Finite Group Databasebeta

← C23×#10322 #19157948

Subgroups — C23×#10322 #19157948

Order 5888.

988 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3187C211yesmaximal, characteristic, complemented
2C23×#2848C211yesmaximal, characteristic, complemented
3C23×#2847C211yesmaximal, characteristic
4C23×#1779C211yesmaximal, characteristic, complemented
5C23×#1520C211yesmaximal, characteristic, complemented
6C23×#1519C211yesmaximal, characteristic, complemented
7C23×#1289C211yesmaximal, characteristic, complemented
8C24×C4×C23C2211yescharacteristic
9C2×C23×C22⋊Q8—21no—
10C2×C23×C22⋊Q8C2211yescharacteristic
11C2×C23×#121—21no—
12C2×C23×#121C2211yescharacteristic
13C2×C23×#120—21no—
14C23×#483—21no—
15C23×#481—21no—
16C23×#465—22no—
17C23×#450—21no—
18C23×#448—21no—
19C2×C23×C22⋊C8—22no—
20C2×C23×C22⋊C8C2211yescharacteristic
21C2×C23×C22⋊C8C2211yescharacteristic
22C23×#386—22no—
23C23×#386—21no—
24C23×#385—22no—
25C23×#385—21no—
26C23×#380—21no—
27C23×#380—21no—
28C23×#380C2211yescharacteristic
29C23×#380C2211yescharacteristic
30C23×#379—21no—
31C23×#328—22no—
32C23×#327—22no—
33C25×C23D411yescharacteristic
34C22×Q8×C23D411yescharacteristic, complemented
35C22×D4×C23—22no—
36C22×D4×C23—21no—
37C22×D4×C23D411yescharacteristic
38C23×C4×C23—21no—
39C23×C4×C23—21no—
40C23×C4×C23—21no—
41C23×C4×C23—21no—
42C23×C4×C23—21no—
43C23×C4×C23—21no—
44C23×C4×C23—44no—
45C23×C4×C23D411yescharacteristic
46C23×C4×C23C2311yescharacteristic
47C2×SD16×C23—22no—
48C22×C8×C23—21no—
49C22×C8×C23D412yescomplemented
50C23×C22⋊Q8—21no—
51C23×C22⋊Q8—21no—
52C23×C22⋊Q8—21no—
53C23×C22⋊Q8—21no—
54C23×C22⋊Q8—44no—
55C23×#121—21no—
56C23×#121—21no—
57C23×#121—21no—
58C23×#121—21no—
59C23×#121—44no—
60C23×#120—42no—
61C23×#120—42no—
62C2×C4⋊C4×C23—22no—
63C2×C4⋊C4×C23—21no—
64C2×C4⋊C4×C23—21no—
65C2×C4⋊C4×C23—21no—
66C2×C4⋊C4×C23D411yescharacteristic
67C2×C4⋊C4×C23D411yescharacteristic, complemented
68C2×C22⋊C4×C23—22no—
69C2×C22⋊C4×C23—22no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—21no—
74C2×C22⋊C4×C23—21no—
75C2×C22⋊C4×C23—21no—
76C23×#106—22no—
77C23×#103—22no—
78C23×#102—22no—
79C23×C22⋊C8—21no—
80C23×C22⋊C8—21no—
81C23×C22⋊C8—21no—
82C23×C22⋊C8—21no—
83C23×C22⋊C8—44no—
84C23×#95—22no—
85C23×#95—22no—
86C23×#95D412yes—
87C23×#95D412yescomplemented
88C24×C23—21no—
89C24×C23—21no—
90C24×C23—21no—
91C24×C23—21no—
92C24×C23—21no—
93C24×C23—21no—
94C24×C23—21no—
95C24×C23—42no—
96C24×C23—42no—
97C24×C23SD1612yescomplemented
98C24×C23C2×D411yescharacteristic
99C2×Q8×C23—21no—
100C2×Q8×C23—21no—
101C2×Q8×C23—42no—
102C2×D4×C23—21no—
103C2×D4×C23—21no—
104C2×D4×C23—42no—
105C2×D4×C23—44no—
106C2×D4×C23—44no—
107C2×D4×C23—44no—
108C2×D4×C23—41no—
109C2×D4×C23—41no—
110C2×D4×C23—41no—
111C2×D4×C23—41no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C22×C4×C23—24no—
115C22×C4×C23—24no—
116C22×C4×C23—24no—
117C22×C4×C23—21no—
118C22×C4×C23—21no—
119C22×C4×C23—21no—
120C22×C4×C23—21no—
121C22×C4×C23—22no—
122C22×C4×C23—21no—
123C22×C4×C23—22no—
124C22×C4×C23—21no—
125C22×C4×C23—42no—
126C22×C4×C23—42no—
127C22×C4×C23—42no—
128C22×C4×C23—42no—
129C22×C4×C23—42no—
130C22×C4×C23—42no—
131C22×C4×C23—42no—
132C22×C4×C23—42no—
133C22×C4×C23—42no—
134C22×C4×C23—42no—
135C22×C4×C23—42no—
136C22×C4×C23—41no—
137C22×C4×C23—41no—
138C22×C4×C23—41no—
139C22×C4×C23—41no—
140C22×C4×C23C2×D411yescharacteristic
141C22×C4×C23C2×D411yescharacteristic
142C22×C4×C23C2×D411yescharacteristic
143C22×C4×C23C2×D411yescharacteristic
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23#4111yescharacteristic
146SD16×C23—82no—
147C2×C8×C23—22no—
148C2×C8×C23—42no—
149C2×C8×C23—42no—
150C2×C8×C23—41no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—21no—
153C4⋊C4×C23—21no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—42no—
156C4⋊C4×C23—42no—
157C4⋊C4×C23—44no—
158C4⋊C4×C23—41no—
159C4⋊C4×C23—41no—
160C22⋊C4×C23—44no—
161C22⋊C4×C23—44no—
162C22⋊C4×C23—44no—
163C22⋊C4×C23—44no—
164C22⋊C4×C23—41no—
165C22⋊C4×C23—41no—
166C22⋊C4×C23—41no—
167C22⋊C4×C23—41no—
168C22⋊C4×C23—41no—
169C22⋊C4×C23—41no—
170C22⋊C4×C23—41no—
171C22⋊C4×C23—41no—
172#10322C2311yesmaximal, characteristic, direct product factor, Sylow
173C23×C23—22no—
174C23×C23—22no—
175C23×C23—22no—
176C23×C23—22no—
177C23×C23—22no—
178C23×C23—22no—
179C23×C23—22no—
180C23×C23—22no—
181C23×C23—22no—
182C23×C23—41no—
183C23×C23—44no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—41no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23—44no—
192C23×C23—44no—
193C23×C23—44no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—41no—
200C23×C23—41no—
201C23×C23—41no—
202C23×C23—41no—
203C23×C23#12011yescharacteristic
204C23×C23#12011yescharacteristic
205C23×C23#12111yescharacteristic
206C23×C23C2×SD1611yescharacteristic
207C23×C23#13511yescharacteristic
208C23×C23AGL(1,ℤ/8)11yescharacteristic
209C23×C23#13711yescharacteristic
210Q8×C23—42no—
211Q8×C23—81no—
212D4×C23—42no—
213D4×C23—81no—
214D4×C23—88no—
215D4×C23—81no—
216D4×C23—81no—
217D4×C23—81no—
218D4×C23—81no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—22no—
223C2×C4×C23—22no—
224C2×C4×C23—22no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23—44no—
229C2×C4×C23—44no—
230C2×C4×C23—44no—
231C2×C4×C23—41no—
232C2×C4×C23—41no—
233C2×C4×C23—41no—
234C2×C4×C23—41no—
235C2×C4×C23—44no—
236C2×C4×C23—42no—
237C2×C4×C23—42no—
238C2×C4×C23—42no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—41no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—41no—
248C2×C4×C23—41no—
249C2×C4×C23#12011yescharacteristic
250C2×C4×C23#12111yescharacteristic
251C2×C4×C23#12111yescharacteristic
252C2×C4×C23#12711yescharacteristic
253C8×C23—42no—
254C8×C23—81no—
255#3187C2×C2311yescharacteristic, complemented
256#2848C2×C2311yescharacteristic, complemented
257#2847C2×C2311yescharacteristic
258#1779C2×C2311yescharacteristic, complemented
259#1520C2×C2311yescharacteristic, complemented
260#1519C2×C2311yescharacteristic, complemented
261#1289C2×C2311yescharacteristic, complemented
262C22×C23—21no—
263C22×C23—21no—
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—21no—
267C22×C23—21no—
268C22×C23—24no—
269C22×C23—24no—
270C22×C23—24no—
271C22×C23—41no—
272C22×C23—41no—
273C22×C23—41no—
274C22×C23—41no—
275C22×C23—42no—
276C22×C23—42no—
277C22×C23—42no—
278C22×C23—42no—
279C22×C23—42no—
280C22×C23—42no—
281C22×C23—42no—
282C22×C23—42no—
283C22×C23—42no—
284C22×C23—42no—
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—82no—
288C22×C23—82no—
289C22×C23—81no—
290C22×C23—81no—
291C22×C23—81no—
292C22×C23#39211yescharacteristic
293C22×C23#44811yescharacteristic
294C22×C23#44911yescharacteristic
295C22×C23#45011yescharacteristic
296C22×C23#45211yescharacteristic
297C22×C23#46511yescharacteristic
298C22×C23#46811yescharacteristic
299C4×C23—21no—
300C4×C23—21no—
301C4×C23—21no—
302C4×C23—42no—
303C4×C23—82no—
304C4×C23—82no—
305C4×C23—81no—
306C4×C23—81no—
307C4×C23—81no—
308C4×C23B(4,𝔽2)12yes—
309C24×C4C22×C2311yescharacteristic
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8C22×C2311yescharacteristic
312C2×#121—21no—
313C2×#121C22×C2311yescharacteristic
314C2×#120—21no—
315#483—21no—
316#481—21no—
317#465—22no—
318#450—21no—
319#448—21no—
320C2×C22⋊C8—22no—
321C2×C22⋊C8C22×C2311yescharacteristic
322C2×C22⋊C8C22×C2311yescharacteristic
323#386—21no—
324#386—22no—
325#385—21no—
326#385—22no—
327#380—21no—
328#380—21no—
329#380C22×C2311yescharacteristic
330#380C22×C2311yescharacteristic
331#379—21no—
332#328—22no—
333#327—22no—
334C2×C23—21no—
335C2×C23—21no—
336C2×C23—21no—
337C2×C23—21no—
338C2×C23—21no—
339C2×C23—21no—
340C2×C23—44no—
341C2×C23—161no—
342C2×C23#200011yescharacteristic
343C2×C23#201011yescharacteristic
344C2×C23#202111yescharacteristic
345C25D4×C2311yescharacteristic
346C22×Q8D4×C2311yescharacteristic, complemented
347C22×D4—22no—
348C22×D4—21no—
349C22×D4D4×C2311yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—21no—
355C23×C4—21no—
356C23×C4—44no—
357C23×C4D4×C2311yescharacteristic
358C23×C4C23×C2311yescharacteristic
359C2×SD16—22no—
360C22×C8—21no—
361C22×C8D4×C2312yescomplemented
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—44no—
367#121—21no—
368#121—21no—
369#121—21no—
370#121—21no—
371#121—44no—
372#120—42no—
373#120—42no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C2311yescharacteristic
379C2×C4⋊C4D4×C2311yescharacteristic, complemented
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388#106—22no—
389#103—22no—
390#102—22no—
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8—21no—
394C22⋊C8—21no—
395C22⋊C8—44no—
396#95—22no—
397#95—22no—
398#95D4×C2312yes—
399#95D4×C2312yescomplemented
400C23#1032211yescharacteristic, direct product factor, Sylow
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—42no—
409C24—42no—
410C24SD16×C2312yescomplemented
411C24C2×D4×C2311yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4C2×D4×C2311yescharacteristic
454C22×C4C2×D4×C2311yescharacteristic
455C22×C4C2×D4×C2311yescharacteristic
456C22×C4C2×D4×C2311yescharacteristic
457C22×C4C2×D4×C2311yescharacteristic
458C22×C4#41×C2311yescharacteristic
459SD16—82no—
460C2×C8—22no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—41no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—44no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C23×#12011yescharacteristic
516C23C23×#12011yescharacteristic
517C23C23×#12111yescharacteristic
518C23C2×SD16×C2311yescharacteristic
519C23C23×#13511yescharacteristic
520C23C23×AGL(1,ℤ/8)11yescharacteristic
521C23C23×#13711yescharacteristic
522Q8—42no—
523Q8—81no—
524D4—42no—
525D4—81no—
526D4—88no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4C23×#12011yescharacteristic
562C2×C4C23×#12111yescharacteristic
563C2×C4C23×#12111yescharacteristic
564C2×C4C23×#12711yescharacteristic
565C8—42no—
566C8—81no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22C23×#39211yescharacteristic
598C22C23×#44811yescharacteristic
599C22C23×#44911yescharacteristic
600C22C23×#45011yescharacteristic
601C22C23×#45211yescharacteristic
602C22C23×#46511yescharacteristic
603C22C23×#46811yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4C23×B(4,𝔽2)12yes—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—44no—
621C2—161no—
622C2C23×#200011yescharacteristic
623C2C23×#201011yescharacteristic
624C2C23×#202111yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2847C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3187C2semidirect
2. C23×#2848C2semidirect
4. C23×#1779C2semidirect
5. C23×#1520C2semidirect
6. C23×#1519C2semidirect
7. C23×#1289C2semidirect
34. C22×Q8×C23D4semidirect
49. C22×C8×C23D4semidirect
67. C2×C4⋊C4×C23D4semidirect
87. C23×#95D4semidirect
97. C24×C23SD16semidirect
172. #10322C23direct (N × H)
255. #3187C2×C23semidirect
256. #2848C2×C23semidirect
258. #1779C2×C23semidirect
259. #1520C2×C23semidirect
260. #1519C2×C23semidirect
261. #1289C2×C23semidirect
346. C22×Q8D4×C23semidirect
361. C22×C8D4×C23semidirect
379. C2×C4⋊C4D4×C23semidirect
399. #95D4×C23semidirect
400. C23#10322direct (N × H)
410. C24SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#31872
C23×#28482
C23×#28472
C23×#17792
C23×#15202
C23×#15192
C23×#12892
#1032223