Finite Group Databasebeta

← #36365059

Subgroups — #36365059

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×#19149704C211yesmaximal, characteristic, complemented
2C22×#14583813C211yesmaximal, characteristic, complemented
3D4×D692C211yesmaximal, characteristic, complemented
4#26011881C211yesmaximal, characteristic, complemented
5#26011739C211yesmaximal, characteristic, complemented
6#26011725C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011691C211yesmaximal, characteristic, complemented
9#26011641C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011630C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13#26011589C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×D4×D173—21no—
19#19149704—24no—
20C2×#14583813—24no—
21C2×#14583813C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23C2×#14583813C2211yescharacteristic, complemented
24C2×#14583813C2211yescharacteristic, complemented
25C2×#14583813C2211yescharacteristic, complemented
26C2×#14583813C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583811C2211yescharacteristic, complemented
30C2×#14583810C2211yescharacteristic, complemented
31C22×#13063436C2211yescharacteristic, complemented
32#19149676C2211yescharacteristic
33#19149654—21no—
34#19149653C2211yescharacteristic
35#19149653C2211yescharacteristic
36#19149649—21no—
37#19149649—21no—
38C4×D692C2211yescharacteristic
39#19149612C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic, complemented
44#19149599C2211yescharacteristic
45#19149599C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic, complemented
48C22⋊#13063436C2211yescharacteristic, complemented
49#19149590C2211yescharacteristic, complemented
50C2×#41×C173C2211yescharacteristic, complemented
51C22×Q8×C173C2211yescharacteristic, complemented
52#124×C173—21no—
53#124×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173—21no—
55C22⋊Q8×C173C2211yescharacteristic, complemented
56#121×C173—21no—
57#121×C173C2211yescharacteristic, complemented
58C4×Q8×C173C2211yescharacteristic, complemented
59C4×D4×C173—21no—
60C4×D4×C173C2211yescharacteristic
61C23×D173—21no—
62C23×D173—21no—
63C2×D692—22no—
64C2×D692—22no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692—41no—
70C2×D692—41no—
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×D692C2311yescharacteristic
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76D4×D173—44no—
77C2×C4×D173—21no—
78#14583813—22no—
79#14583813—22no—
80#14583813—22no—
81#14583813—22no—
82#14583813—41no—
83#14583813D414yescomplemented
84#14583811—22no—
85#14583811—22no—
86#14583811—41no—
87#14583810—22no—
88C2×#13063436—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C2×#13063436C2311yescharacteristic
93C4⋊Dic173C2311yescharacteristic
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173—21no—
100C2×Q8×C173—22no—
101C2×Q8×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×D4×C173—21no—
105C2×D4×C173—21no—
106C2×D4×C173D411yescharacteristic
107C2×D4×C173C2311yescharacteristic
108C22×C4×C173—21no—
109C22×C4×C173—21no—
110C22×C4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C4⋊C4×C173—21no—
113C4⋊C4×C173—21no—
114C4⋊C4×C173D411yescharacteristic
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C42×C173—21no—
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—21no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173—42no—
130C22×D173—42no—
131C22×D173—81no—
132C22×D173—81no—
133D692—21no—
134D692—21no—
135D692—21no—
136D692—21no—
137D692—21no—
138D692—42no—
139D692—42no—
140D692—42no—
141D692—81no—
142D692C2×D412yes—
143C22⋊D173—42no—
144C22⋊D173—42no—
145C2×Dic173—21no—
146C4×D173—42no—
147#13063436—21no—
148#13063436—21no—
149#13063436—21no—
150#13063436C2×D412yescomplemented
151C23×C173—21no—
152C23×C173C2×D411yescharacteristic
153C23×C173C2×D411yescharacteristic
154Q8×C173—21no—
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×C173—42no—
158Q8×C173C2×D412yescomplemented
159D4×C173—22no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—42no—
163D4×C173—42no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—42no—
172C2×C4×C173—42no—
173C2×C4×C173C2×D411yescharacteristic
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2×D411yescharacteristic
176C2×C4×C173C2×D411yescharacteristic
177C2×C4×C173C2411yescharacteristic
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—21no—
181C22×C173—21no—
182C22×C173—21no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173AGL(1,ℤ/8)12yescomplemented
187C22×C173C22×D411yescharacteristic
188C2×D173—22no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—41no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—84no—
198C2×D173—84no—
199C4×C173—21no—
200C4×C173—21no—
201C4×C173—22no—
202C4×C173—21no—
203C4×C173—21no—
204C4×C173—41no—
205C4×C173—41no—
206C4×C173C22×D411yescharacteristic
207C4×C173H(5,𝔽2)11yescharacteristic
208Dic173—41no—
209C2×C173—22no—
210C2×C173—41no—
211C2×C173—81no—
212C2×C173D4211yescharacteristic
213C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C173#57711yescharacteristic
215D173—42no—
216D173—81no—
217D173—81no—
218C173#326511yescharacteristic, complemented, Sylow
219#3265—1731nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1731no—
221C22×SD16—1731no—
222#548D17311yescharacteristic, complemented
223D42—1731no—
224#496—1731no—
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×#41C2×D17311yescharacteristic, complemented
236C22×Q8C2×D17311yescharacteristic, complemented
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×SD16—1731no—
248C2×D8—1731no—
249C2×D8—1731no—
250C2×D8—1731no—
251C2×M16—1731no—
252C22×C8—1731no—
253#127—1731no—
254#124—21no—
255#124C2×D17311yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D17311yescharacteristic, complemented
258#121—21no—
259#121—3461no—
260#121—1731no—
261#121—1731no—
262#121C2×D17311yescharacteristic, complemented
263#120—3461no—
264#120—3461no—
265C4×Q8C2×D17311yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1731no—
268C4×D4C2×D17311yescharacteristic
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#97—1731no—
280C24—3461no—
281C24—3461no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D17311yescharacteristic
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—1731no—
302C2×D4—6921no—
303C2×D4—6921no—
304C2×D4—6924no—
305C2×D4C22⋊D17311yescharacteristic
306C2×D4C22×D17311yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—3461no—
310C22×C4C22×D17311yescharacteristic
311C22×C4C22×D17311yescharacteristic
312SD16—3462no—
313SD16—3462no—
314SD16—3462no—
315SD16—3462no—
316SD16—1734no—
317SD16—6921no—
318D8—3462no—
319D8—3462no—
320D8—6921no—
321M16—3462no—
322C2×C8—3462no—
323C2×C8—1731no—
324C2×C8—1731no—
325C2×C8—1731no—
326C2×C8—1731no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1731no—
330C4⋊C4C22⋊D17311yescharacteristic
331C4⋊C4C22×D17311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—3461no—
337C22⋊C4—3461no—
338C22⋊C4—3461no—
339C22⋊C4C22⋊D17311yescharacteristic
340C22⋊C4C22⋊D17311yescharacteristic
341C42—21no—
342C42C22×D17311yescharacteristic
343C23—21no—
344C23—3461no—
345C23—3461no—
346C23—3461no—
347C23—13841no—
348C23—13841no—
349C23—6922no—
350C23—6922no—
351C23—6922no—
352C23—6922no—
353C23C2×C22⋊D17311yescharacteristic
354C23C2×C22⋊D17311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D17312yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—3461no—
366D4—3461no—
367D4—3461no—
368D4—3461no—
369D4—3461no—
370D4—13841no—
371D4—1732no—
372D4—6922no—
373D4—6922no—
374D4—6922no—
375D4—6922no—
376D4—6922no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—3461no—
387C2×C4—6922no—
388C2×C4C2×C22⋊D17311yescharacteristic
389C2×C4C2×C22⋊D17311yescharacteristic
390C2×C4C2×C22⋊D17311yescharacteristic
391C2×C4C2×C22⋊D17311yescharacteristic
392C2×C4C23×D17311yescharacteristic
393C8—3461no—
394C8—3461no—
395C8—3461no—
396C8—1732no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—3462no—
406C22—13844no—
407C22—13844no—
408C22—6921no—
409C22—6921no—
410C22—6921no—
411C22—6921no—
412C22—6921no—
413C22—6921no—
414C22—6921no—
415C22—6921no—
416C22#1914970412yescomplemented
417C22C22×C22⋊D17311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—6921no—
426C4C2×D4×D17311yescharacteristic
427C4#1914972411yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—13841no—
432C2—13841no—
433C2—6922no—
434C2D4×C22⋊D17311yescharacteristic
435C2C2×#1914970411yescharacteristic
436C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#548×C173C2
socleC22×C173C22×D4
hypercenter#548D173
hypocenterC173#3265
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19149704C2semidirect
2. C22×#14583813C2semidirect
3. D4×D692C2semidirect
4. #26011881C2semidirect
5. #26011739C2semidirect
6. #26011725C2semidirect
7. #26011703C2semidirect
8. #26011691C2semidirect
9. #26011641C2semidirect
10. #26011636C2semidirect
11. #26011630C2semidirect
12. #26011624C2semidirect
13. #26011589C2semidirect
14. #26011563C2semidirect
15. #548×C173C2semidirect
23. C2×#14583813C22semidirect
24. C2×#14583813C22semidirect
25. C2×#14583813C22semidirect
26. C2×#14583813C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583811C22semidirect
30. C2×#14583810C22semidirect
31. C22×#13063436C22semidirect
39. #19149612C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
43. #19149600C22semidirect
45. #19149599C22semidirect
46. #19149599C22semidirect
47. C22⋊#13063436C22semidirect
48. C22⋊#13063436C22semidirect
49. #19149590C22semidirect
50. C2×#41×C173C22semidirect
51. C22×Q8×C173C22semidirect
53. #124×C173C22semidirect
55. C22⋊Q8×C173C22semidirect
57. #121×C173C22semidirect
58. C4×Q8×C173C22semidirect
83. #14583813D4semidirect
150. #13063436C2×D4semidirect
158. Q8×C173C2×D4semidirect
186. C22×C173AGL(1,ℤ/8)semidirect
218. C173#3265semidirect
222. #548D173semidirect
235. C2×#41C2×D173semidirect
236. C22×Q8C2×D173semidirect
255. #124C2×D173semidirect
257. C22⋊Q8C2×D173semidirect
262. #121C2×D173semidirect
265. C4×Q8C2×D173semidirect
359. Q8D4×D173semidirect
416. C22#19149704semidirect

Maximal subgroups

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

subgroupindex
C2×#191497042
C22×#145838132
D4×D6922
#260118812
#260117392
#260117252
#260117032
#260116912
#260116412
#260116362
#260116302
#260116242
#260115892
#260115632
#548×C1732
#3265173