Finite Group Databasebeta

← #36362865

Subgroups — #36362865

Order 22144.

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×#19149725C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3C2×#19149700C211yesmaximal, characteristic, complemented
4#26012070C211yesmaximal, characteristic, complemented
5#26011656C211yesmaximal, characteristic, complemented
6#26011653C211yesmaximal, characteristic, complemented
7#26011644C211yesmaximal, characteristic, complemented
8#26011643C211yesmaximal, characteristic, complemented
9#26011635C211yesmaximal, characteristic, complemented
10#26011624C211yesmaximal, characteristic, complemented
11#26011623C211yesmaximal, characteristic, complemented
12#26011491C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16#19149725—24no—
17#19149725—24no—
18C2×#14583827C2211yescharacteristic
19C2×#14583827C2211yescharacteristic
20C2×#14583826—21no—
21C2×#14583826C2211yescharacteristic
22C2×#14583825C2211yescharacteristic, complemented
23C22×D692—21no—
24C22×D692C2211yescharacteristic
25C2×D4×D173—21no—
26C2×D4×D173—21no—
27C2×D4×D173C2211yescharacteristic
28C2×D4×D173C2211yescharacteristic
29#19149703—24no—
30#19149700—24no—
31C2×#14583816C2211yescharacteristic, complemented
32C2×#14583815C2211yescharacteristic, complemented
33C2×#14583815C2211yescharacteristic, complemented
34C2×#14583814C2211yescharacteristic, complemented
35C2×#14583811C2211yescharacteristic, complemented
36C2×#14583811C2211yescharacteristic, complemented
37C2×#14583809C2211yescharacteristic, complemented
38C2×C8×D173C2211yescharacteristic, complemented
39#19149659C2211yescharacteristic, complemented
40#19149658C2211yescharacteristic, complemented
41#19149656—21no—
42#19149652—21no—
43#19149651—21no—
44#19149650—21no—
45#19149650C2211yescharacteristic, complemented
46C4⋊C4×D173C2211yescharacteristic, complemented
47#19149603C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic, complemented
52#19149599C2211yescharacteristic, complemented
53C22⋊#13063436C2211yescharacteristic, complemented
54#19149591C2211yescharacteristic, complemented
55#19149591C2211yescharacteristic, complemented
56C2×#41×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic, complemented
58C2×D8×C173C2211yescharacteristic, complemented
59#121×C173C2211yescharacteristic, complemented
60#103×C173C2211yescharacteristic, complemented
61#102×C173C2211yescharacteristic, complemented
62C22⋊C8×C173C2211yescharacteristic, complemented
63C23×D173—21no—
64C23×D173—21no—
65C23×D173—21no—
66#14583827—22no—
67#14583827—22no—
68#14583827—22no—
69#14583827D414yescomplemented
70#14583826—28no—
71#14583826—22no—
72#14583826—22no—
73#14583825—22no—
74#14583825—22no—
75C2×Dic346D411yescharacteristic
76C2×Dic346C2311yescharacteristic
77C2×D692—28no—
78C2×D692—21no—
79C2×D692—21no—
80C2×D692—22no—
81C2×D692—22no—
82C2×D692—21no—
83C2×D692D411yescharacteristic
84C2×D692C2311yescharacteristic
85C2×D692C2311yescharacteristic
86C2×C22⋊D173—21no—
87C2×C22⋊D173—21no—
88C2×C22⋊D173—21no—
89C2×C22⋊D173—21no—
90C2×C22⋊D173D411yescharacteristic
91C2×C22⋊D173D411yescharacteristic
92D4×D173—28no—
93D4×D173—28no—
94D4×D173—22no—
95D4×D173—22no—
96D4×D173—22no—
97D4×D173D414yescomplemented
98C22×Dic173—21no—
99C2×C4×D173—21no—
100C2×C4×D173—21no—
101C2×C4×D173—21no—
102C2×C4×D173C2311yescharacteristic
103C2×C4×D173C2311yescharacteristic
104#14583816—22no—
105#14583815—22no—
106#14583815—22no—
107#14583814—22no—
108#14583811—22no—
109#14583811—22no—
110#14583809—22no—
111C2×#13063436C2311yescharacteristic
112C2×#13063436C2311yescharacteristic
113C8×D173—22no—
114C4⋊Dic173—21no—
115C4⋊Dic173C2311yescharacteristic
116#14583805—21no—
117#14583805—21no—
118C22⋊Dic173—21no—
119C22⋊Dic173—21no—
120#14583803—21no—
121#41×C173—22no—
122#41×C173—22no—
123C2×Q8×C173C2311yescharacteristic
124C2×D4×C173—21no—
125C2×D4×C173—21no—
126C2×D4×C173C2311yescharacteristic
127C2×D4×C173C2311yescharacteristic
128C22×C4×C173—21no—
129C22×C4×C173C2311yescharacteristic
130SD16×C173—22no—
131D8×C173—22no—
132C2×C8×C173C2311yescharacteristic
133C2×C8×C173C2311yescharacteristic
134C4⋊C4×C173C2311yescharacteristic
135C22⋊C4×C173—21no—
136C22×D173—21no—
137C22×D173—21no—
138C22×D173—24no—
139C22×D173—24no—
140C22×D173—24no—
141C22×D173—81no—
142C22×D173—81no—
143C22×D173—81no—
144C22×D173—81no—
145C22×D173—81no—
146C22×D173—81no—
147C22×D173C2×D411yescharacteristic
148C22×D173C2×D411yescharacteristic
149Dic346—22no—
150Dic346—21no—
151Dic346—21no—
152D692—24no—
153D692—24no—
154D692—24no—
155D692—22no—
156D692—21no—
157D692—21no—
158D692C2×D412yes—
159D692C2×D412yes—
160C22⋊D173—24no—
161C22⋊D173—22no—
162C22⋊D173—22no—
163C22⋊D173—24no—
164C22⋊D173—42no—
165C22⋊D173—81no—
166C2×Dic173—21no—
167C2×Dic173—21no—
168C2×Dic173—42no—
169C2×Dic173C2×D411yescharacteristic
170C2×Dic173C2×D411yescharacteristic
171C4×D173—24no—
172C4×D173—21no—
173C4×D173—21no—
174C4×D173—24no—
175C4×D173—81no—
176C4×D173C2×D412yes—
177C4×D173C2×D412yes—
178#13063436—21no—
179#13063436—21no—
180C23×C173—21no—
181C23×C173—21no—
182C23×C173C2×D411yescharacteristic
183Q8×C173—21no—
184Q8×C173C2×D412yes—
185D4×C173—24no—
186D4×C173—21no—
187D4×C173—21no—
188D4×C173—21no—
189D4×C173—81no—
190D4×C173C2×D412yes—
191C2×C4×C173—24no—
192C2×C4×C173—21no—
193C2×C4×C173—21no—
194C2×C4×C173—21no—
195C2×C4×C173—21no—
196C2×C4×C173C2×D411yescharacteristic
197C2×C4×C173C2411yescharacteristic
198C8×C173—21no—
199C8×C173—21no—
200C22×C173—21no—
201C22×C173—22no—
202C22×C173—21no—
203C22×C173—21no—
204C22×C173—41no—
205C22×C173—41no—
206C22×C173—41no—
207C22×C173—41no—
208C22×C173—41no—
209C22×C173C22×D411yescharacteristic
210C2×D173—21no—
211C2×D173—21no—
212C2×D173—21no—
213C2×D173—22no—
214C2×D173—22no—
215C2×D173—21no—
216C2×D173—21no—
217C2×D173—41no—
218C2×D173—41no—
219C2×D173—41no—
220C2×D173—41no—
221C2×D173—84no—
222C2×D173—84no—
223C2×D173—84no—
224C2×D173#12012yes—
225C4×C173—21no—
226C4×C173—22no—
227C4×C173—41no—
228C4×C173C22×D411yescharacteristic
229C4×C173C22×D411yescharacteristic
230Dic173—21no—
231Dic173—41no—
232Dic173—41no—
233Dic173#12012yes—
234C2×C173—41no—
235C2×C173—42no—
236C2×C173—81no—
237C2×C173C2×#12011yescharacteristic
238C2×C173#57611yescharacteristic
239C2×C173#57711yescharacteristic
240D173—22no—
241D173—41no—
242D173—42no—
243D173—42no—
244C173#301211yescharacteristic, complemented, Sylow
245#3012—1731nomaximal, Sylow
246C2×H(5,𝔽2)—1731no—
247C2×AGL(1,ℤ/8)—1731no—
248C2×#135—1731no—
249#537—1731no—
250#452—1731no—
251#452—1731no—
252#450—1731no—
253#450—1731no—
254#449—1731no—
255#449D17311yescharacteristic, complemented
256#447—1731no—
257#447—1731no—
258#417—1731no—
259C2×#102—1731no—
260#408—1731no—
261H(5,𝔽2)—3464no—
262H(5,𝔽2)—3464no—
263C2×#41—3461no—
264C2×#41—1731no—
265C2×#41—1731no—
266C2×#41—1731no—
267C2×#41—1731no—
268C2×#41C2×D17311yescharacteristic
269C22×D4—3461no—
270C22×D4—3461no—
271C22×D4—3461no—
272C22×D4—1731no—
273C22×D4—1731no—
274C22×D4—1731no—
275AGL(1,ℤ/8)—3464no—
276#135—3464no—
277C2×Q16—1731no—
278C2×SD16—1731no—
279C2×SD16—1731no—
280C2×SD16—1731no—
281C2×SD16C2×D17311yescharacteristic, complemented
282C2×D8—1731no—
283C2×D8—1731no—
284C2×D8C2×D17311yescharacteristic, complemented
285C2×M16—1731no—
286C22×C8—1731no—
287C22⋊Q8—3461no—
288C22⋊Q8—1731no—
289C22⋊Q8—1731no—
290#121—3461no—
291#121—3461no—
292#121—3461no—
293#121—1731no—
294#121C2×D17311yescharacteristic, complemented
295C2×C4⋊C4—1731no—
296#103—1731no—
297#103C2×D17311yescharacteristic, complemented
298#102—1731no—
299#102—1731no—
300#102—1731no—
301#102—1731no—
302#102—1731no—
303#102C2×D17311yescharacteristic, complemented
304C22⋊C8—1731no—
305C22⋊C8—1731no—
306C22⋊C8—1731no—
307C22⋊C8C2×D17311yescharacteristic, complemented
308C24—3461no—
309C24—3461no—
310C24—3461no—
311#41—22no—
312#41—22no—
313#41—3468no—
314#41—3462no—
315#41—3462no—
316#41—3462no—
317#41—3462no—
318#41—3462no—
319#41—3462no—
320#41—3462no—
321#41—1734no—
322C2×Q8—1731no—
323C2×Q8—1731no—
324C2×Q8C22×D17311yescharacteristic
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—3468no—
328C2×D4—3461no—
329C2×D4—3461no—
330C2×D4—3462no—
331C2×D4—3462no—
332C2×D4—3468no—
333C2×D4—3468no—
334C2×D4—3462no—
335C2×D4—3462no—
336C2×D4—3461no—
337C2×D4—3461no—
338C2×D4—3462no—
339C2×D4—3461no—
340C2×D4—3461no—
341C2×D4—3461no—
342C2×D4—1731no—
343C2×D4—1731no—
344C2×D4—1734no—
345C2×D4—1731no—
346C2×D4—1731no—
347C2×D4—1731no—
348C2×D4C22×D17311yescharacteristic
349C2×D4C22×D17311yescharacteristic
350C22×C4—21no—
351C22×C4—3461no—
352C22×C4—3461no—
353C22×C4—3461no—
354C22×C4—3461no—
355C22×C4—1731no—
356C22×C4—1731no—
357C22×C4C22×D17311yescharacteristic
358Q16—3462no—
359SD16—22no—
360SD16—3462no—
361SD16—3462no—
362SD16—3462no—
363D8—22no—
364D8—3462no—
365D8—3462no—
366M16—3462no—
367C2×C8—3462no—
368C2×C8—1731no—
369C2×C8—1731no—
370C2×C8C22×D17311yescharacteristic
371C2×C8C22×D17311yescharacteristic
372C4⋊C4—3461no—
373C4⋊C4—3461no—
374C4⋊C4—3461no—
375C4⋊C4—1731no—
376C4⋊C4C22×D17311yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—3461no—
379C22⋊C4—3461no—
380C22⋊C4—3461no—
381C23—21no—
382C23—21no—
383C23—3464no—
384C23—3464no—
385C23—3464no—
386C23—3461no—
387C23—3461no—
388C23—13841no—
389C23—13841no—
390C23—13841no—
391C23—13841no—
392C23—13841no—
393C23—13841no—
394C23—1731no—
395C23—1731no—
396C23D4×D17311yescharacteristic
397Q8—21no—
398Q8—3462no—
399Q8—3461no—
400Q8—3461no—
401Q8D4×D17312yes—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—81no—
407D4—3464no—
408D4—3464no—
409D4—3464no—
410D4—3462no—
411D4—3464no—
412D4—3462no—
413D4—3462no—
414D4—3464no—
415D4—3461no—
416D4—3461no—
417D4—13841no—
418D4—1732no—
419D4—1732no—
420D4—6922no—
421D4D4×D17312yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—3464no—
428C2×C4—3464no—
429C2×C4—3461no—
430C2×C4—3461no—
431C2×C4—3461no—
432C2×C4—3461no—
433C2×C4—13841no—
434C2×C4—1731no—
435C2×C4—1731no—
436C2×C4—1732no—
437C2×C4—1732no—
438C2×C4—6922no—
439C2×C4D4×D17311yescharacteristic
440C2×C4C23×D17311yescharacteristic
441C8—21no—
442C8—21no—
443C8—3461no—
444C8—3461no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—3461no—
455C22—3461no—
456C22—3461no—
457C22—3462no—
458C22—3462no—
459C22—3461no—
460C22—3461no—
461C22—13844no—
462C22—13844no—
463C22—13844no—
464C22—1732no—
465C22—6921no—
466C22—6921no—
467C22—6921no—
468C22—6921no—
469C22C2×D4×D17311yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—3461no—
474C4—1732no—
475C4—6921no—
476C4—6921no—
477C4C2×D4×D17311yescharacteristic
478C4C2×D4×D17311yescharacteristic
479C2—41no—
480C2—42no—
481C2—81no—
482C2—3462no—
483C2—6921no—
484C2—6922no—
485C2—6922no—
486C2#120×D17311yescharacteristic
487C2#2601229511yescharacteristic
488C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3012
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2semidirect
2. C2×#19149703C2semidirect
3. C2×#19149700C2semidirect
4. #26012070C2semidirect
5. #26011656C2semidirect
6. #26011653C2semidirect
7. #26011644C2semidirect
8. #26011643C2semidirect
9. #26011635C2semidirect
10. #26011624C2semidirect
11. #26011623C2semidirect
12. #26011491C2semidirect
13. #102×D173C2semidirect
14. #26011456C2semidirect
15. #449×C173C2semidirect
22. C2×#14583825C22semidirect
31. C2×#14583816C22semidirect
32. C2×#14583815C22semidirect
33. C2×#14583815C22semidirect
34. C2×#14583814C22semidirect
35. C2×#14583811C22semidirect
36. C2×#14583811C22semidirect
37. C2×#14583809C22semidirect
38. C2×C8×D173C22semidirect
39. #19149659C22semidirect
40. #19149658C22semidirect
45. #19149650C22semidirect
46. C4⋊C4×D173C22semidirect
47. #19149603C22semidirect
48. #19149601C22semidirect
49. #19149601C22semidirect
50. #19149600C22semidirect
51. #19149599C22semidirect
52. #19149599C22semidirect
53. C22⋊#13063436C22semidirect
54. #19149591C22semidirect
55. #19149591C22semidirect
57. C2×SD16×C173C22semidirect
58. C2×D8×C173C22semidirect
59. #121×C173C22semidirect
60. #103×C173C22semidirect
61. #102×C173C22semidirect
62. C22⋊C8×C173C22semidirect
69. #14583827D4semidirect
97. D4×D173D4semidirect
244. C173#3012semidirect
255. #449D173semidirect
281. C2×SD16C2×D173semidirect
284. C2×D8C2×D173semidirect
294. #121C2×D173semidirect
297. #103C2×D173semidirect
303. #102C2×D173semidirect
307. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497032
C2×#191497002
#260120702
#260116562
#260116532
#260116442
#260116432
#260116352
#260116242
#260116232
#260114912
#102×D1732
#260114562
#449×C1732
#3012173