Finite Group Databasebeta

← C23×#18432 #19166058

Subgroups — C23×#18432 #19166058

Order 5888.

940 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic, complemented
2C23×#3217C211yesmaximal, characteristic
3C23×#3208C211yesmaximal, characteristic, complemented
4C23×#2641C211yesmaximal, characteristic, complemented
5C23×#1599C214yesmaximal, complemented
6C23×#1597C214yesmaximal, complemented
7C23×#1506C211yesmaximal, characteristic, complemented
8C23×#1500C211yesmaximal, characteristic, complemented
9C23×#1470C211yesmaximal, characteristic, complemented
10C23×#564—21no—
11C23×#563—21no—
12C23×#557—21no—
13C23×#554—21no—
14C23×#552—21no—
15C23×#552—21no—
16C23×#552—22no—
17C23×#544—21no—
18C23×#544C2212yes—
19C23×#541—21no—
20C23×#541—22no—
21C23×#539C2212yes—
22C23×#533C2211yescharacteristic
23C2×C23×#127C2211yescharacteristic
24C23×#529C2211yescharacteristic
25C2×C23×#125C2211yescharacteristic
26C23×#496—21no—
27C23×#492—21no—
28C23×#490—21no—
29C23×#486—21no—
30C23×#460—22no—
31C23×#460—21no—
32C23×#459—21no—
33C23×#450C2214yes—
34C23×#450C2212yes—
35C23×#447C2212yes—
36C23×#432C2211yescharacteristic
37C23×#431C2211yescharacteristic
38C23×#394—22no—
39C23×#394C2214yes—
40C23×#390—21no—
41C23×#390—21no—
42C23×#376C2211yescharacteristic
43C23×#328C2212yes—
44C23×#328C2214yes—
45C23×#327C2212yes—
46C23×#324C2214yes—
47C22×D4×C23—21no—
48C22×D4×C23—21no—
49C22×D4×C23C2312yes—
50C2×SD16×C23—24no—
51C2×SD16×C23—22no—
52C2×D8×C23—22no—
53C23×C4⋊Q8—21no—
54C23×C4⋊Q8—22no—
55C23×C4⋊Q8—22no—
56C23×C4⋊Q8D411yescharacteristic
57C23×C4⋊Q8D411yescharacteristic
58C23×#127—21no—
59C23×#127—21no—
60C23×#127—21no—
61C23×#127—21no—
62C23×#126—22no—
63C23×#126—22no—
64C23×#126—22no—
65C23×#126—22no—
66C23×#126—21no—
67C23×#125—22no—
68C23×#125—21no—
69C23×#125—22no—
70C23×#125—22no—
71C23×#125—22no—
72C23×#125—22no—
73C23×#125—22no—
74C23×#125—22no—
75C23×#125—22no—
76C23×#125—21no—
77C23×#125—21no—
78C23×#125—21no—
79C23×#125—21no—
80C23×#125—21no—
81C23×#125D411yescharacteristic
82C23×#124D411yescharacteristic
83C23×#123—22no—
84C23×#123—21no—
85C23×#123—22no—
86C23×C22⋊Q8—21no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—22no—
93C23×C22⋊Q8C2311yescharacteristic
94C23×C22⋊Q8C2312yes—
95C23×#121C2311yescharacteristic
96C4×Q8×C23—21no—
97C4×Q8×C23—22no—
98C4×Q8×C23—21no—
99C4×Q8×C23—21no—
100C4×Q8×C23D411yescharacteristic
101C4×Q8×C23D412yes—
102C4×D4×C23—21no—
103C4×D4×C23—22no—
104C4×D4×C23D411yescharacteristic
105C23×#117—22no—
106C23×#117—21no—
107C23×#117—21no—
108C23×#117—22no—
109C23×#117—21no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23D411yescharacteristic
112C2×C4⋊C4×C23D412yes—
113C2×C4⋊C4×C23D411yescharacteristic
114C2×C4⋊C4×C23C2312yes—
115C2×C22⋊C4×C23—24no—
116C2×C22⋊C4×C23—24no—
117C2×C42×C23—21no—
118C2×C42×C23C2311yescharacteristic
119C23×C4⋊C8—21no—
120C23×C4⋊C8—21no—
121C23×#102—22no—
122C23×#102—22no—
123C23×#102—24no—
124C23×C22⋊C8C2312yes—
125C23×C22⋊C8C2312yes—
126C23×#97—21no—
127C4×C8×C23—21no—
128C23×#95—21no—
129C23×#95C2312yes—
130C24×C23—22no—
131C2×Q8×C23—22no—
132C2×Q8×C23—21no—
133C2×Q8×C23C2×D412yes—
134C2×Q8×C23C2×D411yescharacteristic
135C2×D4×C23—21no—
136C2×D4×C23—22no—
137C2×D4×C23—22no—
138C2×D4×C23—44no—
139C2×D4×C23—41no—
140C2×D4×C23—41no—
141C2×D4×C23—41no—
142C2×D4×C23—41no—
143C2×D4×C23—82no—
144C2×D4×C23—82no—
145C2×D4×C23C2×D411yescharacteristic
146C22×C4×C23—22no—
147C22×C4×C23—21no—
148C22×C4×C23—22no—
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D411yescharacteristic
152C22×C4×C23C2×D412yes—
153C22×C4×C23C2411yescharacteristic
154SD16×C23—82no—
155SD16×C23—84no—
156D8×C23—82no—
157C2×C8×C23—22no—
158C2×C8×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—21no—
168C4⋊C4×C23—21no—
169C4⋊C4×C23—21no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—21no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23C2×D411yescharacteristic
185C4⋊C4×C23C2×D411yescharacteristic
186C4⋊C4×C23C2×D412yes—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—21no—
192C22⋊C4×C23—21no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—21no—
195C22⋊C4×C23—21no—
196C22⋊C4×C23—84no—
197C22⋊C4×C23—84no—
198C42×C23—22no—
199C42×C23—22no—
200C42×C23—22no—
201C42×C23—22no—
202C42×C23—21no—
203C42×C23—42no—
204C42×C23C2×D411yescharacteristic
205C42×C23C2×D411yescharacteristic
206C42×C23C2×D411yescharacteristic
207C42×C23C2×D411yescharacteristic
208#18432C2311yesmaximal, characteristic, direct product factor, Sylow
209C23×C23—21no—
210C23×C23—44no—
211C23×C23—44no—
212C23×C23—42no—
213C23×C23—44no—
214C23×C23C22×D411yescharacteristic
215Q8×C23—41no—
216Q8×C23—42no—
217Q8×C23—42no—
218Q8×C23—82no—
219D4×C23—41no—
220D4×C23—42no—
221D4×C23—42no—
222D4×C23—44no—
223D4×C23—42no—
224D4×C23—82no—
225D4×C23—82no—
226D4×C23—82no—
227D4×C23—82no—
228C2×C4×C23—22no—
229C2×C4×C23—22no—
230C2×C4×C23—22no—
231C2×C4×C23—21no—
232C2×C4×C23—21no—
233C2×C4×C23—22no—
234C2×C4×C23—21no—
235C2×C4×C23—22no—
236C2×C4×C23—22no—
237C2×C4×C23—24no—
238C2×C4×C23—41no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—82no—
245C2×C4×C23#12011yescharacteristic
246C2×C4×C23#12011yescharacteristic
247C2×C4×C23#12011yescharacteristic
248C2×C4×C23#12011yescharacteristic
249C2×C4×C23C22×D411yescharacteristic
250C2×C4×C23C22×D411yescharacteristic
251C8×C23—42no—
252C8×C23—42no—
253#3563C2×C2311yescharacteristic, complemented
254#3217C2×C2311yescharacteristic
255#3208C2×C2311yescharacteristic, complemented
256#2641C2×C2311yescharacteristic, complemented
257#1599C2×C2314yescomplemented
258#1597C2×C2314yescomplemented
259#1506C2×C2311yescharacteristic, complemented
260#1500C2×C2311yescharacteristic, complemented
261#1470C2×C2311yescharacteristic, complemented
262C22×C23—21no—
263C22×C23—21no—
264C22×C23—41no—
265C22×C23—41no—
266C22×C23—41no—
267C22×C23—88no—
268C22×C23—82no—
269C22×C23—82no—
270C22×C23—82no—
271C22×C23C2×#12011yescharacteristic
272C22×C23#57712yes—
273C4×C23—21no—
274C4×C23—21no—
275C4×C23—21no—
276C4×C23—21no—
277C4×C23—21no—
278C4×C23—42no—
279C4×C23—41no—
280C4×C23—41no—
281C4×C23—42no—
282C4×C23—42no—
283C4×C23—42no—
284C4×C23—41no—
285C4×C23—82no—
286C4×C23#45412yes—
287#564—21no—
288#563—21no—
289#557—21no—
290#554—21no—
291#552—21no—
292#552—21no—
293#552—22no—
294#544—21no—
295#544C22×C2312yes—
296#541—21no—
297#541—22no—
298#539C22×C2312yes—
299#533C22×C2311yescharacteristic
300C2×#127C22×C2311yescharacteristic
301#529C22×C2311yescharacteristic
302C2×#125C22×C2311yescharacteristic
303#496—21no—
304#492—21no—
305#490—21no—
306#486—21no—
307#460—22no—
308#460—21no—
309#459—21no—
310#450C22×C2314yes—
311#450C22×C2312yes—
312#447C22×C2312yes—
313#432C22×C2311yescharacteristic
314#431C22×C2311yescharacteristic
315#394—22no—
316#394C22×C2314yes—
317#390—21no—
318#390—21no—
319#376C22×C2311yescharacteristic
320#328C22×C2314yes—
321#328C22×C2312yes—
322#327C22×C2312yes—
323#324C22×C2314yes—
324C2×C23—22no—
325C2×C23—81no—
326C2×C23—162no—
327C2×C23#301011yescharacteristic
328C2×C23C2×#45411yescharacteristic
329C2×C23#302711yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4C23×C2312yes—
333C2×SD16—24no—
334C2×SD16—22no—
335C2×D8—22no—
336C4⋊Q8—21no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8D4×C2311yescharacteristic
340C4⋊Q8D4×C2311yescharacteristic
341#127—21no—
342#127—21no—
343#127—21no—
344#127—21no—
345#126—22no—
346#126—22no—
347#126—22no—
348#126—22no—
349#126—21no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125—22no—
354#125—21no—
355#125—22no—
356#125—22no—
357#125—22no—
358#125—22no—
359#125—21no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125—21no—
364#125D4×C2311yescharacteristic
365#124D4×C2311yescharacteristic
366#123—22no—
367#123—21no—
368#123—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8C23×C2311yescharacteristic
377C22⋊Q8C23×C2312yes—
378#121C23×C2311yescharacteristic
379C4×Q8—21no—
380C4×Q8—22no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8D4×C2311yescharacteristic
384C4×Q8D4×C2312yes—
385C4×D4—21no—
386C4×D4—22no—
387C4×D4D4×C2311yescharacteristic
388#117—22no—
389#117—21no—
390#117—21no—
391#117—22no—
392#117—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C2311yescharacteristic
395C2×C4⋊C4D4×C2312yes—
396C2×C4⋊C4D4×C2311yescharacteristic
397C2×C4⋊C4C23×C2312yes—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C42—21no—
401C2×C42C23×C2311yescharacteristic
402C4⋊C8—21no—
403C4⋊C8—21no—
404#102—22no—
405#102—22no—
406#102—24no—
407C22⋊C8C23×C2312yes—
408C22⋊C8C23×C2312yes—
409#97—21no—
410C4×C8—21no—
411#95—21no—
412#95C23×C2312yes—
413C23#1843211yescharacteristic, direct product factor, Sylow
414C24—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8C2×D4×C2312yes—
418C2×Q8C2×D4×C2311yescharacteristic
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—44no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4C2×D4×C2311yescharacteristic
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4C2×D4×C2312yes—
434C22×C4C2×D4×C2311yescharacteristic
435C22×C4C2×D4×C2311yescharacteristic
436C22×C4C2×D4×C2312yes—
437C22×C4C24×C2311yescharacteristic
438SD16—82no—
439SD16—84no—
440D8—82no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4C2×D4×C2311yescharacteristic
469C4⋊C4C2×D4×C2311yescharacteristic
470C4⋊C4C2×D4×C2312yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—42no—
488C42C2×D4×C2311yescharacteristic
489C42C2×D4×C2311yescharacteristic
490C42C2×D4×C2311yescharacteristic
491C42C2×D4×C2311yescharacteristic
492C23—21no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23C22×D4×C2311yescharacteristic
498Q8—41no—
499Q8—42no—
500Q8—42no—
501Q8—82no—
502D4—41no—
503D4—42no—
504D4—42no—
505D4—44no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—82no—
510D4—82no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4C23×#12011yescharacteristic
529C2×C4C23×#12011yescharacteristic
530C2×C4C23×#12011yescharacteristic
531C2×C4C23×#12011yescharacteristic
532C2×C4C22×D4×C2311yescharacteristic
533C2×C4C22×D4×C2311yescharacteristic
534C8—42no—
535C8—42no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—88no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×C23×#12011yescharacteristic
546C22C23×#57712yes—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—82no—
560C4C23×#45412yes—
561C2—22no—
562C2—81no—
563C2—162no—
564C2C23×#301011yescharacteristic
565C2C2×C23×#45411yescharacteristic
566C2C23×#302711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3217C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3563C2semidirect
3. C23×#3208C2semidirect
4. C23×#2641C2semidirect
5. C23×#1599C2semidirect
6. C23×#1597C2semidirect
7. C23×#1506C2semidirect
8. C23×#1500C2semidirect
9. C23×#1470C2semidirect
208. #18432C23direct (N × H)
253. #3563C2×C23semidirect
255. #3208C2×C23semidirect
256. #2641C2×C23semidirect
257. #1599C2×C23semidirect
258. #1597C2×C23semidirect
259. #1506C2×C23semidirect
260. #1500C2×C23semidirect
261. #1470C2×C23semidirect
413. C23#18432direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#32172
C23×#32082
C23×#26412
C23×#15992
C23×#15972
C23×#15062
C23×#15002
C23×#14702
#1843223