Finite Group Databasebeta

← C23×#8481 #19156107

Subgroups — C23×#8481 #19156107

Order 5888.

844 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3536C211yesmaximal, characteristic
2C23×#1964C211yesmaximal, characteristic, complemented
3C23×#1900C211yesmaximal, characteristic, complemented
4C23×#1894C211yesmaximal, characteristic, complemented
5C23×#1871C211yesmaximal, characteristic, complemented
6C23×#1513C211yesmaximal, characteristic, complemented
7C23×#1508C211yesmaximal, characteristic, complemented
8C22×M16×C23C2211yescharacteristic
9C23×#562—22no—
10C23×#554—21no—
11C23×#548—21no—
12C23×#547—21no—
13C23×#546—21no—
14C23×#546—22no—
15C23×#543C412yescomplemented
16C23×#542—21no—
17C23×#539—21no—
18C23×#538C412yescomplemented
19C23×#537—21no—
20C23×#537—21no—
21C23×#525C2211yescharacteristic
22C2×C23×C22⋊Q8C2211yescharacteristic
23C2×C23×#121C2211yescharacteristic
24C2×C23×#104—21no—
25C2×C23×#103—21no—
26C2×C23×#100—21no—
27C2×C23×#99—22no—
28C23×#407C2211yescharacteristic
29C23×#380C2211yescharacteristic
30C4×C22⋊C4×C23C2211yescharacteristic
31C23×#343—21no—
32C23×#340—21no—
33C23×#339—21no—
34C23×#328—22no—
35C23×#328—21no—
36C23×#327—21no—
37C2×#41×C23—22no—
38C2×#41×C23—21no—
39C2×#41×C23D411yescharacteristic
40C22×Q8×C23D411yescharacteristic
41C22×D4×C23D412yes—
42C22×D4×C23D411yescharacteristic
43C23×C4×C23C2311yescharacteristic
44C2×M16×C23—21no—
45C2×M16×C23—21no—
46C2×M16×C23—21no—
47C2×M16×C23—42no—
48C22×C8×C23—21no—
49C23×C4⋊Q8—21no—
50C23×#126—22no—
51C23×#125—22no—
52C23×#124—21no—
53C23×#124—22no—
54C23×#124—22no—
55C23×#123—22no—
56C23×#123—22no—
57C23×#123—22no—
58C23×#123—21no—
59C23×#123—22no—
60C23×C22⋊Q8—21no—
61C23×C22⋊Q8—22no—
62C23×C22⋊Q8—22no—
63C23×C22⋊Q8—22no—
64C23×C22⋊Q8—22no—
65C23×C22⋊Q8—21no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8C2×C412yes—
70C23×C22⋊Q8C2×C411yescharacteristic
71C23×C22⋊Q8D412yes—
72C23×C22⋊Q8D411yescharacteristic
73C23×#121—21no—
74C23×#121—22no—
75C23×#121—22no—
76C23×#121—22no—
77C23×#121—22no—
78C23×#121—22no—
79C23×#121—21no—
80C23×#121—21no—
81C23×#121—21no—
82C23×#121—21no—
83C23×#121C2×C411yescharacteristic
84C23×#121D411yescharacteristic
85C23×#120—22no—
86C23×#120—21no—
87C4×Q8×C23—21no—
88C4×D4×C23—21no—
89C4×D4×C23—22no—
90C4×D4×C23—22no—
91C4×D4×C23—21no—
92C4×D4×C23—21no—
93C23×#117D411yescharacteristic
94C2×C4⋊C4×C23—21no—
95C2×C4⋊C4×C23—21no—
96C2×C4⋊C4×C23—21no—
97C2×C4⋊C4×C23D411yescharacteristic
98C2×C4⋊C4×C23D411yescharacteristic
99C2×C22⋊C4×C23—22no—
100C2×C22⋊C4×C23—21no—
101C2×C22⋊C4×C23—21no—
102C2×C22⋊C4×C23—21no—
103C2×C22⋊C4×C23C2×C412yes—
104C2×C42×C23—21no—
105C23×#104—42no—
106C23×#103—42no—
107C23×#100—42no—
108C23×#99—44no—
109C23×C22⋊C8—21no—
110C23×C22⋊C8—21no—
111C23×#95—21no—
112C23×#95—21no—
113C24×C23C22⋊C412yes—
114C24×C23C2×D411yescharacteristic
115#41×C23—41no—
116#41×C23—41no—
117#41×C23—82no—
118#41×C23—82no—
119#41×C23—81no—
120#41×C23—81no—
121C2×Q8×C23—22no—
122C2×Q8×C23—21no—
123C2×Q8×C23—41no—
124C2×Q8×C23—41no—
125C2×Q8×C23C22⋊C411yescharacteristic
126C2×Q8×C23C22⋊C412yes—
127C2×D4×C23—21no—
128C2×D4×C23—21no—
129C2×D4×C23—22no—
130C2×D4×C23—22no—
131C2×D4×C23—22no—
132C2×D4×C23—21no—
133C2×D4×C23—22no—
134C2×D4×C23—21no—
135C2×D4×C23—21no—
136C2×D4×C23—41no—
137C2×D4×C23—41no—
138C2×D4×C23—42no—
139C2×D4×C23—42no—
140C2×D4×C23C22⋊C411yescharacteristic
141C22×C4×C23—21no—
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—22no—
145C22×C4×C23—21no—
146C22×C4×C23—21no—
147C22×C4×C23—41no—
148C22×C4×C23—41no—
149C22×C4×C23—41no—
150C22×C4×C23—41no—
151C22×C4×C23C22⋊C412yes—
152C22×C4×C23C22×C411yescharacteristic
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2×D411yescharacteristic
155C22×C4×C23C2×D411yescharacteristic
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158M16×C23—22no—
159M16×C23—22no—
160M16×C23—41no—
161M16×C23—41no—
162M16×C23—41no—
163M16×C23—41no—
164C2×C8×C23—22no—
165C2×C8×C23—41no—
166C2×C8×C23—41no—
167C2×C8×C23—41no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—41no—
180C4⋊C4×C23—41no—
181C4⋊C4×C23C22⋊C412yes—
182C4⋊C4×C23C22⋊C411yescharacteristic
183C4⋊C4×C23C22⋊C411yescharacteristic
184C22⋊C4×C23—22no—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—21no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—21no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—21no—
197C22⋊C4×C23—21no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—41no—
200C22⋊C4×C23—42no—
201C42×C23—22no—
202C42×C23—21no—
203C42×C23—41no—
204C42×C23—41no—
205#8481C2311yesmaximal, characteristic, direct product factor, Sylow
206C23×C23—21no—
207C23×C23—21no—
208C23×C23—22no—
209C23×C23—21no—
210C23×C23—41no—
211C23×C23—41no—
212C23×C23—42no—
213C23×C23—42no—
214C23×C23—42no—
215C23×C23C2×C22⋊C411yescharacteristic
216C23×C23#12011yescharacteristic
217C23×C23#12011yescharacteristic
218Q8×C23—42no—
219Q8×C23—41no—
220Q8×C23—41no—
221Q8×C23—41no—
222Q8×C23—82no—
223D4×C23—41no—
224D4×C23—42no—
225D4×C23—42no—
226D4×C23—41no—
227D4×C23—42no—
228D4×C23—41no—
229D4×C23—41no—
230D4×C23—41no—
231D4×C23—81no—
232D4×C23—82no—
233D4×C23—81no—
234D4×C23—82no—
235D4×C23—82no—
236C2×C4×C23—21no—
237C2×C4×C23—21no—
238C2×C4×C23—21no—
239C2×C4×C23—21no—
240C2×C4×C23—21no—
241C2×C4×C23—21no—
242C2×C4×C23—21no—
243C2×C4×C23—22no—
244C2×C4×C23—21no—
245C2×C4×C23—22no—
246C2×C4×C23—21no—
247C2×C4×C23—22no—
248C2×C4×C23—41no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—81no—
258C2×C4×C23—82no—
259C2×C4×C23—82no—
260C2×C4×C23—82no—
261C2×C4×C23—81no—
262C2×C4×C23—81no—
263C2×C4×C23—81no—
264C2×C4×C23—81no—
265C2×C4×C23—81no—
266C2×C4×C23C2×C22⋊C411yescharacteristic
267C2×C4×C23C2×C22⋊C411yescharacteristic
268C2×C4×C23#12011yescharacteristic
269C2×C4×C23#12011yescharacteristic
270C8×C23—42no—
271C8×C23—81no—
272#3536C2×C2311yescharacteristic
273#1964C2×C2311yescharacteristic, complemented
274#1900C2×C2311yescharacteristic, complemented
275#1894C2×C2311yescharacteristic, complemented
276#1871C2×C2311yescharacteristic, complemented
277#1513C2×C2311yescharacteristic, complemented
278#1508C2×C2311yescharacteristic, complemented
279C22×C23—21no—
280C22×C23—21no—
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—21no—
286C22×C23—21no—
287C22×C23—41no—
288C22×C23—42no—
289C22×C23—41no—
290C22×C23—42no—
291C22×C23—41no—
292C22×C23—42no—
293C22×C23—81no—
294C22×C23—81no—
295C22×C23—162no—
296C22×C23#37911yescharacteristic
297C22×C23#45311yescharacteristic
298C22×C23#45511yescharacteristic
299C4×C23—21no—
300C4×C23—21no—
301C4×C23—21no—
302C4×C23—42no—
303C4×C23—42no—
304C4×C23—41no—
305C4×C23—41no—
306C4×C23—41no—
307C4×C23—42no—
308C4×C23—82no—
309C4×C23—161no—
310C4×C23B(4,𝔽2)11yescharacteristic
311C4×C23#45811yescharacteristic
312C22×M16C22×C2311yescharacteristic
313#562—22no—
314#554—21no—
315#548—21no—
316#547—21no—
317#546—21no—
318#546—22no—
319#543C4×C2312yescomplemented
320#542—21no—
321#539—21no—
322#538C4×C2312yescomplemented
323#537—21no—
324#537—21no—
325#525C22×C2311yescharacteristic
326C2×C22⋊Q8C22×C2311yescharacteristic
327C2×#121C22×C2311yescharacteristic
328C2×#104—21no—
329C2×#103—21no—
330C2×#100—21no—
331C2×#99—22no—
332#407C22×C2311yescharacteristic
333#380C22×C2311yescharacteristic
334C4×C22⋊C4C22×C2311yescharacteristic
335#343—21no—
336#340—21no—
337#339—21no—
338#328—22no—
339#328—21no—
340#327—21no—
341C2×C23—21no—
342C2×C23—21no—
343C2×C23—41no—
344C2×C23—41no—
345C2×C23—81no—
346C2×C23—82no—
347C2×C23#179011yescharacteristic
348C2×C23#179211yescharacteristic
349C2×C23#179611yescharacteristic
350C2×#41—22no—
351C2×#41—21no—
352C2×#41D4×C2311yescharacteristic
353C22×Q8D4×C2311yescharacteristic
354C22×D4D4×C2312yes—
355C22×D4D4×C2311yescharacteristic
356C23×C4C23×C2311yescharacteristic
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—42no—
361C22×C8—21no—
362C4⋊Q8—21no—
363#126—22no—
364#125—22no—
365#124—21no—
366#124—22no—
367#124—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—21no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8C2×C4×C2312yes—
383C22⋊Q8C2×C4×C2311yescharacteristic
384C22⋊Q8D4×C2312yes—
385C22⋊Q8D4×C2311yescharacteristic
386#121—21no—
387#121—22no—
388#121—22no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—21no—
394#121—21no—
395#121—21no—
396#121C2×C4×C2311yescharacteristic
397#121D4×C2311yescharacteristic
398#120—22no—
399#120—21no—
400C4×Q8—21no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4—21no—
406#117D4×C2311yescharacteristic
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4D4×C2311yescharacteristic
411C2×C4⋊C4D4×C2311yescharacteristic
412C2×C22⋊C4—22no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4C2×C4×C2312yes—
417C2×C42—21no—
418#104—42no—
419#103—42no—
420#100—42no—
421#99—44no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424#95—21no—
425#95—21no—
426C23#848111yescharacteristic, direct product factor, Sylow
427C24C22⋊C4×C2312yes—
428C24C2×D4×C2311yescharacteristic
429#41—41no—
430#41—41no—
431#41—82no—
432#41—82no—
433#41—81no—
434#41—81no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—41no—
438C2×Q8—41no—
439C2×Q8C22⋊C4×C2311yescharacteristic
440C2×Q8C22⋊C4×C2312yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C22⋊C4×C2311yescharacteristic
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4C22⋊C4×C2312yes—
466C22×C4C22×C4×C2311yescharacteristic
467C22×C4C2×D4×C2311yescharacteristic
468C22×C4C2×D4×C2311yescharacteristic
469C22×C4C2×D4×C2311yescharacteristic
470C22×C4C2×D4×C2311yescharacteristic
471C22×C4C2×D4×C2311yescharacteristic
472M16—22no—
473M16—22no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4C22⋊C4×C2312yes—
496C4⋊C4C22⋊C4×C2311yescharacteristic
497C4⋊C4C22⋊C4×C2311yescharacteristic
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C42—22no—
516C42—21no—
517C42—41no—
518C42—41no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—21no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C2×C22⋊C4×C2311yescharacteristic
529C23C23×#12011yescharacteristic
530C23C23×#12011yescharacteristic
531Q8—42no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—82no—
536D4—41no—
537D4—42no—
538D4—42no—
539D4—41no—
540D4—42no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—81no—
545D4—82no—
546D4—81no—
547D4—82no—
548D4—82no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4C2×C22⋊C4×C2311yescharacteristic
580C2×C4C2×C22⋊C4×C2311yescharacteristic
581C2×C4C23×#12011yescharacteristic
582C2×C4C23×#12011yescharacteristic
583C8—42no—
584C8—81no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—162no—
602C22C23×#37911yescharacteristic
603C22C23×#45311yescharacteristic
604C22C23×#45511yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—41no—
614C4—82no—
615C4—161no—
616C4C23×B(4,𝔽2)11yescharacteristic
617C4C23×#45811yescharacteristic
618C2—21no—
619C2—21no—
620C2—41no—
621C2—41no—
622C2—82no—
623C2—81no—
624C2C23×#179011yescharacteristic
625C2C23×#179211yescharacteristic
626C2C23×#179611yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3536C2
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×#1964C2semidirect
3. C23×#1900C2semidirect
4. C23×#1894C2semidirect
5. C23×#1871C2semidirect
6. C23×#1513C2semidirect
7. C23×#1508C2semidirect
15. C23×#543C4semidirect
18. C23×#538C4semidirect
205. #8481C23direct (N × H)
273. #1964C2×C23semidirect
274. #1900C2×C23semidirect
275. #1894C2×C23semidirect
276. #1871C2×C23semidirect
277. #1513C2×C23semidirect
278. #1508C2×C23semidirect
319. #543C4×C23semidirect
322. #538C4×C23semidirect
426. C23#8481direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35362
C23×#19642
C23×#19002
C23×#18942
C23×#18712
C23×#15132
C23×#15082
#848123