Finite Group Databasebeta

← C23×#10394 #19158020

Subgroups — C23×#10394 #19158020

Order 5888.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2617C211yesmaximal, characteristic
2C23×#1874C211yesmaximal, characteristic, complemented
3C23×#1852C211yesmaximal, characteristic, complemented
4C2×C23×#327C211yesmaximal, characteristic, complemented
5C2×C23×#323C211yesmaximal, characteristic, complemented
6C23×#1439C211yesmaximal, characteristic, complemented
7C23×#1294C211yesmaximal, characteristic, complemented
8C2×C23×#121C2211yescharacteristic
9C2×C23×#120—21no—
10C22×C4⋊C4×C23C2211yescharacteristic
11C22×C22⋊C4×C23—21no—
12C22×C22⋊C4×C23C2211yescharacteristic
13C2×C23×#107—21no—
14C2×C23×#102—21no—
15C2×C23×C22⋊C8C2211yescharacteristic
16C2×C23×C22⋊C8C2211yescharacteristic
17C23×#397—21no—
18C23×#394—21no—
19C23×#394—21no—
20C23×#392—21no—
21C23×#386—21no—
22C23×#380C2211yescharacteristic
23C2×C23×#95—21no—
24C2×C23×#95C2211yescharacteristic
25C23×#340—21no—
26C23×#340—21no—
27C23×#327—22no—
28C23×#323—22no—
29C25×C23D411yescharacteristic
30C22×D4×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23D411yescharacteristic
34C23×C4×C23—21no—
35C23×C4×C23—21no—
36C23×C4×C23D411yescharacteristic
37C23×C4×C23C2311yescharacteristic
38C22×C8×C23—21no—
39C22×C8×C23—21no—
40C23×#121—21no—
41C23×#121—21no—
42C23×#121—21no—
43C23×#121—21no—
44C23×#120—42no—
45C23×#120—42no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—42no—
49C2×C4⋊C4×C23D411yescharacteristic
50C2×C4⋊C4×C23D411yescharacteristic
51C2×C4⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—42no—
65C2×C22⋊C4×C23—42no—
66C2×C22⋊C4×C23—42no—
67C23×#107—42no—
68C23×#102—42no—
69C23×C22⋊C8—21no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8D412yescomplemented
73C23×#95—21no—
74C23×#95—21no—
75C23×#95—21no—
76C23×#95—21no—
77C23×#95—21no—
78C23×#95—42no—
79C24×C23—21no—
80C24×C23—21no—
81C24×C23—21no—
82C24×C23—21no—
83C24×C23—21no—
84C24×C23—21no—
85C24×C23—21no—
86C24×C23—41no—
87C24×C23—41no—
88C24×C23—41no—
89C24×C23—41no—
90C24×C23D812yes—
91C24×C23C2×D411yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—41no—
95C2×D4×C23—41no—
96C2×D4×C23—42no—
97C2×D4×C23—42no—
98C2×D4×C23—41no—
99C2×D4×C23—41no—
100C2×D4×C23—41no—
101C2×D4×C23—41no—
102C2×D4×C23—41no—
103C2×D4×C23—41no—
104C2×D4×C23—81no—
105C2×D4×C23—81no—
106C2×D4×C23—81no—
107C2×D4×C23—81no—
108C22×C4×C23—22no—
109C22×C4×C23—22no—
110C22×C4×C23—21no—
111C22×C4×C23—21no—
112C22×C4×C23—21no—
113C22×C4×C23—21no—
114C22×C4×C23—21no—
115C22×C4×C23—21no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23—41no—
121C22×C4×C23—41no—
122C22×C4×C23—41no—
123C22×C4×C23—41no—
124C22×C4×C23—41no—
125C22×C4×C23—41no—
126C22×C4×C23—41no—
127C22×C4×C23—41no—
128C22×C4×C23—41no—
129C22×C4×C23—41no—
130C22×C4×C23—41no—
131C22×C4×C23—41no—
132C22×C4×C23C2×D411yescharacteristic
133C22×C4×C23C2×D411yescharacteristic
134C22×C4×C23C2×D411yescharacteristic
135C22×C4×C23#4111yescharacteristic
136C22×C4×C23#4111yescharacteristic
137C22×C4×C23#4111yescharacteristic
138C2×C8×C23—22no—
139C2×C8×C23—41no—
140C2×C8×C23—41no—
141C2×C8×C23—41no—
142C2×C8×C23—41no—
143C2×C8×C23—41no—
144C4⋊C4×C23—22no—
145C4⋊C4×C23—22no—
146C4⋊C4×C23—21no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—41no—
149C4⋊C4×C23—41no—
150C4⋊C4×C23—41no—
151C4⋊C4×C23—41no—
152C22⋊C4×C23—42no—
153C22⋊C4×C23—42no—
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—81no—
164C22⋊C4×C23—81no—
165#10394C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—21no—
169C23×C23—21no—
170C23×C23—22no—
171C23×C23—22no—
172C23×C23—22no—
173C23×C23—42no—
174C23×C23—42no—
175C23×C23—42no—
176C23×C23—42no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23—41no—
180C23×C23—41no—
181C23×C23—41no—
182C23×C23—41no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—41no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23—41no—
192C23×C23—41no—
193C23×C23—41no—
194C23×C23—81no—
195C23×C23—81no—
196C23×C23—81no—
197C23×C23—81no—
198C23×C23—81no—
199C23×C23—81no—
200C23×C23#12011yescharacteristic
201C23×C23#12111yescharacteristic
202C23×C23#12311yescharacteristic
203C23×C23C2×D811yescharacteristic
204C23×C23#13511yescharacteristic
205C23×C23AGL(1,ℤ/8)11yescharacteristic
206C23×C23#13711yescharacteristic
207D4×C23—42no—
208D4×C23—81no—
209D4×C23—84no—
210D4×C23—84no—
211D4×C23—81no—
212D4×C23—81no—
213D4×C23—81no—
214D4×C23—81no—
215C2×C4×C23—21no—
216C2×C4×C23—21no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—41no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—41no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—42no—
228C2×C4×C23—42no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—41no—
232C2×C4×C23—41no—
233C2×C4×C23—41no—
234C2×C4×C23—41no—
235C2×C4×C23—41no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—42no—
240C2×C4×C23—41no—
241C2×C4×C23—41no—
242C2×C4×C23—81no—
243C2×C4×C23—81no—
244C2×C4×C23—81no—
245C2×C4×C23—81no—
246C2×C4×C23—81no—
247C2×C4×C23—81no—
248C2×C4×C23—81no—
249C2×C4×C23—81no—
250C2×C4×C23—81no—
251C2×C4×C23#12111yescharacteristic
252C2×C4×C23#12111yescharacteristic
253C2×C4×C23#12311yescharacteristic
254C2×C4×C23#12411yescharacteristic
255C8×C23—42no—
256C8×C23—81no—
257#2617C2×C2311yescharacteristic
258#1874C2×C2311yescharacteristic, complemented
259#1852C2×C2311yescharacteristic, complemented
260C2×#327C2×C2311yescharacteristic, complemented
261C2×#323C2×C2311yescharacteristic, complemented
262#1439C2×C2311yescharacteristic, complemented
263#1294C2×C2311yescharacteristic, complemented
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—21no—
267C22×C23—21no—
268C22×C23—21no—
269C22×C23—21no—
270C22×C23—21no—
271C22×C23—21no—
272C22×C23—21no—
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—21no—
276C22×C23—44no—
277C22×C23—42no—
278C22×C23—42no—
279C22×C23—42no—
280C22×C23—82no—
281C22×C23—82no—
282C22×C23—82no—
283C22×C23—81no—
284C22×C23—81no—
285C22×C23—81no—
286C22×C23—81no—
287C22×C23—81no—
288C22×C23—81no—
289C22×C23—81no—
290C22×C23—81no—
291C22×C23—81no—
292C22×C23—81no—
293C22×C23—81no—
294C22×C23#39411yescharacteristic
295C22×C23#44711yescharacteristic
296C22×C23#45211yescharacteristic
297C22×C23#45311yescharacteristic
298C22×C23#45511yescharacteristic
299C22×C23B(4,𝔽2)11yescharacteristic
300C22×C23#45811yescharacteristic
301C22×C23#46611yescharacteristic
302C22×C23#46811yescharacteristic
303C22×C23#48011yescharacteristic
304C22×C23#48211yescharacteristic
305C4×C23—21no—
306C4×C23—21no—
307C4×C23—21no—
308C4×C23—21no—
309C4×C23—82no—
310C4×C23—82no—
311C4×C23—81no—
312C4×C23—81no—
313C4×C23—81no—
314C4×C23—81no—
315C2×#121C22×C2311yescharacteristic
316C2×#120—21no—
317C22×C4⋊C4C22×C2311yescharacteristic
318C22×C22⋊C4—21no—
319C22×C22⋊C4C22×C2311yescharacteristic
320C2×#107—21no—
321C2×#102—21no—
322C2×C22⋊C8C22×C2311yescharacteristic
323C2×C22⋊C8C22×C2311yescharacteristic
324#397—21no—
325#394—21no—
326#394—21no—
327#392—21no—
328#386—21no—
329#380C22×C2311yescharacteristic
330C2×#95—21no—
331C2×#95C22×C2311yescharacteristic
332#340—21no—
333#340—21no—
334#327—22no—
335#323—22no—
336C2×C23—21no—
337C2×C23—21no—
338C2×C23—21no—
339C2×C23—21no—
340C2×C23—82no—
341C2×C23—161no—
342C2×C23#159511yescharacteristic
343C2×C23#160611yescharacteristic
344C2×C23#160711yescharacteristic
345C2×C23#161811yescharacteristic
346C2×C23#203311yescharacteristic
347C2×C23#203711yescharacteristic
348C2×C23#204411yescharacteristic
349C25D4×C2311yescharacteristic
350C22×D4—21no—
351C22×D4—21no—
352C22×D4—21no—
353C22×D4D4×C2311yescharacteristic
354C23×C4—21no—
355C23×C4—21no—
356C23×C4D4×C2311yescharacteristic
357C23×C4C23×C2311yescharacteristic
358C22×C8—21no—
359C22×C8—21no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#120—42no—
365#120—42no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—42no—
369C2×C4⋊C4D4×C2311yescharacteristic
370C2×C4⋊C4D4×C2311yescharacteristic
371C2×C4⋊C4D4×C2311yescharacteristic
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—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387#107—42no—
388#102—42no—
389C22⋊C8—21no—
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8D4×C2312yescomplemented
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—42no—
399C23#1039411yescharacteristic, direct product factor, Sylow
400C24—21no—
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24D8×C2312yes—
412C24C2×D4×C2311yescharacteristic
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
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×C4#41×C2311yescharacteristic
457C22×C4#41×C2311yescharacteristic
458C22×C4#41×C2311yescharacteristic
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—48no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23C23×#12011yescharacteristic
521C23C23×#12111yescharacteristic
522C23C23×#12311yescharacteristic
523C23C2×D8×C2311yescharacteristic
524C23C23×#13511yescharacteristic
525C23C23×AGL(1,ℤ/8)11yescharacteristic
526C23C23×#13711yescharacteristic
527D4—42no—
528D4—81no—
529D4—84no—
530D4—84no—
531D4—81no—
532D4—81no—
533D4—81no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4C23×#12111yescharacteristic
572C2×C4C23×#12111yescharacteristic
573C2×C4C23×#12311yescharacteristic
574C2×C4C23×#12411yescharacteristic
575C8—42no—
576C8—81no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C23×#39411yescharacteristic
608C22C23×#44711yescharacteristic
609C22C23×#45211yescharacteristic
610C22C23×#45311yescharacteristic
611C22C23×#45511yescharacteristic
612C22C23×B(4,𝔽2)11yescharacteristic
613C22C23×#45811yescharacteristic
614C22C23×#46611yescharacteristic
615C22C23×#46811yescharacteristic
616C22C23×#48011yescharacteristic
617C22C23×#48211yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—81no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—82no—
633C2—161no—
634C2C23×#159511yescharacteristic
635C2C23×#160611yescharacteristic
636C2C23×#160711yescharacteristic
637C2C23×#161811yescharacteristic
638C2C23×#203311yescharacteristic
639C2C23×#203711yescharacteristic
640C2C23×#204411yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2617C2
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
2. C23×#1874C2semidirect
3. C23×#1852C2semidirect
4. C2×C23×#327C2semidirect
5. C2×C23×#323C2semidirect
6. C23×#1439C2semidirect
7. C23×#1294C2semidirect
72. C23×C22⋊C8D4semidirect
165. #10394C23direct (N × H)
258. #1874C2×C23semidirect
259. #1852C2×C23semidirect
260. C2×#327C2×C23semidirect
261. C2×#323C2×C23semidirect
262. #1439C2×C23semidirect
263. #1294C2×C23semidirect
392. C22⋊C8D4×C23semidirect
399. C23#10394direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26172
C23×#18742
C23×#18522
C2×C23×#3272
C2×C23×#3232
C23×#14392
C23×#12942
#1039423