Finite Group Databasebeta

← C23×#18538 #19166164

Subgroups — C23×#18538 #19166164

Order 5888.

922 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3482C211yesmaximal, characteristic
2C23×#3139C211yesmaximal, characteristic, complemented
3C23×#3079C211yesmaximal, characteristic, complemented
4C23×#3079C211yesmaximal, characteristic, complemented
5C23×#1638C212yesmaximal, complemented
6C23×#1637C212yesmaximal, complemented
7C23×#1636C212yesmaximal, complemented
8C23×#1635C212yesmaximal, complemented
9C23×#1561C211yesmaximal, characteristic, complemented
10C23×#1489C211yesmaximal, characteristic, complemented
11C23×#1489C211yesmaximal, characteristic, complemented
12C23×#557C2211yescharacteristic
13C23×#556—21no—
14C23×#556C2211yescharacteristic
15C23×#552—22no—
16C23×#552—21no—
17D4×Q8×C23C2211yescharacteristic
18C23×#548—21no—
19C23×#547—21no—
20C23×#547C2211yescharacteristic
21C23×#533—21no—
22C23×#533C2211yescharacteristic
23C2×C23×C4⋊Q8C2211yescharacteristic
24C23×#529—21no—
25C23×#529—22no—
26C4×#41×C23—21no—
27C4×#41×C23C2211yescharacteristic
28C4×#41×C23C2211yescharacteristic
29C2×C4×Q8×C23—21no—
30C23×#491—22no—
31C23×#487C2212yes—
32C23×#487C2212yes—
33C23×#477C2212yes—
34C23×#477C2212yes—
35C23×#476C2212yes—
36C23×#476C2212yes—
37C23×#475—21no—
38C23×#475—21no—
39C23×#474—21no—
40C23×#474—21no—
41C23×#428—21no—
42C23×#428—21no—
43C23×#423C2211yescharacteristic
44C23×#423C2211yescharacteristic
45C23×#419—22no—
46C23×#417—21no—
47C23×#417—21no—
48C4×M16×C23C2211yescharacteristic
49C23×#332C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×D4⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C2×#41×C23—21no—
56C2×#41×C23D411yescharacteristic
57C2×#41×C23D411yescharacteristic
58C22×Q8×C23—21no—
59C22×Q8×C23—21no—
60C2×M16×C23—21no—
61C2×M16×C23—21no—
62C2×M16×C23—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8—21no—
67C23×C4⋊Q8—21no—
68C23×C4⋊Q8C2312yes—
69C23×C4⋊Q8C2311yescharacteristic
70C23×C4⋊Q8C2311yescharacteristic
71C23×#126—22no—
72C23×#126—22no—
73C23×#126—22no—
74C23×#125—22no—
75C23×#125—22no—
76C23×#125—21no—
77C23×#125D412yes—
78C23×#124—22no—
79C23×#124—21no—
80C23×#123—22no—
81C23×#123—22no—
82C23×#123—22no—
83C23×C22⋊Q8—22no—
84C23×C22⋊Q8—21no—
85C23×C22⋊Q8—21no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—22no—
90C23×#121—22no—
91C23×#121—21no—
92C4×Q8×C23—22no—
93C4×Q8×C23—22no—
94C4×Q8×C23—21no—
95C4×Q8×C23—21no—
96C4×Q8×C23—21no—
97C4×Q8×C23—21no—
98C4×Q8×C23—21no—
99C4×Q8×C23D411yescharacteristic
100C4×Q8×C23D411yescharacteristic
101C4×Q8×C23C2311yescharacteristic
102C4×Q8×C23C2311yescharacteristic
103C4×D4×C23—22no—
104C4×D4×C23—22no—
105C4×D4×C23—21no—
106C4×D4×C23—21no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23—21no—
110C4×D4×C23D411yescharacteristic
111C4×D4×C23D411yescharacteristic
112C4×D4×C23C2311yescharacteristic
113C4×D4×C23C2311yescharacteristic
114C23×#117—22no—
115C23×#117—22no—
116C23×#117—21no—
117C23×#117—21no—
118C23×#117—21no—
119C23×#117D411yescharacteristic
120C23×#117D411yescharacteristic
121C23×#117D412yes—
122C2×C4⋊C4×C23—22no—
123C2×C4⋊C4×C23—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C42×C23—22no—
127C2×C42×C23—21no—
128C2×C42×C23—21no—
129C2×C42×C23C2311yescharacteristic
130C23×#107—22no—
131C23×#107—22no—
132C23×#106—22no—
133C23×#106—22no—
134C23×C4⋊C8—21no—
135C23×C4⋊C8—21no—
136C23×C4⋊C8C2312yes—
137C23×C4⋊C8C2312yes—
138C23×#103—22no—
139C23×#103—22no—
140C23×#103—24no—
141C23×#103—24no—
142C23×#102—22no—
143C23×#102—22no—
144C23×#97—21no—
145C4×C8×C23C2312yes—
146#41×C23—22no—
147#41×C23—22no—
148#41×C23—41no—
149#41×C23—41no—
150#41×C23—42no—
151#41×C23—42no—
152C2×Q8×C23—22no—
153C2×Q8×C23—21no—
154C2×Q8×C23—21no—
155C2×Q8×C23—22no—
156C2×Q8×C23—44no—
157C2×Q8×C23—42no—
158C2×Q8×C23—42no—
159C2×Q8×C23—41no—
160C2×Q8×C23C2×D411yescharacteristic
161C2×Q8×C23C2×D411yescharacteristic
162C2×D4×C23—22no—
163C2×D4×C23—21no—
164C2×D4×C23—21no—
165C2×D4×C23C2×D411yescharacteristic
166C2×D4×C23C2×D411yescharacteristic
167C22×C4×C23—22no—
168C22×C4×C23—22no—
169C22×C4×C23—21no—
170C22×C4×C23—21no—
171C22×C4×C23—21no—
172C22×C4×C23—21no—
173C22×C4×C23—21no—
174C22×C4×C23—21no—
175C22×C4×C23—41no—
176C22×C4×C23C2×D411yescharacteristic
177C22×C4×C23C2×D411yescharacteristic
178C22×C4×C23C2×D411yescharacteristic
179M16×C23—42no—
180M16×C23—42no—
181M16×C23—42no—
182C2×C8×C23—22no—
183C2×C8×C23—22no—
184C2×C8×C23—22no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—22no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—22no—
190C4⋊C4×C23—22no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—21no—
198C4⋊C4×C23—21no—
199C4⋊C4×C23—22no—
200C4⋊C4×C23—22no—
201C4⋊C4×C23—41no—
202C4⋊C4×C23—41no—
203C4⋊C4×C23C2×D412yes—
204C4⋊C4×C23C2×D412yes—
205C4⋊C4×C23C2×D411yescharacteristic
206C4⋊C4×C23C2×D411yescharacteristic
207C4⋊C4×C23C2×D411yescharacteristic
208C4⋊C4×C23C2×D411yescharacteristic
209C22⋊C4×C23—22no—
210C22⋊C4×C23—22no—
211C22⋊C4×C23—22no—
212C22⋊C4×C23—22no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—21no—
215C22⋊C4×C23—22no—
216C22⋊C4×C23—21no—
217C22⋊C4×C23—21no—
218C22⋊C4×C23—21no—
219C22⋊C4×C23—21no—
220C42×C23—22no—
221C42×C23—22no—
222C42×C23—21no—
223C42×C23—21no—
224C42×C23—21no—
225C42×C23—22no—
226C42×C23—21no—
227C42×C23—21no—
228C42×C23—21no—
229C42×C23C2×D411yescharacteristic
230C42×C23C2×D411yescharacteristic
231C42×C23C2×D411yescharacteristic
232C42×C23C2411yescharacteristic
233#18538C2311yesmaximal, characteristic, direct product factor, Sylow
234C23×C23—21no—
235C23×C23—21no—
236C23×C23#12011yescharacteristic
237Q8×C23—21no—
238Q8×C23—21no—
239Q8×C23—21no—
240Q8×C23—44no—
241Q8×C23—44no—
242Q8×C23—41no—
243Q8×C23—41no—
244Q8×C23—41no—
245Q8×C23—41no—
246Q8×C23—82no—
247Q8×C23#13712yes—
248D4×C23—21no—
249D4×C23—21no—
250D4×C23—21no—
251D4×C23—42no—
252D4×C23—42no—
253D4×C23—44no—
254D4×C23#13712yes—
255C2×C4×C23—21no—
256C2×C4×C23—21no—
257C2×C4×C23—21no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—22no—
261C2×C4×C23—21no—
262C2×C4×C23—21no—
263C2×C4×C23—21no—
264C2×C4×C23—22no—
265C2×C4×C23—21no—
266C2×C4×C23—21no—
267C2×C4×C23—22no—
268C2×C4×C23—42no—
269C2×C4×C23—42no—
270C2×C4×C23—44no—
271C2×C4×C23—44no—
272C2×C4×C23—41no—
273C2×C4×C23—41no—
274C2×C4×C23—41no—
275C2×C4×C23—41no—
276C2×C4×C23—42no—
277C2×C4×C23—42no—
278C2×C4×C23—42no—
279C2×C4×C23—82no—
280C2×C4×C23#12011yescharacteristic
281C2×C4×C23#12011yescharacteristic
282C2×C4×C23#12011yescharacteristic
283C2×C4×C23C22×D411yescharacteristic
284C2×C4×C23C22×D411yescharacteristic
285C2×C4×C23C22×D411yescharacteristic
286C8×C23—42no—
287C8×C23—42no—
288C8×C23—42no—
289#3482C2×C2311yescharacteristic
290#3139C2×C2311yescharacteristic, complemented
291#3079C2×C2311yescharacteristic, complemented
292#3079C2×C2311yescharacteristic, complemented
293#1638C2×C2312yescomplemented
294#1637C2×C2312yescomplemented
295#1636C2×C2312yescomplemented
296#1635C2×C2312yescomplemented
297#1561C2×C2311yescharacteristic, complemented
298#1489C2×C2311yescharacteristic, complemented
299#1489C2×C2311yescharacteristic, complemented
300C22×C23—21no—
301C22×C23—21no—
302C22×C23—22no—
303C22×C23—21no—
304C22×C23—41no—
305C22×C23—41no—
306C22×C23—41no—
307C22×C23—41no—
308C22×C23—41no—
309C22×C23C2×#12011yescharacteristic
310C4×C23—21no—
311C4×C23—21no—
312C4×C23—22no—
313C4×C23—21no—
314C4×C23—41no—
315C4×C23—41no—
316C4×C23—42no—
317C4×C23—42no—
318C4×C23—41no—
319C4×C23—41no—
320C4×C23—41no—
321C4×C23—82no—
322C4×C23#45412yes—
323C4×C23C2×#13711yescharacteristic
324C4×C23C2×#13711yescharacteristic
325C4×C23#57511yescharacteristic
326C4×C23#57511yescharacteristic
327#557C22×C2311yescharacteristic
328#556—21no—
329#556C22×C2311yescharacteristic
330#552—22no—
331#552—21no—
332D4×Q8C22×C2311yescharacteristic
333#548—21no—
334#547—21no—
335#547C22×C2311yescharacteristic
336#533—21no—
337#533C22×C2311yescharacteristic
338C2×C4⋊Q8C22×C2311yescharacteristic
339#529—21no—
340#529—22no—
341C4×#41—21no—
342C4×#41C22×C2311yescharacteristic
343C4×#41C22×C2311yescharacteristic
344C2×C4×Q8—21no—
345#491—22no—
346#487C22×C2312yes—
347#487C22×C2312yes—
348#477C22×C2312yes—
349#477C22×C2312yes—
350#476C22×C2312yes—
351#476C22×C2312yes—
352#475—21no—
353#475—21no—
354#474—21no—
355#474—21no—
356#428—21no—
357#428—21no—
358#423C22×C2311yescharacteristic
359#423C22×C2311yescharacteristic
360#419—22no—
361#417—21no—
362#417—21no—
363C4×M16C22×C2311yescharacteristic
364#332C22×C2312yes—
365#332C22×C2312yes—
366Q8⋊C8C22×C2312yes—
367Q8⋊C8C22×C2312yes—
368D4⋊C8C22×C2312yes—
369D4⋊C8C22×C2312yes—
370C2×C23—41no—
371C2×C23—42no—
372C2×C23—81no—
373C2×C23#300711yescharacteristic
374C2×C23#300711yescharacteristic
375C2×C23C2×#45411yescharacteristic
376C2×#41—21no—
377C2×#41D4×C2311yescharacteristic
378C2×#41D4×C2311yescharacteristic
379C22×Q8—21no—
380C22×Q8—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8C23×C2312yes—
390C4⋊Q8C23×C2311yescharacteristic
391C4⋊Q8C23×C2311yescharacteristic
392#126—22no—
393#126—22no—
394#126—22no—
395#125—22no—
396#125—22no—
397#125—21no—
398#125D4×C2312yes—
399#124—22no—
400#124—21no—
401#123—22no—
402#123—22no—
403#123—22no—
404C22⋊Q8—22no—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411#121—22no—
412#121—21no—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8D4×C2311yescharacteristic
421C4×Q8D4×C2311yescharacteristic
422C4×Q8C23×C2311yescharacteristic
423C4×Q8C23×C2311yescharacteristic
424C4×D4—22no—
425C4×D4—22no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4D4×C2311yescharacteristic
432C4×D4D4×C2311yescharacteristic
433C4×D4C23×C2311yescharacteristic
434C4×D4C23×C2311yescharacteristic
435#117—22no—
436#117—22no—
437#117—21no—
438#117—21no—
439#117—21no—
440#117D4×C2311yescharacteristic
441#117D4×C2311yescharacteristic
442#117D4×C2312yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C42—22no—
448C2×C42—21no—
449C2×C42—21no—
450C2×C42C23×C2311yescharacteristic
451#107—22no—
452#107—22no—
453#106—22no—
454#106—22no—
455C4⋊C8—21no—
456C4⋊C8—21no—
457C4⋊C8C23×C2312yes—
458C4⋊C8C23×C2312yes—
459#103—22no—
460#103—22no—
461#103—24no—
462#103—24no—
463#102—22no—
464#102—22no—
465#97—21no—
466C4×C8C23×C2312yes—
467C23#1853811yescharacteristic, direct product factor, Sylow
468#41—22no—
469#41—22no—
470#41—41no—
471#41—41no—
472#41—42no—
473#41—42no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—44no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—41no—
482C2×Q8C2×D4×C2311yescharacteristic
483C2×Q8C2×D4×C2311yescharacteristic
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4C2×D4×C2311yescharacteristic
488C2×D4C2×D4×C2311yescharacteristic
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—41no—
498C22×C4C2×D4×C2311yescharacteristic
499C22×C4C2×D4×C2311yescharacteristic
500C22×C4C2×D4×C2311yescharacteristic
501M16—42no—
502M16—42no—
503M16—42no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4C2×D4×C2312yes—
526C4⋊C4C2×D4×C2312yes—
527C4⋊C4C2×D4×C2311yescharacteristic
528C4⋊C4C2×D4×C2311yescharacteristic
529C4⋊C4C2×D4×C2311yescharacteristic
530C4⋊C4C2×D4×C2311yescharacteristic
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42C2×D4×C2311yescharacteristic
552C42C2×D4×C2311yescharacteristic
553C42C2×D4×C2311yescharacteristic
554C42C24×C2311yescharacteristic
555C23—21no—
556C23—21no—
557C23C23×#12011yescharacteristic
558Q8—21no—
559Q8—21no—
560Q8—21no—
561Q8—44no—
562Q8—44no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—82no—
568Q8C23×#13712yes—
569D4—21no—
570D4—21no—
571D4—21no—
572D4—42no—
573D4—42no—
574D4—44no—
575D4C23×#13712yes—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4C23×#12011yescharacteristic
602C2×C4C23×#12011yescharacteristic
603C2×C4C23×#12011yescharacteristic
604C2×C4C22×D4×C2311yescharacteristic
605C2×C4C22×D4×C2311yescharacteristic
606C2×C4C22×D4×C2311yescharacteristic
607C8—42no—
608C8—42no—
609C8—42no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—21no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22C2×C23×#12011yescharacteristic
620C4—21no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4C23×#45412yes—
633C4C2×C23×#13711yescharacteristic
634C4C2×C23×#13711yescharacteristic
635C4C23×#57511yescharacteristic
636C4C23×#57511yescharacteristic
637C2—41no—
638C2—42no—
639C2—81no—
640C2C23×#300711yescharacteristic
641C2C23×#300711yescharacteristic
642C2C2×C23×#45411yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3482C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3139C2semidirect
3. C23×#3079C2semidirect
4. C23×#3079C2semidirect
5. C23×#1638C2semidirect
6. C23×#1637C2semidirect
7. C23×#1636C2semidirect
8. C23×#1635C2semidirect
9. C23×#1561C2semidirect
10. C23×#1489C2semidirect
11. C23×#1489C2semidirect
233. #18538C23direct (N × H)
290. #3139C2×C23semidirect
291. #3079C2×C23semidirect
292. #3079C2×C23semidirect
293. #1638C2×C23semidirect
294. #1637C2×C23semidirect
295. #1636C2×C23semidirect
296. #1635C2×C23semidirect
297. #1561C2×C23semidirect
298. #1489C2×C23semidirect
299. #1489C2×C23semidirect
467. C23#18538direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34822
C23×#31392
C23×#30792
C23×#16382
C23×#16372
C23×#16362
C23×#16352
C23×#15612
C23×#14892
#1853823