Finite Group Databasebeta

← #36365057

Subgroups — #36365057

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
4#26012126C211yesmaximal, characteristic, complemented
5#26011734C211yesmaximal, characteristic, complemented
6#26011724C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011688C211yesmaximal, characteristic, complemented
9#26011640C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011628C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011593C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic, complemented
17C2×Q8×D173C2211yescharacteristic, complemented
18C2×D4×D173—21no—
19C2×D4×D173C2211yescharacteristic
20C2×D4×D173C2211yescharacteristic
21#19149703—24no—
22C2×#14583815C2211yescharacteristic
23C2×#14583814C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic, complemented
25C2×#14583813C2211yescharacteristic
26SD16×D173—24no—
27C2×#14583811C2211yescharacteristic
28C2×#14583809C2211yescharacteristic, complemented
29C2×C8×D173C2211yescharacteristic, complemented
30#19149665—21no—
31#19149656—21no—
32#19149655C2211yescharacteristic, complemented
33#19149652—21no—
34#19149650—21no—
35#19149650C2211yescharacteristic
36#19149650C2211yescharacteristic
37#19149650C2211yescharacteristic, complemented
38#19149647—21no—
39#19149647—21no—
40C4×Dic346C2211yescharacteristic, complemented
41C4×D692C2211yescharacteristic
42C4×D692C2211yescharacteristic
43#19149639—21no—
44#19149613C2211yescharacteristic, complemented
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149604C2211yescharacteristic, complemented
47#19149603C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic
49#19149600C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149591C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic, complemented
56C2×D8×C173C2211yescharacteristic, complemented
57C2×D8×C173C2211yescharacteristic, complemented
58#127×C173C2211yescharacteristic
59#124×C173C2211yescharacteristic, complemented
60#97×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×D692D411yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77D4×D173—22no—
78D4×D173—22no—
79D4×D173—41no—
80D4×D173—41no—
81D4×D173—44no—
82C22×Dic173—21no—
83C2×C4×D173—21no—
84C2×C4×D173—21no—
85C2×C4×D173C2311yescharacteristic
86C2×C4×D173C2311yescharacteristic
87#14583815D414yescomplemented
88#14583814—22no—
89#14583814—22no—
90#14583813—22no—
91#14583811—22no—
92#14583809—22no—
93C2×#13063436C2311yescharacteristic
94C2×#13063436C2311yescharacteristic
95C8×D173—22no—
96C4⋊Dic173D411yescharacteristic
97C4⋊Dic173C2311yescharacteristic
98C4⋊Dic173C2311yescharacteristic
99#14583805—21no—
100#14583805—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107#14583803D411yescharacteristic
108C4×Dic173—21no—
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173—21no—
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115SD16×C173—22no—
116SD16×C173—41no—
117D8×C173—22no—
118D8×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C22⋊C4×C173—21no—
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—81no—
130C22×D173—81no—
131C22×D173C2×D411yescharacteristic
132C22×D173C2×D411yescharacteristic
133Dic346—21no—
134Dic346—42no—
135Dic346C2×D412yescomplemented
136D692—22no—
137D692—21no—
138D692C2×D412yes—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C22⋊D173—42no—
143C22⋊D173—42no—
144C22⋊D173—42no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—21no—
148C2×Dic173—42no—
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C4×D173—21no—
152C4×D173—21no—
153C4×D173—21no—
154C4×D173—21no—
155C4×D173—42no—
156C4×D173—42no—
157#13063436—21no—
158#13063436—21no—
159C23×C173—21no—
160C23×C173—21no—
161C23×C173—21no—
162Q8×C173—21no—
163Q8×C173—21no—
164D4×C173—21no—
165D4×C173—21no—
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—42no—
171D4×C173—81no—
172C2×C4×C173—21no—
173C2×C4×C173C2×D411yescharacteristic
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2411yescharacteristic
176C8×C173—21no—
177C8×C173C2×D412yescomplemented
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173C22×D411yescharacteristic
188C2×D173—21no—
189C2×D173—21no—
190C2×D173—21no—
191C2×D173—22no—
192C2×D173—21no—
193C2×D173—21no—
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×C173C22×D411yescharacteristic
203C4×C173H(5,𝔽2)11yescharacteristic
204Dic173—21no—
205Dic173—21no—
206Dic173—22no—
207Dic173—41no—
208Dic173—41no—
209C2×C173—81no—
210C2×C173—81no—
211C2×C173—81no—
212C2×C173D4211yescharacteristic
213C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C173#57711yescharacteristic
215D173—22no—
216D173—41no—
217D173—42no—
218C173#326511yescharacteristic, complemented, Sylow
219#3265—1731nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1731no—
221C22×SD16—1731no—
222#548—1731no—
223D42—1731no—
224#496D17311yescharacteristic, complemented
225#468—1731no—
226#466—1731no—
227#463—1731no—
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×SD16C2×D17311yescharacteristic, complemented
247C2×SD16C2×D17311yescharacteristic, complemented
248C2×D8—1731no—
249C2×D8C2×D17311yescharacteristic, complemented
250C2×D8C2×D17311yescharacteristic, complemented
251C2×M16—1731no—
252C22×C8—1731no—
253#127C2×D17311yescharacteristic
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×D4—1731no—
269#107—1731no—
270C4⋊C8—1731no—
271#103—1731no—
272#103—1731no—
273#102—1731no—
274#102—1731no—
275#102—1731no—
276#102—1731no—
277C22⋊C8—1731no—
278C22⋊C8—1731no—
279#97C2×D17311yescharacteristic, complemented
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—21no—
290C2×D4—21no—
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—6921no—
302C2×D4—6921no—
303C2×D4—6924no—
304C2×D4C22×D17311yescharacteristic
305C2×D4C22×D17311yescharacteristic
306C2×D4C22×D17311yescharacteristic
307C22×C4—3461no—
308C22×C4—3461no—
309C22×C4—3461no—
310C22×C4—1731no—
311C22×C4—1731no—
312SD16—22no—
313SD16—41no—
314SD16—3462no—
315SD16—3462no—
316SD16—3462no—
317SD16—1734no—
318D8—22no—
319D8—41no—
320D8—3462no—
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⋊C4—1731no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3461no—
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—21no—
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—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—3462no—
369D4—3461no—
370D4—1732no—
371D4—6922no—
372D4—6922no—
373D4—6922no—
374D4—6922no—
375D4—6922no—
376D4—6922no—
377C2×C4—21no—
378C2×C4—3461no—
379C2×C4—3461no—
380C2×C4—3461no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—1731no—
386C2×C4—1731no—
387C2×C4—6922no—
388C2×C4—6922no—
389C2×C4—6922no—
390C2×C4D4×D17311yescharacteristic
391C2×C4D4×D17311yescharacteristic
392C2×C4C23×D17311yescharacteristic
393C8—21no—
394C8—3461no—
395C8—3461no—
396C8D4×D17312yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3461no—
407C22—3461no—
408C22—3461no—
409C22—3461no—
410C22—3461no—
411C22—3462no—
412C22—13844no—
413C22—13844no—
414C22—1732no—
415C22—6921no—
416C22—6921no—
417C22C2×D4×D17311yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—3461no—
422C4—3461no—
423C4—3462no—
424C4—6921no—
425C4—6921no—
426C4C2×D4×D17311yescharacteristic
427C4#1914972411yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—3462no—
432C2—6921no—
433C2—6922no—
434C2#2601212611yescharacteristic
435C2AGL(1,ℤ/8)×D17311yescharacteristic
436C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3265
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×SD16×D173C2semidirect
3. #26012149C2semidirect
4. #26012126C2semidirect
5. #26011734C2semidirect
6. #26011724C2semidirect
7. #26011703C2semidirect
8. #26011688C2semidirect
9. #26011640C2semidirect
10. #26011633C2semidirect
11. #26011628C2semidirect
12. #26011623C2semidirect
13. #26011593C2semidirect
14. #26011564C2semidirect
15. #496×C173C2semidirect
16. C2×#14583825C22semidirect
17. C2×Q8×D173C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583814C22semidirect
28. C2×#14583809C22semidirect
29. C2×C8×D173C22semidirect
32. #19149655C22semidirect
37. #19149650C22semidirect
40. C4×Dic346C22semidirect
44. #19149613C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149604C22semidirect
47. #19149603C22semidirect
49. #19149600C22semidirect
50. #19149600C22semidirect
51. #19149600C22semidirect
52. #19149591C22semidirect
53. #19149591C22semidirect
54. C2×SD16×C173C22semidirect
55. C2×SD16×C173C22semidirect
56. C2×D8×C173C22semidirect
57. C2×D8×C173C22semidirect
59. #124×C173C22semidirect
60. #97×C173C22semidirect
87. #14583815D4semidirect
135. Dic346C2×D4semidirect
177. C8×C173C2×D4semidirect
198. C2×D173AGL(1,ℤ/8)semidirect
218. C173#3265semidirect
224. #496D173semidirect
246. C2×SD16C2×D173semidirect
247. C2×SD16C2×D173semidirect
249. C2×D8C2×D173semidirect
250. C2×D8C2×D173semidirect
255. #124C2×D173semidirect
279. #97C2×D173semidirect
396. C8D4×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
#260121262
#260117342
#260117242
#260117032
#260116882
#260116402
#260116332
#260116282
#260116232
#260115932
#260115642
#496×C1732
#3265173