Finite Group Databasebeta

← #15682705

Subgroups — #15682705

Order 2944.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625160C211yesmaximal, characteristic, complemented
2C2×#10625085C211yesmaximal, characteristic, complemented
3C2×#10625090C211yesmaximal, characteristic, complemented
4#13092997C211yesmaximal, characteristic, complemented
5#13092578C211yesmaximal, characteristic, complemented
6#13092567C211yesmaximal, characteristic, complemented
7#13092566C211yesmaximal, characteristic, complemented
8#13092565C211yesmaximal, characteristic, complemented
9#13092562C211yesmaximal, characteristic, complemented
10#13092553C211yesmaximal, characteristic, complemented
11#13092552C211yesmaximal, characteristic, complemented
12#13092417C211yesmaximal, characteristic, complemented
13#102×D23C211yesmaximal, characteristic, complemented
14#13092382C211yesmaximal, characteristic, complemented
15C23×#452C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17#10625160—24no—
18#10625160—24no—
19C2×#67292C2211yescharacteristic
20C2×#67292C2211yescharacteristic
21C2×#67292C2211yescharacteristic, complemented
22C2×D4×D23—21no—
23C2×D4×D23—21no—
24C2×D4×D23C2211yescharacteristic
25C2×D4×D23C2211yescharacteristic
26C2×#67288—21no—
27C2×#67288C2211yescharacteristic
28C22×D92—21no—
29C22×D92C2211yescharacteristic
30C2×Q16×C23C2211yescharacteristic, complemented
31C2×SD16×C23C2211yescharacteristic, complemented
32C23×C22⋊Q8C2211yescharacteristic, complemented
33#10625117C2211yescharacteristic, complemented
34C2×#67274C2211yescharacteristic, complemented
35C2×#67274C2211yescharacteristic, complemented
36C2×#67274C2211yescharacteristic, complemented
37C2×#67272C2211yescharacteristic, complemented
38#10625101C2211yescharacteristic, complemented
39#10625090—24no—
40#10625085—24no—
41C2×D184C2211yescharacteristic, complemented
42C2×D184C2211yescharacteristic, complemented
43C2×#67262C2211yescharacteristic, complemented
44C2×C8×D23C2211yescharacteristic, complemented
45#10625067—21no—
46#10625066—21no—
47#10625066C2211yescharacteristic, complemented
48C4⋊C4×D23C2211yescharacteristic, complemented
49#10625055—21no—
50#10625055—21no—
51C23×#103C2211yescharacteristic, complemented
52C23×#102C2211yescharacteristic, complemented
53C23×C22⋊C8C2211yescharacteristic, complemented
54#10625017C2211yescharacteristic, complemented
55#10625014C2211yescharacteristic, complemented
56C22⋊(C23⋊C8)C2211yescharacteristic, complemented
57#10625003C2211yescharacteristic, complemented
58#10625003C2211yescharacteristic, complemented
59(C2×D23)⋊C8C2211yescharacteristic, complemented
60(C2×D23)⋊C8C2211yescharacteristic, complemented
61#10624991C2211yescharacteristic, complemented
62#10624991C2211yescharacteristic, complemented
63C23×D23—21no—
64C23×D23—21no—
65C23×D23—21no—
66#41×C23—22no—
67#41×C23—22no—
68C2×Q8×C23C2311yescharacteristic
69C2×Q8×C23C2311yescharacteristic
70C2×D4×C23—21no—
71C2×D4×C23C2311yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23C2311yescharacteristic
74C2×C22⋊D23—21no—
75C2×C22⋊D23—21no—
76C2×C22⋊D23—21no—
77C2×C22⋊D23D411yescharacteristic
78C2×C22⋊D23D411yescharacteristic
79#67292—22no—
80#67292—22no—
81#67292—22no—
82#67292—22no—
83#67292—22no—
84#67292D414yescomplemented
85D4×D23—28no—
86D4×D23—28no—
87D4×D23—22no—
88D4×D23—22no—
89D4×D23—22no—
90D4×D23D414yescomplemented
91#67288—28no—
92#67288—22no—
93#67288—22no—
94C2×D92—28no—
95C2×D92—21no—
96C2×D92—21no—
97C2×D92—21no—
98C2×D92—22no—
99C2×D92—22no—
100C2×D92—21no—
101C2×D92—21no—
102C2×D92D411yescharacteristic
103C2×D92C2311yescharacteristic
104C2×D92C2311yescharacteristic
105C2×D92C2311yescharacteristic
106C2×C4×D23—21no—
107C2×C4×D23—21no—
108C2×C4×D23—21no—
109C2×C4×D23—21no—
110C2×C4×D23C2311yescharacteristic
111C2×C4×D23C2311yescharacteristic
112C2×Dic46D411yescharacteristic
113Q16×C23—22no—
114SD16×C23—22no—
115C2×C8×C23C2311yescharacteristic
116C2×C8×C23C2311yescharacteristic
117C4⋊C4×C23—21no—
118C4⋊C4×C23C2311yescharacteristic
119C22⋊C4×C23—21no—
120#67274—22no—
121#67274—22no—
122#67274—22no—
123#67272—22no—
124(C2×D23)⋊C4—21no—
125(C2×D23)⋊C4—21no—
126(C2×D23)⋊C4—21no—
127C4⋊Dic23C2311yescharacteristic
128Dic23⋊C4—21no—
129Dic23⋊C4—21no—
130C2×C23⋊C8C2311yescharacteristic
131C2×C23⋊C8C2311yescharacteristic
132D184—22no—
133D184—22no—
134#67262—22no—
135C8×D23—22no—
136C23×C23—21no—
137C23×C23C2×D411yescharacteristic
138C22×D23—21no—
139C22×D23—21no—
140C22×D23—24no—
141C22×D23—24no—
142C22×D23—24no—
143C22×D23—21no—
144C22×D23—81no—
145C22×D23—81no—
146C22×D23—81no—
147C22×D23—81no—
148C22×D23—81no—
149C22×D23—81no—
150C22×D23C2×D411yescharacteristic
151C22×D23C2×D411yescharacteristic
152Q8×C23—21no—
153Q8×C23—21no—
154Q8×C23—21no—
155Q8×C23C2×D412yes—
156D4×C23—24no—
157D4×C23—21no—
158D4×C23C2×D412yes—
159C2×C4×C23—24no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23C2×D411yescharacteristic
166C2×C4×C23C2411yescharacteristic
167C22⋊D23—24no—
168C22⋊D23—22no—
169C22⋊D23—22no—
170C22⋊D23—24no—
171C22⋊D23—81no—
172C2×Dic23—21no—
173C2×Dic23C2×D411yescharacteristic
174C2×Dic23C2×D411yescharacteristic
175D92—24no—
176D92—24no—
177D92—22no—
178D92—24no—
179D92—21no—
180D92—21no—
181D92—21no—
182D92—21no—
183D92—42no—
184D92—81no—
185D92C2×D412yes—
186D92C2×D412yes—
187C4×D23—24no—
188C4×D23—21no—
189C4×D23—21no—
190C4×D23—24no—
191C4×D23—42no—
192C4×D23—81no—
193C4×D23C2×D412yes—
194C4×D23C2×D412yes—
195Dic46—22no—
196C8×C23—21no—
197C8×C23—21no—
198C23⋊C8—21no—
199C23⋊C8—21no—
200#3012—231nomaximal, Sylow
201C22×C23—21no—
202C22×C23—22no—
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—41no—
206C22×C23—41no—
207C22×C23C22×D411yescharacteristic
208C2×D23—21no—
209C2×D23—21no—
210C2×D23—21no—
211C2×D23—22no—
212C2×D23—22no—
213C2×D23—21no—
214C2×D23—21no—
215C2×D23—41no—
216C2×D23—41no—
217C2×D23—41no—
218C2×D23—41no—
219C2×D23—41no—
220C2×D23—41no—
221C2×D23—41no—
222C2×D23—84no—
223C2×D23—84no—
224C2×D23—84no—
225C2×D23#12012yes—
226C4×C23—21no—
227C4×C23—22no—
228C4×C23—41no—
229C4×C23—41no—
230C4×C23C22×D411yescharacteristic
231C4×C23C22×D411yescharacteristic
232Dic23—21no—
233Dic23—41no—
234Dic23#12012yes—
235C2×H(5,𝔽2)—231no—
236C2×AGL(1,ℤ/8)—231no—
237C2×#135—231no—
238#537—231no—
239#452—231no—
240#452D2311yescharacteristic, complemented
241#450—231no—
242#450—231no—
243#449—231no—
244#449—231no—
245#447—231no—
246#447—231no—
247#417—231no—
248C2×#102—231no—
249#408—231no—
250C2×C23—41no—
251C2×C23—42no—
252C2×C23C2×#12011yescharacteristic
253C2×C23#57611yescharacteristic
254C2×C23#57711yescharacteristic
255D23—22no—
256D23—41no—
257D23—42no—
258D23—42no—
259D23—81no—
260H(5,𝔽2)—464no—
261H(5,𝔽2)—464no—
262C2×#41—231no—
263C2×#41—231no—
264C2×#41—231no—
265C2×#41—231no—
266C2×#41—461no—
267C2×#41C2×D2311yescharacteristic
268C22×D4—231no—
269C22×D4—231no—
270C22×D4—231no—
271C22×D4—461no—
272C22×D4—461no—
273C22×D4—461no—
274AGL(1,ℤ/8)—464no—
275#135—464no—
276C2×Q16C2×D2311yescharacteristic, complemented
277C2×SD16—231no—
278C2×SD16—231no—
279C2×SD16—231no—
280C2×SD16C2×D2311yescharacteristic, complemented
281C2×D8—231no—
282C2×D8—231no—
283C2×D8—231no—
284C2×M16—231no—
285C22×C8—231no—
286C22⋊Q8—231no—
287C22⋊Q8—461no—
288C22⋊Q8C2×D2311yescharacteristic, complemented
289#121—231no—
290#121—231no—
291#121—461no—
292#121—461no—
293#121—461no—
294C2×C4⋊C4—231no—
295#103—231no—
296#103C2×D2311yescharacteristic, complemented
297#102—231no—
298#102—231no—
299#102—231no—
300#102—231no—
301#102—231no—
302#102C2×D2311yescharacteristic, complemented
303C22⋊C8—231no—
304C22⋊C8—231no—
305C22⋊C8—231no—
306C22⋊C8C2×D2311yescharacteristic, complemented
307C23#301211yescharacteristic, complemented, Sylow
308C24—461no—
309C24—461no—
310C24—461no—
311#41—22no—
312#41—22no—
313#41—234no—
314#41—462no—
315#41—462no—
316#41—468no—
317#41—462no—
318#41—462no—
319#41—462no—
320#41—462no—
321#41—462no—
322C2×Q8—231no—
323C2×Q8C22×D2311yescharacteristic
324C2×Q8C22×D2311yescharacteristic
325C2×D4—21no—
326C2×D4—234no—
327C2×D4—231no—
328C2×D4—231no—
329C2×D4—231no—
330C2×D4—231no—
331C2×D4—231no—
332C2×D4—231no—
333C2×D4—462no—
334C2×D4—462no—
335C2×D4—461no—
336C2×D4—461no—
337C2×D4—468no—
338C2×D4—461no—
339C2×D4—461no—
340C2×D4—462no—
341C2×D4—461no—
342C2×D4—461no—
343C2×D4—462no—
344C2×D4—468no—
345C2×D4—468no—
346C2×D4—462no—
347C2×D4—461no—
348C2×D4—461no—
349C2×D4C22×D2311yescharacteristic
350C22×C4—21no—
351C22×C4—231no—
352C22×C4—231no—
353C22×C4—461no—
354C22×C4—461no—
355C22×C4—461no—
356C22×C4—461no—
357C22×C4C22×D2311yescharacteristic
358Q16—22no—
359SD16—22no—
360SD16—462no—
361SD16—462no—
362SD16—462no—
363D8—462no—
364D8—462no—
365D8—462no—
366M16—462no—
367C2×C8—231no—
368C2×C8—231no—
369C2×C8—462no—
370C2×C8C22×D2311yescharacteristic
371C2×C8C22×D2311yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—231no—
374C4⋊C4—461no—
375C4⋊C4—461no—
376C4⋊C4C22×D2311yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—461no—
379C22⋊C4—461no—
380C22⋊C4—461no—
381C23—21no—
382C23—231no—
383C23—231no—
384C23—464no—
385C23—464no—
386C23—464no—
387C23—461no—
388C23—461no—
389C23—461no—
390C23—1841no—
391C23—1841no—
392C23—1841no—
393C23—1841no—
394C23—1841no—
395C23—1841no—
396C23D4×D2311yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—21no—
400Q8—462no—
401Q8D4×D2312yes—
402D4—24no—
403D4—21no—
404D4—232no—
405D4—232no—
406D4—462no—
407D4—464no—
408D4—464no—
409D4—464no—
410D4—462no—
411D4—464no—
412D4—462no—
413D4—464no—
414D4—461no—
415D4—461no—
416D4—461no—
417D4—461no—
418D4—922no—
419D4—1841no—
420D4—1841no—
421D4D4×D2312yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—231no—
429C2×C4—232no—
430C2×C4—232no—
431C2×C4—231no—
432C2×C4—464no—
433C2×C4—464no—
434C2×C4—461no—
435C2×C4—461no—
436C2×C4—461no—
437C2×C4—922no—
438C2×C4—1841no—
439C2×C4D4×D2311yescharacteristic
440C2×C4C23×D2311yescharacteristic
441C8—21no—
442C8—21no—
443C8—461no—
444C8—461no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—232no—
452C22—461no—
453C22—461no—
454C22—462no—
455C22—462no—
456C22—461no—
457C22—461no—
458C22—461no—
459C22—921no—
460C22—921no—
461C22—921no—
462C22—921no—
463C22—921no—
464C22—921no—
465C22—921no—
466C22—1844no—
467C22—1844no—
468C22—1844no—
469C22C2×D4×D2311yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—232no—
475C4—461no—
476C4—921no—
477C4C2×D4×D2311yescharacteristic
478C4C2×D4×D2311yescharacteristic
479C2—41no—
480C2—42no—
481C2—462no—
482C2—921no—
483C2—922no—
484C2—922no—
485C2—1841no—
486C2#120×D2311yescharacteristic
487C2#1309321911yescharacteristic
488C2#1309322611yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#452C2
socleC22×C23C22×D4
hypercenter#452D23
hypocenterC23#3012
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#10625160C2semidirect
2. C2×#10625085C2semidirect
3. C2×#10625090C2semidirect
4. #13092997C2semidirect
5. #13092578C2semidirect
6. #13092567C2semidirect
7. #13092566C2semidirect
8. #13092565C2semidirect
9. #13092562C2semidirect
10. #13092553C2semidirect
11. #13092552C2semidirect
12. #13092417C2semidirect
13. #102×D23C2semidirect
14. #13092382C2semidirect
15. C23×#452C2semidirect
21. C2×#67292C22semidirect
30. C2×Q16×C23C22semidirect
31. C2×SD16×C23C22semidirect
32. C23×C22⋊Q8C22semidirect
33. #10625117C22semidirect
34. C2×#67274C22semidirect
35. C2×#67274C22semidirect
36. C2×#67274C22semidirect
37. C2×#67272C22semidirect
38. #10625101C22semidirect
41. C2×D184C22semidirect
42. C2×D184C22semidirect
43. C2×#67262C22semidirect
44. C2×C8×D23C22semidirect
47. #10625066C22semidirect
48. C4⋊C4×D23C22semidirect
51. C23×#103C22semidirect
52. C23×#102C22semidirect
53. C23×C22⋊C8C22semidirect
54. #10625017C22semidirect
55. #10625014C22semidirect
56. C22⋊(C23⋊C8)C22semidirect
57. #10625003C22semidirect
58. #10625003C22semidirect
59. (C2×D23)⋊C8C22semidirect
60. (C2×D23)⋊C8C22semidirect
61. #10624991C22semidirect
62. #10624991C22semidirect
84. #67292D4semidirect
90. D4×D23D4semidirect
240. #452D23semidirect
276. C2×Q16C2×D23semidirect
280. C2×SD16C2×D23semidirect
288. C22⋊Q8C2×D23semidirect
296. #103C2×D23semidirect
302. #102C2×D23semidirect
306. C22⋊C8C2×D23semidirect
307. C23#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#106251602
C2×#106250852
C2×#106250902
#130929972
#130925782
#130925672
#130925662
#130925652
#130925622
#130925532
#130925522
#130924172
#102×D232
#130923822
C23×#4522
#301223