Finite Group Databasebeta

← #36365039

Subgroups — #36365039

Order 22144.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149705C211yesmaximal, characteristic, complemented
2C22×#14583811C211yesmaximal, characteristic, complemented
3#26012137C211yesmaximal, characteristic, complemented
4#26011878C211yesmaximal, characteristic, complemented
5#26011742C211yesmaximal, characteristic, complemented
6#26011718C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011683C211yesmaximal, characteristic, complemented
9#26011642C211yesmaximal, characteristic, complemented
10#26011634C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011624C211yesmaximal, characteristic, complemented
13#26011601C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15D42×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C22×D692C2211yescharacteristic, complemented
18#19149705—24no—
19C2×#14583814C2211yescharacteristic, complemented
20C2×#14583814C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583811—24no—
23C2×#14583811C2211yescharacteristic
24C2×#14583811C2211yescharacteristic
25C2×#14583811C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583810C2211yescharacteristic, complemented
30C22×#13063436C2211yescharacteristic, complemented
31#19149667C2211yescharacteristic, complemented
32#19149664—21no—
33#19149658C2211yescharacteristic, complemented
34#19149653C2211yescharacteristic, complemented
35#19149652—21no—
36#19149647—21no—
37C4×D692C2211yescharacteristic, complemented
38C22⋊C4×D173—21no—
39#19149610C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149602C2211yescharacteristic, complemented
42#19149600C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic, complemented
44#19149600C2211yescharacteristic, complemented
45#19149599C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic, complemented
48C22⋊#13063436C2211yescharacteristic, complemented
49#19149590C2211yescharacteristic, complemented
50C22×D4×C173—21no—
51C22×D4×C173C2211yescharacteristic
52C22×D4×C173C2211yescharacteristic, complemented
53#127×C173C2211yescharacteristic, complemented
54#121×C173—21no—
55#121×C173C2211yescharacteristic
56#121×C173C2211yescharacteristic, complemented
57#120×C173—21no—
58#120×C173—21no—
59C4×D4×C173C2211yescharacteristic
60C4×D4×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692—22no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74#14583814—22no—
75#14583814—22no—
76#14583814—41no—
77#14583811—22no—
78#14583811—22no—
79#14583811—22no—
80#14583811—22no—
81#14583811—41no—
82#14583811D414yescomplemented
83#14583810—22no—
84C2×#13063436—22no—
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C2×#13063436C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C24×C173—21no—
97C24×C173—21no—
98C2×D4×C173—22no—
99C2×D4×C173—22no—
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173—21no—
105C2×D4×C173—21no—
106C2×D4×C173—41no—
107C2×D4×C173—41no—
108C2×D4×C173—44no—
109C2×D4×C173D411yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116C4⋊C4×C173D411yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22⋊C4×C173D411yescharacteristic
122C22⋊C4×C173D411yescharacteristic
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—81no—
129Dic346—21no—
130Dic346—21no—
131D692—21no—
132D692—21no—
133D692—21no—
134D692—42no—
135D692C2×D412yescomplemented
136C22⋊D173—42no—
137C22⋊D173—81no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C4×D173—42no—
141C4×D173—81no—
142#13063436—21no—
143#13063436—21no—
144#13063436—21no—
145#13063436C2×D412yescomplemented
146C23×C173—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173—42no—
150C23×C173—42no—
151C23×C173—42no—
152C23×C173—42no—
153C23×C173—81no—
154C23×C173—81no—
155C23×C173C2×D411yescharacteristic
156C23×C173C2×D411yescharacteristic
157D4×C173—22no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—42no—
164D4×C173—42no—
165D4×C173—42no—
166D4×C173—42no—
167D4×C173—42no—
168D4×C173—81no—
169D4×C173C2×D412yes—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—42no—
176C2×C4×C173C2×D411yescharacteristic
177C2×C4×C173C2×D411yescharacteristic
178C2×C4×C173C2×D411yescharacteristic
179C2×C4×C173C2×D411yescharacteristic
180C2×C4×C173C2411yescharacteristic
181C22×C173—21no—
182C22×C173—21no—
183C22×C173—21no—
184C22×C173—22no—
185C22×C173—21no—
186C22×C173—21no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—41no—
191C22×C173—41no—
192C22×C173—41no—
193C22×C173—41no—
194C22×C173—41no—
195C22×C173—84no—
196C22×C173—84no—
197C22×C173AGL(1,ℤ/8)12yescomplemented
198C22×C173C22×D411yescharacteristic
199C2×D173—22no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—41no—
204C2×D173—41no—
205C2×D173—84no—
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—21no—
209C4×C173—21no—
210C4×C173—41no—
211C4×C173C22×D411yescharacteristic
212C4×C173H(5,𝔽2)11yescharacteristic
213Dic173—41no—
214Dic173—41no—
215C2×C173—22no—
216C2×C173—41no—
217C2×C173—42no—
218C2×C173—81no—
219C2×C173—81no—
220C2×C173D4211yescharacteristic
221C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C173#57611yescharacteristic
223D173—42no—
224D173—81no—
225C173#326411yescharacteristic, complemented, Sylow
226#3264—1731nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1731no—
228C22×D8—1731no—
229#546—1731no—
230D42D17311yescharacteristic, complemented
231#496—1731no—
232#469—1731no—
233#465—1731no—
234#463—1731no—
235#459—1731no—
236#450—1731no—
237#449—1731no—
238#447—1731no—
239#447—1731no—
240#442—1731no—
241#435—1731no—
242C2×#41—1731no—
243C22×D4—21no—
244C22×D4—1731no—
245C22×D4C2×D17311yescharacteristic
246C22×D4C2×D17311yescharacteristic, complemented
247AGL(1,ℤ/8)—3464no—
248C2×SD16—1731no—
249C2×SD16—1731no—
250C2×SD16—1731no—
251C2×D8—3464no—
252C2×D8—1731no—
253C2×D8—1731no—
254C2×D8—1731no—
255C2×D8—1731no—
256C2×D8—1731no—
257C2×D8—1731no—
258C2×M16—1731no—
259C22×C8—1731no—
260#127C2×D17311yescharacteristic, complemented
261#124—1731no—
262#123—3461no—
263C22⋊Q8—1731no—
264#121—21no—
265#121—3461no—
266#121—1731no—
267#121C2×D17311yescharacteristic
268#121C2×D17311yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—3461no—
272C4×D4—1731no—
273C4×D4C2×D17311yescharacteristic
274C4×D4C2×D17311yescharacteristic, complemented
275C2×C22⋊C4—3461no—
276#106—1731no—
277C4⋊C8—1731no—
278#103—1731no—
279#102—1731no—
280#102—1731no—
281#102—1731no—
282#102—1731no—
283#102—1731no—
284C22⋊C8—1731no—
285C22⋊C8—1731no—
286#97—1731no—
287C24—21no—
288C24—21no—
289C24—3461no—
290#41—3462no—
291#41—6921no—
292C2×Q8—1731no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—3462no—
305C2×D4—3461no—
306C2×D4—3461no—
307C2×D4—3461no—
308C2×D4—1731no—
309C2×D4—1731no—
310C2×D4—6921no—
311C2×D4C22⋊D17311yescharacteristic
312C2×D4C22×D17311yescharacteristic
313C2×D4C22×D17311yescharacteristic
314C2×D4C22×D17311yescharacteristic
315C22×C4—21no—
316C22×C4—3461no—
317C22×C4—3461no—
318C22×C4C22×D17311yescharacteristic
319C22×C4C22×D17311yescharacteristic
320SD16—3462no—
321SD16—3462no—
322SD16—6921no—
323D8—3462no—
324D8—3462no—
325D8—3462no—
326D8—3462no—
327D8—1734no—
328D8—6921no—
329M16—3462no—
330C2×C8—3462no—
331C2×C8—1731no—
332C2×C8—1731no—
333C2×C8—1731no—
334C2×C8—1731no—
335C4⋊C4—3461no—
336C4⋊C4—1731no—
337C4⋊C4C22⋊D17311yescharacteristic
338C4⋊C4C22×D17311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—3461no—
343C22⋊C4—3461no—
344C22⋊C4—3461no—
345C22⋊C4—3461no—
346C22⋊C4—3461no—
347C22⋊C4C22⋊D17311yescharacteristic
348C22⋊C4C22⋊D17311yescharacteristic
349C42C22×D17311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—3461no—
360C23—3461no—
361C23—13841no—
362C23—6922no—
363C23—6922no—
364C23C2×C22⋊D17311yescharacteristic
365C23C2×C22⋊D17311yescharacteristic
366Q8—3461no—
367Q8—3461no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—3461no—
381D4—3461no—
382D4—3461no—
383D4—13841no—
384D4—1732no—
385D4—6922no—
386D4—6922no—
387D4D4×D17312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—13841no—
397C2×C4—6922no—
398C2×C4C2×C22⋊D17311yescharacteristic
399C2×C4C2×C22⋊D17311yescharacteristic
400C2×C4C2×C22⋊D17311yescharacteristic
401C2×C4C2×C22⋊D17311yescharacteristic
402C2×C4C23×D17311yescharacteristic
403C8—3461no—
404C8—3461no—
405C8—3461no—
406C8—1732no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—3462no—
424C22—13844no—
425C22—6921no—
426C22—6921no—
427C22—6921no—
428C22—6921no—
429C22—6921no—
430C22#1914970512yescomplemented
431C22C22×C22⋊D17311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—6921no—
438C4—6921no—
439C4C2×D4×D17311yescharacteristic
440C4#1914972411yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—13841no—
447C2—6922no—
448C2D4×C22⋊D17311yescharacteristic
449C2C2×#1914970511yescharacteristic
450C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingD42×C173C2
socleC22×C173C22×D4
hypercenterD42D173
hypocenterC173#3264
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C22×#14583811C2semidirect
3. #26012137C2semidirect
4. #26011878C2semidirect
5. #26011742C2semidirect
6. #26011718C2semidirect
7. #26011706C2semidirect
8. #26011683C2semidirect
9. #26011642C2semidirect
10. #26011634C2semidirect
11. #26011625C2semidirect
12. #26011624C2semidirect
13. #26011601C2semidirect
14. #26011563C2semidirect
15. D42×C173C2semidirect
16. C2×#14583826C22semidirect
17. C22×D692C22semidirect
19. C2×#14583814C22semidirect
20. C2×#14583814C22semidirect
21. C2×#14583814C22semidirect
25. C2×#14583811C22semidirect
26. C2×#14583811C22semidirect
27. C2×#14583811C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583810C22semidirect
30. C22×#13063436C22semidirect
31. #19149667C22semidirect
33. #19149658C22semidirect
34. #19149653C22semidirect
37. C4×D692C22semidirect
39. #19149610C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149602C22semidirect
42. #19149600C22semidirect
43. #19149600C22semidirect
44. #19149600C22semidirect
45. #19149599C22semidirect
46. #19149599C22semidirect
47. C22⋊#13063436C22semidirect
48. C22⋊#13063436C22semidirect
49. #19149590C22semidirect
52. C22×D4×C173C22semidirect
53. #127×C173C22semidirect
56. #121×C173C22semidirect
60. C4×D4×C173C22semidirect
82. #14583811D4semidirect
135. D692C2×D4semidirect
145. #13063436C2×D4semidirect
197. C22×C173AGL(1,ℤ/8)semidirect
225. C173#3264semidirect
230. D42D173semidirect
246. C22×D4C2×D173semidirect
260. #127C2×D173semidirect
268. #121C2×D173semidirect
274. C4×D4C2×D173semidirect
430. C22#19149705semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C22×#145838112
#260121372
#260118782
#260117422
#260117182
#260117062
#260116832
#260116422
#260116342
#260116252
#260116242
#260116012
#260115632
D42×C1732
#3264173