×#38611yescharacteristic
649C22C23×#38612yes—
650C22C23×#38611yescharacteristic
651C22C23×#39012yes—
652C22C23×#39212yes—
653C22C23×#39212yes—
654C22C23×#39412yes—
655C22C23×#39412yes—
656C22C23×B(4,𝔽2)14yes—
657C4—44no—
658C4—42no—
659C4—84no—
660C4—84no—
661C2—21no—
662C2—22no—
663C2—22no—
664C2—21no—
665C2—22no—
666C2—22no—
667C2—22no—
668C2—48no—
669C2—82no—
670C2C23×#143811yescharacteristic
671C2C23×#188912yes—
672C2C23×#202114yes—
subgroupG/H
center, socleC23×C23#120
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC22×C23×#120C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C23×#99C2semidirect
3. C23×#2308C2semidirect
4. C2×C23×#379C2semidirect
5. C2×C23×#342C2semidirect
12. C2×C23×#120C4semidirect
52. C23×C4×C23D4semidirect
76. C2×C22⋊C4×C23D4semidirect
116. C24×C23C22⋊C4semidirect
117. C24×C23C22⋊C4semidirect
161. C22×C4×C23C22⋊C4semidirect
175. C22⋊C4×C23C22⋊C4semidirect
176. #10991C23direct (N × H)
274. C22×#99C2×C23semidirect
275. #2308C2×C23semidirect
276. C2×#379C2×C23semidirect
277. C2×#342C2×C23semidirect
336. C2×#120C4×C23semidirect
388. C23×C4D4×C23semidirect
412. C2×C22⋊C4D4×C23semidirect
415. C23#10991direct (N × H)
453. C24C22⋊C4×C23semidirect
454. C24C22⋊C4×C23semidirect
498. C22×C4C22⋊C4×C23semidirect
512. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1202
C22×C23×#992
C23×#23082
C2×C23×#3792
C2×C23×#3422
#1099123