Finite Group Databasebeta

← D23×#388 #15675825

Subgroups — D23×#388 #15675825

Order 2944.

928 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D23C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D23C211yesmaximal, characteristic
3C2×C22⋊C4×D23C211yesmaximal, characteristic
4C2×C22⋊C4×D23C211yesmaximal, characteristic
5C2×C42×D23C211yesmaximal, characteristic, complemented
6#13092311C211yesmaximal, characteristic, complemented
7#13092310C211yesmaximal, characteristic, complemented
8#13092309C211yesmaximal, characteristic, complemented
9#13092308C211yesmaximal, characteristic, complemented
10#13092307C211yesmaximal, characteristic, complemented
11#13092306C211yesmaximal, characteristic, complemented
12#13092305C211yesmaximal, characteristic, complemented
13#95×D23C211yesmaximal, characteristic, complemented
14#95×D23C211yesmaximal, characteristic, complemented
15C23×#388C211yesmaximal, characteristic, complemented
16C24×D23C2211yescharacteristic
17C22×C4×D23C2211yescharacteristic
18C22×C4×D23C2211yescharacteristic
19C22×C4×D23C2211yescharacteristic
20C22×C4×D23C2211yescharacteristic
21C22×C4×D23C2211yescharacteristic
22C22×C4×D23C2211yescharacteristic
23C2×C4⋊C4×C23C2211yescharacteristic, complemented
24C2×C22⋊C4×C23C2211yescharacteristic
25C2×C22⋊C4×C23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic
27C2×C42×C23C2211yescharacteristic, complemented
28C2×C22⋊Dic23C2211yescharacteristic
29C2×C22⋊Dic23C2211yescharacteristic
30C2×C22⋊Dic23C2211yescharacteristic
31C2×(C2×D23)⋊C4C2211yescharacteristic
32C2×(C2×D23)⋊C4C2211yescharacteristic
33C2×(C2×D23)⋊C4C2211yescharacteristic
34C2×(C2×D23)⋊C4C2211yescharacteristic
35C2×(C2×D23)⋊C4C2211yescharacteristic
36C2×(C2×D23)⋊C4C2211yescharacteristic
37C2×C4⋊Dic23C2211yescharacteristic, complemented
38C2×Dic23⋊C4C2211yescharacteristic, complemented
39C2×Dic23⋊C4C2211yescharacteristic, complemented
40C2×C4×Dic23C2211yescharacteristic, complemented
41C2×C4×Dic23C2211yescharacteristic, complemented
42C2×C4×Dic23C2211yescharacteristic, complemented
43C4⋊C4×D23—24no—
44C22⋊C4×D23—24no—
45C22⋊C4×D23—24no—
46C22⋊C4×D23C418yescomplemented
47C42×D23—24no—
48C23×#95C2211yescharacteristic, complemented
49C23×#95C2211yescharacteristic, complemented
50C23⋊#95C2211yescharacteristic, complemented
51C23⋊#95C2211yescharacteristic, complemented
52C23⋊#95C2211yescharacteristic, complemented
53C23⋊#95C2211yescharacteristic, complemented
54C23⋊#95C2211yescharacteristic, complemented
55C23⋊#95C2211yescharacteristic, complemented
56C24×C23C2311yescharacteristic
57C23×D23—22no—
58C23×D23—22no—
59C23×D23—22no—
60C23×D23—22no—
61C23×D23—22no—
62C23×D23—22no—
63C23×D23C2×C414yes—
64C23×D23C2311yescharacteristic
65C23×D23C2311yescharacteristic
66C22×C4×C23C2311yescharacteristic
67C22×C4×C23C2311yescharacteristic
68C22×C4×C23C2311yescharacteristic
69C22×C4×C23C2311yescharacteristic
70C22×C4×C23C2311yescharacteristic
71C22×C4×C23C2311yescharacteristic
72C22×Dic23C2311yescharacteristic
73C22×Dic23C2311yescharacteristic
74C22×Dic23C2311yescharacteristic
75C22×Dic23C2311yescharacteristic
76C22×Dic23C2311yescharacteristic
77C22×Dic23C2311yescharacteristic
78C2×C4×D23—22no—
79C2×C4×D23—22no—
80C2×C4×D23—22no—
81C2×C4×D23—22no—
82C2×C4×D23—22no—
83C2×C4×D23—22no—
84C2×C4×D23—22no—
85C2×C4×D23—22no—
86C2×C4×D23—22no—
87C2×C4×D23—22no—
88C2×C4×D23—22no—
89C2×C4×D23—22no—
90C2×C4×D23—22no—
91C2×C4×D23—22no—
92C2×C4×D23C2×C414yes—
93C2×C4×D23C2×C414yes—
94C2×C4×D23D414yes—
95C2×C4×D23D414yes—
96C4⋊C4×C23—22no—
97C22⋊C4×C23—22no—
98C22⋊C4×C23—22no—
99C22⋊C4×C23C2×C414yescomplemented
100C42×C23—22no—
101C22⋊Dic23—22no—
102C22⋊Dic23—22no—
103C22⋊Dic23C2×C414yescomplemented
104(C2×D23)⋊C4—22no—
105(C2×D23)⋊C4—22no—
106(C2×D23)⋊C4—22no—
107(C2×D23)⋊C4—22no—
108(C2×D23)⋊C4C2×C414yescomplemented
109(C2×D23)⋊C4C2×C414yescomplemented
110C4⋊Dic23—22no—
111Dic23⋊C4—22no—
112Dic23⋊C4—22no—
113C4×Dic23—22no—
114C4×Dic23—22no—
115C4×Dic23—22no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—21no—
120C23×C23—21no—
121C23×C23—21no—
122C23×C23C22×C412yes—
123C23×C23C2411yescharacteristic
124C22×D23—21no—
125C22×D23—21no—
126C22×D23—21no—
127C22×D23—21no—
128C22×D23—21no—
129C22×D23—21no—
130C22×D23—28no—
131C22×D23—28no—
132C22×D23—28no—
133C22×D23—44no—
134C22×D23—44no—
135C22×D23—44no—
136C22×D23—44no—
137C22×D23C22×C412yes—
138C22×D23C22×C412yes—
139C22×D23C2×D412yes—
140C22×D23C2×D412yes—
141C22×D23#4112yes—
142C22×D23#4112yes—
143C22×D23#4112yes—
144C22×D23#4112yes—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23C22×C412yes—
160C2×C4×C23C22×C412yes—
161C2×C4×C23C2×D412yes—
162C2×C4×C23C2×D412yes—
163C2×Dic23—21no—
164C2×Dic23—21no—
165C2×Dic23—21no—
166C2×Dic23—21no—
167C2×Dic23—21no—
168C2×Dic23—21no—
169C2×Dic23—21no—
170C2×Dic23—21no—
171C2×Dic23—21no—
172C2×Dic23—21no—
173C2×Dic23—21no—
174C2×Dic23—21no—
175C2×Dic23—21no—
176C2×Dic23—21no—
177C2×Dic23C22×C412yes—
178C2×Dic23C22×C412yes—
179C2×Dic23C2×D412yes—
180C2×Dic23C2×D412yes—
181C4×D23—28no—
182C4×D23—28no—
183C4×D23—28no—
184C4×D23—28no—
185C4×D23—44no—
186C4×D23—44no—
187C2×#388—231nomaximal, Sylow
188C22×C23—22no—
189C22×C23—22no—
190C22×C23—22no—
191C22×C23—41no—
192C22×C23—41no—
193C22×C23—41no—
194C22×C23—41no—
195C22×C23C23×C411yescharacteristic
196C22×C23C22×D411yescharacteristic
197C22×C23C22×D411yescharacteristic
198C22×C23C2×#4111yescharacteristic
199C22×C23C2×#4111yescharacteristic
200C22×C23C2×#4111yescharacteristic
201C22×C23C2×#4111yescharacteristic
202C2×D23—22no—
203C2×D23—22no—
204C2×D23—22no—
205C2×D23—41no—
206C2×D23—41no—
207C2×D23—41no—
208C2×D23—41no—
209C2×D23—416no—
210C2×D23#11714yes—
211C2×D23C4×D414yes—
212C2×D23C4×D414yes—
213C2×D23#12114yes—
214C2×D23#12314yes—
215C2×D23#12414yes—
216C2×D23#12614yes—
217C4×C23—22no—
218C4×C23—22no—
219C4×C23—22no—
220C4×C23—22no—
221C4×C23—41no—
222C4×C23—41no—
223Dic23—22no—
224Dic23—22no—
225Dic23—22no—
226Dic23—22no—
227Dic23—41no—
228Dic23—41no—
229C22×C4⋊C4—231no—
230C22×C22⋊C4—231no—
231C22×C22⋊C4—231no—
232C22×C22⋊C4—231no—
233C22×C42—231no—
234#388—231no—
235#388—231no—
236#388—231no—
237#388—231no—
238#388—231no—
239#388—231no—
240#388—231no—
241#388D2311yescharacteristic, direct product factor
242C2×#95—231no—
243C2×#95—231no—
244C2×C23—42no—
245C2×C23C2×#11711yescharacteristic
246C2×C23C2×C4×D411yescharacteristic
247C2×C23C2×C4×D411yescharacteristic
248C2×C23C2×#12111yescharacteristic
249C2×C23C2×#12311yescharacteristic
250C2×C23C2×#12411yescharacteristic
251C2×C23C2×#12611yescharacteristic
252D23—42no—
253D23#38818yesdirect product factor
254C25—231no—
255C23×C4—231no—
256C23×C4—231no—
257C23×C4—231no—
258C23×C4—231no—
259C23×C4—231no—
260C23×C4—231no—
261C2×C4⋊C4—231no—
262C2×C4⋊C4—231no—
263C2×C4⋊C4—231no—
264C2×C4⋊C4—464no—
265C2×C4⋊C4C2×D2311yescharacteristic, complemented
266C2×C22⋊C4—231no—
267C2×C22⋊C4—231no—
268C2×C22⋊C4—231no—
269C2×C22⋊C4—238no—
270C2×C22⋊C4—231no—
271C2×C22⋊C4—231no—
272C2×C22⋊C4—231no—
273C2×C22⋊C4—231no—
274C2×C22⋊C4—231no—
275C2×C22⋊C4—231no—
276C2×C22⋊C4—464no—
277C2×C22⋊C4—464no—
278C2×C22⋊C4C2×D2311yescharacteristic
279C2×C22⋊C4C2×D2311yescharacteristic
280C2×C22⋊C4C2×D2311yescharacteristic
281C2×C42—231no—
282C2×C42—231no—
283C2×C42—231no—
284C2×C42—464no—
285C2×C42C2×D2311yescharacteristic, complemented
286#95—231no—
287#95—231no—
288#95—231no—
289#95—231no—
290#95—231no—
291#95—231no—
292#95C2×D2311yescharacteristic, complemented
293#95C2×D2311yescharacteristic, complemented
294C23C2×#38811yescharacteristic, complemented, Sylow
295C24—231no—
296C24—234no—
297C24—231no—
298C24—462no—
299C24—462no—
300C24—462no—
301C24—462no—
302C24—462no—
303C24—462no—
304C24C22×D2311yescharacteristic
305C22×C4—231no—
306C22×C4—231no—
307C22×C4—231no—
308C22×C4—231no—
309C22×C4—234no—
310C22×C4—234no—
311C22×C4—234no—
312C22×C4—234no—
313C22×C4—231no—
314C22×C4—231no—
315C22×C4—462no—
316C22×C4—462no—
317C22×C4—462no—
318C22×C4—462no—
319C22×C4—462no—
320C22×C4—462no—
321C22×C4—462no—
322C22×C4—462no—
323C22×C4—462no—
324C22×C4—462no—
325C22×C4—462no—
326C22×C4—462no—
327C22×C4—462no—
328C22×C4—462no—
329C22×C4C22×D2311yescharacteristic
330C22×C4C22×D2311yescharacteristic
331C22×C4C22×D2311yescharacteristic
332C22×C4C22×D2311yescharacteristic
333C22×C4C22×D2311yescharacteristic
334C22×C4C22×D2311yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—462no—
337C4⋊C4—462no—
338C4⋊C4—462no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—234no—
342C22⋊C4—234no—
343C22⋊C4—234no—
344C22⋊C4—462no—
345C22⋊C4—462no—
346C22⋊C4—462no—
347C22⋊C4—462no—
348C22⋊C4—462no—
349C22⋊C4—462no—
350C22⋊C4C4×D2314yescomplemented
351C42—22no—
352C42—462no—
353C42—462no—
354C42—462no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—232no—
362C23—232no—
363C23—232no—
364C23—232no—
365C23—232no—
366C23—232no—
367C23—232no—
368C23—232no—
369C23—468no—
370C23—468no—
371C23—468no—
372C23—461no—
373C23—461no—
374C23—461no—
375C23—461no—
376C23—461no—
377C23—461no—
378C23—924no—
379C23—924no—
380C23—924no—
381C23—924no—
382C23C2×C4×D2312yes—
383C23C23×D2311yescharacteristic
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—232no—
399C2×C4—232no—
400C2×C4—232no—
401C2×C4—232no—
402C2×C4—468no—
403C2×C4—468no—
404C2×C4—468no—
405C2×C4—468no—
406C2×C4—461no—
407C2×C4—461no—
408C2×C4—461no—
409C2×C4—461no—
410C2×C4—461no—
411C2×C4—461no—
412C2×C4—461no—
413C2×C4—461no—
414C2×C4—461no—
415C2×C4—461no—
416C2×C4—461no—
417C2×C4—461no—
418C2×C4—461no—
419C2×C4—461no—
420C2×C4—924no—
421C2×C4—924no—
422C2×C4C2×C4×D2312yes—
423C2×C4C2×C4×D2312yes—
424C2×C4D4×D2312yes—
425C2×C4D4×D2312yes—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—234no—
434C22—234no—
435C22—234no—
436C22—234no—
437C22—234no—
438C22—234no—
439C22—234no—
440C22—462no—
441C22—462no—
442C22—462no—
443C22—9216no—
444C22—921no—
445C22—921no—
446C22—921no—
447C22—921no—
448C22C22×C4×D2311yescharacteristic
449C22C2×D4×D2311yescharacteristic
450C22C2×D4×D2311yescharacteristic
451C22#41×D2311yescharacteristic
452C22#41×D2311yescharacteristic
453C22#41×D2311yescharacteristic
454C22#41×D2311yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—462no—
462C4—462no—
463C4—462no—
464C4—462no—
465C4—921no—
466C4—921no—
467C2—42no—
468C2—238no—
469C2—922no—
470C2#117×D2311yescharacteristic
471C2C4×D4×D2311yescharacteristic
472C2C4×D4×D2311yescharacteristic
473C2#121×D2311yescharacteristic
474C2#123×D2311yescharacteristic
475C2#124×D2311yescharacteristic
476C2#126×D2311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC23×#388C2
socle, cosocleC23×C23C24
prime radicalC24×D23C22
hypercenter#388D23
hypocenterC23C2×#388
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D23C2semidirect
5. C2×C42×D23C2semidirect
6. #13092311C2semidirect
7. #13092310C2semidirect
8. #13092309C2semidirect
9. #13092308C2semidirect
10. #13092307C2semidirect
11. #13092306C2semidirect
12. #13092305C2semidirect
13. #95×D23C2semidirect
14. #95×D23C2semidirect
15. C23×#388C2semidirect
23. C2×C4⋊C4×C23C22semidirect
27. C2×C42×C23C22semidirect
37. C2×C4⋊Dic23C22semidirect
38. C2×Dic23⋊C4C22semidirect
39. C2×Dic23⋊C4C22semidirect
40. C2×C4×Dic23C22semidirect
41. C2×C4×Dic23C22semidirect
42. C2×C4×Dic23C22semidirect
46. C22⋊C4×D23C4semidirect
48. C23×#95C22semidirect
49. C23×#95C22semidirect
50. C23⋊#95C22semidirect
51. C23⋊#95C22semidirect
52. C23⋊#95C22semidirect
53. C23⋊#95C22semidirect
54. C23⋊#95C22semidirect
55. C23⋊#95C22semidirect
99. C22⋊C4×C23C2×C4semidirect
103. C22⋊Dic23C2×C4semidirect
108. (C2×D23)⋊C4C2×C4semidirect
109. (C2×D23)⋊C4C2×C4semidirect
241. #388D23direct (N × H)
253. D23#388direct (N × H)
265. C2×C4⋊C4C2×D23semidirect
285. C2×C42C2×D23semidirect
292. #95C2×D23semidirect
293. #95C2×D23semidirect
294. C23C2×#388semidirect
350. C22⋊C4C4×D23semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D232
C2×C22⋊C4×D232
C2×C42×D232
#130923112
#130923102
#130923092
#130923082
#130923072
#130923062
#130923052
#95×D232
C23×#3882
C2×#38823