Finite Group Databasebeta

← C23×#8869 #19156495

Subgroups — C23×#8869 #19156495

Order 5888.

658 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2325C211yesmaximal, characteristic
2C23×#1874C211yesmaximal, characteristic
3C23×#1852C211yesmaximal, characteristic, complemented
4C2×C23×#327C211yesmaximal, characteristic
5C23×#1442C211yesmaximal, characteristic, complemented
6C23×#1294C211yesmaximal, characteristic, complemented
7C23×#1280C211yesmaximal, characteristic, complemented
8C2×C23×#121C2211yescharacteristic
9C2×C4×D4×C23—21no—
10C22×C4⋊C4×C23C2211yescharacteristic
11C2×C23×#107—21no—
12C2×C23×#102—21no—
13C2×C23×C22⋊C8C2211yescharacteristic
14C2×C23×C22⋊C8C2211yescharacteristic
15C23×#388—21no—
16C23×#387—21no—
17C23×#386—21no—
18C23×#386—21no—
19C23×#385—21no—
20C23×#380C2211yescharacteristic
21C2×C23×#95—21no—
22C2×C23×#95—21no—
23C2×C23×#95C2211yescharacteristic
24C2×C23×#95C2211yescharacteristic
25C23×#340—21no—
26C23×#340—21no—
27C23×#327C414yescomplemented
28C22×D4×C23—21no—
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C23×C4×C23—21no—
32C23×C4×C23—21no—
33C23×C4×C23D411yescharacteristic
34C23×C4×C23D411yescharacteristic
35C23×C4×C23C2311yescharacteristic
36C22×C8×C23—21no—
37C22×C8×C23—21no—
38C23×#121—21no—
39C23×#121—21no—
40C23×#121C2×C412yes—
41C23×#121Q812yes—
42C4×D4×C23—42no—
43C4×D4×C23—42no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—42no—
48C2×C4⋊C4×C23D411yescharacteristic
49C2×C4⋊C4×C23D411yescharacteristic
50C2×C4⋊C4×C23D411yescharacteristic
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C42×C23—21no—
57C23×#107—42no—
58C23×#102—42no—
59C23×C22⋊C8—21no—
60C23×C22⋊C8—21no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8C2×C412yescomplemented
63C23×#95—21no—
64C23×#95—21no—
65C23×#95—21no—
66C23×#95—21no—
67C23×#95—21no—
68C23×#95—21no—
69C23×#95—42no—
70C23×#95—42no—
71C23×#95C2×C412yescomplemented
72C24×C23—21no—
73C24×C23C2×D411yescharacteristic
74C2×D4×C23—22no—
75C2×D4×C23—22no—
76C2×D4×C23—21no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23C4⋊C412yes—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—22no—
88C22×C4×C23—22no—
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23—21no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—42no—
97C22×C4×C23—41no—
98C22×C4×C23—41no—
99C22×C4×C23—41no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23—41no—
106C22×C4×C23—41no—
107C22×C4×C23—41no—
108C22×C4×C23—41no—
109C22×C4×C23—41no—
110C22×C4×C23—41no—
111C22×C4×C23—41no—
112C22×C4×C23—41no—
113C22×C4×C23—81no—
114C22×C4×C23—81no—
115C22×C4×C23D812yes—
116C22×C4×C23C22×C411yescharacteristic
117C22×C4×C23C2×D411yescharacteristic
118C22×C4×C23C2×D411yescharacteristic
119C22×C4×C23C2×Q811yescharacteristic
120C22×C4×C23#4111yescharacteristic
121C22×C4×C23#4111yescharacteristic
122C2×C8×C23—22no—
123C2×C8×C23—41no—
124C2×C8×C23—41no—
125C2×C8×C23—41no—
126C2×C8×C23—41no—
127C2×C8×C23—41no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—22no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—41no—
132C4⋊C4×C23—41no—
133C4⋊C4×C23—41no—
134C4⋊C4×C23—41no—
135C4⋊C4×C23—42no—
136C4⋊C4×C23C4⋊C412yes—
137C22⋊C4×C23—22no—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—42no—
140C22⋊C4×C23—41no—
141C22⋊C4×C23—41no—
142C22⋊C4×C23—81no—
143C22⋊C4×C23—81no—
144C42×C23—42no—
145#8869C2311yesmaximal, characteristic, direct product factor, Sylow
146C23×C23—21no—
147C23×C23—21no—
148C23×C23—21no—
149C23×C23—21no—
150C23×C23—22no—
151C23×C23—41no—
152C23×C23—41no—
153C23×C23—41no—
154C23×C23—41no—
155C23×C23—41no—
156C23×C23—41no—
157C23×C23C4×D411yescharacteristic
158C23×C23#12011yescharacteristic
159C23×C23C22⋊Q811yescharacteristic
160C23×C23C2×D811yescharacteristic
161C23×C23#13511yescharacteristic
162C23×C23AGL(1,ℤ/8)11yescharacteristic
163C23×C23#13711yescharacteristic
164D4×C23—42no—
165D4×C23—42no—
166D4×C23—42no—
167D4×C23—42no—
168D4×C23—42no—
169D4×C23—81no—
170C2×C4×C23—21no—
171C2×C4×C23—21no—
172C2×C4×C23—21no—
173C2×C4×C23—21no—
174C2×C4×C23—22no—
175C2×C4×C23—22no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—41no—
179C2×C4×C23—41no—
180C2×C4×C23—41no—
181C2×C4×C23—41no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—42no—
185C2×C4×C23—42no—
186C2×C4×C23—42no—
187C2×C4×C23—42no—
188C2×C4×C23—42no—
189C2×C4×C23—42no—
190C2×C4×C23—42no—
191C2×C4×C23—41no—
192C2×C4×C23—41no—
193C2×C4×C23—41no—
194C2×C4×C23—41no—
195C2×C4×C23—41no—
196C2×C4×C23—41no—
197C2×C4×C23—41no—
198C2×C4×C23—41no—
199C2×C4×C23—42no—
200C2×C4×C23—41no—
201C2×C4×C23—81no—
202C2×C4×C23—84no—
203C2×C4×C23—81no—
204C2×C4×C23—81no—
205C2×C4×C23—81no—
206C2×C4×C23—81no—
207C2×C4×C23—81no—
208C2×C4×C23—81no—
209C2×C4×C23—81no—
210C2×C4×C23—81no—
211C2×C4×C23—81no—
212C2×C4×C23C2×C4⋊C411yescharacteristic
213C2×C4×C23C4×D411yescharacteristic
214C2×C4×C23C22⋊Q811yescharacteristic
215C2×C4×C23#12311yescharacteristic
216C8×C23—42no—
217C8×C23—81no—
218#2325C2×C2311yescharacteristic
219#1874C2×C2311yescharacteristic
220#1852C2×C2311yescharacteristic, complemented
221C2×#327C2×C2311yescharacteristic
222#1442C2×C2311yescharacteristic, complemented
223#1294C2×C2311yescharacteristic, complemented
224#1280C2×C2311yescharacteristic, complemented
225C22×C23—21no—
226C22×C23—21no—
227C22×C23—21no—
228C22×C23—21no—
229C22×C23—21no—
230C22×C23—21no—
231C22×C23—21no—
232C22×C23—21no—
233C22×C23—21no—
234C22×C23—21no—
235C22×C23—21no—
236C22×C23—21no—
237C22×C23—42no—
238C22×C23—42no—
239C22×C23—42no—
240C22×C23—81no—
241C22×C23—81no—
242C22×C23—81no—
243C22×C23—81no—
244C22×C23#38511yescharacteristic
245C22×C23C4×D811yescharacteristic
246C22×C23#44011yescharacteristic
247C22×C23#44711yescharacteristic
248C22×C23#45211yescharacteristic
249C22×C23#45411yescharacteristic
250C22×C23#45511yescharacteristic
251C22×C23B(4,𝔽2)11yescharacteristic
252C22×C23#45811yescharacteristic
253C22×C23#47411yescharacteristic
254C22×C23#47811yescharacteristic
255C4×C23—21no—
256C4×C23—21no—
257C4×C23—21no—
258C4×C23—21no—
259C4×C23—42no—
260C4×C23—42no—
261C4×C23—82no—
262C4×C23—81no—
263C4×C23—82no—
264C4×C23—81no—
265C4×C23—81no—
266C2×#121C22×C2311yescharacteristic
267C2×C4×D4—21no—
268C22×C4⋊C4C22×C2311yescharacteristic
269C2×#107—21no—
270C2×#102—21no—
271C2×C22⋊C8C22×C2311yescharacteristic
272C2×C22⋊C8C22×C2311yescharacteristic
273#388—21no—
274#387—21no—
275#386—21no—
276#386—21no—
277#385—21no—
278#380C22×C2311yescharacteristic
279C2×#95—21no—
280C2×#95—21no—
281C2×#95C22×C2311yescharacteristic
282C2×#95C22×C2311yescharacteristic
283#340—21no—
284#340—21no—
285#327C4×C2314yescomplemented
286C2×C23—21no—
287C2×C23—21no—
288C2×C23—21no—
289C2×C23—21no—
290C2×C23—82no—
291C2×C23#159811yescharacteristic
292C2×C23#160611yescharacteristic
293C2×C23#160711yescharacteristic
294C2×C23#161511yescharacteristic
295C2×C23#186011yescharacteristic
296C2×C23#186211yescharacteristic
297C2×C23#186711yescharacteristic
298C22×D4—21no—
299C22×D4D4×C2311yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—21no—
303C23×C4D4×C2311yescharacteristic
304C23×C4D4×C2311yescharacteristic
305C23×C4C23×C2311yescharacteristic
306C22×C8—21no—
307C22×C8—21no—
308#121—21no—
309#121—21no—
310#121C2×C4×C2312yes—
311#121Q8×C2312yes—
312C4×D4—42no—
313C4×D4—42no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4D4×C2311yescharacteristic
319C2×C4⋊C4D4×C2311yescharacteristic
320C2×C4⋊C4D4×C2311yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C42—21no—
327#107—42no—
328#102—42no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8C2×C4×C2312yescomplemented
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—42no—
340#95—42no—
341#95C2×C4×C2312yescomplemented
342C23#886911yescharacteristic, direct product factor, Sylow
343C24—21no—
344C24C2×D4×C2311yescharacteristic
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4C4⋊C4×C2312yes—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4D8×C2312yes—
387C22×C4C22×C4×C2311yescharacteristic
388C22×C4C2×D4×C2311yescharacteristic
389C22×C4C2×D4×C2311yescharacteristic
390C22×C4C2×Q8×C2311yescharacteristic
391C22×C4#41×C2311yescharacteristic
392C22×C4#41×C2311yescharacteristic
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4C4⋊C4×C2312yes—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—42no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C42—42no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C4×D4×C2311yescharacteristic
428C23C23×#12011yescharacteristic
429C23C23×C22⋊Q811yescharacteristic
430C23C2×D8×C2311yescharacteristic
431C23C23×#13511yescharacteristic
432C23C23×AGL(1,ℤ/8)11yescharacteristic
433C23C23×#13711yescharacteristic
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—81no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4—84no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4C2×C4⋊C4×C2311yescharacteristic
483C2×C4C4×D4×C2311yescharacteristic
484C2×C4C23×C22⋊Q811yescharacteristic
485C2×C4C23×#12311yescharacteristic
486C8—42no—
487C8—81no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22C23×#38511yescharacteristic
508C22C4×D8×C2311yescharacteristic
509C22C23×#44011yescharacteristic
510C22C23×#44711yescharacteristic
511C22C23×#45211yescharacteristic
512C22C23×#45411yescharacteristic
513C22C23×#45511yescharacteristic
514C22C23×B(4,𝔽2)11yescharacteristic
515C22C23×#45811yescharacteristic
516C22C23×#47411yescharacteristic
517C22C23×#47811yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—42no—
524C4—82no—
525C4—82no—
526C4—81no—
527C4—81no—
528C4—81no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—82no—
534C2C23×#159811yescharacteristic
535C2C23×#160611yescharacteristic
536C2C23×#160711yescharacteristic
537C2C23×#161511yescharacteristic
538C2C23×#186011yescharacteristic
539C2C23×#186211yescharacteristic
540C2C23×#186711yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1852C2semidirect
5. C23×#1442C2semidirect
6. C23×#1294C2semidirect
7. C23×#1280C2semidirect
27. C23×#327C4semidirect
62. C23×C22⋊C8C2×C4semidirect
71. C23×#95C2×C4semidirect
145. #8869C23direct (N × H)
220. #1852C2×C23semidirect
222. #1442C2×C23semidirect
223. #1294C2×C23semidirect
224. #1280C2×C23semidirect
285. #327C4×C23semidirect
332. C22⋊C8C2×C4×C23semidirect
341. #95C2×C4×C23semidirect
342. C23#8869direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23252
C23×#18742
C23×#18522
C2×C23×#3272
C23×#14422
C23×#12942
C23×#12802
#886923