Finite Group Databasebeta

← #36365056

Subgroups — #36365056

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×#19149705C211yesmaximal, characteristic, complemented
2C2×SD16×D173C211yesmaximal, characteristic, complemented
3#26012151C211yesmaximal, characteristic, complemented
4D4×C22⋊D173C211yesmaximal, characteristic, complemented
5#26011877C211yesmaximal, characteristic, complemented
6#26011739C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011638C211yesmaximal, characteristic, complemented
11#26011628C211yesmaximal, characteristic, complemented
12#26011626C211yesmaximal, characteristic, complemented
13#26011588C211yesmaximal, characteristic, complemented
14#26011565C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C2×Q8×D173C2211yescharacteristic, complemented
18C22×C22⋊D173—21no—
19C2×D4×D173C2211yescharacteristic
20#19149705—24no—
21C2×#14583815C2211yescharacteristic, complemented
22C2×#14583814C2211yescharacteristic
23C2×#14583814C2211yescharacteristic, complemented
24C2×#14583813C2211yescharacteristic, complemented
25SD16×D173—24no—
26C2×D1384C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583810C2211yescharacteristic, complemented
30C2×C8×D173C2211yescharacteristic, complemented
31#19149675C2211yescharacteristic
32#19149668C2211yescharacteristic, complemented
33#19149665—21no—
34#19149657—21no—
35#19149654C2211yescharacteristic, complemented
36#19149653C2211yescharacteristic
37#19149652—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40#19149648—21no—
41#19149647—21no—
42#19149644C2211yescharacteristic, complemented
43D4×Dic173C2211yescharacteristic
44#19149641—21no—
45C4×C22⋊D173C2211yescharacteristic
46#19149612C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149599C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149589C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C2×SD16×C173C2211yescharacteristic, complemented
57C22⋊Q8×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic, complemented
60C22⋊C8×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—41no—
68C2×D692—21no—
69C2×D692C2311yescharacteristic
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—
77C2×C22⋊D173—44no—
78C2×C22⋊D173D411yescharacteristic
79D4×D173—22no—
80D4×D173—41no—
81C22×Dic173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173C2311yescharacteristic
85#14583815—22no—
86#14583814—22no—
87#14583814D414yescomplemented
88#14583813—22no—
89D1384—41no—
90#14583811—22no—
91#14583811—22no—
92#14583810—22no—
93C2×#13063436C2311yescharacteristic
94C2×#13063436C2311yescharacteristic
95C8×D173—22no—
96C4⋊Dic173C2311yescharacteristic
97#14583805—21no—
98#14583805D411yescharacteristic
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173D411yescharacteristic
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C4×Dic173C2311yescharacteristic
109C24×C173—21no—
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—22no—
112C2×D4×C173—21no—
113C2×D4×C173—41no—
114C2×D4×C173C2311yescharacteristic
115C2×D4×C173C2311yescharacteristic
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118SD16×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C4⋊C4×C173—21no—
122C4⋊C4×C173C2311yescharacteristic
123C22⋊C4×C173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—81no—
129C22×D173C2×D411yescharacteristic
130Dic346—21no—
131Dic346—42no—
132Dic346C2×D412yescomplemented
133D692—21no—
134D692—21no—
135D692—21no—
136D692—21no—
137D692—42no—
138C22⋊D173—22no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C22⋊D173—42no—
143C22⋊D173—42no—
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—
154C4×D173—42no—
155#13063436—21no—
156#13063436C2×D412yescomplemented
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—42no—
160C23×C173—42no—
161C23×C173—81no—
162C23×C173C2×D411yescharacteristic
163Q8×C173—21no—
164Q8×C173—21no—
165D4×C173—21no—
166D4×C173—21no—
167D4×C173—21no—
168D4×C173—42no—
169D4×C173C2×D412yes—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2411yescharacteristic
176C8×C173—21no—
177C8×C173—21no—
178C22×C173—21no—
179C22×C173—22no—
180C22×C173—21no—
181C22×C173—21no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—84no—
188C22×C173C22×D411yescharacteristic
189C2×D173—21no—
190C2×D173—21no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—41no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—84no—
198C2×D173AGL(1,ℤ/8)12yescomplemented
199C4×C173—21no—
200C4×C173—41no—
201C4×C173—41no—
202C4×C173C22×D411yescharacteristic
203C4×C173H(5,𝔽2)11yescharacteristic
204Dic173—21no—
205Dic173—22no—
206Dic173—21no—
207Dic173—21no—
208Dic173—41no—
209C2×C173—42no—
210C2×C173—41no—
211C2×C173—81no—
212C2×C173D4211yescharacteristic
213C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C173#57711yescharacteristic
215D173—22no—
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#463—1731no—
228#460—1731no—
229#450D17311yescharacteristic, complemented
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×D4C2×D17311yescharacteristic
240AGL(1,ℤ/8)—3464no—
241C2×SD16—3464no—
242C2×SD16—1731no—
243C2×SD16—1731no—
244C2×SD16—1731no—
245C2×SD16—1731no—
246C2×SD16C2×D17311yescharacteristic
247C2×SD16C2×D17311yescharacteristic, complemented
248C2×D8—1731no—
249C2×D8—1731no—
250C2×D8—1731no—
251C2×M16—1731no—
252C22×C8—1731no—
253#127—1731no—
254#124—3461no—
255#124—1731no—
256C22⋊Q8—3461no—
257C22⋊Q8C2×D17311yescharacteristic, complemented
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#102C2×D17311yescharacteristic
276#102C2×D17311yescharacteristic, complemented
277C22⋊C8—1731no—
278C22⋊C8C2×D17311yescharacteristic, complemented
279#97—1731no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3461no—
293C2×D4—3462no—
294C2×D4—3461no—
295C2×D4—3461no—
296C2×D4—3461no—
297C2×D4—3461no—
298C2×D4—3461no—
299C2×D4—3461no—
300C2×D4—1731no—
301C2×D4—1731no—
302C2×D4—1731no—
303C2×D4—6921no—
304C2×D4—6924no—
305C2×D4C22×D17311yescharacteristic
306C2×D4C22×D17311yescharacteristic
307C22×C4—3461no—
308C22×C4—3461no—
309C22×C4—3461no—
310C22×C4—1731no—
311C22×C4C22×D17311yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—3462no—
315SD16—3462no—
316SD16—3462no—
317SD16—1734no—
318D8—3462no—
319D8—3462no—
320D8—6921no—
321M16—3462no—
322C2×C8—3462no—
323C2×C8—1731no—
324C2×C8—1731no—
325C2×C8C22×D17311yescharacteristic
326C2×C8C22×D17311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3461no—
329C4⋊C4—1731no—
330C4⋊C4—1731no—
331C4⋊C4C22×D17311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3461no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4—3461no—
337C22⋊C4—3461no—
338C22⋊C4—3461no—
339C22⋊C4—1731no—
340C22⋊C4—1731no—
341C42—3461no—
342C42—1731no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—3461no—
349C23—3461no—
350C23—13841no—
351C23—1731no—
352C23—6922no—
353C23—6922no—
354C23D4×D17311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3461no—
358Q8—1732no—
359Q8—6922no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—3462no—
365D4—3461no—
366D4—3461no—
367D4—3461no—
368D4—3461no—
369D4—13841no—
370D4—6922no—
371D4—6922no—
372D4—6922no—
373D4—6922no—
374D4—6922no—
375D4—6922no—
376D4D4×D17312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—1731no—
386C2×C4—1731no—
387C2×C4—1731no—
388C2×C4—6922no—
389C2×C4—6922no—
390C2×C4—6922no—
391C2×C4D4×D17311yescharacteristic
392C2×C4C23×D17311yescharacteristic
393C8—21no—
394C8—21no—
395C8—3461no—
396C8—1732no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—3461no—
408C22—3461no—
409C22—13844no—
410C22—1732no—
411C22—6921no—
412C22—6921no—
413C22—6921no—
414C22—6921no—
415C22—6921no—
416C22—6921no—
417C22C2×D4×D17311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—3461no—
422C4—3461no—
423C4—3461no—
424C4—3462no—
425C4—6921no—
426C4C2×D4×D17311yescharacteristic
427C4#1914972411yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—3462no—
432C2—13841no—
433C2—13841no—
434C2#2601212311yescharacteristic
435C2AGL(1,ℤ/8)×D17311yescharacteristic
436C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
hypercenter#450D173
hypocenterC173#3265
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C2×SD16×D173C2semidirect
3. #26012151C2semidirect
4. D4×C22⋊D173C2semidirect
5. #26011877C2semidirect
6. #26011739C2semidirect
7. #26011726C2semidirect
8. #26011708C2semidirect
9. #26011690C2semidirect
10. #26011638C2semidirect
11. #26011628C2semidirect
12. #26011626C2semidirect
13. #26011588C2semidirect
14. #26011565C2semidirect
15. #450×C173C2semidirect
16. C2×#14583826C22semidirect
17. C2×Q8×D173C22semidirect
21. C2×#14583815C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583813C22semidirect
26. C2×D1384C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583810C22semidirect
30. C2×C8×D173C22semidirect
32. #19149668C22semidirect
35. #19149654C22semidirect
42. #19149644C22semidirect
46. #19149612C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149601C22semidirect
51. #19149599C22semidirect
52. #19149591C22semidirect
53. #19149589C22semidirect
56. C2×SD16×C173C22semidirect
57. C22⋊Q8×C173C22semidirect
59. #102×C173C22semidirect
60. C22⋊C8×C173C22semidirect
87. #14583814D4semidirect
132. Dic346C2×D4semidirect
156. #13063436C2×D4semidirect
198. C2×D173AGL(1,ℤ/8)semidirect
218. C173#3265semidirect
229. #450D173semidirect
247. C2×SD16C2×D173semidirect
257. C22⋊Q8C2×D173semidirect
276. #102C2×D173semidirect
278. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C2×SD16×D1732
#260121512
D4×C22⋊D1732
#260118772
#260117392
#260117262
#260117082
#260116902
#260116382
#260116282
#260116262
#260115882
#260115652
#450×C1732
#3265173