Finite Group Databasebeta

← #15676377

Subgroups — #15676377

Order 2944.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625108C211yesmaximal, characteristic
2C2×D4×Dic23C211yesmaximal, characteristic, complemented
3C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic, complemented
4C22×(C2×D23)⋊C4C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D23C211yesmaximal, characteristic, complemented
6#13092317C211yesmaximal, characteristic, complemented
7#13092309C211yesmaximal, characteristic, complemented
8#13092306C211yesmaximal, characteristic, complemented
9#13092294C211yesmaximal, characteristic, complemented
10#13092293C211yesmaximal, characteristic, complemented
11#13092291C211yesmaximal, characteristic, complemented
12#13092289C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic23C211yesmaximal, characteristic, complemented
14#13092264C211yesmaximal, characteristic, complemented
15C23×#386C211yesmaximal, characteristic, complemented
16C24×D23C2211yescharacteristic
17C22×D4×C23C2211yescharacteristic
18C23×C4×C23C2211yescharacteristic
19C22×C22⋊D23C2211yescharacteristic
20C22×C22⋊D23C2211yescharacteristic
21C23×Dic23C2211yescharacteristic
22C23×Dic23C2211yescharacteristic, complemented
23C22×C4×D23C2211yescharacteristic
24C22×C4×D23C2211yescharacteristic
25C2×C22⋊C4×C23C2211yescharacteristic
26C2×C22⋊C4×C23C2211yescharacteristic, complemented
27C2×C22⋊C4×C23C2211yescharacteristic, complemented
28C2×C22⋊Dic23C2211yescharacteristic
29C2×C22⋊Dic23C2211yescharacteristic
30C2×C22⋊Dic23C2211yescharacteristic, complemented
31#10625108C418yescomplemented
32D4×Dic23—24no—
33C2×(C2×D23)⋊C4—24no—
34C2×(C2×D23)⋊C4C2211yescharacteristic
35C2×(C2×D23)⋊C4C2211yescharacteristic
36C2×(C2×D23)⋊C4C2211yescharacteristic
37C2×(C2×D23)⋊C4C2211yescharacteristic
38C2×(C2×D23)⋊C4C2211yescharacteristic
39C2×(C2×D23)⋊C4C2211yescharacteristic
40C2×(C2×D23)⋊C4C2211yescharacteristic
41C2×(C2×D23)⋊C4C2211yescharacteristic
42C2×C4⋊Dic23C2211yescharacteristic, complemented
43C2×Dic23⋊C4C2211yescharacteristic, complemented
44C2×Dic23⋊C4C2211yescharacteristic, complemented
45C2×C4×Dic23C2211yescharacteristic, complemented
46C2×C4×Dic23C2211yescharacteristic, complemented
47C2×C4×Dic23C2211yescharacteristic, complemented
48(C22⋊D23)⋊C4—24no—
49C22⋊C4×D23—24no—
50C23×#95C2211yescharacteristic, complemented
51C23×#95C2211yescharacteristic, complemented
52C23⋊#95C2211yescharacteristic, complemented
53C23⋊#95C2211yescharacteristic, complemented
54C23⋊#95C2211yescharacteristic, complemented
55C23⋊#95C2211yescharacteristic, complemented
56C24×C23C2311yescharacteristic
57C24×C23C2311yescharacteristic
58C23×D23—22no—
59C23×D23—22no—
60C23×D23—22no—
61C23×D23—22no—
62C23×D23—41no—
63C23×D23—41no—
64C23×D23C2×C414yescomplemented
65C23×D23C2311yescharacteristic
66C23×D23C2311yescharacteristic
67C2×D4×C23—22no—
68C2×D4×C23—22no—
69C2×D4×C23C2×C414yescomplemented
70C22×C4×C23—22no—
71C22×C4×C23—22no—
72C22×C4×C23—41no—
73C22×C4×C23C2311yescharacteristic
74C22×C4×C23C2311yescharacteristic
75C22×C4×C23C2311yescharacteristic
76C22×C4×C23C2311yescharacteristic
77C22×C4×C23C2311yescharacteristic
78C2×C22⋊D23—22no—
79C2×C22⋊D23—22no—
80C2×C22⋊D23—41no—
81C2×C22⋊D23—41no—
82C2×C22⋊D23C2×C414yescomplemented
83C2×C22⋊D23C2×C414yescomplemented
84C22×Dic23—22no—
85C22×Dic23—22no—
86C22×Dic23—22no—
87C22×Dic23—22no—
88C22×Dic23—41no—
89C22×Dic23—41no—
90C22×Dic23C2311yescharacteristic
91C22×Dic23C2311yescharacteristic
92C22×Dic23C2311yescharacteristic
93C22×Dic23C2311yescharacteristic
94C22×Dic23C2311yescharacteristic
95C22×Dic23C2311yescharacteristic
96C2×C4×D23—22no—
97C2×C4×D23—22no—
98C2×C4×D23—22no—
99C2×C4×D23—22no—
100C2×C4×D23—41no—
101C2×C4×D23—41no—
102C22⋊C4×C23—22no—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—41no—
105C22⋊Dic23—22no—
106C22⋊Dic23C2×C414yescomplemented
107C22⋊Dic23D414yescomplemented
108(C2×D23)⋊C4—22no—
109(C2×D23)⋊C4—22no—
110(C2×D23)⋊C4—22no—
111(C2×D23)⋊C4—22no—
112(C2×D23)⋊C4—22no—
113(C2×D23)⋊C4C2×C414yescomplemented
114(C2×D23)⋊C4C2×C414yescomplemented
115(C2×D23)⋊C4D414yescomplemented
116C4⋊Dic23—22no—
117Dic23⋊C4—22no—
118Dic23⋊C4—41no—
119C4×Dic23—22no—
120C4×Dic23—22no—
121C4×Dic23—41no—
122C23×C23—21no—
123C23×C23—21no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—21no—
128C23×C23—21no—
129C23×C23—21no—
130C23×C23—21no—
131C23×C23—21no—
132C23×C23C22×C412yes—
133C23×C23C22×C412yes—
134C23×C23C2×D412yes—
135C23×C23#4112yes—
136C23×C23C2411yescharacteristic
137C22×D23—21no—
138C22×D23—21no—
139C22×D23—21no—
140C22×D23—21no—
141C22×D23—28no—
142C22×D23—21no—
143C22×D23—21no—
144C22×D23—21no—
145C22×D23—21no—
146C22×D23—21no—
147C22×D23—21no—
148C22×D23—28no—
149C22×D23—44no—
150C22×D23—44no—
151C22×D23—44no—
152C22×D23—44no—
153C22×D23—44no—
154C22×D23C22×C412yes—
155C22×D23C22×C412yes—
156C22×D23C2×D412yes—
157C22×D23#4112yes—
158D4×C23—28no—
159C2×C4×C23—21no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23—21no—
170C2×C4×C23—21no—
171C2×C4×C23—21no—
172C2×C4×C23—44no—
173C2×C4×C23C22×C412yes—
174C2×C4×C23C2×D412yes—
175C22⋊D23—28no—
176C22⋊D23—44no—
177C2×Dic23—21no—
178C2×Dic23—21no—
179C2×Dic23—21no—
180C2×Dic23—21no—
181C2×Dic23—21no—
182C2×Dic23—21no—
183C2×Dic23—21no—
184C2×Dic23—21no—
185C2×Dic23—21no—
186C2×Dic23—21no—
187C2×Dic23—21no—
188C2×Dic23—21no—
189C2×Dic23—21no—
190C2×Dic23—44no—
191C2×Dic23—44no—
192C2×Dic23C22×C412yes—
193C2×Dic23C22×C412yes—
194C2×Dic23C2×D412yes—
195C2×Dic23C2×D412yes—
196C2×Dic23C2×D412yes—
197C4×D23—44no—
198C4×D23—44no—
199#2358—231nomaximal, Sylow
200C22×C23—22no—
201C22×C23—22no—
202C22×C23—22no—
203C22×C23—22no—
204C22×C23—22no—
205C22×C23—22no—
206C22×C23—22no—
207C22×C23—22no—
208C22×C23—22no—
209C22×C23—41no—
210C22×C23—41no—
211C22×C23—41no—
212C22×C23—41no—
213C22×C23C4×D414yescomplemented
214C22×C23C23×C411yescharacteristic
215C22×C23C22×D411yescharacteristic
216C22×C23C22×D411yescharacteristic
217C22×C23C2×#4111yescharacteristic
218C22×C23C2×#4111yescharacteristic
219C22×C23H(5,𝔽2)11yescharacteristic
220C22×C23H(5,𝔽2)11yescharacteristic
221C2×D23—22no—
222C2×D23—22no—
223C2×D23—22no—
224C2×D23—22no—
225C2×D23—22no—
226C2×D23—22no—
227C2×D23—22no—
228C2×D23—22no—
229C2×D23—22no—
230C2×D23—416no—
231C2×D23—41no—
232C2×D23—41no—
233C2×D23—41no—
234C2×D23—41no—
235C2×D23C4×D414yescomplemented
236C4×C23—22no—
237C4×C23—22no—
238C4×C23—41no—
239C4×C23—41no—
240C4×C23—41no—
241Dic23—22no—
242Dic23—22no—
243Dic23—22no—
244Dic23—22no—
245Dic23—22no—
246Dic23—41no—
247C2×#120—231no—
248C2×C4×D4—231no—
249C2×C4×D4—231no—
250C22×C22⋊C4—231no—
251C22×C22⋊C4—231no—
252#390—231no—
253#388—231no—
254#388—231no—
255#386—231no—
256#386—231no—
257#386D2311yescharacteristic, complemented
258#385—231no—
259#385—231no—
260C4×C22⋊C4—231no—
261C4×C22⋊C4—231no—
262C2×C23—24no—
263C2×C23—42no—
264C2×C23C2×C4×D411yescharacteristic
265C2×C23C2×C4×D411yescharacteristic
266C2×C23#51811yescharacteristic
267C2×C23D4211yescharacteristic
268C2×C23#54611yescharacteristic
269C2×C23#54611yescharacteristic
270C2×C23#55111yescharacteristic
271D23—24no—
272D23—42no—
273C25—231no—
274C22×D4—231no—
275C22×D4—231no—
276C22×D4C2×D2311yescharacteristic
277C23×C4—231no—
278C23×C4—231no—
279C23×C4—231no—
280C23×C4—231no—
281C23×C4C2×D2311yescharacteristic
282#120—238no—
283C4×D4—464no—
284C4×D4—464no—
285C2×C4⋊C4—231no—
286C2×C4⋊C4—231no—
287C2×C4⋊C4—231no—
288C2×C22⋊C4—231no—
289C2×C22⋊C4—231no—
290C2×C22⋊C4—231no—
291C2×C22⋊C4—231no—
292C2×C22⋊C4—231no—
293C2×C22⋊C4—231no—
294C2×C22⋊C4—231no—
295C2×C22⋊C4—231no—
296C2×C22⋊C4—231no—
297C2×C22⋊C4—231no—
298C2×C22⋊C4—231no—
299C2×C22⋊C4—464no—
300C2×C22⋊C4—464no—
301C2×C22⋊C4C2×D2311yescharacteristic
302C2×C22⋊C4C2×D2311yescharacteristic, complemented
303C2×C22⋊C4C2×D2311yescharacteristic, complemented
304C2×C42—231no—
305C2×C42—231no—
306C2×C42—231no—
307#95—231no—
308#95—231no—
309#95—231no—
310#95—231no—
311#95C2×D2311yescharacteristic, complemented
312#95C2×D2311yescharacteristic, complemented
313C23#235811yescharacteristic, complemented, Sylow
314C24—231no—
315C24—231no—
316C24—234no—
317C24—462no—
318C24—462no—
319C24—462no—
320C24—462no—
321C24—921no—
322C24—921no—
323C24C22×D2311yescharacteristic
324C24C22×D2311yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—234no—
328C2×D4—234no—
329C2×D4—462no—
330C2×D4—462no—
331C2×D4—921no—
332C2×D4—921no—
333C2×D4C4×D2314yescomplemented
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—231no—
338C22×C4—231no—
339C22×C4—231no—
340C22×C4—231no—
341C22×C4—231no—
342C22×C4—231no—
343C22×C4—462no—
344C22×C4—462no—
345C22×C4—462no—
346C22×C4—462no—
347C22×C4—462no—
348C22×C4—462no—
349C22×C4—462no—
350C22×C4—462no—
351C22×C4—921no—
352C22×C4—921no—
353C22×C4—921no—
354C22×C4—921no—
355C22×C4C22×D2311yescharacteristic
356C22×C4C22×D2311yescharacteristic
357C22×C4C22×D2311yescharacteristic
358C22×C4C22×D2311yescharacteristic
359C22×C4C22×D2311yescharacteristic
360C4⋊C4—462no—
361C4⋊C4—462no—
362C4⋊C4—921no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—41no—
366C22⋊C4—234no—
367C22⋊C4—234no—
368C22⋊C4—234no—
369C22⋊C4—234no—
370C22⋊C4—234no—
371C22⋊C4—462no—
372C22⋊C4—462no—
373C22⋊C4—462no—
374C22⋊C4—462no—
375C22⋊C4—462no—
376C22⋊C4—462no—
377C42—462no—
378C42—462no—
379C42—921no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—232no—
391C23—232no—
392C23—232no—
393C23—232no—
394C23—468no—
395C23—461no—
396C23—461no—
397C23—461no—
398C23—461no—
399C23—468no—
400C23—461no—
401C23—461no—
402C23—461no—
403C23—461no—
404C23—461no—
405C23—461no—
406C23—924no—
407C23—924no—
408C23—924no—
409C23—924no—
410C23—924no—
411C23C2×C4×D2312yes—
412C23C2×C4×D2312yes—
413C23D4×D2312yes—
414C23#6729012yes—
415C23C23×D2311yescharacteristic
416D4—28no—
417D4—468no—
418D4—924no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—44no—
433C2×C4—232no—
434C2×C4—232no—
435C2×C4—232no—
436C2×C4—232no—
437C2×C4—232no—
438C2×C4—461no—
439C2×C4—461no—
440C2×C4—461no—
441C2×C4—461no—
442C2×C4—461no—
443C2×C4—461no—
444C2×C4—461no—
445C2×C4—461no—
446C2×C4—461no—
447C2×C4—461no—
448C2×C4—461no—
449C2×C4—461no—
450C2×C4—461no—
451C2×C4—924no—
452C2×C4—924no—
453C2×C4—924no—
454C2×C4—924no—
455C2×C4C2×C4×D2312yes—
456C2×C4D4×D2312yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—234no—
471C22—462no—
472C22—462no—
473C22—462no—
474C22—462no—
475C22—462no—
476C22—462no—
477C22—462no—
478C22—462no—
479C22—462no—
480C22—9216no—
481C22—921no—
482C22—921no—
483C22—921no—
484C22—921no—
485C22(C22⋊D23)⋊C414yescomplemented
486C22C22×C4×D2311yescharacteristic
487C22C2×D4×D2311yescharacteristic
488C22C2×D4×D2311yescharacteristic
489C22C2×#6729011yescharacteristic
490C22#1062515511yescharacteristic
491C22#41×D2311yescharacteristic
492C22#1062516011yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—462no—
499C4—462no—
500C4—462no—
501C4—462no—
502C4—462no—
503C4—921no—
504C2—24no—
505C2—42no—
506C2—464no—
507C2—922no—
508C2C4×D4×D2311yescharacteristic
509C2C2×(C22⋊D23)⋊C411yescharacteristic
510C2#1309288911yescharacteristic
511C2#1309305111yescharacteristic
512C2#1309305411yescharacteristic
513C2#1309305611yescharacteristic
514C2#1309309011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC23×#386C2
socle, cosocleC23×C23C24
prime radicalC2×#10625108C2
hypercenter#386D23
hypocenterC23#2358
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic23C2semidirect
3. C2×(C22⋊D23)⋊C4C2semidirect
4. C22×(C2×D23)⋊C4C2semidirect
5. C2×C22⋊C4×D23C2semidirect
6. #13092317C2semidirect
7. #13092309C2semidirect
8. #13092306C2semidirect
9. #13092294C2semidirect
10. #13092293C2semidirect
11. #13092291C2semidirect
12. #13092289C2semidirect
13. C4×C22⋊Dic23C2semidirect
14. #13092264C2semidirect
15. C23×#386C2semidirect
22. C23×Dic23C22semidirect
26. C2×C22⋊C4×C23C22semidirect
27. C2×C22⋊C4×C23C22semidirect
30. C2×C22⋊Dic23C22semidirect
31. #10625108C4semidirect
42. C2×C4⋊Dic23C22semidirect
43. C2×Dic23⋊C4C22semidirect
44. C2×Dic23⋊C4C22semidirect
45. C2×C4×Dic23C22semidirect
46. C2×C4×Dic23C22semidirect
47. C2×C4×Dic23C22semidirect
50. C23×#95C22semidirect
51. C23×#95C22semidirect
52. C23⋊#95C22semidirect
53. C23⋊#95C22semidirect
54. C23⋊#95C22semidirect
55. C23⋊#95C22semidirect
64. C23×D23C2×C4semidirect
69. C2×D4×C23C2×C4semidirect
82. C2×C22⋊D23C2×C4semidirect
83. C2×C22⋊D23C2×C4semidirect
106. C22⋊Dic23C2×C4semidirect
107. C22⋊Dic23D4semidirect
113. (C2×D23)⋊C4C2×C4semidirect
114. (C2×D23)⋊C4C2×C4semidirect
115. (C2×D23)⋊C4D4semidirect
213. C22×C23C4×D4semidirect
235. C2×D23C4×D4semidirect
257. #386D23semidirect
302. C2×C22⋊C4C2×D23semidirect
303. C2×C22⋊C4C2×D23semidirect
311. #95C2×D23semidirect
312. #95C2×D23semidirect
313. C23#2358semidirect
333. C2×D4C4×D23semidirect
485. C22(C22⋊D23)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#106251082
C2×D4×Dic232
C2×(C22⋊D23)⋊C42
C22×(C2×D23)⋊C42
C2×C22⋊C4×D232
#130923172
#130923092
#130923062
#130922942
#130922932
#130922912
#130922892
C4×C22⋊Dic232
#130922642
C23×#3862
#235823