Finite Group Databasebeta

← C23×#8328 #19155954

Subgroups — C23×#8328 #19155954

Order 5888.

1620 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#121C211yesmaximal, characteristic
2C23×#1779C211yesmaximal, characteristic, complemented
3C2×C23×#327C214yesmaximal, complemented
4C23×C22⋊#95C211yesmaximal, characteristic, complemented
5C23×D4×C23—21no—
6C23×D4×C23C411yescharacteristic, complemented
7C24×C4×C23C2211yescharacteristic
8C2×C23×#121—21no—
9C2×C23×#121—21no—
10C2×C23×#121—28no—
11C2×C23×#121C412yescomplemented
12C2×C23×#121C2212yes—
13C22×C4⋊C4×C23C411yescharacteristic, complemented
14C22×C22⋊C4×C23—21no—
15C22×C22⋊C4×C23—21no—
16C2×C23×C22⋊C8—21no—
17C2×C23×C22⋊C8—21no—
18C2×C23×C22⋊C8C2212yescomplemented
19C2×C23×#95—21no—
20C2×C23×#95C2212yescomplemented
21C23×#327—28no—
22C25×C23—21no—
23C25×C23D411yescharacteristic
24C22×D4×C23—28no—
25C22×D4×C23—22no—
26C22×D4×C23—22no—
27C22×D4×C23—24no—
28C22×D4×C23—21no—
29C22×D4×C23—42no—
30C22×D4×C23—42no—
31C22×D4×C23—44no—
32C22×D4×C23—42no—
33C22×D4×C23—42no—
34C22×D4×C23C2×C412yes—
35C23×C4×C23—21no—
36C23×C4×C23—24no—
37C23×C4×C23—22no—
38C23×C4×C23—42no—
39C23×C4×C23—42no—
40C23×C4×C23C2×C411yescharacteristic
41C23×C4×C23C2×C411yescharacteristic
42C23×C4×C23D411yescharacteristic
43C23×C4×C23D411yescharacteristic
44C23×C4×C23D411yescharacteristic
45C23×C4×C23C2311yescharacteristic
46C22×C8×C23—22no—
47C23×#121—28no—
48C23×#121—28no—
49C23×#121—24no—
50C23×#121—24no—
51C23×#121—24no—
52C23×#121D418yes—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—24no—
55C2×C4⋊C4×C23C2×C412yes—
56C2×C22⋊C4×C23—28no—
57C2×C22⋊C4×C23—22no—
58C2×C22⋊C4×C23—22no—
59C2×C22⋊C4×C23—24no—
60C2×C22⋊C4×C23—42no—
61C2×C22⋊C4×C23—42no—
62C23×C22⋊C8—24no—
63C23×C22⋊C8—42no—
64C23×C22⋊C8—42no—
65C23×#95—24no—
66C23×#95—42no—
67C24×C23—22no—
68C24×C23—22no—
69C24×C23—22no—
70C24×C23—22no—
71C24×C23—24no—
72C24×C23—22no—
73C24×C23—41no—
74C24×C23—41no—
75C24×C23—42no—
76C24×C23—42no—
77C24×C23—42no—
78C24×C23—42no—
79C24×C23—42no—
80C24×C23—42no—
81C24×C23C22⋊C411yescharacteristic
82C24×C23C22⋊C411yescharacteristic
83C24×C23D812yes—
84C24×C23SD1612yes—
85C24×C23C2×D411yescharacteristic
86C2×D4×C23—24no—
87C2×D4×C23—28no—
88C2×D4×C23—28no—
89C2×D4×C23—22no—
90C2×D4×C23—42no—
91C2×D4×C23—42no—
92C2×D4×C23—416no—
93C2×D4×C23—42no—
94C2×D4×C23—416no—
95C2×D4×C23—416no—
96C2×D4×C23—44no—
97C2×D4×C23—44no—
98C2×D4×C23—42no—
99C2×D4×C23—42no—
100C2×D4×C23—44no—
101C2×D4×C23—42no—
102C2×D4×C23C22⋊C414yescomplemented
103C22×C4×C23—22no—
104C22×C4×C23—22no—
105C22×C4×C23—22no—
106C22×C4×C23—24no—
107C22×C4×C23—22no—
108C22×C4×C23—22no—
109C22×C4×C23—22no—
110C22×C4×C23—22no—
111C22×C4×C23—22no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—24no—
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—416no—
124C22×C4×C23—42no—
125C22×C4×C23—42no—
126C22×C4×C23—42no—
127C22×C4×C23—42no—
128C22×C4×C23—42no—
129C22×C4×C23—42no—
130C22×C4×C23—42no—
131C22×C4×C23—42no—
132C22×C4×C23C22⋊C411yescharacteristic
133C22×C4×C23C22⋊C411yescharacteristic
134C22×C4×C23C22×C411yescharacteristic
135C22×C4×C23C2×D411yescharacteristic
136C22×C4×C23C2×D414yes—
137C2×C8×C23—42no—
138C2×C8×C23—42no—
139C2×C8×C23—42no—
140C4⋊C4×C23—24no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23C22⋊C414yescomplemented
143C22⋊C4×C23—28no—
144C22⋊C4×C23—28no—
145C22⋊C4×C23—48no—
146#8328C2311yesmaximal, characteristic, direct product factor, Sylow
147C23×C23—21no—
148C23×C23—21no—
149C23×C23—21no—
150C23×C23—21no—
151C23×C23—21no—
152C23×C23—21no—
153C23×C23—21no—
154C23×C23—21no—
155C23×C23—24no—
156C23×C23—24no—
157C23×C23—21no—
158C23×C23—21no—
159C23×C23—21no—
160C23×C23—21no—
161C23×C23—21no—
162C23×C23—21no—
163C23×C23—24no—
164C23×C23—24no—
165C23×C23—24no—
166C23×C23—42no—
167C23×C23—42no—
168C23×C23—44no—
169C23×C23—44no—
170C23×C23—44no—
171C23×C23—44no—
172C23×C23—48no—
173C23×C23—48no—
174C23×C23—48no—
175C23×C23—42no—
176C23×C23—42no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—84no—
182C23×C23—84no—
183C23×C23—84no—
184C23×C23—84no—
185C23×C23#9912yes—
186C23×C23#10212yes—
187C23×C23#10212yes—
188C23×C23#10412yes—
189C23×C23C2×C22⋊C411yescharacteristic
190C23×C23#12012yes—
191C23×C23C2×D811yescharacteristic
192C23×C23C2×SD1611yescharacteristic
193C23×C23AGL(1,ℤ/8)12yes—
194D4×C23—44no—
195D4×C23—48no—
196D4×C23—44no—
197D4×C23—416no—
198D4×C23—48no—
199D4×C23—48no—
200C2×C4×C23—28no—
201C2×C4×C23—21no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—21no—
205C2×C4×C23—21no—
206C2×C4×C23—21no—
207C2×C4×C23—21no—
208C2×C4×C23—24no—
209C2×C4×C23—21no—
210C2×C4×C23—21no—
211C2×C4×C23—44no—
212C2×C4×C23—44no—
213C2×C4×C23—44no—
214C2×C4×C23—44no—
215C2×C4×C23—44no—
216C2×C4×C23—48no—
217C2×C4×C23—42no—
218C2×C4×C23—42no—
219C2×C4×C23—44no—
220C2×C4×C23—44no—
221C2×C4×C23—44no—
222C2×C4×C23—82no—
223C2×C4×C23—82no—
224C2×C4×C23—82no—
225C2×C4×C23—82no—
226C2×C4×C23#9912yes—
227C2×C4×C23C2×C22⋊C412yes—
228C2×C4×C23#12012yes—
229C8×C23—82no—
230C22×#121C2×C2311yescharacteristic
231#1779C2×C2311yescharacteristic, complemented
232C2×#327C2×C2314yescomplemented
233C22⋊#95C2×C2311yescharacteristic, complemented
234C22×C23—21no—
235C22×C23—21no—
236C22×C23—22no—
237C22×C23—22no—
238C22×C23—22no—
239C22×C23—22no—
240C22×C23—21no—
241C22×C23—22no—
242C22×C23—21no—
243C22×C23—22no—
244C22×C23—22no—
245C22×C23—22no—
246C22×C23—22no—
247C22×C23—22no—
248C22×C23—22no—
249C22×C23—22no—
250C22×C23—22no—
251C22×C23—48no—
252C22×C23—48no—
253C22×C23—41no—
254C22×C23—41no—
255C22×C23—41no—
256C22×C23—41no—
257C22×C23—44no—
258C22×C23—44no—
259C22×C23—44no—
260C22×C23—816no—
261C22×C23—82no—
262C22×C23—82no—
263C22×C23—82no—
264C22×C23—82no—
265C22×C23#32714yescomplemented
266C22×C23#37911yescharacteristic
267C22×C23C2×#9911yescharacteristic
268C22×C23C2×#9911yescharacteristic
269C22×C23C2×#10211yescharacteristic
270C22×C23#41811yescharacteristic
271C22×C23C2×#10411yescharacteristic
272C22×C23#42111yescharacteristic
273C22×C23#44712yes—
274C22×C23#45012yes—
275C4×C23—22no—
276C4×C23—22no—
277C4×C23—22no—
278C4×C23—41no—
279C4×C23—44no—
280C4×C23—84no—
281C23×D4—21no—
282C23×D4C4×C2311yescharacteristic, complemented
283C24×C4C22×C2311yescharacteristic
284C2×#121—21no—
285C2×#121—21no—
286C2×#121—28no—
287C2×#121C4×C2312yescomplemented
288C2×#121C22×C2312yes—
289C22×C4⋊C4C4×C2311yescharacteristic, complemented
290C22×C22⋊C4—21no—
291C22×C22⋊C4—21no—
292C2×C22⋊C8—21no—
293C2×C22⋊C8—21no—
294C2×C22⋊C8C22×C2312yescomplemented
295C2×#95—21no—
296C2×#95C22×C2312yescomplemented
297#327—28no—
298C2×C23—22no—
299C2×C23—24no—
300C2×C23—22no—
301C2×C23—42no—
302C2×C23—84no—
303C2×C23C2×#32711yescharacteristic
304C2×C23#150711yescharacteristic
305C2×C23#151011yescharacteristic
306C2×C23#151911yescharacteristic
307C2×C23#178111yescharacteristic
308C2×C23#178611yescharacteristic
309C2×C23#178911yescharacteristic
310C25—21no—
311C25D4×C2311yescharacteristic
312C22×D4—28no—
313C22×D4—22no—
314C22×D4—22no—
315C22×D4—24no—
316C22×D4—21no—
317C22×D4—42no—
318C22×D4—42no—
319C22×D4—44no—
320C22×D4—42no—
321C22×D4—42no—
322C22×D4C2×C4×C2312yes—
323C23×C4—21no—
324C23×C4—24no—
325C23×C4—22no—
326C23×C4—42no—
327C23×C4—42no—
328C23×C4C2×C4×C2311yescharacteristic
329C23×C4C2×C4×C2311yescharacteristic
330C23×C4D4×C2311yescharacteristic
331C23×C4D4×C2311yescharacteristic
332C23×C4D4×C2311yescharacteristic
333C23×C4C23×C2311yescharacteristic
334C22×C8—22no—
335#121—28no—
336#121—28no—
337#121—24no—
338#121—24no—
339#121—24no—
340#121D4×C2318yes—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×C4×C2312yes—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C22⋊C8—24no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353#95—24no—
354#95—42no—
355C23#832811yescharacteristic, direct product factor, Sylow
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—41no—
363C24—41no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24C22⋊C4×C2311yescharacteristic
371C24C22⋊C4×C2311yescharacteristic
372C24D8×C2312yes—
373C24SD16×C2312yes—
374C24C2×D4×C2311yescharacteristic
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—416no—
382C2×D4—42no—
383C2×D4—416no—
384C2×D4—416no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4C22⋊C4×C2314yescomplemented
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—416no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C2311yescharacteristic
422C22×C4C22⋊C4×C2311yescharacteristic
423C22×C4C22×C4×C2311yescharacteristic
424C22×C4C2×D4×C2311yescharacteristic
425C22×C4C2×D4×C2314yes—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4C22⋊C4×C2314yescomplemented
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—48no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—48no—
461C23—48no—
462C23—48no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23C23×#9912yes—
474C23C23×#10212yes—
475C23C23×#10212yes—
476C23C23×#10412yes—
477C23C2×C22⋊C4×C2311yescharacteristic
478C23C23×#12012yes—
479C23C2×D8×C2311yescharacteristic
480C23C2×SD16×C2311yescharacteristic
481C23C23×AGL(1,ℤ/8)12yes—
482D4—44no—
483D4—48no—
484D4—44no—
485D4—416no—
486D4—48no—
487D4—48no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4C23×#9912yes—
515C2×C4C2×C22⋊C4×C2312yes—
516C2×C4C23×#12012yes—
517C8—82no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—48no—
536C22—48no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—816no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C23×#32714yescomplemented
550C22C23×#37911yescharacteristic
551C22C2×C23×#9911yescharacteristic
552C22C2×C23×#9911yescharacteristic
553C22C2×C23×#10211yescharacteristic
554C22C23×#41811yescharacteristic
555C22C2×C23×#10411yescharacteristic
556C22C23×#42111yescharacteristic
557C22C23×#44712yes—
558C22C23×#45012yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—41no—
563C4—44no—
564C4—84no—
565C2—22no—
566C2—24no—
567C2—22no—
568C2—42no—
569C2—84no—
570C2C2×C23×#32711yescharacteristic
571C2C23×#150711yescharacteristic
572C2C23×#151011yescharacteristic
573C2C23×#151911yescharacteristic
574C2C23×#178111yescharacteristic
575C2C23×#178611yescharacteristic
576C2C23×#178911yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC22×C23×#121C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1779C2semidirect
3. C2×C23×#327C2semidirect
4. C23×C22⋊#95C2semidirect
6. C23×D4×C23C4semidirect
11. C2×C23×#121C4semidirect
13. C22×C4⋊C4×C23C4semidirect
18. C2×C23×C22⋊C8C22semidirect
20. C2×C23×#95C22semidirect
102. C2×D4×C23C22⋊C4semidirect
142. C4⋊C4×C23C22⋊C4semidirect
146. #8328C23direct (N × H)
231. #1779C2×C23semidirect
232. C2×#327C2×C23semidirect
233. C22⋊#95C2×C23semidirect
265. C22×C23#327semidirect
282. C23×D4C4×C23semidirect
287. C2×#121C4×C23semidirect
289. C22×C4⋊C4C4×C23semidirect
294. C2×C22⋊C8C22×C23semidirect
296. C2×#95C22×C23semidirect
355. C23#8328direct (N × H)
391. C2×D4C22⋊C4×C23semidirect
431. C4⋊C4C22⋊C4×C23semidirect
549. C22C23×#327semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1212
C23×#17792
C2×C23×#3272
C23×C22⋊#952
#832823