Finite Group Databasebeta

← C23×#8538 #19156164

Subgroups — C23×#8538 #19156164

Order 5888.

820 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3505C211yesmaximal, characteristic
2C23×#1901C211yesmaximal, characteristic, complemented
3C23×#1900C211yesmaximal, characteristic, complemented
4C23×#1870C211yesmaximal, characteristic, complemented
5C23×#1472C211yesmaximal, characteristic, complemented
6C23×#1471C211yesmaximal, characteristic, complemented
7C23×#1465C211yesmaximal, characteristic, complemented
8C22×M16×C23C2211yescharacteristic
9C23×#553—22no—
10C23×#553—21no—
11C23×#552—21no—
12C23×#551—22no—
13C23×#548—21no—
14C23×#546—21no—
15C23×#539—21no—
16C23×#538—21no—
17C23×#534C412yes—
18C23×#529—21no—
19C23×#525—21no—
20C2×C23×C22⋊Q8C2211yescharacteristic
21C2×C23×#121C2211yescharacteristic
22C23×#518C412yes—
23C2×C4×D4×C23C2211yescharacteristic
24C2×C23×#101—21no—
25C2×C23×#100—21no—
26C23×#407C2211yescharacteristic
27C23×#407C2211yescharacteristic
28C2×C23×C22⋊C8—21no—
29C2×C23×C22⋊C8C2211yescharacteristic
30C23×#343—21no—
31C23×#343—21no—
32C23×#336—21no—
33C23×#324—21no—
34C23×#324—22no—
35C23×#323—21no—
36C23×#323—22no—
37C2×#41×C23—21no—
38C22×Q8×C23D411yescharacteristic
39C22×D4×C23C2×C411yescharacteristic
40C22×D4×C23C2×C412yes—
41C22×D4×C23D411yescharacteristic
42C23×C4×C23D411yescharacteristic
43C23×C4×C23C2311yescharacteristic
44C2×M16×C23—21no—
45C2×M16×C23—21no—
46C2×M16×C23—21no—
47C2×M16×C23—21no—
48C2×M16×C23—42no—
49C22×C8×C23—21no—
50C22×C8×C23—21no—
51C23×#126—22no—
52C23×#126—22no—
53C23×#126—22no—
54C23×#125—21no—
55C23×#125—22no—
56C23×#124—22no—
57C23×#124—21no—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×#123—22no—
63C23×C22⋊Q8—22no—
64C23×C22⋊Q8—21no—
65C23×C22⋊Q8—21no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—21no—
68C23×C22⋊Q8—21no—
69C23×#121—21no—
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—22no—
73C23×#121—21no—
74C23×#121—21no—
75C23×#121—21no—
76C23×#121—21no—
77C23×#120—22no—
78C23×#120—22no—
79C4×Q8×C23—21no—
80C4×D4×C23—22no—
81C4×D4×C23—21no—
82C4×D4×C23—22no—
83C4×D4×C23—22no—
84C4×D4×C23—22no—
85C4×D4×C23—22no—
86C4×D4×C23—21no—
87C4×D4×C23—21no—
88C4×D4×C23—21no—
89C4×D4×C23—21no—
90C23×#117—22no—
91C23×#117—21no—
92C23×#117—22no—
93C2×C4⋊C4×C23D411yescharacteristic
94C2×C4⋊C4×C23D412yes—
95C2×C4⋊C4×C23D411yescharacteristic
96C2×C4⋊C4×C23D411yescharacteristic
97C2×C22⋊C4×C23—22no—
98C2×C22⋊C4×C23C2×C411yescharacteristic
99C2×C22⋊C4×C23C2×C412yes—
100C2×C22⋊C4×C23D412yes—
101C2×C22⋊C4×C23D411yescharacteristic
102C2×C42×C23D411yescharacteristic
103C23×#101—42no—
104C23×#100—42no—
105C23×C22⋊C8—21no—
106C23×C22⋊C8—21no—
107C23×C22⋊C8—21no—
108C23×C22⋊C8—21no—
109C23×C22⋊C8—21no—
110C23×C22⋊C8—21no—
111C23×C22⋊C8—42no—
112C24×C23C22⋊C411yescharacteristic
113C24×C23C22⋊C412yes—
114C24×C23C22×C411yescharacteristic
115#41×C23—81no—
116#41×C23—81no—
117C2×Q8×C23—21no—
118C2×Q8×C23—21no—
119C2×Q8×C23—41no—
120C2×Q8×C23—41no—
121C2×D4×C23—21no—
122C2×D4×C23—22no—
123C2×D4×C23—22no—
124C2×D4×C23—21no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—21no—
128C2×D4×C23—21no—
129C2×D4×C23—41no—
130C2×D4×C23—41no—
131C2×D4×C23—41no—
132C2×D4×C23—41no—
133C2×D4×C23—42no—
134C2×D4×C23—42no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—41no—
140C22×C4×C23—41no—
141C22×C4×C23—41no—
142C22×C4×C23—41no—
143C22×C4×C23C22⋊C411yescharacteristic
144C22×C4×C23C22⋊C411yescharacteristic
145C22×C4×C23C22⋊C412yes—
146C22×C4×C23C22⋊C411yescharacteristic
147C22×C4×C23C22⋊C412yes—
148C22×C4×C23C22⋊C412yes—
149C22×C4×C23C2×D411yescharacteristic
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D411yescharacteristic
152C22×C4×C23C2×D411yescharacteristic
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2×D411yescharacteristic
155M16×C23—22no—
156M16×C23—22no—
157M16×C23—41no—
158M16×C23—41no—
159M16×C23—41no—
160M16×C23—41no—
161M16×C23—41no—
162M16×C23—41no—
163C2×C8×C23—22no—
164C2×C8×C23—41no—
165C2×C8×C23—41no—
166C2×C8×C23—41no—
167C2×C8×C23—41no—
168C2×C8×C23—41no—
169C2×C8×C23—41no—
170C2×C8×C23—41no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—21no—
183C22⋊C4×C23—22no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—21no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—21no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—21no—
195C22⋊C4×C23—21no—
196C22⋊C4×C23—22no—
197C42×C23—22no—
198C42×C23—22no—
199C42×C23—21no—
200C42×C23—21no—
201#8538C2311yesmaximal, characteristic, direct product factor, Sylow
202C23×C23—21no—
203C23×C23—21no—
204C23×C23—21no—
205C23×C23—22no—
206C23×C23—21no—
207C23×C23—21no—
208C23×C23—41no—
209C23×C23—41no—
210C23×C23—41no—
211C23×C23—41no—
212C23×C23—42no—
213C23×C23—42no—
214C23×C23—42no—
215C23×C23C2×C22⋊C411yescharacteristic
216C23×C23C2×C22⋊C411yescharacteristic
217C23×C23C2×C22⋊C411yescharacteristic
218Q8×C23—41no—
219Q8×C23—41no—
220Q8×C23—81no—
221D4×C23—42no—
222D4×C23—42no—
223D4×C23—44no—
224D4×C23—42no—
225D4×C23—42no—
226D4×C23—41no—
227D4×C23—41no—
228D4×C23—81no—
229D4×C23—82no—
230D4×C23—82no—
231C2×C4×C23—21no—
232C2×C4×C23—21no—
233C2×C4×C23—21no—
234C2×C4×C23—21no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—21no—
238C2×C4×C23—21no—
239C2×C4×C23—22no—
240C2×C4×C23—21no—
241C2×C4×C23—22no—
242C2×C4×C23—22no—
243C2×C4×C23—21no—
244C2×C4×C23—21no—
245C2×C4×C23—41no—
246C2×C4×C23—41no—
247C2×C4×C23—41no—
248C2×C4×C23—41no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—81no—
256C2×C4×C23—82no—
257C2×C4×C23—82no—
258C2×C4×C23#12011yescharacteristic
259C2×C4×C23#12011yescharacteristic
260C2×C4×C23#12011yescharacteristic
261C2×C4×C23#12011yescharacteristic
262C8×C23—42no—
263C8×C23—81no—
264C8×C23—81no—
265C8×C23—81no—
266#3505C2×C2311yescharacteristic
267#1901C2×C2311yescharacteristic, complemented
268#1900C2×C2311yescharacteristic, complemented
269#1870C2×C2311yescharacteristic, complemented
270#1472C2×C2311yescharacteristic, complemented
271#1471C2×C2311yescharacteristic, complemented
272#1465C2×C2311yescharacteristic, complemented
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—21no—
276C22×C23—21no—
277C22×C23—21no—
278C22×C23—21no—
279C22×C23—21no—
280C22×C23—21no—
281C22×C23—42no—
282C22×C23—41no—
283C22×C23—42no—
284C22×C23—41no—
285C22×C23—42no—
286C22×C23—42no—
287C22×C23—41no—
288C22×C23—81no—
289C22×C23—81no—
290C22×C23—81no—
291C22×C23—162no—
292C22×C23#37911yescharacteristic
293C22×C23#41311yescharacteristic
294C22×C23#41311yescharacteristic
295C4×C23—21no—
296C4×C23—21no—
297C4×C23—21no—
298C4×C23—41no—
299C4×C23—42no—
300C4×C23—41no—
301C4×C23—42no—
302C4×C23—41no—
303C4×C23—42no—
304C4×C23B(4,𝔽2)11yescharacteristic
305C4×C23#45811yescharacteristic
306C22×M16C22×C2311yescharacteristic
307#553—22no—
308#553—21no—
309#552—21no—
310#551—22no—
311#548—21no—
312#546—21no—
313#539—21no—
314#538—21no—
315#534C4×C2312yes—
316#529—21no—
317#525—21no—
318C2×C22⋊Q8C22×C2311yescharacteristic
319C2×#121C22×C2311yescharacteristic
320#518C4×C2312yes—
321C2×C4×D4C22×C2311yescharacteristic
322C2×#101—21no—
323C2×#100—21no—
324#407C22×C2311yescharacteristic
325#407C22×C2311yescharacteristic
326C2×C22⋊C8—21no—
327C2×C22⋊C8C22×C2311yescharacteristic
328#343—21no—
329#343—21no—
330#336—21no—
331#324—21no—
332#324—22no—
333#323—21no—
334#323—22no—
335C2×C23—21no—
336C2×C23—21no—
337C2×C23—41no—
338C2×C23—41no—
339C2×C23—81no—
340C2×C23—82no—
341C2×C23#178511yescharacteristic
342C2×C23#179011yescharacteristic
343C2×C23#179111yescharacteristic
344C2×#41—21no—
345C22×Q8D4×C2311yescharacteristic
346C22×D4C2×C4×C2311yescharacteristic
347C22×D4C2×C4×C2312yes—
348C22×D4D4×C2311yescharacteristic
349C23×C4D4×C2311yescharacteristic
350C23×C4C23×C2311yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—42no—
356C22×C8—21no—
357C22×C8—21no—
358#126—22no—
359#126—22no—
360#126—22no—
361#125—21no—
362#125—22no—
363#124—21no—
364#124—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—21no—
377#121—22no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121—21no—
382#121—21no—
383#121—21no—
384#120—22no—
385#120—22no—
386C4×Q8—21no—
387C4×D4—22no—
388C4×D4—21no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397#117—21no—
398#117—22no—
399#117—22no—
400C2×C4⋊C4D4×C2311yescharacteristic
401C2×C4⋊C4D4×C2311yescharacteristic
402C2×C4⋊C4D4×C2312yes—
403C2×C4⋊C4D4×C2311yescharacteristic
404C2×C22⋊C4—22no—
405C2×C22⋊C4C2×C4×C2311yescharacteristic
406C2×C22⋊C4C2×C4×C2312yes—
407C2×C22⋊C4D4×C2311yescharacteristic
408C2×C22⋊C4D4×C2312yes—
409C2×C42D4×C2311yescharacteristic
410#101—42no—
411#100—42no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—42no—
419C23#853811yescharacteristic, direct product factor, Sylow
420C24C22⋊C4×C2311yescharacteristic
421C24C22⋊C4×C2312yes—
422C24C22×C4×C2311yescharacteristic
423#41—81no—
424#41—81no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—42no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4C22⋊C4×C2311yescharacteristic
452C22×C4C22⋊C4×C2311yescharacteristic
453C22×C4C22⋊C4×C2312yes—
454C22×C4C22⋊C4×C2311yescharacteristic
455C22×C4C22⋊C4×C2312yes—
456C22×C4C22⋊C4×C2312yes—
457C22×C4C2×D4×C2311yescharacteristic
458C22×C4C2×D4×C2311yescharacteristic
459C22×C4C2×D4×C2311yescharacteristic
460C22×C4C2×D4×C2311yescharacteristic
461C22×C4C2×D4×C2311yescharacteristic
462C22×C4C2×D4×C2311yescharacteristic
463M16—22no—
464M16—22no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471C2×C8—22no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23C2×C22⋊C4×C2311yescharacteristic
523C23C2×C22⋊C4×C2311yescharacteristic
524C23C2×C22⋊C4×C2311yescharacteristic
525Q8—41no—
526Q8—41no—
527Q8—81no—
528D4—42no—
529D4—42no—
530D4—44no—
531D4—42no—
532D4—42no—
533D4—41no—
534D4—41no—
535D4—81no—
536D4—82no—
537D4—82no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4C23×#12011yescharacteristic
566C2×C4C23×#12011yescharacteristic
567C2×C4C23×#12011yescharacteristic
568C2×C4C23×#12011yescharacteristic
569C8—42no—
570C8—81no—
571C8—81no—
572C8—81no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—162no—
592C22C23×#37911yescharacteristic
593C22C23×#41311yescharacteristic
594C22C23×#41311yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4C23×B(4,𝔽2)11yescharacteristic
605C4C23×#45811yescharacteristic
606C2—21no—
607C2—21no—
608C2—41no—
609C2—41no—
610C2—81no—
611C2—82no—
612C2C23×#178511yescharacteristic
613C2C23×#179011yescharacteristic
614C2C23×#179111yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3505C2
LCSG ▷ C24 ▷ 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×#1901C2semidirect
3. C23×#1900C2semidirect
4. C23×#1870C2semidirect
5. C23×#1472C2semidirect
6. C23×#1471C2semidirect
7. C23×#1465C2semidirect
201. #8538C23direct (N × H)
267. #1901C2×C23semidirect
268. #1900C2×C23semidirect
269. #1870C2×C23semidirect
270. #1472C2×C23semidirect
271. #1471C2×C23semidirect
272. #1465C2×C23semidirect
419. C23#8538direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35052
C23×#19012
C23×#19002
C23×#18702
C23×#14722
C23×#14712
C23×#14652
#853823