Finite Group Databasebeta

← C23×#18961 #19166587

Subgroups — C23×#18961 #19166587

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3555C211yesmaximal, characteristic
2C23×#3238C211yesmaximal, characteristic
3C23×#3230C211yesmaximal, characteristic, complemented
4C23×#3209C211yesmaximal, characteristic, complemented
5C23×#1638C212yesmaximal, complemented
6C23×#1636C212yesmaximal, complemented
7C23×#1633C212yesmaximal, complemented
8C23×#1632C212yesmaximal, complemented
9C23×#1554C211yesmaximal, characteristic, complemented
10C23×#1497C211yesmaximal, characteristic, complemented
11C23×#1482C211yesmaximal, characteristic
12C23×#563—22no—
13Q82×C23C2211yescharacteristic
14C23×#557—21no—
15C23×#557C2211yescharacteristic
16C23×#556—21no—
17C23×#556—21no—
18C23×#556C2211yescharacteristic
19C23×#554C2211yescharacteristic
20C23×#552—21no—
21C23×#544—21no—
22C23×#542—21no—
23C23×#541—21no—
24C23×#541—21no—
25C23×#541—22no—
26C23×#533C2211yescharacteristic
27C23×#533C2211yescharacteristic
28C2×C23×C4⋊Q8C2211yescharacteristic
29C23×#529C2211yescharacteristic
30C23×#501—21no—
31C23×#500—21no—
32C23×#494C2212yes—
33C23×#489—21no—
34C23×#487—21no—
35C23×#487C2212yes—
36C23×#484—21no—
37C23×#484—21no—
38C23×#483—21no—
39C23×#480—21no—
40C23×#477C2212yes—
41C23×#477C2212yes—
42C23×#477C2212yes—
43C23×#476C2212yes—
44C23×#462—22no—
45C23×#451—22no—
46C23×#432C2211yescharacteristic
47C23×#431C2211yescharacteristic
48C23×#431C2211yescharacteristic
49C23×#333C2212yes—
50C23×#332C2212yes—
51C23×Q8⋊C8C2212yes—
52C23×Q8⋊C8C2212yes—
53C23×Q8⋊C8C2212yes—
54C23×D4⋊C8C2212yes—
55C22×Q8×C23—21no—
56C2×Q16×C23—24no—
57C23×C4⋊Q8—22no—
58C23×C4⋊Q8—21no—
59C23×C4⋊Q8—22no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8—21no—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8D411yescharacteristic
66C23×C4⋊Q8D411yescharacteristic
67C23×C4⋊Q8C2312yes—
68C23×C4⋊Q8C2311yescharacteristic
69C23×C4⋊Q8C2311yescharacteristic
70C23×#126—22no—
71C23×#126—22no—
72C23×#126—22no—
73C23×#126—22no—
74C23×#126—22no—
75C23×#126—22no—
76C23×#126—21no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—22no—
80C23×#125—22no—
81C23×#125—21no—
82C23×#125—21no—
83C23×#125—22no—
84C23×#125—21no—
85C23×#125—21no—
86C23×#125D411yescharacteristic
87C23×#124—22no—
88C23×#124—21no—
89C23×#124D411yescharacteristic
90C23×#123—22no—
91C23×#123—21no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—21no—
95C23×C22⋊Q8—21no—
96C23×C22⋊Q8—22no—
97C23×C22⋊Q8—21no—
98C23×C22⋊Q8—21no—
99C23×C22⋊Q8D411yescharacteristic
100C23×C22⋊Q8D411yescharacteristic
101C23×C22⋊Q8D412yes—
102C23×C22⋊Q8D411yescharacteristic
103C23×#121D411yescharacteristic
104C4×Q8×C23—22no—
105C4×Q8×C23—21no—
106C4×Q8×C23—21no—
107C4×Q8×C23—21no—
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23—21no—
111C4×Q8×C23—21no—
112C4×Q8×C23D412yes—
113C4×Q8×C23C2311yescharacteristic
114C4×Q8×C23C2311yescharacteristic
115C4×Q8×C23C2311yescharacteristic
116C4×D4×C23—21no—
117C4×D4×C23—21no—
118C4×D4×C23C2311yescharacteristic
119C23×#117—22no—
120C23×#117—21no—
121C23×#117—21no—
122C23×#117—21no—
123C23×#117—21no—
124C2×C4⋊C4×C23—21no—
125C2×C4⋊C4×C23—21no—
126C2×C4⋊C4×C23—21no—
127C2×C42×C23C2311yescharacteristic
128C23×#107—22no—
129C23×#107—22no—
130C23×#107—22no—
131C23×#106—22no—
132C23×C4⋊C8—21no—
133C23×C4⋊C8C2312yes—
134C23×C4⋊C8C2312yes—
135C23×#103—22no—
136C23×#103—22no—
137C23×#103—22no—
138C23×#103—24no—
139C23×#102—22no—
140C23×C22⋊C8—21no—
141C23×C22⋊C8—21no—
142C23×C22⋊C8—21no—
143C23×#97—21no—
144C4×C8×C23—21no—
145C4×C8×C23C2312yes—
146C2×Q8×C23—22no—
147C2×Q8×C23—21no—
148C2×Q8×C23—21no—
149C2×Q8×C23—22no—
150C2×Q8×C23—41no—
151C2×Q8×C23—82no—
152C2×Q8×C23C2×D411yescharacteristic
153C2×Q8×C23C2×D411yescharacteristic
154C2×Q8×C23C2×D411yescharacteristic
155C2×Q8×C23C2×D412yes—
156C2×D4×C23—21no—
157C2×D4×C23C2×D411yescharacteristic
158C22×C4×C23—22no—
159C22×C4×C23—21no—
160C22×C4×C23—41no—
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2×D411yescharacteristic
164Q16×C23—84no—
165C2×C8×C23—22no—
166C2×C8×C23—22no—
167C2×C8×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—21no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—22no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—21no—
188C4⋊C4×C23—21no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—21no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—22no—
194C4⋊C4×C23—41no—
195C4⋊C4×C23—41no—
196C4⋊C4×C23C2×D412yes—
197C4⋊C4×C23C2×D411yescharacteristic
198C4⋊C4×C23C2×D411yescharacteristic
199C4⋊C4×C23C2×D411yescharacteristic
200C4⋊C4×C23C2×D411yescharacteristic
201C22⋊C4×C23—22no—
202C22⋊C4×C23—22no—
203C22⋊C4×C23—22no—
204C22⋊C4×C23—21no—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—21no—
207C22⋊C4×C23—21no—
208C22⋊C4×C23—21no—
209C22⋊C4×C23—21no—
210C22⋊C4×C23—21no—
211C22⋊C4×C23—21no—
212C42×C23—22no—
213C42×C23—22no—
214C42×C23—22no—
215C42×C23—22no—
216C42×C23—21no—
217C42×C23—21no—
218C42×C23—21no—
219C42×C23C2×D411yescharacteristic
220C42×C23C2×D411yescharacteristic
221C42×C23C2×D411yescharacteristic
222C42×C23C2411yescharacteristic
223#18961C2311yesmaximal, characteristic, direct product factor, Sylow
224C23×C23—21no—
225C23×C23#12011yescharacteristic
226Q8×C23—22no—
227Q8×C23—22no—
228Q8×C23—22no—
229Q8×C23—42no—
230Q8×C23—44no—
231Q8×C23—42no—
232Q8×C23—81no—
233Q8×C23—82no—
234Q8×C23—82no—
235D4×C23—22no—
236D4×C23—81no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—21no—
240C2×C4×C23—21no—
241C2×C4×C23—21no—
242C2×C4×C23—22no—
243C2×C4×C23—21no—
244C2×C4×C23—21no—
245C2×C4×C23—21no—
246C2×C4×C23—22no—
247C2×C4×C23—21no—
248C2×C4×C23—41no—
249C2×C4×C23—42no—
250C2×C4×C23—42no—
251C2×C4×C23—82no—
252C2×C4×C23—82no—
253C2×C4×C23#12011yescharacteristic
254C2×C4×C23#12011yescharacteristic
255C2×C4×C23#12011yescharacteristic
256C2×C4×C23C22×D411yescharacteristic
257C2×C4×C23C22×D411yescharacteristic
258C2×C4×C23C22×D411yescharacteristic
259C8×C23—42no—
260C8×C23—42no—
261C8×C23—42no—
262#3555C2×C2311yescharacteristic
263#3238C2×C2311yescharacteristic
264#3230C2×C2311yescharacteristic, complemented
265#3209C2×C2311yescharacteristic, complemented
266#1638C2×C2312yescomplemented
267#1636C2×C2312yescomplemented
268#1633C2×C2312yescomplemented
269#1632C2×C2312yescomplemented
270#1554C2×C2311yescharacteristic, complemented
271#1497C2×C2311yescharacteristic, complemented
272#1482C2×C2311yescharacteristic
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—21no—
276C22×C23—41no—
277C22×C23—41no—
278C22×C23—41no—
279C22×C23C2×#12011yescharacteristic
280C4×C23—21no—
281C4×C23—21no—
282C4×C23—21no—
283C4×C23—21no—
284C4×C23—41no—
285C4×C23—41no—
286C4×C23—41no—
287C4×C23—41no—
288C4×C23—42no—
289C4×C23—41no—
290C4×C23—42no—
291C4×C23—41no—
292C4×C23—41no—
293C4×C23—82no—
294C4×C23#57812yes—
295C4×C23#57812yes—
296#563—22no—
297Q82C22×C2311yescharacteristic
298#557—21no—
299#557C22×C2311yescharacteristic
300#556—21no—
301#556—21no—
302#556C22×C2311yescharacteristic
303#554C22×C2311yescharacteristic
304#552—21no—
305#544—21no—
306#542—21no—
307#541—21no—
308#541—21no—
309#541—22no—
310#533C22×C2311yescharacteristic
311#533C22×C2311yescharacteristic
312C2×C4⋊Q8C22×C2311yescharacteristic
313#529C22×C2311yescharacteristic
314#501—21no—
315#500—21no—
316#494C22×C2312yes—
317#489—21no—
318#487—21no—
319#487C22×C2312yes—
320#484—21no—
321#484—21no—
322#483—21no—
323#480—21no—
324#477C22×C2312yes—
325#477C22×C2312yes—
326#477C22×C2312yes—
327#476C22×C2312yes—
328#462—22no—
329#451—22no—
330#432C22×C2311yescharacteristic
331#431C22×C2311yescharacteristic
332#431C22×C2311yescharacteristic
333#333C22×C2312yes—
334#332C22×C2312yes—
335Q8⋊C8C22×C2312yes—
336Q8⋊C8C22×C2312yes—
337Q8⋊C8C22×C2312yes—
338D4⋊C8C22×C2312yes—
339C2×C23—41no—
340C2×C23—81no—
341C2×C23#301111yescharacteristic
342C2×C23#301111yescharacteristic
343C2×C23#302111yescharacteristic
344C22×Q8—21no—
345C2×Q16—24no—
346C4⋊Q8—22no—
347C4⋊Q8—21no—
348C4⋊Q8—22no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8D4×C2311yescharacteristic
355C4⋊Q8D4×C2311yescharacteristic
356C4⋊Q8C23×C2312yes—
357C4⋊Q8C23×C2311yescharacteristic
358C4⋊Q8C23×C2311yescharacteristic
359#126—22no—
360#126—22no—
361#126—22no—
362#126—22no—
363#126—22no—
364#126—22no—
365#126—21no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—21no—
371#125—22no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125D4×C2311yescharacteristic
376#124—22no—
377#124—21no—
378#124D4×C2311yescharacteristic
379#123—22no—
380#123—21no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8D4×C2311yescharacteristic
389C22⋊Q8D4×C2311yescharacteristic
390C22⋊Q8D4×C2311yescharacteristic
391C22⋊Q8D4×C2312yes—
392#121D4×C2311yescharacteristic
393C4×Q8—22no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8D4×C2312yes—
402C4×Q8C23×C2311yescharacteristic
403C4×Q8C23×C2311yescharacteristic
404C4×Q8C23×C2311yescharacteristic
405C4×D4—21no—
406C4×D4—21no—
407C4×D4C23×C2311yescharacteristic
408#117—22no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C42C23×C2311yescharacteristic
417#107—22no—
418#107—22no—
419#107—22no—
420#106—22no—
421C4⋊C8—21no—
422C4⋊C8C23×C2312yes—
423C4⋊C8C23×C2312yes—
424#103—22no—
425#103—22no—
426#103—22no—
427#103—24no—
428#102—22no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432#97—21no—
433C4×C8—21no—
434C4×C8C23×C2312yes—
435C23#1896111yescharacteristic, direct product factor, Sylow
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—41no—
441C2×Q8—82no—
442C2×Q8C2×D4×C2311yescharacteristic
443C2×Q8C2×D4×C2311yescharacteristic
444C2×Q8C2×D4×C2311yescharacteristic
445C2×Q8C2×D4×C2312yes—
446C2×D4—21no—
447C2×D4C2×D4×C2311yescharacteristic
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4C2×D4×C2311yescharacteristic
452C22×C4C2×D4×C2311yescharacteristic
453C22×C4C2×D4×C2311yescharacteristic
454Q16—84no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4C2×D4×C2312yes—
487C4⋊C4C2×D4×C2311yescharacteristic
488C4⋊C4C2×D4×C2311yescharacteristic
489C4⋊C4C2×D4×C2311yescharacteristic
490C4⋊C4C2×D4×C2311yescharacteristic
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42C2×D4×C2311yescharacteristic
510C42C2×D4×C2311yescharacteristic
511C42C2×D4×C2311yescharacteristic
512C42C24×C2311yescharacteristic
513C23—21no—
514C23C23×#12011yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—42no—
519Q8—44no—
520Q8—42no—
521Q8—81no—
522Q8—82no—
523Q8—82no—
524D4—22no—
525D4—81no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4C23×#12011yescharacteristic
543C2×C4C23×#12011yescharacteristic
544C2×C4C23×#12011yescharacteristic
545C2×C4C22×D4×C2311yescharacteristic
546C2×C4C22×D4×C2311yescharacteristic
547C2×C4C22×D4×C2311yescharacteristic
548C8—42no—
549C8—42no—
550C8—42no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C23×#12011yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—82no—
572C4C23×#57812yes—
573C4C23×#57812yes—
574C2—41no—
575C2—81no—
576C2C23×#301111yescharacteristic
577C2C23×#301111yescharacteristic
578C2C23×#302111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#121D4
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
3. C23×#3230C2semidirect
4. C23×#3209C2semidirect
5. C23×#1638C2semidirect
6. C23×#1636C2semidirect
7. C23×#1633C2semidirect
8. C23×#1632C2semidirect
9. C23×#1554C2semidirect
10. C23×#1497C2semidirect
223. #18961C23direct (N × H)
264. #3230C2×C23semidirect
265. #3209C2×C23semidirect
266. #1638C2×C23semidirect
267. #1636C2×C23semidirect
268. #1633C2×C23semidirect
269. #1632C2×C23semidirect
270. #1554C2×C23semidirect
271. #1497C2×C23semidirect
435. C23#18961direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35552
C23×#32382
C23×#32302
C23×#32092
C23×#16382
C23×#16362
C23×#16332
C23×#16322
C23×#15542
C23×#14972
C23×#14822
#1896123