Finite Group Databasebeta

← #36362828

Subgroups — #36362828

Order 22144.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149725C211yesmaximal, characteristic, complemented
2C2×#19149706C211yesmaximal, characteristic, complemented
3C2×D8×D173C211yesmaximal, characteristic, complemented
4#26012057C211yesmaximal, characteristic, complemented
5#26011639C211yesmaximal, characteristic, complemented
6#26011636C211yesmaximal, characteristic, complemented
7#26011635C211yesmaximal, characteristic, complemented
8#26011627C211yesmaximal, characteristic, complemented
9#26011626C211yesmaximal, characteristic, complemented
10#26011624C211yesmaximal, characteristic, complemented
11#26011623C211yesmaximal, characteristic, complemented
12#26011494C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#449×C173C211yesmaximal, characteristic, complemented
16#19149725—24no—
17#19149725—24no—
18C2×#14583827C2211yescharacteristic
19C2×#14583827C2211yescharacteristic, complemented
20C2×#14583826—21no—
21C2×#14583826C2211yescharacteristic
22C22×D692—21no—
23C22×D692C2211yescharacteristic, complemented
24C2×D4×D173—21no—
25C2×D4×D173—21no—
26C2×D4×D173C2211yescharacteristic
27C2×D4×D173C2211yescharacteristic, complemented
28C2×D4×D173C2211yescharacteristic, complemented
29#19149706—24no—
30C2×#14583813C2211yescharacteristic, complemented
31C2×D1384C2211yescharacteristic, complemented
32C2×D1384C2211yescharacteristic, complemented
33C2×#14583811C2211yescharacteristic, complemented
34C2×#14583811C2211yescharacteristic, complemented
35C2×#14583811C2211yescharacteristic, complemented
36D8×D173—24no—
37C2×#14583809C2211yescharacteristic, complemented
38C2×C8×D173C2211yescharacteristic, complemented
39#19149675—21no—
40#19149665—21no—
41#19149654C2211yescharacteristic, complemented
42#19149653C2211yescharacteristic, complemented
43#19149650C2211yescharacteristic, complemented
44#19149649—21no—
45#19149647—21no—
46#19149637C2211yescharacteristic, 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, complemented
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—
66C23×D173—21no—
67#14583827—22no—
68#14583827—22no—
69#14583827—22no—
70#14583827D414yescomplemented
71#14583826—28no—
72#14583826—22no—
73#14583826—22no—
74C2×Dic346D411yescharacteristic
75C2×D692—28no—
76C2×D692—21no—
77C2×D692—21no—
78C2×D692—22no—
79C2×D692—22no—
80C2×D692—21no—
81C2×D692—21no—
82C2×D692D411yescharacteristic
83C2×D692C2311yescharacteristic
84C2×D692C2311yescharacteristic
85C2×D692C2311yescharacteristic
86C2×C22⋊D173—21no—
87C2×C22⋊D173—21no—
88C2×C22⋊D173—21no—
89C2×C22⋊D173—21no—
90C2×C22⋊D173—21no—
91C2×C22⋊D173D411yescharacteristic
92C2×C22⋊D173D411yescharacteristic
93D4×D173—28no—
94D4×D173—28no—
95D4×D173—22no—
96D4×D173—22no—
97D4×D173—22no—
98D4×D173—22no—
99D4×D173—22no—
100D4×D173D414yescomplemented
101C2×C4×D173—21no—
102C2×C4×D173—21no—
103C2×C4×D173C2311yescharacteristic
104C2×C4×D173C2311yescharacteristic
105#14583813—22no—
106D1384—22no—
107D1384—22no—
108#14583811—22no—
109#14583811—22no—
110#14583811—22no—
111#14583809—22no—
112C2×#13063436C2311yescharacteristic
113C2×#13063436C2311yescharacteristic
114C8×D173—22no—
115C4⋊Dic173C2311yescharacteristic
116C22⋊Dic173—21no—
117#14583803—21no—
118#14583803—21no—
119#14583803—21no—
120C4×Dic173—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—21no—
142C22×D173—42no—
143C22×D173—81no—
144C22×D173—81no—
145C22×D173—81no—
146C22×D173—81no—
147C22×D173—81no—
148C22×D173—81no—
149C22×D173—81no—
150C22×D173—81no—
151C22×D173C2×D411yescharacteristic
152C22×D173C2×D411yescharacteristic
153Dic346—22no—
154D692—24no—
155D692—24no—
156D692—24no—
157D692—22no—
158D692—21no—
159D692—21no—
160D692—21no—
161D692—21no—
162D692—81no—
163D692C2×D412yes—
164D692C2×D412yes—
165C22⋊D173—24no—
166C22⋊D173—22no—
167C22⋊D173—22no—
168C22⋊D173—24no—
169C22⋊D173—42no—
170C22⋊D173—81no—
171C22⋊D173—81no—
172C2×Dic173—21no—
173C2×Dic173C2×D411yescharacteristic
174C2×Dic173C2×D411yescharacteristic
175C4×D173—24no—
176C4×D173—21no—
177C4×D173—21no—
178C4×D173—24no—
179C4×D173C2×D412yes—
180C4×D173C2×D412yes—
181#13063436—21no—
182#13063436—21no—
183C23×C173—21no—
184C23×C173—21no—
185C23×C173C2×D411yescharacteristic
186Q8×C173—21no—
187Q8×C173C2×D412yes—
188D4×C173—24no—
189D4×C173—21no—
190D4×C173—21no—
191D4×C173—21no—
192D4×C173—81no—
193D4×C173C2×D412yes—
194C2×C4×C173—24no—
195C2×C4×C173—21no—
196C2×C4×C173—21no—
197C2×C4×C173—21no—
198C2×C4×C173—21no—
199C2×C4×C173C2×D411yescharacteristic
200C2×C4×C173C2411yescharacteristic
201C8×C173—21no—
202C8×C173—21no—
203C22×C173—21no—
204C22×C173—22no—
205C22×C173—21no—
206C22×C173—21no—
207C22×C173—41no—
208C22×C173—41no—
209C22×C173—41no—
210C22×C173—41no—
211C22×C173—41no—
212C22×C173C22×D411yescharacteristic
213C2×D173—21no—
214C2×D173—21no—
215C2×D173—21no—
216C2×D173—22no—
217C2×D173—22no—
218C2×D173—21no—
219C2×D173—21no—
220C2×D173—41no—
221C2×D173—41no—
222C2×D173—41no—
223C2×D173—41no—
224C2×D173—41no—
225C2×D173—41no—
226C2×D173—41no—
227C2×D173—84no—
228C2×D173—84no—
229C2×D173—84no—
230C2×D173—84no—
231C2×D173#12012yes—
232C4×C173—21no—
233C4×C173—22no—
234C4×C173—41no—
235C4×C173C22×D411yescharacteristic
236C4×C173C22×D411yescharacteristic
237Dic173—21no—
238Dic173—41no—
239Dic173#12012yes—
240C2×C173—41no—
241C2×C173—42no—
242C2×C173—81no—
243C2×C173C2×#12011yescharacteristic
244C2×C173#57611yescharacteristic
245C2×C173#57611yescharacteristic
246D173—22no—
247D173—41no—
248D173—42no—
249D173—42no—
250D173—81no—
251C173#300811yescharacteristic, complemented, Sylow
252#3008—1731nomaximal, Sylow
253C2×H(5,𝔽2)—1731no—
254C2×AGL(1,ℤ/8)—1731no—
255C22×D8—1731no—
256#535—1731no—
257#449—1731no—
258#449—1731no—
259#449—1731no—
260#449D17311yescharacteristic, complemented
261#447—1731no—
262#447—1731no—
263#447—1731no—
264#447—1731no—
265#418—1731no—
266#416—1731no—
267#408—1731no—
268H(5,𝔽2)—3464no—
269H(5,𝔽2)—3464no—
270C2×#41—3461no—
271C2×#41—1731no—
272C2×#41—1731no—
273C2×#41—1731no—
274C2×#41C2×D17311yescharacteristic, complemented
275C22×D4—3461no—
276C22×D4—3461no—
277C22×D4—3461no—
278C22×D4—1731no—
279C22×D4—1731no—
280C22×D4—1731no—
281C22×D4—1731no—
282AGL(1,ℤ/8)—3464no—
283C2×SD16—1731no—
284C2×SD16C2×D17311yescharacteristic, complemented
285C2×D8—3464no—
286C2×D8—1731no—
287C2×D8—1731no—
288C2×D8—1731no—
289C2×D8—1731no—
290C2×D8—1731no—
291C2×D8C2×D17311yescharacteristic, complemented
292C2×M16—1731no—
293C22×C8—1731no—
294#127—3461no—
295#124—3461no—
296#121—1731no—
297#121—1731no—
298#121—1731no—
299#121C2×D17311yescharacteristic, complemented
300#120—3461no—
301#120—3461no—
302#117—1731no—
303#103—1731no—
304#103C2×D17311yescharacteristic, complemented
305#102—1731no—
306#102—1731no—
307#102—1731no—
308#102—1731no—
309#102—1731no—
310#102C2×D17311yescharacteristic, complemented
311C22⋊C8—1731no—
312C22⋊C8—1731no—
313C22⋊C8—1731no—
314C22⋊C8C2×D17311yescharacteristic, complemented
315C24—3461no—
316C24—3461no—
317C24—3461no—
318C24—3461no—
319#41—22no—
320#41—22no—
321#41—3468no—
322#41—3462no—
323#41—3462no—
324#41—3462no—
325#41—3462no—
326#41—3462no—
327#41—1734no—
328C2×Q8—1731no—
329C2×Q8C22×D17311yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—3468no—
333C2×D4—3461no—
334C2×D4—3461no—
335C2×D4—3462no—
336C2×D4—3462no—
337C2×D4—3461no—
338C2×D4—3468no—
339C2×D4—3468no—
340C2×D4—3462no—
341C2×D4—3462no—
342C2×D4—3461no—
343C2×D4—3461no—
344C2×D4—3462no—
345C2×D4—3462no—
346C2×D4—3462no—
347C2×D4—3461no—
348C2×D4—3461no—
349C2×D4—3461no—
350C2×D4—3461no—
351C2×D4—1731no—
352C2×D4—1731no—
353C2×D4—1734no—
354C2×D4—1731no—
355C2×D4—1731no—
356C2×D4—1731no—
357C2×D4—1731no—
358C2×D4C22×D17311yescharacteristic
359C2×D4C22×D17311yescharacteristic
360C22×C4—21no—
361C22×C4—3461no—
362C22×C4—3461no—
363C22×C4—1731no—
364C22×C4—1731no—
365C22×C4C22×D17311yescharacteristic
366SD16—22no—
367SD16—3462no—
368D8—22no—
369D8—3462no—
370D8—3462no—
371D8—3462no—
372D8—3462no—
373D8—3462no—
374M16—3462no—
375C2×C8—3462no—
376C2×C8—1731no—
377C2×C8—1731no—
378C2×C8C22×D17311yescharacteristic
379C2×C8C22×D17311yescharacteristic
380C4⋊C4—1731no—
381C4⋊C4C22×D17311yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—3461no—
384C22⋊C4—3461no—
385C22⋊C4—3461no—
386C22⋊C4—3461no—
387C42—3461no—
388C23—21no—
389C23—21no—
390C23—3464no—
391C23—3464no—
392C23—3464no—
393C23—3461no—
394C23—3461no—
395C23—3461no—
396C23—13841no—
397C23—13841no—
398C23—13841no—
399C23—13841no—
400C23—13841no—
401C23—13841no—
402C23—13841no—
403C23—13841no—
404C23—1731no—
405C23—1731no—
406C23—6922no—
407C23D4×D17311yescharacteristic
408Q8—21no—
409Q8—3462no—
410Q8D4×D17312yes—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—81no—
416D4—3464no—
417D4—3464no—
418D4—3464no—
419D4—3462no—
420D4—3464no—
421D4—3462no—
422D4—3462no—
423D4—3464no—
424D4—3461no—
425D4—3461no—
426D4—3461no—
427D4—3461no—
428D4—13841no—
429D4—13841no—
430D4—13841no—
431D4—1732no—
432D4—1732no—
433D4—6922no—
434D4D4×D17312yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—3464no—
441C2×C4—3464no—
442C2×C4—3461no—
443C2×C4—3461no—
444C2×C4—3461no—
445C2×C4—1731no—
446C2×C4—1731no—
447C2×C4—1732no—
448C2×C4—1732no—
449C2×C4D4×D17311yescharacteristic
450C2×C4C23×D17311yescharacteristic
451C8—21no—
452C8—21no—
453C8—3461no—
454C8—3461no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—3461no—
465C22—3461no—
466C22—3461no—
467C22—3462no—
468C22—3462no—
469C22—3461no—
470C22—3461no—
471C22—13844no—
472C22—13844no—
473C22—13844no—
474C22—13844no—
475C22—1732no—
476C22—6921no—
477C22—6921no—
478C22—6921no—
479C22—6921no—
480C22—6921no—
481C22—6921no—
482C22—6921no—
483C22C2×D4×D17311yescharacteristic
484C4—21no—
485C4—22no—
486C4—41no—
487C4—3461no—
488C4—1732no—
489C4—6921no—
490C4C2×D4×D17311yescharacteristic
491C4C2×D4×D17311yescharacteristic
492C2—41no—
493C2—42no—
494C2—81no—
495C2—3462no—
496C2—13841no—
497C2—6921no—
498C2—6922no—
499C2—6922no—
500C2#120×D17311yescharacteristic
501C2#2601229311yescharacteristic
502C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#449×C173C2
socleC22×C173C22×D4
hypercenter#449D173
hypocenterC173#3008
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×#19149706C2semidirect
3. C2×D8×D173C2semidirect
4. #26012057C2semidirect
5. #26011639C2semidirect
6. #26011636C2semidirect
7. #26011635C2semidirect
8. #26011627C2semidirect
9. #26011626C2semidirect
10. #26011624C2semidirect
11. #26011623C2semidirect
12. #26011494C2semidirect
13. #26011486C2semidirect
14. #26011456C2semidirect
15. #449×C173C2semidirect
19. C2×#14583827C22semidirect
23. C22×D692C22semidirect
27. C2×D4×D173C22semidirect
28. C2×D4×D173C22semidirect
30. C2×#14583813C22semidirect
31. C2×D1384C22semidirect
32. C2×D1384C22semidirect
33. C2×#14583811C22semidirect
34. C2×#14583811C22semidirect
35. C2×#14583811C22semidirect
37. C2×#14583809C22semidirect
38. C2×C8×D173C22semidirect
41. #19149654C22semidirect
42. #19149653C22semidirect
43. #19149650C22semidirect
46. #19149637C22semidirect
47. #19149603C22semidirect
48. #19149601C22semidirect
49. #19149601C22semidirect
50. #19149600C22semidirect
51. #19149599C22semidirect
52. #19149599C22semidirect
53. C22⋊#13063436C22semidirect
54. #19149591C22semidirect
55. #19149591C22semidirect
56. C2×#41×C173C22semidirect
57. C2×SD16×C173C22semidirect
58. C2×D8×C173C22semidirect
59. #121×C173C22semidirect
60. #103×C173C22semidirect
61. #102×C173C22semidirect
62. C22⋊C8×C173C22semidirect
70. #14583827D4semidirect
100. D4×D173D4semidirect
251. C173#3008semidirect
260. #449D173semidirect
274. C2×#41C2×D173semidirect
284. C2×SD16C2×D173semidirect
291. C2×D8C2×D173semidirect
299. #121C2×D173semidirect
304. #103C2×D173semidirect
310. #102C2×D173semidirect
314. 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×#191497062
C2×D8×D1732
#260120572
#260116392
#260116362
#260116352
#260116272
#260116262
#260116242
#260116232
#260114942
#260114862
#260114562
#449×C1732
#3008173