Finite Group Databasebeta

← #36365177

Subgroups — #36365177

Order 22144.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2#26012160C211yesmaximal, characteristic, complemented
3D4×C22⋊D173C211yesmaximal, characteristic, complemented
4#26011866C211yesmaximal, characteristic, complemented
5#26011744C211yesmaximal, characteristic, complemented
6#26011741C211yesmaximal, characteristic, complemented
7#26011689C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011635C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13SD16×Dic173C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic, complemented
17C22×C22⋊D173—21no—
18C2×D4×D173C2211yescharacteristic
19#19149706—24no—
20C2×#14583813C2211yescharacteristic, complemented
21C2×D1384C2211yescharacteristic, complemented
22C2×#14583811C2211yescharacteristic
23C2×#14583811C2211yescharacteristic, complemented
24C2×#14583811C2211yescharacteristic, complemented
25C2×#14583809C2211yescharacteristic, complemented
26#19149675—21no—
27#19149675C2211yescharacteristic
28#19149675C2211yescharacteristic, complemented
29#19149653—21no—
30#19149653C2211yescharacteristic
31#19149653C2211yescharacteristic, complemented
32#19149651—21no—
33#19149650—21no—
34#19149650C2211yescharacteristic
35#19149650C2211yescharacteristic, complemented
36#19149648—21no—
37#19149647—21no—
38Q8×Dic173C2211yescharacteristic, complemented
39D4×Dic173C2211yescharacteristic
40C4×C22⋊D173—21no—
41C4×C22⋊D173C2211yescharacteristic
42#19149611C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50C22⋊#13063436C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C8×Dic173C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55AGL(1,ℤ/8)×C173—24no—
56C2×SD16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic, complemented
58C2×D8×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic, complemented
60C2×M16×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—44no—
77C2×C22⋊D173D411yescharacteristic
78D4×D173—22no—
79D4×D173—41no—
80C22×Dic173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83#14583813—22no—
84D1384—22no—
85#14583811—22no—
86#14583811—41no—
87#14583811—41no—
88#14583809—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93C4⋊Dic173C2311yescharacteristic
94#14583805D411yescharacteristic
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173D411yescharacteristic
99#14583803—21no—
100#14583803—21no—
101#14583803D411yescharacteristic
102C4×Dic173—21no—
103C4×Dic173C2311yescharacteristic
104C24×C173—21no—
105#41×C173—22no—
106#41×C173—41no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—22no—
109C2×D4×C173—21no—
110C2×D4×C173—21no—
111C2×D4×C173—41no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118SD16×C173D414yescomplemented
119D8×C173—22no—
120D8×C173—22no—
121M16×C173—22no—
122C2×C8×C173C2311yescharacteristic
123C2×C8×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—81no—
129C22×D173C2×D411yescharacteristic
130D692—21no—
131D692—21no—
132D692—21no—
133D692—21no—
134D692—42no—
135C22⋊D173—22no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—81no—
142C22⋊D173—81no—
143C22⋊D173—81no—
144C22⋊D173—81no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—42no—
148C2×Dic173C2×D411yescharacteristic
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C4×D173—21no—
152C4×D173—21no—
153C4×D173—42no—
154#13063436—21no—
155#13063436—21no—
156C23×C173—21no—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—42no—
160C23×C173—42no—
161C23×C173—81no—
162C23×C173C2×D411yescharacteristic
163Q8×C173—21no—
164Q8×C173C2×D412yescomplemented
165D4×C173—21no—
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—42no—
171D4×C173—42no—
172D4×C173C2×D412yes—
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173—42no—
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2411yescharacteristic
179C8×C173—21no—
180C8×C173C2×D412yescomplemented
181C22×C173—21no—
182C22×C173—22no—
183C22×C173—21no—
184C22×C173—21no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—41no—
191C22×C173—41no—
192C22×C173—41no—
193C22×C173—84no—
194C22×C173C22×D411yescharacteristic
195C2×D173—21no—
196C2×D173—21no—
197C2×D173—21no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—41no—
204C2×D173—84no—
205C4×C173—21no—
206C4×C173—22no—
207C4×C173C22×D411yescharacteristic
208C4×C173H(5,𝔽2)11yescharacteristic
209Dic173—21no—
210Dic173—21no—
211Dic173—41no—
212Dic173—41no—
213Dic173AGL(1,ℤ/8)12yescomplemented
214C2×C173—42no—
215C2×C173—41no—
216C2×C173—81no—
217C2×C173—81no—
218C2×C173D4211yescharacteristic
219C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C173#57611yescharacteristic
221D173—41no—
222D173—81no—
223D173—81no—
224C173#327411yescharacteristic, complemented, Sylow
225#3274—1731nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1731no—
227C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
228#550—1731no—
229D42—1731no—
230#493—1731no—
231#469—1731no—
232#469—1731no—
233#460—1731no—
234#459—1731no—
235#449—1731no—
236#449—1731no—
237#447—1731no—
238#447—1731no—
239C4×SD16—1731no—
240#436—1731no—
241C2×#41—1731no—
242C2×#41C2×D17311yescharacteristic, complemented
243C22×D4—3461no—
244C22×D4—1731no—
245C22×D4C2×D17311yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—3464no—
248C2×SD16—1731no—
249C2×SD16C2×D17311yescharacteristic
250C2×SD16C2×D17311yescharacteristic, complemented
251C2×D8—1731no—
252C2×D8—1731no—
253C2×D8—1731no—
254C2×D8—1731no—
255C2×D8C2×D17311yescharacteristic
256C2×D8C2×D17311yescharacteristic, complemented
257C2×M16—1731no—
258C2×M16C2×D17311yescharacteristic, complemented
259#127—3461no—
260#127—1731no—
261#127—1731no—
262#121—3461no—
263#121—3461no—
264#121—3461no—
265#121—1731no—
266#121—1731no—
267#121—1731no—
268#121—1731no—
269#120—3461no—
270#120—3461no—
271C4×Q8—1731no—
272C4×D4—3461no—
273C4×D4—1731no—
274C4×D4—1731no—
275#106—1731no—
276C4⋊C8—1731no—
277#103—1731no—
278#102—1731no—
279#102—1731no—
280#102—1731no—
281#102—1731no—
282#102—1731no—
283C22⋊C8—1731no—
284C22⋊C8—1731no—
285C4×C8—1731no—
286C24—21no—
287C24—3461no—
288#41—22no—
289#41—41no—
290#41—3462no—
291#41—6921no—
292C2×Q8C22×D17311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—3461no—
298C2×D4—3462no—
299C2×D4—3461no—
300C2×D4—3461no—
301C2×D4—3461no—
302C2×D4—3461no—
303C2×D4—3461no—
304C2×D4—3461no—
305C2×D4—3461no—
306C2×D4—3461no—
307C2×D4—3461no—
308C2×D4—1731no—
309C2×D4—1731no—
310C2×D4—1731no—
311C2×D4—6921no—
312C2×D4—6924no—
313C2×D4C22×D17311yescharacteristic
314C2×D4C22×D17311yescharacteristic
315C2×D4C22×D17311yescharacteristic
316C22×C4—21no—
317C22×C4—3461no—
318C22×C4—3461no—
319C22×C4—1731no—
320C22×C4C22×D17311yescharacteristic
321SD16—22no—
322SD16—3462no—
323SD16C22⋊D17314yescomplemented
324D8—22no—
325D8—22no—
326D8—3462no—
327D8—3462no—
328D8—6921no—
329D8—6921no—
330M16—22no—
331M16—3462no—
332C2×C8—1731no—
333C2×C8—1731no—
334C2×C8C22×D17311yescharacteristic
335C2×C8C22×D17311yescharacteristic
336C4⋊C4—3461no—
337C4⋊C4—1731no—
338C4⋊C4—1731no—
339C4⋊C4—1731no—
340C22⋊C4—3461no—
341C22⋊C4—3461no—
342C22⋊C4—3461no—
343C22⋊C4—3461no—
344C22⋊C4—3461no—
345C22⋊C4—1731no—
346C22⋊C4—1731no—
347C42—3461no—
348C42—1731no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—3461no—
356C23—3461no—
357C23—13841no—
358C23—1731no—
359C23—6922no—
360C23—6922no—
361C23D4×D17311yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D17312yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—3462no—
372D4—3461no—
373D4—3461no—
374D4—3461no—
375D4—3461no—
376D4—13841no—
377D4—13841no—
378D4—13841no—
379D4—13841no—
380D4—6922no—
381D4—6922no—
382D4—6922no—
383D4—6922no—
384D4—6922no—
385D4—6922no—
386D4C2×C22⋊D17312yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—1731no—
396C2×C4—1731no—
397C2×C4—1731no—
398C2×C4—6922no—
399C2×C4—6922no—
400C2×C4D4×D17311yescharacteristic
401C2×C4C23×D17311yescharacteristic
402C8—21no—
403C8—3461no—
404C8—3461no—
405C8C2×C22⋊D17312yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—3461no—
420C22—3461no—
421C22—3461no—
422C22—13844no—
423C22—6921no—
424C22—6921no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22C2×D4×D17311yescharacteristic
430C4—21no—
431C4—22no—
432C4—3461no—
433C4—3461no—
434C4—1732no—
435C4—6921no—
436C4—6921no—
437C4C22×C22⋊D17311yescharacteristic
438C4#1914972411yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—13841no—
444C2—13841no—
445C2—6921no—
446C2D4×C22⋊D17311yescharacteristic
447C2AGL(1,ℤ/8)×D17311yescharacteristic
448C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3274
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012160C2semidirect
3. D4×C22⋊D173C2semidirect
4. #26011866C2semidirect
5. #26011744C2semidirect
6. #26011741C2semidirect
7. #26011689C2semidirect
8. #26011684C2semidirect
9. #26011636C2semidirect
10. #26011635C2semidirect
11. #26011625C2semidirect
12. #26011623C2semidirect
13. SD16×Dic173C2semidirect
14. #26011567C2semidirect
15. C2×AGL(1,ℤ/8)×C173C2semidirect
16. C2×#14583827C22semidirect
20. C2×#14583813C22semidirect
21. C2×D1384C22semidirect
23. C2×#14583811C22semidirect
24. C2×#14583811C22semidirect
25. C2×#14583809C22semidirect
28. #19149675C22semidirect
31. #19149653C22semidirect
35. #19149650C22semidirect
38. Q8×Dic173C22semidirect
42. #19149611C22semidirect
43. #19149608C22semidirect
44. #19149603C22semidirect
45. #19149601C22semidirect
46. #19149601C22semidirect
48. #19149600C22semidirect
49. #19149600C22semidirect
50. C22⋊#13063436C22semidirect
51. #19149591C22semidirect
52. C8×Dic173C22semidirect
53. C2×#41×C173C22semidirect
57. C2×SD16×C173C22semidirect
59. C2×D8×C173C22semidirect
60. C2×M16×C173C22semidirect
118. SD16×C173D4semidirect
164. Q8×C173C2×D4semidirect
180. C8×C173C2×D4semidirect
213. Dic173AGL(1,ℤ/8)semidirect
224. C173#3274semidirect
227. C2×AGL(1,ℤ/8)D173semidirect
242. C2×#41C2×D173semidirect
250. C2×SD16C2×D173semidirect
256. C2×D8C2×D173semidirect
258. C2×M16C2×D173semidirect
323. SD16C22⋊D173semidirect
363. Q8C2×C22⋊D173semidirect
405. C8C2×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121602
D4×C22⋊D1732
#260118662
#260117442
#260117412
#260116892
#260116842
#260116362
#260116352
#260116252
#260116232
SD16×Dic1732
#260115672
C2×AGL(1,ℤ/8)×C1732
#3274173