Finite Group Databasebeta

← #36365064

Subgroups — #36365064

Order 22144.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic, complemented
2C2×SD16×D173C211yesmaximal, characteristic, complemented
3#26012149C211yesmaximal, characteristic, complemented
4D4×D692C211yesmaximal, characteristic, complemented
5#26011881C211yesmaximal, characteristic, complemented
6#26011740C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011692C211yesmaximal, characteristic, complemented
9#26011640C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011631C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011561C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic, complemented
17C2×Q8×D173C2211yescharacteristic, complemented
18C22×D692—21no—
19C2×D4×D173C2211yescharacteristic
20C2×D4×D173C2211yescharacteristic
21#19149703—24no—
22C2×#14583815C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic
25C2×#14583813C2211yescharacteristic, complemented
26C2×#14583813C2211yescharacteristic, complemented
27SD16×D173—24no—
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583811C2211yescharacteristic, complemented
30C2×#14583809C2211yescharacteristic, complemented
31C2×C8×D173C2211yescharacteristic, complemented
32#19149676C2211yescharacteristic
33#19149665—21no—
34#19149656—21no—
35#19149655C2211yescharacteristic, complemented
36#19149654C2211yescharacteristic
37#19149654C2211yescharacteristic
38#19149653—21no—
39#19149652—21no—
40#19149650C2211yescharacteristic, complemented
41#19149649—21no—
42#19149649—21no—
43C4×Dic346C2211yescharacteristic, complemented
44C4×D692C2211yescharacteristic
45#19149639—21no—
46#19149612C2211yescharacteristic, complemented
47#19149602C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic
51#19149591C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149590C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173—21no—
63#14583825—22no—
64#14583825—41no—
65C2×Dic346—21no—
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—41no—
69C2×D692—21no—
70C2×D692—21no—
71C2×D692—21no—
72C2×D692—21no—
73C2×D692—44no—
74C2×D692D411yescharacteristic
75C2×D692C2311yescharacteristic
76C2×D692C2311yescharacteristic
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173—21no—
81D4×D173—22no—
82D4×D173—22no—
83D4×D173—41no—
84D4×D173—41no—
85C22×Dic173—21no—
86C2×C4×D173—21no—
87C2×C4×D173C2311yescharacteristic
88C2×C4×D173C2311yescharacteristic
89#14583815—22no—
90#14583815—22no—
91#14583814D414yescomplemented
92#14583813—22no—
93#14583813—41no—
94#14583811—22no—
95#14583811—41no—
96#14583809—22no—
97C2×#13063436C2311yescharacteristic
98C2×#13063436C2311yescharacteristic
99C8×D173—22no—
100C4⋊Dic173D411yescharacteristic
101C4⋊Dic173C2311yescharacteristic
102#14583805—21no—
103#14583805—21no—
104C22⋊Dic173—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803D411yescharacteristic
109#14583803D411yescharacteristic
110C4×Dic173—21no—
111C2×Q8×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115SD16×C173—22no—
116D8×C173—22no—
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C4⋊C4×C173C2311yescharacteristic
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—81no—
131C22×D173—81no—
132C22×D173C2×D411yescharacteristic
133C22×D173C2×D411yescharacteristic
134Dic346—21no—
135Dic346—42no—
136Dic346C2×D412yescomplemented
137D692—22no—
138D692—21no—
139D692—21no—
140D692—21no—
141D692—21no—
142D692—42no—
143D692—42no—
144D692—42no—
145D692—81no—
146C22⋊D173—42no—
147C22⋊D173—42no—
148C22⋊D173—42no—
149C22⋊D173—42no—
150C2×Dic173—21no—
151C2×Dic173—21no—
152C2×Dic173—42no—
153C2×Dic173C2×D411yescharacteristic
154C2×Dic173C2×D411yescharacteristic
155C4×D173—21no—
156C4×D173—21no—
157C4×D173—21no—
158C4×D173—21no—
159C4×D173—42no—
160#13063436—21no—
161#13063436C2×D412yescomplemented
162C23×C173—21no—
163C23×C173—21no—
164Q8×C173—21no—
165Q8×C173—21no—
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173C2×D412yes—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—42no—
173C2×C4×C173C2×D411yescharacteristic
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2411yescharacteristic
176C8×C173—21no—
177C8×C173—21no—
178C22×C173—22no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173C22×D411yescharacteristic
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—21no—
188C2×D173—21no—
189C2×D173—21no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—41no—
195C2×D173—41no—
196C2×D173—84no—
197C2×D173—84no—
198C2×D173AGL(1,ℤ/8)12yescomplemented
199C4×C173—21no—
200C4×C173—21no—
201C4×C173—41no—
202C4×C173—41no—
203C4×C173C22×D411yescharacteristic
204C4×C173H(5,𝔽2)11yescharacteristic
205Dic173—21no—
206Dic173—21no—
207Dic173—22no—
208Dic173—41no—
209C2×C173—42no—
210C2×C173—81no—
211C2×C173D4211yescharacteristic
212C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C173#57711yescharacteristic
214D173—22no—
215D173—41no—
216D173—81no—
217D173—81no—
218C173#326511yescharacteristic, complemented, Sylow
219#3265—1731nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1731no—
221C22×SD16—1731no—
222#548—1731no—
223D42—1731no—
224#496—1731no—
225#468—1731no—
226#466—1731no—
227#463D17311yescharacteristic, complemented
228#460—1731no—
229#450—1731no—
230#449—1731no—
231#448—1731no—
232#447—1731no—
233#440—1731no—
234#435—1731no—
235C2×#41—1731no—
236C22×Q8—1731no—
237C22×D4—3461no—
238C22×D4—1731no—
239C22×D4—1731no—
240AGL(1,ℤ/8)—3464no—
241C2×SD16—3464no—
242C2×SD16—1731no—
243C2×SD16—1731no—
244C2×SD16—1731no—
245C2×SD16—1731no—
246C2×SD16—1731no—
247C2×SD16C2×D17311yescharacteristic
248C2×D8—1731no—
249C2×D8—1731no—
250C2×D8C2×D17311yescharacteristic
251C2×M16—1731no—
252C22×C8—1731no—
253#127—1731no—
254#124—3461no—
255#124C2×D17311yescharacteristic, complemented
256C22⋊Q8—3461no—
257C22⋊Q8—1731no—
258#121—3461no—
259#121—3461no—
260#121—1731no—
261#121—1731no—
262#121—1731no—
263#120—3461no—
264#120—3461no—
265C4×Q8—1731no—
266C4×D4—3461no—
267C4×D4—1731no—
268C4×D4C2×D17311yescharacteristic
269#107—1731no—
270C4⋊C8C2×D17311yescharacteristic, complemented
271#103—1731no—
272#103C2×D17311yescharacteristic, complemented
273#102—1731no—
274#102—1731no—
275#102—1731no—
276#102C2×D17311yescharacteristic, complemented
277C22⋊C8—1731no—
278C22⋊C8—1731no—
279#97—1731no—
280C24—3461no—
281C24—3461no—
282#41—3462no—
283#41—6921no—
284C2×Q8—3461no—
285C2×Q8—3462no—
286C2×Q8—1731no—
287C2×Q8—6921no—
288C2×Q8C22×D17311yescharacteristic
289C2×D4—3461no—
290C2×D4—3461no—
291C2×D4—3462no—
292C2×D4—3462no—
293C2×D4—3461no—
294C2×D4—3461no—
295C2×D4—3461no—
296C2×D4—3461no—
297C2×D4—3461no—
298C2×D4—3461no—
299C2×D4—1731no—
300C2×D4—1731no—
301C2×D4—1731no—
302C2×D4—6924no—
303C2×D4—6921no—
304C2×D4—6921no—
305C2×D4C22×D17311yescharacteristic
306C2×D4C22×D17311yescharacteristic
307C22×C4—21no—
308C22×C4—3461no—
309C22×C4—3461no—
310C22×C4—1731no—
311C22×C4—1731no—
312SD16—22no—
313SD16—3462no—
314SD16—3462no—
315SD16—3462no—
316SD16—1734no—
317SD16—6921no—
318D8—22no—
319D8—3462no—
320D8—6921no—
321M16—3462no—
322C2×C8—3462no—
323C2×C8—1731no—
324C2×C8—1731no—
325C2×C8C22×D17311yescharacteristic
326C2×C8C22×D17311yescharacteristic
327C4⋊C4—3461no—
328C4⋊C4—3461no—
329C4⋊C4—1731no—
330C4⋊C4—1731no—
331C4⋊C4C22×D17311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3461no—
336C22⋊C4—3461no—
337C22⋊C4—3461no—
338C22⋊C4—3461no—
339C22⋊C4—1731no—
340C22⋊C4—1731no—
341C42—3461no—
342C42C22×D17311yescharacteristic
343C23—21no—
344C23—21no—
345C23—3461no—
346C23—3461no—
347C23—13841no—
348C23—13841no—
349C23—1731no—
350C23—1731no—
351C23—6922no—
352C23—6922no—
353C23—6922no—
354C23—6922no—
355Q8—21no—
356Q8—21no—
357Q8—3461no—
358Q8—1732no—
359Q8—6922no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—3462no—
364D4—3461no—
365D4—3461no—
366D4—3461no—
367D4—3461no—
368D4—13841no—
369D4—6922no—
370D4—6922no—
371D4—6922no—
372D4—6922no—
373D4—6922no—
374D4—6922no—
375D4—6922no—
376D4D4×D17312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—3461no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—3461no—
386C2×C4—1731no—
387C2×C4—1731no—
388C2×C4—6922no—
389C2×C4—6922no—
390C2×C4D4×D17311yescharacteristic
391C2×C4D4×D17311yescharacteristic
392C2×C4C23×D17311yescharacteristic
393C8—21no—
394C8—21no—
395C8—3461no—
396C8—1732no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—3461no—
404C22—3461no—
405C22—3461no—
406C22—3461no—
407C22—3461no—
408C22—13844no—
409C22—13844no—
410C22—1732no—
411C22—6921no—
412C22—6921no—
413C22—6921no—
414C22—6921no—
415C22—6921no—
416C22—6921no—
417C22C2×D4×D17311yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—3461no—
423C4—3461no—
424C4—3462no—
425C4—6921no—
426C4C2×D4×D17311yescharacteristic
427C4#1914972511yescharacteristic
428C2—42no—
429C2—81no—
430C2—3462no—
431C2—13841no—
432C2—13841no—
433C2—6921no—
434C2#2601212511yescharacteristic
435C2AGL(1,ℤ/8)×D17311yescharacteristic
436C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
hypercenter#463D173
hypocenterC173#3265
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×SD16×D173C2semidirect
3. #26012149C2semidirect
4. D4×D692C2semidirect
5. #26011881C2semidirect
6. #26011740C2semidirect
7. #26011726C2semidirect
8. #26011692C2semidirect
9. #26011640C2semidirect
10. #26011637C2semidirect
11. #26011631C2semidirect
12. #26011623C2semidirect
13. #26011591C2semidirect
14. #26011561C2semidirect
15. #463×C173C2semidirect
16. C2×#14583825C22semidirect
17. C2×Q8×D173C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583815C22semidirect
25. C2×#14583813C22semidirect
26. C2×#14583813C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583811C22semidirect
30. C2×#14583809C22semidirect
31. C2×C8×D173C22semidirect
35. #19149655C22semidirect
40. #19149650C22semidirect
43. C4×Dic346C22semidirect
46. #19149612C22semidirect
47. #19149602C22semidirect
48. #19149601C22semidirect
49. #19149601C22semidirect
51. #19149591C22semidirect
52. #19149591C22semidirect
53. #19149590C22semidirect
56. #124×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
91. #14583814D4semidirect
136. Dic346C2×D4semidirect
161. #13063436C2×D4semidirect
198. C2×D173AGL(1,ℤ/8)semidirect
218. C173#3265semidirect
227. #463D173semidirect
255. #124C2×D173semidirect
270. C4⋊C8C2×D173semidirect
272. #103C2×D173semidirect
276. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×SD16×D1732
#260121492
D4×D6922
#260118812
#260117402
#260117262
#260116922
#260116402
#260116372
#260116312
#260116232
#260115912
#260115612
#463×C1732
#3265173