Finite Group Databasebeta

← #37583996

Subgroups — #37583996

Order 24448.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259348C211yesmaximal, characteristic, complemented
2C22×#15690272C211yesmaximal, characteristic, complemented
3#27133309C211yesmaximal, characteristic, complemented
4#27133050C211yesmaximal, characteristic, complemented
5#27132914C211yesmaximal, characteristic, complemented
6#27132890C211yesmaximal, characteristic, complemented
7#27132878C211yesmaximal, characteristic, complemented
8#27132855C211yesmaximal, characteristic, complemented
9#27132814C211yesmaximal, characteristic, complemented
10#27132806C211yesmaximal, characteristic, complemented
11#27132797C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132773C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15D42×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic, complemented
17C22×D764C2211yescharacteristic, complemented
18#20259348—24no—
19C2×#15690275C2211yescharacteristic, complemented
20C2×#15690275C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690272—24no—
23C2×#15690272C2211yescharacteristic
24C2×#15690272C2211yescharacteristic
25C2×#15690272C2211yescharacteristic, complemented
26C2×#15690272C2211yescharacteristic, complemented
27C2×#15690272C2211yescharacteristic, complemented
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690271C2211yescharacteristic, complemented
30C22×C191⋊C8C2211yescharacteristic, complemented
31#20259310C2211yescharacteristic, complemented
32#20259307—21no—
33#20259301C2211yescharacteristic, complemented
34#20259296C2211yescharacteristic, complemented
35#20259295—21no—
36#20259290—21no—
37C4×D764C2211yescharacteristic, complemented
38C22⋊C4×D191—21no—
39#20259253C2211yescharacteristic, complemented
40C4⋊(C191⋊C8)C2211yescharacteristic, complemented
41#20259245C2211yescharacteristic, complemented
42#20259243C2211yescharacteristic, complemented
43#20259243C2211yescharacteristic, complemented
44#20259243C2211yescharacteristic, complemented
45#20259242C2211yescharacteristic, complemented
46#20259242C2211yescharacteristic, complemented
47C22⋊(C191⋊C8)C2211yescharacteristic, complemented
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49#20259233C2211yescharacteristic, complemented
50C22×D4×C191—21no—
51C22×D4×C191C2211yescharacteristic
52C22×D4×C191C2211yescharacteristic, complemented
53#127×C191C2211yescharacteristic, complemented
54#121×C191—21no—
55#121×C191C2211yescharacteristic
56#121×C191C2211yescharacteristic, complemented
57#120×C191—21no—
58#120×C191—21no—
59C4×D4×C191C2211yescharacteristic
60C4×D4×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690287—22no—
63#15690287—41no—
64C2×Dic382C2311yescharacteristic
65C2×D764—22no—
66C2×D764—21no—
67C2×D764—41no—
68C2×D764C2311yescharacteristic
69C2×D764C2311yescharacteristic
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191—21no—
74#15690275—22no—
75#15690275—22no—
76#15690275—41no—
77#15690272—22no—
78#15690272—22no—
79#15690272—22no—
80#15690272—22no—
81#15690272—41no—
82#15690272D414yescomplemented
83#15690271—22no—
84C2×C191⋊C8—22no—
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C4⋊Dic191C2311yescharacteristic
90Dic191⋊C4—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96C24×C191—21no—
97C24×C191—21no—
98C2×D4×C191—22no—
99C2×D4×C191—22no—
100C2×D4×C191—21no—
101C2×D4×C191—21no—
102C2×D4×C191—21no—
103C2×D4×C191—21no—
104C2×D4×C191—21no—
105C2×D4×C191—21no—
106C2×D4×C191—41no—
107C2×D4×C191—41no—
108C2×D4×C191—44no—
109C2×D4×C191D411yescharacteristic
110C2×D4×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191—21no—
114C22×C4×C191C2311yescharacteristic
115C22×C4×C191C2311yescharacteristic
116C4⋊C4×C191D411yescharacteristic
117C4⋊C4×C191C2311yescharacteristic
118C22⋊C4×C191—21no—
119C22⋊C4×C191—21no—
120C22⋊C4×C191—21no—
121C22⋊C4×C191D411yescharacteristic
122C22⋊C4×C191D411yescharacteristic
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—42no—
127C22×D191—42no—
128C22×D191—81no—
129Dic382—21no—
130Dic382—21no—
131D764—21no—
132D764—21no—
133D764—21no—
134D764—42no—
135D764C2×D412yescomplemented
136C22⋊D191—42no—
137C22⋊D191—81no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C4×D191—42no—
141C4×D191—81no—
142C191⋊C8—21no—
143C191⋊C8—21no—
144C191⋊C8—21no—
145C191⋊C8C2×D412yescomplemented
146C23×C191—21no—
147C23×C191—21no—
148C23×C191—21no—
149C23×C191—42no—
150C23×C191—42no—
151C23×C191—42no—
152C23×C191—42no—
153C23×C191—81no—
154C23×C191—81no—
155C23×C191C2×D411yescharacteristic
156C23×C191C2×D411yescharacteristic
157D4×C191—22no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—42no—
164D4×C191—42no—
165D4×C191—42no—
166D4×C191—42no—
167D4×C191—42no—
168D4×C191—81no—
169D4×C191C2×D412yes—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—21no—
173C2×C4×C191—21no—
174C2×C4×C191—21no—
175C2×C4×C191—42no—
176C2×C4×C191C2×D411yescharacteristic
177C2×C4×C191C2×D411yescharacteristic
178C2×C4×C191C2×D411yescharacteristic
179C2×C4×C191C2×D411yescharacteristic
180C2×C4×C191C2411yescharacteristic
181C22×C191—21no—
182C22×C191—21no—
183C22×C191—21no—
184C22×C191—22no—
185C22×C191—21no—
186C22×C191—21no—
187C22×C191—41no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191—41no—
191C22×C191—41no—
192C22×C191—41no—
193C22×C191—41no—
194C22×C191—41no—
195C22×C191—84no—
196C22×C191—84no—
197C22×C191AGL(1,ℤ/8)12yescomplemented
198C22×C191C22×D411yescharacteristic
199C2×D191—22no—
200C2×D191—41no—
201C2×D191—41no—
202C2×D191—41no—
203C2×D191—41no—
204C2×D191—41no—
205C2×D191—84no—
206C4×C191—21no—
207C4×C191—21no—
208C4×C191—21no—
209C4×C191—21no—
210C4×C191—41no—
211C4×C191C22×D411yescharacteristic
212C4×C191H(5,𝔽2)11yescharacteristic
213Dic191—41no—
214Dic191—41no—
215C2×C191—22no—
216C2×C191—41no—
217C2×C191—42no—
218C2×C191—81no—
219C2×C191—81no—
220C2×C191D4211yescharacteristic
221C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C191#57611yescharacteristic
223D191—42no—
224D191—81no—
225C191#326411yescharacteristic, complemented, Sylow
226#3264—1911nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1911no—
228C22×D8—1911no—
229#546—1911no—
230D42D19111yescharacteristic, complemented
231#496—1911no—
232#469—1911no—
233#465—1911no—
234#463—1911no—
235#459—1911no—
236#450—1911no—
237#449—1911no—
238#447—1911no—
239#447—1911no—
240#442—1911no—
241#435—1911no—
242C2×#41—1911no—
243C22×D4—21no—
244C22×D4—1911no—
245C22×D4C2×D19111yescharacteristic
246C22×D4C2×D19111yescharacteristic, complemented
247AGL(1,ℤ/8)—3824no—
248C2×SD16—1911no—
249C2×SD16—1911no—
250C2×SD16—1911no—
251C2×D8—3824no—
252C2×D8—1911no—
253C2×D8—1911no—
254C2×D8—1911no—
255C2×D8—1911no—
256C2×D8—1911no—
257C2×D8—1911no—
258C2×M16—1911no—
259C22×C8—1911no—
260#127C2×D19111yescharacteristic, complemented
261#124—1911no—
262#123—3821no—
263C22⋊Q8—1911no—
264#121—21no—
265#121—3821no—
266#121—1911no—
267#121C2×D19111yescharacteristic
268#121C2×D19111yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—3821no—
272C4×D4—1911no—
273C4×D4C2×D19111yescharacteristic
274C4×D4C2×D19111yescharacteristic, complemented
275C2×C22⋊C4—3821no—
276#106—1911no—
277C4⋊C8—1911no—
278#103—1911no—
279#102—1911no—
280#102—1911no—
281#102—1911no—
282#102—1911no—
283#102—1911no—
284C22⋊C8—1911no—
285C22⋊C8—1911no—
286#97—1911no—
287C24—21no—
288C24—21no—
289C24—3821no—
290#41—3822no—
291#41—7641no—
292C2×Q8—1911no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—3822no—
305C2×D4—3821no—
306C2×D4—3821no—
307C2×D4—3821no—
308C2×D4—1911no—
309C2×D4—1911no—
310C2×D4—7641no—
311C2×D4C22⋊D19111yescharacteristic
312C2×D4C22×D19111yescharacteristic
313C2×D4C22×D19111yescharacteristic
314C2×D4C22×D19111yescharacteristic
315C22×C4—21no—
316C22×C4—3821no—
317C22×C4—3821no—
318C22×C4C22×D19111yescharacteristic
319C22×C4C22×D19111yescharacteristic
320SD16—3822no—
321SD16—3822no—
322SD16—7641no—
323D8—3822no—
324D8—3822no—
325D8—3822no—
326D8—3822no—
327D8—1914no—
328D8—7641no—
329M16—3822no—
330C2×C8—3822no—
331C2×C8—1911no—
332C2×C8—1911no—
333C2×C8—1911no—
334C2×C8—1911no—
335C4⋊C4—3821no—
336C4⋊C4—1911no—
337C4⋊C4C22⋊D19111yescharacteristic
338C4⋊C4C22×D19111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—3821no—
343C22⋊C4—3821no—
344C22⋊C4—3821no—
345C22⋊C4—3821no—
346C22⋊C4—3821no—
347C22⋊C4C22⋊D19111yescharacteristic
348C22⋊C4C22⋊D19111yescharacteristic
349C42C22×D19111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—3821no—
360C23—3821no—
361C23—15281no—
362C23—7642no—
363C23—7642no—
364C23C2×C22⋊D19111yescharacteristic
365C23C2×C22⋊D19111yescharacteristic
366Q8—3821no—
367Q8—3821no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—3821no—
381D4—3821no—
382D4—3821no—
383D4—1912no—
384D4—15281no—
385D4—7642no—
386D4—7642no—
387D4D4×D19112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—3821no—
395C2×C4—3821no—
396C2×C4—15281no—
397C2×C4—7642no—
398C2×C4C2×C22⋊D19111yescharacteristic
399C2×C4C2×C22⋊D19111yescharacteristic
400C2×C4C2×C22⋊D19111yescharacteristic
401C2×C4C2×C22⋊D19111yescharacteristic
402C2×C4C23×D19111yescharacteristic
403C8—3821no—
404C8—3821no—
405C8—3821no—
406C8—1912no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—3822no—
424C22—15284no—
425C22—7641no—
426C22—7641no—
427C22—7641no—
428C22—7641no—
429C22—7641no—
430C22#2025934812yescomplemented
431C22C22×C22⋊D19111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—7641no—
438C4—7641no—
439C4C2×D4×D19111yescharacteristic
440C4#2025936711yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—15281no—
447C2—7642no—
448C2D4×C22⋊D19111yescharacteristic
449C2C2×#2025934811yescharacteristic
450C2#2713346611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingD42×C191C2
socleC22×C191C22×D4
hypercenterD42D191
hypocenterC191#3264
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C22×#15690272C2semidirect
3. #27133309C2semidirect
4. #27133050C2semidirect
5. #27132914C2semidirect
6. #27132890C2semidirect
7. #27132878C2semidirect
8. #27132855C2semidirect
9. #27132814C2semidirect
10. #27132806C2semidirect
11. #27132797C2semidirect
12. #27132796C2semidirect
13. #27132773C2semidirect
14. #27132735C2semidirect
15. D42×C191C2semidirect
16. C2×#15690287C22semidirect
17. C22×D764C22semidirect
19. C2×#15690275C22semidirect
20. C2×#15690275C22semidirect
21. C2×#15690275C22semidirect
25. C2×#15690272C22semidirect
26. C2×#15690272C22semidirect
27. C2×#15690272C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690271C22semidirect
30. C22×C191⋊C8C22semidirect
31. #20259310C22semidirect
33. #20259301C22semidirect
34. #20259296C22semidirect
37. C4×D764C22semidirect
39. #20259253C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259245C22semidirect
42. #20259243C22semidirect
43. #20259243C22semidirect
44. #20259243C22semidirect
45. #20259242C22semidirect
46. #20259242C22semidirect
47. C22⋊(C191⋊C8)C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. #20259233C22semidirect
52. C22×D4×C191C22semidirect
53. #127×C191C22semidirect
56. #121×C191C22semidirect
60. C4×D4×C191C22semidirect
82. #15690272D4semidirect
135. D764C2×D4semidirect
145. C191⋊C8C2×D4semidirect
197. C22×C191AGL(1,ℤ/8)semidirect
225. C191#3264semidirect
230. D42D191semidirect
246. C22×D4C2×D191semidirect
260. #127C2×D191semidirect
268. #121C2×D191semidirect
274. C4×D4C2×D191semidirect
430. C22#20259348semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C22×#156902722
#271333092
#271330502
#271329142
#271328902
#271328782
#271328552
#271328142
#271328062
#271327972
#271327962
#271327732
#271327352
D42×C1912
#3264191