Finite Group Databasebeta

← C23×#16604 #19164230

Subgroups — C23×#16604 #19164230

Order 5888.

1578 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#125C211yesmaximal, characteristic, complemented
2C2×C23×#396C212yesmaximal
3C2×C23×#388C211yesmaximal, characteristic
4C23×#1444C211yesmaximal, characteristic, complemented
5C23×#1444C212yesmaximal, complemented
6C23×#1442C211yesmaximal, characteristic, complemented
7C23×#1441C212yesmaximal, complemented
8C23×#1439C211yesmaximal, characteristic
9C23×#1439C211yesmaximal, characteristic
10C23×#1439C212yesmaximal
11C23×C4⋊#95C211yesmaximal, characteristic, complemented
12C2×C23×#125—28no—
13C2×C23×#125—24no—
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2212yes—
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C4⋊C4×C23C2212yes—
18C22×C4⋊C4×C23C2212yes—
19C22×C4⋊C4×C23C2211yescharacteristic
20C22×C4⋊C4×C23C2212yes—
21C22×C22⋊C4×C23C2211yescharacteristic
22C22×C22⋊C4×C23C2212yes—
23C22×C22⋊C4×C23C2211yescharacteristic
24C22×C22⋊C4×C23C2211yescharacteristic
25C22×C22⋊C4×C23C2212yes—
26C22×C42×C23C2211yescharacteristic
27C23×#396—28no—
28C23×#388—24no—
29C2×C23×#95C2211yescharacteristic
30C2×C23×#95C2211yescharacteristic
31C2×C23×#95C2212yes—
32C2×C23×#95C2211yescharacteristic
33C2×C23×#95C2211yescharacteristic
34C2×C23×#95C2212yes—
35C2×C23×#95C2212yes—
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2212yes—
39C25×C23C2311yescharacteristic
40C23×C4×C23C2311yescharacteristic
41C23×C4×C23C2312yes—
42C23×C4×C23C2311yescharacteristic
43C23×C4×C23C2312yes—
44C23×C4×C23C2311yescharacteristic
45C23×C4×C23C2312yes—
46C23×C4×C23C2311yescharacteristic
47C23×C4×C23C2312yes—
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2311yescharacteristic
50C23×#125—232no—
51C2×C4⋊C4×C23—24no—
52C2×C4⋊C4×C23—24no—
53C2×C4⋊C4×C23—24no—
54C2×C4⋊C4×C23—24no—
55C2×C4⋊C4×C23—24no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—22no—
58C2×C4⋊C4×C23—24no—
59C2×C4⋊C4×C23—24no—
60C2×C4⋊C4×C23—22no—
61C2×C4⋊C4×C23—24no—
62C2×C4⋊C4×C23—24no—
63C2×C4⋊C4×C23—24no—
64C2×C4⋊C4×C23—42no—
65C2×C4⋊C4×C23—42no—
66C2×C4⋊C4×C23—42no—
67C2×C4⋊C4×C23—42no—
68C2×C4⋊C4×C23D418yes—
69C2×C22⋊C4×C23—22no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—22no—
72C2×C22⋊C4×C23—22no—
73C2×C22⋊C4×C23—24no—
74C2×C22⋊C4×C23—24no—
75C2×C22⋊C4×C23—24no—
76C2×C22⋊C4×C23—24no—
77C2×C22⋊C4×C23—42no—
78C2×C22⋊C4×C23—42no—
79C2×C22⋊C4×C23—42no—
80C2×C22⋊C4×C23—42no—
81C2×C22⋊C4×C23Q814yes—
82C2×C42×C23—24no—
83C2×C42×C23D414yes—
84C23×#95—22no—
85C23×#95—22no—
86C23×#95—24no—
87C23×#95—24no—
88C23×#95—22no—
89C23×#95—22no—
90C23×#95—42no—
91C23×#95—42no—
92C23×#95—42no—
93C23×#95—42no—
94C24×C23—21no—
95C24×C23—22no—
96C24×C23—21no—
97C24×C23—22no—
98C24×C23—21no—
99C24×C23—21no—
100C24×C23—22no—
101C24×C23—21no—
102C24×C23—21no—
103C24×C23—22no—
104C24×C23C2×Q812yes—
105C24×C23C2411yescharacteristic
106C22×C4×C23—22no—
107C22×C4×C23—22no—
108C22×C4×C23—21no—
109C22×C4×C23—22no—
110C22×C4×C23—21no—
111C22×C4×C23—22no—
112C22×C4×C23—22no—
113C22×C4×C23—21no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
117C22×C4×C23—22no—
118C22×C4×C23—21no—
119C22×C4×C23—22no—
120C22×C4×C23—22no—
121C22×C4×C23—22no—
122C22×C4×C23—22no—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—22no—
128C22×C4×C23—22no—
129C22×C4×C23—22no—
130C22×C4×C23—22no—
131C22×C4×C23—21no—
132C22×C4×C23—22no—
133C22×C4×C23—21no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—21no—
138C22×C4×C23—22no—
139C22×C4×C23—22no—
140C22×C4×C23—21no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—22no—
145C22×C4×C23—22no—
146C22×C4×C23—21no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—21no—
150C22×C4×C23—22no—
151C22×C4×C23—21no—
152C22×C4×C23—21no—
153C22×C4×C23—22no—
154C22×C4×C23C2×D412yes—
155C22×C4×C23C2×D414yes—
156C22×C4×C23C2×D414yes—
157C22×C4×C23C2×D414yes—
158C22×C4×C23C2×D412yes—
159C22×C4×C23C2×D412yes—
160C22×C4×C23C2×Q812yes—
161C22×C4×C23C2×Q812yes—
162C22×C4×C23#4114yes—
163C22×C4×C23#4112yes—
164C4⋊C4×C23—28no—
165C4⋊C4×C23—28no—
166C4⋊C4×C23—216no—
167C4⋊C4×C23—216no—
168C4⋊C4×C23—48no—
169C4⋊C4×C23—44no—
170C4⋊C4×C23—48no—
171C22⋊C4×C23—28no—
172C22⋊C4×C23—48no—
173C22⋊C4×C23—44no—
174C22⋊C4×C23—48no—
175C22⋊C4×C23—44no—
176C42×C23—28no—
177#16604C2311yesmaximal, characteristic, direct product factor, Sylow
178C23×C23—22no—
179C23×C23—22no—
180C23×C23—22no—
181C23×C23—24no—
182C23×C23—24no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—42no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—42no—
191C23×C23—42no—
192C23×C23—41no—
193C23×C23—41no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—42no—
197C23×C23—42no—
198C23×C23—42no—
199C23×C23—42no—
200C23×C23—42no—
201C23×C23C22×D411yescharacteristic
202C23×C23C22×D412yes—
203C23×C23C22×Q811yescharacteristic
204C23×C23C2×#4111yescharacteristic
205C23×C23C2×#4112yes—
206C23×C23H(5,𝔽2)11yescharacteristic
207C23×C23H(5,𝔽2)12yes—
208C23×C23H(5,𝔽2)11yescharacteristic
209C23×C232−1+412yes—
210C23×C232−1+411yescharacteristic
211C23×C232−1+411yescharacteristic
212C2×C4×C23—22no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—22no—
216C2×C4×C23—24no—
217C2×C4×C23—24no—
218C2×C4×C23—24no—
219C2×C4×C23—24no—
220C2×C4×C23—22no—
221C2×C4×C23—24no—
222C2×C4×C23—24no—
223C2×C4×C23—22no—
224C2×C4×C23—22no—
225C2×C4×C23—24no—
226C2×C4×C23—22no—
227C2×C4×C23—24no—
228C2×C4×C23—24no—
229C2×C4×C23—24no—
230C2×C4×C23—24no—
231C2×C4×C23—24no—
232C2×C4×C23—24no—
233C2×C4×C23—24no—
234C2×C4×C23—41no—
235C2×C4×C23—42no—
236C2×C4×C23—41no—
237C2×C4×C23—42no—
238C2×C4×C23—42no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—42no—
248C2×C4×C23—42no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—42no—
254C2×C4×C23—41no—
255C2×C4×C23—42no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—42no—
260C2×C4×C23—42no—
261C2×C4×C23—42no—
262C2×C4×C23—42no—
263C2×C4×C23—42no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23#12014yes—
269C2×C4×C23C22⋊Q814yes—
270C22×#125C2×C2311yescharacteristic, complemented
271C2×#396C2×C2312yes—
272C2×#388C2×C2311yescharacteristic
273#1444C2×C2311yescharacteristic, complemented
274#1444C2×C2312yescomplemented
275#1442C2×C2311yescharacteristic, complemented
276#1441C2×C2312yescomplemented
277#1439C2×C2311yescharacteristic
278#1439C2×C2312yes—
279#1439C2×C2311yescharacteristic
280C4⋊#95C2×C2311yescharacteristic, complemented
281C22×C23—42no—
282C22×C23—42no—
283C22×C23—42no—
284C22×C23—44no—
285C22×C23—44no—
286C22×C23—81no—
287C22×C23—81no—
288C22×C23—81no—
289C22×C23—81no—
290C22×C23—82no—
291C22×C23—82no—
292C22×C23C2×#12011yescharacteristic
293C22×C23C2×C22⋊Q811yescharacteristic
294C22×C23#53211yescharacteristic
295C22×C23#53511yescharacteristic
296C22×C23#53512yes—
297C22×C23#53611yescharacteristic
298C22×C23#53711yescharacteristic
299C22×C23#53711yescharacteristic
300C22×C23#53712yes—
301C22×C23#53912yes—
302C22×C23#54011yescharacteristic
303C22×C23#54111yescharacteristic
304C22×C23#54112yes—
305C22×C23#54411yescharacteristic
306C22×C23#54411yescharacteristic
307C22×C23#54612yes—
308C22×C23#54712yes—
309C22×C23#54812yes—
310C22×C23D4×Q812yes—
311C22×C23#55312yes—
312C22×C23#55412yes—
313C22×C23#56212yes—
314C22×C23#56312yes—
315C4×C23—24no—
316C4×C23—24no—
317C4×C23—42no—
318C4×C23—42no—
319C4×C23—44no—
320C4×C23—44no—
321C4×C23—44no—
322C4×C23—81no—
323C4×C23—81no—
324C4×C23—82no—
325C2×#125—28no—
326C2×#125—24no—
327C22×C4⋊C4C22×C2311yescharacteristic
328C22×C4⋊C4C22×C2311yescharacteristic
329C22×C4⋊C4C22×C2312yes—
330C22×C4⋊C4C22×C2312yes—
331C22×C4⋊C4C22×C2312yes—
332C22×C4⋊C4C22×C2312yes—
333C22×C4⋊C4C22×C2311yescharacteristic
334C22×C22⋊C4C22×C2311yescharacteristic
335C22×C22⋊C4C22×C2312yes—
336C22×C22⋊C4C22×C2311yescharacteristic
337C22×C22⋊C4C22×C2312yes—
338C22×C22⋊C4C22×C2311yescharacteristic
339C22×C42C22×C2311yescharacteristic
340#396—28no—
341#388—24no—
342C2×#95C22×C2312yes—
343C2×#95C22×C2311yescharacteristic
344C2×#95C22×C2311yescharacteristic
345C2×#95C22×C2312yes—
346C2×#95C22×C2311yescharacteristic
347C2×#95C22×C2311yescharacteristic
348C2×#95C22×C2312yes—
349C2×#95C22×C2312yes—
350C2×#95C22×C2312yes—
351C2×#95C22×C2312yes—
352C2×C23—82no—
353C2×C23#240911yescharacteristic
354C2×C23#243011yescharacteristic
355C2×C23#256611yescharacteristic
356C2×C23#261511yescharacteristic
357C2×C23#262911yescharacteristic
358C2×C23#266411yescharacteristic
359C2×C23#270412yes—
360C2×C23#271112yes—
361C2×C23#272012yes—
362C2×C23#272512yes—
363C2×C23#282011yescharacteristic
364C25C23×C2311yescharacteristic
365C23×C4C23×C2311yescharacteristic
366C23×C4C23×C2312yes—
367C23×C4C23×C2311yescharacteristic
368C23×C4C23×C2312yes—
369C23×C4C23×C2312yes—
370C23×C4C23×C2311yescharacteristic
371C23×C4C23×C2311yescharacteristic
372C23×C4C23×C2312yes—
373C23×C4C23×C2311yescharacteristic
374C23×C4C23×C2311yescharacteristic
375#125—232no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—42no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—42no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4D4×C2318yes—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4Q8×C2314yes—
407C2×C42—24no—
408C2×C42D4×C2314yes—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—22no—
414#95—22no—
415#95—42no—
416#95—42no—
417#95—42no—
418#95—42no—
419C23#1660411yescharacteristic, direct product factor, Sylow
420C24—22no—
421C24—21no—
422C24—22no—
423C24—21no—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24C2×Q8×C2312yes—
431C24C24×C2311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4C2×D4×C2314yes—
481C22×C4C2×D4×C2312yes—
482C22×C4C2×D4×C2314yes—
483C22×C4C2×D4×C2312yes—
484C22×C4C2×D4×C2314yes—
485C22×C4C2×D4×C2312yes—
486C22×C4C2×Q8×C2312yes—
487C22×C4C2×Q8×C2312yes—
488C22×C4#41×C2314yes—
489C22×C4#41×C2312yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C22⋊C4—28no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C42—28no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23C22×D4×C2312yes—
527C23C22×D4×C2311yescharacteristic
528C23C22×Q8×C2311yescharacteristic
529C23C2×#41×C2312yes—
530C23C2×#41×C2311yescharacteristic
531C23C23×H(5,𝔽2)12yes—
532C23C23×H(5,𝔽2)11yescharacteristic
533C23C23×H(5,𝔽2)11yescharacteristic
534C23C23×2−1+411yescharacteristic
535C23C23×2−1+412yes—
536C23C23×2−1+411yescharacteristic
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4C23×#12014yes—
594C2×C4C23×C22⋊Q814yes—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—81no—
602C22—82no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22C2×C23×#12011yescharacteristic
607C22C2×C23×C22⋊Q811yescharacteristic
608C22C23×#53211yescharacteristic
609C22C23×#53512yes—
610C22C23×#53511yescharacteristic
611C22C23×#53611yescharacteristic
612C22C23×#53711yescharacteristic
613C22C23×#53711yescharacteristic
614C22C23×#53712yes—
615C22C23×#53912yes—
616C22C23×#54011yescharacteristic
617C22C23×#54112yes—
618C22C23×#54111yescharacteristic
619C22C23×#54411yescharacteristic
620C22C23×#54411yescharacteristic
621C22C23×#54612yes—
622C22C23×#54712yes—
623C22C23×#54812yes—
624C22D4×Q8×C2312yes—
625C22C23×#55312yes—
626C22C23×#55412yes—
627C22C23×#56212yes—
628C22C23×#56312yes—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—82no—
637C4—81no—
638C4—81no—
639C2—82no—
640C2C23×#240911yescharacteristic
641C2C23×#243011yescharacteristic
642C2C23×#256611yescharacteristic
643C2C23×#261511yescharacteristic
644C2C23×#262911yescharacteristic
645C2C23×#266411yescharacteristic
646C2C23×#270412yes—
647C2C23×#271112yes—
648C2C23×#272012yes—
649C2C23×#272512yes—
650C2C23×#282011yescharacteristic
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC24C24×C23
prime radicalC25×C23C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C23×#125C2semidirect
4. C23×#1444C2semidirect
5. C23×#1444C2semidirect
6. C23×#1442C2semidirect
7. C23×#1441C2semidirect
11. C23×C4⋊#95C2semidirect
177. #16604C23direct (N × H)
270. C22×#125C2×C23semidirect
273. #1444C2×C23semidirect
274. #1444C2×C23semidirect
275. #1442C2×C23semidirect
276. #1441C2×C23semidirect
280. C4⋊#95C2×C23semidirect
419. C23#16604direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1252
C2×C23×#3962
C2×C23×#3882
C23×#14442
C23×#14422
C23×#14412
C23×#14392
C23×C4⋊#952
#1660423