Finite Group Databasebeta

← C23×#18763 #19166389

Subgroups — C23×#18763 #19166389

Order 5888.

910 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#556C211yesmaximal, characteristic
2C23×#3230C211yesmaximal, characteristic, complemented
3C23×#3101C211yesmaximal, characteristic
4C23×#3078C211yesmaximal, characteristic, complemented
5C23×#1650C214yesmaximal, complemented
6C23×#1638C214yesmaximal, complemented
7C23×#1549C211yesmaximal, characteristic, complemented
8C23×#1486C211yesmaximal, characteristic
9C23×#1479C211yesmaximal, characteristic, complemented
10C23×#556—22no—
11C23×#556—22no—
12C23×#556—22no—
13C23×#556C2214yes—
14C23×#533C2211yescharacteristic
15C2×C23×C4⋊Q8C2211yescharacteristic
16C2×C23×#126—22no—
17C2×C23×#124—21no—
18C2×C23×C22⋊Q8—21no—
19C2×C4×Q8×C23—21no—
20C2×C4×Q8×C23C2211yescharacteristic
21C2×C4×D4×C23C2211yescharacteristic
22C2×C23×#117—21no—
23C23×#500C2212yescomplemented
24C23×#487C2212yescomplemented
25C23×#484—22no—
26C23×#483—21no—
27C23×#479—22no—
28C23×#477—22no—
29C23×#477C2212yescomplemented
30C23×#476C2212yes—
31C23×#475C2212yescomplemented
32C23×#461C2212yes—
33C23×#450—21no—
34C4×SD16×C23—22no—
35C23×#431C2211yescharacteristic
36C23×#431C2211yescharacteristic
37C23×#428—21no—
38C23×#423C2211yescharacteristic
39C23×#419—21no—
40C23×#332C2214yescomplemented
41C23×Q8⋊C8C2214yescomplemented
42C23×D4⋊C8C2214yes—
43C22×Q8×C23—21no—
44C22×Q8×C23D411yescharacteristic
45C22×D4×C23D411yescharacteristic
46C23×C4×C23—21no—
47C2×SD16×C23—22no—
48C2×M16×C23—21no—
49C23×C4⋊Q8—21no—
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8D412yes—
52C23×C4⋊Q8C2311yescharacteristic
53C23×C4⋊Q8C2311yescharacteristic
54C23×C4⋊Q8C2312yes—
55C23×#126—28no—
56C23×#126—28no—
57C23×#125—21no—
58C23×#124—24no—
59C23×#124—24no—
60C23×C22⋊Q8—24no—
61C23×C22⋊Q8—24no—
62C23×C22⋊Q8—21no—
63C23×C22⋊Q8—21no—
64C4×Q8×C23—24no—
65C4×Q8×C23—24no—
66C4×Q8×C23—21no—
67C4×Q8×C23—21no—
68C4×Q8×C23—21no—
69C4×Q8×C23—21no—
70C4×Q8×C23D412yes—
71C4×Q8×C23C2312yes—
72C4×D4×C23—21no—
73C4×D4×C23—21no—
74C4×D4×C23D412yes—
75C4×D4×C23C2312yes—
76C23×#117—24no—
77C23×#117—24no—
78C23×#117—21no—
79C2×C4⋊C4×C23—22no—
80C2×C4⋊C4×C23—21no—
81C2×C4⋊C4×C23—21no—
82C2×C4⋊C4×C23D411yescharacteristic
83C2×C4⋊C4×C23D411yescharacteristic
84C2×C4⋊C4×C23D412yes—
85C2×C22⋊C4×C23—22no—
86C2×C22⋊C4×C23—22no—
87C2×C22⋊C4×C23—21no—
88C2×C42×C23—22no—
89C2×C42×C23—21no—
90C2×C42×C23C2311yescharacteristic
91C23×#107—22no—
92C23×#107—24no—
93C23×#106—22no—
94C23×#106—22no—
95C23×C4⋊C8—21no—
96C23×C4⋊C8—21no—
97C23×C4⋊C8C2312yes—
98C23×C4⋊C8C2312yes—
99C23×#103—22no—
100C23×#103—22no—
101C23×#103—22no—
102C23×#103—24no—
103C23×#102—22no—
104C23×C22⋊C8—21no—
105C23×C22⋊C8—21no—
106C4×C8×C23—21no—
107C4×C8×C23C2312yes—
108C24×C23—21no—
109C2×Q8×C23—21no—
110C2×Q8×C23—22no—
111C2×Q8×C23—22no—
112C2×Q8×C23—22no—
113C2×Q8×C23—22no—
114C2×Q8×C23—21no—
115C2×Q8×C23—21no—
116C2×Q8×C23—44no—
117C2×Q8×C23C2×D412yes—
118C2×D4×C23—21no—
119C2×D4×C23—42no—
120C2×D4×C23C2×D412yes—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—42no—
128C22×C4×C23—42no—
129C22×C4×C23C2×D411yescharacteristic
130C22×C4×C23C2×D411yescharacteristic
131C22×C4×C23C2×D411yescharacteristic
132SD16×C23—44no—
133M16×C23—42no—
134C2×C8×C23—22no—
135C2×C8×C23—22no—
136C2×C8×C23—22no—
137C4⋊C4×C23—28no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—21no—
141C4⋊C4×C23—21no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—22no—
145C4⋊C4×C23—21no—
146C4⋊C4×C23—21no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—21no—
149C4⋊C4×C23—41no—
150C4⋊C4×C23—41no—
151C4⋊C4×C23C2×D412yes—
152C4⋊C4×C23C2×D414yes—
153C4⋊C4×C23C2×D412yes—
154C22⋊C4×C23—28no—
155C22⋊C4×C23—28no—
156C22⋊C4×C23—22no—
157C22⋊C4×C23—22no—
158C22⋊C4×C23—41no—
159C42×C23—28no—
160C42×C23—22no—
161C42×C23—22no—
162C42×C23—41no—
163C42×C23C2×D411yescharacteristic
164C42×C23C2×D411yescharacteristic
165C42×C23C2×D411yescharacteristic
166C42×C23C2411yescharacteristic
167#18763C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23—22no—
169C23×C23—42no—
170C23×C23—42no—
171C23×C23—42no—
172C23×C23#12011yescharacteristic
173Q8×C23—22no—
174Q8×C23—22no—
175Q8×C23—44no—
176Q8×C23—42no—
177Q8×C23—44no—
178Q8×C23—42no—
179Q8×C23—42no—
180D4×C23—22no—
181D4×C23—22no—
182D4×C23—42no—
183C2×C4×C23—22no—
184C2×C4×C23—22no—
185C2×C4×C23—22no—
186C2×C4×C23—22no—
187C2×C4×C23—24no—
188C2×C4×C23—41no—
189C2×C4×C23—42no—
190C2×C4×C23—44no—
191C2×C4×C23—42no—
192C2×C4×C23—42no—
193C2×C4×C23—42no—
194C2×C4×C23—42no—
195C2×C4×C23—44no—
196C2×C4×C23—41no—
197C2×C4×C23—41no—
198C2×C4×C23#12011yescharacteristic
199C2×C4×C23#12011yescharacteristic
200C2×C4×C23#12011yescharacteristic
201C2×C4×C23#13512yes—
202C2×C4×C23#13712yes—
203C2×C4×C23C22×D411yescharacteristic
204C2×C4×C23C22×D411yescharacteristic
205C2×C4×C23C22×D411yescharacteristic
206C8×C23—42no—
207C8×C23—42no—
208C8×C23—42no—
209C2×#556C2×C2311yescharacteristic
210#3230C2×C2311yescharacteristic, complemented
211#3101C2×C2311yescharacteristic
212#3078C2×C2311yescharacteristic, complemented
213#1650C2×C2314yescomplemented
214#1638C2×C2314yescomplemented
215#1549C2×C2311yescharacteristic, complemented
216#1486C2×C2311yescharacteristic
217#1479C2×C2311yescharacteristic, complemented
218C22×C23—21no—
219C22×C23—21no—
220C22×C23—42no—
221C22×C23—42no—
222C22×C23—42no—
223C22×C23—84no—
224C22×C23#45212yes—
225C22×C23C2×#12011yescharacteristic
226C4×C23—21no—
227C4×C23—21no—
228C4×C23—21no—
229C4×C23—42no—
230C4×C23—42no—
231C4×C23—44no—
232C4×C23—42no—
233C4×C23—81no—
234C4×C23—81no—
235C4×C23#45212yes—
236C4×C23C2×#13511yescharacteristic
237C4×C23C2×#13711yescharacteristic
238C4×C23#57512yes—
239#556—22no—
240#556—22no—
241#556—22no—
242#556C22×C2314yes—
243#533C22×C2311yescharacteristic
244C2×C4⋊Q8C22×C2311yescharacteristic
245C2×#126—22no—
246C2×#124—21no—
247C2×C22⋊Q8—21no—
248C2×C4×Q8—21no—
249C2×C4×Q8C22×C2311yescharacteristic
250C2×C4×D4C22×C2311yescharacteristic
251C2×#117—21no—
252#500C22×C2312yescomplemented
253#487C22×C2312yescomplemented
254#484—22no—
255#483—21no—
256#479—22no—
257#477—22no—
258#477C22×C2312yescomplemented
259#476C22×C2312yes—
260#475C22×C2312yescomplemented
261#461C22×C2312yes—
262#450—21no—
263C4×SD16—22no—
264#431C22×C2311yescharacteristic
265#431C22×C2311yescharacteristic
266#428—21no—
267#423C22×C2311yescharacteristic
268#419—21no—
269#332C22×C2314yescomplemented
270Q8⋊C8C22×C2314yescomplemented
271D4⋊C8C22×C2314yes—
272C2×C23—22no—
273C2×C23—82no—
274C2×C23C2×#45211yescharacteristic
275C2×C23#300511yescharacteristic
276C2×C23#302011yescharacteristic
277C22×Q8—21no—
278C22×Q8D4×C2311yescharacteristic
279C22×D4D4×C2311yescharacteristic
280C23×C4—21no—
281C2×SD16—22no—
282C2×M16—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8D4×C2312yes—
286C4⋊Q8C23×C2311yescharacteristic
287C4⋊Q8C23×C2311yescharacteristic
288C4⋊Q8C23×C2312yes—
289#126—28no—
290#126—28no—
291#125—21no—
292#124—24no—
293#124—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8D4×C2312yes—
305C4×Q8C23×C2312yes—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4D4×C2312yes—
309C4×D4C23×C2312yes—
310#117—24no—
311#117—24no—
312#117—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4D4×C2311yescharacteristic
317C2×C4⋊C4D4×C2311yescharacteristic
318C2×C4⋊C4D4×C2312yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42C23×C2311yescharacteristic
325#107—22no—
326#107—24no—
327#106—22no—
328#106—22no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8C23×C2312yes—
332C4⋊C8C23×C2312yes—
333#103—22no—
334#103—22no—
335#103—22no—
336#103—24no—
337#102—22no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C4×C8—21no—
341C4×C8C23×C2312yes—
342C23#1876311yescharacteristic, direct product factor, Sylow
343C24—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—44no—
352C2×Q8C2×D4×C2312yes—
353C2×D4—21no—
354C2×D4—42no—
355C2×D4C2×D4×C2312yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4C2×D4×C2311yescharacteristic
365C22×C4C2×D4×C2311yescharacteristic
366C22×C4C2×D4×C2311yescharacteristic
367SD16—44no—
368M16—42no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4C2×D4×C2312yes—
387C4⋊C4C2×D4×C2314yes—
388C4⋊C4C2×D4×C2312yes—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—41no—
394C42—28no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42C2×D4×C2311yescharacteristic
399C42C2×D4×C2311yescharacteristic
400C42C2×D4×C2311yescharacteristic
401C42C24×C2311yescharacteristic
402C23—22no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23C23×#12011yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—44no—
410Q8—42no—
411Q8—44no—
412Q8—42no—
413Q8—42no—
414D4—22no—
415D4—22no—
416D4—42no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4C23×#12011yescharacteristic
433C2×C4C23×#12011yescharacteristic
434C2×C4C23×#12011yescharacteristic
435C2×C4C23×#13512yes—
436C2×C4C23×#13712yes—
437C2×C4C22×D4×C2311yescharacteristic
438C2×C4C22×D4×C2311yescharacteristic
439C2×C4C22×D4×C2311yescharacteristic
440C8—42no—
441C8—42no—
442C8—42no—
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—84no—
449C22C23×#45212yes—
450C22C2×C23×#12011yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4C23×#45212yes—
461C4C2×C23×#13511yescharacteristic
462C4C2×C23×#13711yescharacteristic
463C4C23×#57512yes—
464C2—22no—
465C2—82no—
466C2C2×C23×#45211yescharacteristic
467C2C23×#300511yescharacteristic
468C2C23×#302011yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC22×D4×C23D4
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
2. C23×#3230C2semidirect
4. C23×#3078C2semidirect
5. C23×#1650C2semidirect
6. C23×#1638C2semidirect
7. C23×#1549C2semidirect
9. C23×#1479C2semidirect
23. C23×#500C22semidirect
24. C23×#487C22semidirect
29. C23×#477C22semidirect
31. C23×#475C22semidirect
40. C23×#332C22semidirect
41. C23×Q8⋊C8C22semidirect
167. #18763C23direct (N × H)
210. #3230C2×C23semidirect
212. #3078C2×C23semidirect
213. #1650C2×C23semidirect
214. #1638C2×C23semidirect
215. #1549C2×C23semidirect
217. #1479C2×C23semidirect
252. #500C22×C23semidirect
253. #487C22×C23semidirect
258. #477C22×C23semidirect
260. #475C22×C23semidirect
269. #332C22×C23semidirect
270. Q8⋊C8C22×C23semidirect
342. C23#18763direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5562
C23×#32302
C23×#31012
C23×#30782
C23×#16502
C23×#16382
C23×#15492
C23×#14862
C23×#14792
#1876323