Finite Group Databasebeta

← C23×#8404 #19156030

Subgroups — C23×#8404 #19156030

Order 5888.

1148 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#537C211yesmaximal, characteristic
2C23×#1874C212yesmaximal, complemented
3C2×C23×#327C212yesmaximal, complemented
4C2×C23×#324C211yesmaximal, characteristic, complemented
5C23×#1441C211yesmaximal, characteristic, complemented
6C22×#41×C23—21no—
7C23×#537—24no—
8C23×#537—22no—
9C23×#537C414yescomplemented
10C2×C23×C22⋊Q8—22no—
11C2×C23×#121—22no—
12C2×C23×#121—21no—
13C2×C23×#121C2212yes—
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2211yescharacteristic
16C2×C23×#102—22no—
17C2×C23×C22⋊C8C2212yes—
18C2×C23×#95—21no—
19C2×C23×#95—21no—
20C2×C23×#95C2211yescharacteristic
21C23×#340—22no—
22C23×#327—24no—
23C23×#324—22no—
24C2×#41×C23—24no—
25C2×#41×C23—24no—
26C2×#41×C23—42no—
27C2×#41×C23—42no—
28C2×#41×C23—42no—
29C2×#41×C23—42no—
30C22×Q8×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—22no—
33C22×D4×C23—21no—
34C22×D4×C23D412yes—
35C23×C4×C23—22no—
36C23×C4×C23—21no—
37C23×C4×C23—21no—
38C23×C4×C23D411yescharacteristic
39C23×C4×C23D411yescharacteristic
40C23×C4×C23C2311yescharacteristic
41C22×C8×C23—22no—
42C23×C22⋊Q8—28no—
43C23×C22⋊Q8—28no—
44C23×#121—24no—
45C23×#121—28no—
46C23×#121—28no—
47C23×#121—24no—
48C23×#121—22no—
49C23×#121—22no—
50C23×#121C2×C414yes—
51C23×#121D414yes—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—22no—
54C2×C4⋊C4×C23—22no—
55C2×C4⋊C4×C23—22no—
56C2×C4⋊C4×C23—42no—
57C2×C4⋊C4×C23C2×C412yes—
58C2×C4⋊C4×C23D412yes—
59C2×C4⋊C4×C23D412yes—
60C2×C22⋊C4×C23—22no—
61C2×C22⋊C4×C23—22no—
62C23×#102—44no—
63C23×C22⋊C8—22no—
64C23×C22⋊C8—22no—
65C23×#95—22no—
66C23×#95—42no—
67C23×#95—42no—
68C24×C23—22no—
69C24×C23C2×D411yescharacteristic
70#41×C23—48no—
71#41×C23—48no—
72#41×C23—48no—
73#41×C23—48no—
74C2×Q8×C23—24no—
75C2×Q8×C23—42no—
76C2×Q8×C23—42no—
77C2×D4×C23—24no—
78C2×D4×C23—24no—
79C2×D4×C23—24no—
80C2×D4×C23—24no—
81C2×D4×C23—22no—
82C2×D4×C23—42no—
83C2×D4×C23—42no—
84C2×D4×C23—42no—
85C2×D4×C23—42no—
86C2×D4×C23—42no—
87C2×D4×C23—42no—
88C2×D4×C23—42no—
89C2×D4×C23—42no—
90C2×D4×C23—81no—
91C2×D4×C23—81no—
92C2×D4×C23C22⋊C414yes—
93C22×C4×C23—21no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—28no—
98C22×C4×C23—21no—
99C22×C4×C23—22no—
100C22×C4×C23—21no—
101C22×C4×C23—22no—
102C22×C4×C23—22no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23—44no—
106C22×C4×C23—44no—
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—41no—
114C22×C4×C23—41no—
115C22×C4×C23—41no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23—41no—
121C22×C4×C23C22⋊C412yes—
122C22×C4×C23C22⋊C412yes—
123C22×C4×C23D812yes—
124C22×C4×C23SD1612yes—
125C22×C4×C23C22×C411yescharacteristic
126C22×C4×C23C2×D412yes—
127C22×C4×C23C2×D412yes—
128C22×C4×C23C2×D411yescharacteristic
129C2×C8×C23—42no—
130C2×C8×C23—42no—
131C2×C8×C23—42no—
132C4⋊C4×C23—24no—
133C4⋊C4×C23—24no—
134C4⋊C4×C23—24no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—24no—
137C4⋊C4×C23—42no—
138C4⋊C4×C23C22⋊C414yes—
139C22⋊C4×C23—28no—
140C22⋊C4×C23—24no—
141C22⋊C4×C23—24no—
142#8404C2311yesmaximal, characteristic, direct product factor, Sylow
143C23×C23—21no—
144C23×C23—21no—
145C23×C23—21no—
146C23×C23—21no—
147C23×C23—24no—
148C23×C23—42no—
149C23×C23—42no—
150C23×C23—42no—
151C23×C23—42no—
152C23×C23—42no—
153C23×C23—42no—
154C23×C23C2×C22⋊C411yescharacteristic
155C23×C23#12011yescharacteristic
156C23×C23#12011yescharacteristic
157C23×C23C2×D811yescharacteristic
158C23×C23C2×SD1611yescharacteristic
159C23×C23AGL(1,ℤ/8)11yescharacteristic
160C23×C23AGL(1,ℤ/8)11yescharacteristic
161Q8×C23—48no—
162D4×C23—44no—
163D4×C23—48no—
164D4×C23—44no—
165D4×C23—44no—
166D4×C23—44no—
167D4×C23—44no—
168D4×C23—82no—
169D4×C23—84no—
170C2×C4×C23—22no—
171C2×C4×C23—22no—
172C2×C4×C23—22no—
173C2×C4×C23—22no—
174C2×C4×C23—22no—
175C2×C4×C23—22no—
176C2×C4×C23—22no—
177C2×C4×C23—22no—
178C2×C4×C23—22no—
179C2×C4×C23—24no—
180C2×C4×C23—41no—
181C2×C4×C23—41no—
182C2×C4×C23—42no—
183C2×C4×C23—416no—
184C2×C4×C23—42no—
185C2×C4×C23—42no—
186C2×C4×C23—41no—
187C2×C4×C23—41no—
188C2×C4×C23—42no—
189C2×C4×C23—42no—
190C2×C4×C23—42no—
191C2×C4×C23—42no—
192C2×C4×C23—42no—
193C2×C4×C23—42no—
194C2×C4×C23—42no—
195C2×C4×C23—81no—
196C2×C4×C23—81no—
197C2×C4×C23—81no—
198C2×C4×C23—81no—
199C2×C4×C23—81no—
200C2×C4×C23—81no—
201C2×C4×C23—81no—
202C2×C4×C23—81no—
203C2×C4×C23—81no—
204C2×C4×C23#10214yescomplemented
205C2×C4×C23C2×C22⋊C412yes—
206C2×C4×C23#12012yes—
207C8×C23—82no—
208C2×#537C2×C2311yescharacteristic
209#1874C2×C2312yescomplemented
210C2×#327C2×C2312yescomplemented
211C2×#324C2×C2311yescharacteristic, complemented
212#1441C2×C2311yescharacteristic, complemented
213C22×C23—21no—
214C22×C23—21no—
215C22×C23—21no—
216C22×C23—21no—
217C22×C23—21no—
218C22×C23—21no—
219C22×C23—21no—
220C22×C23—21no—
221C22×C23—21no—
222C22×C23—21no—
223C22×C23—21no—
224C22×C23—21no—
225C22×C23—44no—
226C22×C23—44no—
227C22×C23—44no—
228C22×C23—82no—
229C22×C23—82no—
230C22×C23—82no—
231C22×C23—82no—
232C22×C23#37911yescharacteristic
233C22×C23C2×#10211yescharacteristic
234C22×C23#41811yescharacteristic
235C22×C23#44711yescharacteristic
236C22×C23#44711yescharacteristic
237C22×C23#45011yescharacteristic
238C22×C23#45011yescharacteristic
239C22×C23#45411yescharacteristic
240C22×C23#45611yescharacteristic
241C22×C23B(4,𝔽2)11yescharacteristic
242C22×C23#45811yescharacteristic
243C4×C23—24no—
244C4×C23—22no—
245C4×C23—22no—
246C4×C23—44no—
247C4×C23—42no—
248C4×C23—82no—
249C4×C23—82no—
250C22×#41—21no—
251#537—24no—
252#537—22no—
253#537C4×C2314yescomplemented
254C2×C22⋊Q8—22no—
255C2×#121—22no—
256C2×#121—21no—
257C2×#121C22×C2312yes—
258C22×C4⋊C4C22×C2311yescharacteristic
259C22×C4⋊C4C22×C2311yescharacteristic
260C2×#102—22no—
261C2×C22⋊C8C22×C2312yes—
262C2×#95—21no—
263C2×#95—21no—
264C2×#95C22×C2311yescharacteristic
265#340—22no—
266#327—24no—
267#324—22no—
268C2×C23—21no—
269C2×C23—21no—
270C2×C23—21no—
271C2×C23—21no—
272C2×C23—84no—
273C2×C23#159811yescharacteristic
274C2×C23#160411yescharacteristic
275C2×C23#161011yescharacteristic
276C2×C23#161611yescharacteristic
277C2×C23#178611yescharacteristic
278C2×C23#179011yescharacteristic
279C2×C23#179311yescharacteristic
280C2×#41—24no—
281C2×#41—24no—
282C2×#41—42no—
283C2×#41—42no—
284C2×#41—42no—
285C2×#41—42no—
286C22×Q8—21no—
287C22×D4—21no—
288C22×D4—22no—
289C22×D4—21no—
290C22×D4D4×C2312yes—
291C23×C4—22no—
292C23×C4—21no—
293C23×C4—21no—
294C23×C4D4×C2311yescharacteristic
295C23×C4D4×C2311yescharacteristic
296C23×C4C23×C2311yescharacteristic
297C22×C8—22no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300#121—24no—
301#121—28no—
302#121—28no—
303#121—24no—
304#121—22no—
305#121—22no—
306#121C2×C4×C2314yes—
307#121D4×C2314yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4C2×C4×C2312yes—
314C2×C4⋊C4D4×C2312yes—
315C2×C4⋊C4D4×C2312yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318#102—44no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321#95—22no—
322#95—42no—
323#95—42no—
324C23#840411yescharacteristic, direct product factor, Sylow
325C24—22no—
326C24C2×D4×C2311yescharacteristic
327#41—48no—
328#41—48no—
329#41—48no—
330#41—48no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4C22⋊C4×C2314yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—28no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
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×C4C22⋊C4×C2312yes—
379C22×C4C22⋊C4×C2312yes—
380C22×C4D8×C2312yes—
381C22×C4SD16×C2312yes—
382C22×C4C22×C4×C2311yescharacteristic
383C22×C4C2×D4×C2312yes—
384C22×C4C2×D4×C2312yes—
385C22×C4C2×D4×C2311yescharacteristic
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—42no—
395C4⋊C4C22⋊C4×C2314yes—
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23C2×C22⋊C4×C2311yescharacteristic
411C23C23×#12011yescharacteristic
412C23C23×#12011yescharacteristic
413C23C2×D8×C2311yescharacteristic
414C23C2×SD16×C2311yescharacteristic
415C23C23×AGL(1,ℤ/8)11yescharacteristic
416C23C23×AGL(1,ℤ/8)11yescharacteristic
417Q8—48no—
418D4—44no—
419D4—48no—
420D4—44no—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—82no—
425D4—84no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—416no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4C23×#10214yescomplemented
461C2×C4C2×C22⋊C4×C2312yes—
462C2×C4C23×#12012yes—
463C8—82no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22C23×#37911yescharacteristic
484C22C2×C23×#10211yescharacteristic
485C22C23×#41811yescharacteristic
486C22C23×#44711yescharacteristic
487C22C23×#44711yescharacteristic
488C22C23×#45011yescharacteristic
489C22C23×#45011yescharacteristic
490C22C23×#45411yescharacteristic
491C22C23×#45611yescharacteristic
492C22C23×B(4,𝔽2)11yescharacteristic
493C22C23×#45811yescharacteristic
494C4—24no—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—42no—
499C4—82no—
500C4—82no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—84no—
506C2C23×#159811yescharacteristic
507C2C23×#160411yescharacteristic
508C2C23×#161011yescharacteristic
509C2C23×#161611yescharacteristic
510C2C23×#178611yescharacteristic
511C2C23×#179011yescharacteristic
512C2C23×#179311yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#537C2
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
2. C23×#1874C2semidirect
3. C2×C23×#327C2semidirect
4. C2×C23×#324C2semidirect
5. C23×#1441C2semidirect
9. C23×#537C4semidirect
142. #8404C23direct (N × H)
204. C2×C4×C23#102semidirect
209. #1874C2×C23semidirect
210. C2×#327C2×C23semidirect
211. C2×#324C2×C23semidirect
212. #1441C2×C23semidirect
253. #537C4×C23semidirect
324. C23#8404direct (N × H)
460. C2×C4C23×#102semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5372
C23×#18742
C2×C23×#3272
C2×C23×#3242
C23×#14412
#840423