Finite Group Databasebeta

← #36365157

Subgroups — #36365157

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
2#26012140C211yesmaximal, characteristic, complemented
3D4×C22⋊D173C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011737C211yesmaximal, characteristic, complemented
6#26011734C211yesmaximal, characteristic, complemented
7#26011694C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011642C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13D8×Dic173C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic, complemented
17C22×C22⋊D173—21no—
18C2×D4×D173C2211yescharacteristic
19#19149703—24no—
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic
22C2×#14583814C2211yescharacteristic, complemented
23C2×#14583813C2211yescharacteristic, complemented
24C2×#14583811C2211yescharacteristic, complemented
25C2×#14583809C2211yescharacteristic, complemented
26#19149677C2211yescharacteristic, complemented
27#19149675C2211yescharacteristic
28#19149662—21no—
29#19149660—21no—
30#19149658C2211yescharacteristic, complemented
31#19149655C2211yescharacteristic, complemented
32#19149653C2211yescharacteristic
33#19149651—21no—
34#19149651—21no—
35#19149650C2211yescharacteristic
36#19149648—21no—
37#19149647—21no—
38D4×Dic173C2211yescharacteristic
39D4×Dic173C2211yescharacteristic, complemented
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583805—21no—
42#19149613C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50C22⋊#13063436C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C8×Dic173C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55AGL(1,ℤ/8)×C173—24no—
56C2×SD16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic, complemented
58C2×D8×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic, complemented
60C2×M16×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583825—22no—
63#14583825—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—44no—
73C2×C22⋊D173D411yescharacteristic
74D4×D173—22no—
75D4×D173—41no—
76C22×Dic173—21no—
77C22×Dic173—21no—
78C22×Dic173—21no—
79C2×C4×D173C2311yescharacteristic
80#14583815—22no—
81#14583814—22no—
82#14583814—41no—
83#14583813—41no—
84#14583811—22no—
85#14583809—22no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91#14583805—21no—
92#14583805—21no—
93#14583805—21no—
94#14583805D411yescharacteristic
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99C22⋊Dic173D411yescharacteristic
100#14583803—21no—
101#14583803—21no—
102#14583803D411yescharacteristic
103C4×Dic173C2311yescharacteristic
104C24×C173—21no—
105#41×C173—22no—
106#41×C173—41no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—22no—
109C2×D4×C173—21no—
110C2×D4×C173—21no—
111C2×D4×C173—41no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118SD16×C173—22no—
119D8×C173—22no—
120D8×C173D414yescomplemented
121M16×C173—22no—
122C2×C8×C173C2311yescharacteristic
123C2×C8×C173C2311yescharacteristic
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131D692—21no—
132D692—21no—
133C22⋊D173—22no—
134C22⋊D173—42no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—81no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—42no—
144C2×Dic173—42no—
145C2×Dic173—42no—
146C2×Dic173C2×D411yescharacteristic
147C2×Dic173C2×D411yescharacteristic
148C2×Dic173C2×D411yescharacteristic
149C4×D173—21no—
150C4×D173—21no—
151#13063436—21no—
152#13063436—21no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173—21no—
156C23×C173—42no—
157C23×C173—42no—
158C23×C173—81no—
159C23×C173C2×D411yescharacteristic
160Q8×C173—21no—
161Q8×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165D4×C173—21no—
166D4×C173—42no—
167D4×C173—42no—
168D4×C173C2×D412yes—
169D4×C173C2×D412yescomplemented
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—42no—
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2411yescharacteristic
176C8×C173—21no—
177C8×C173C2×D412yescomplemented
178C22×C173—21no—
179C22×C173—22no—
180C22×C173—22no—
181C22×C173—21no—
182C22×C173—21no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—84no—
191C22×C173C22×D411yescharacteristic
192C2×D173—21no—
193C2×D173—21no—
194C2×D173—21no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—41no—
198C2×D173—84no—
199C4×C173—21no—
200C4×C173—41no—
201C4×C173C22×D411yescharacteristic
202C4×C173H(5,𝔽2)11yescharacteristic
203Dic173—21no—
204Dic173—21no—
205Dic173—41no—
206Dic173—41no—
207Dic173—41no—
208Dic173AGL(1,ℤ/8)12yescomplemented
209C2×C173—42no—
210C2×C173—41no—
211C2×C173—42no—
212C2×C173—81no—
213C2×C173D4211yescharacteristic
214C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C173#57711yescharacteristic
216D173—41no—
217D173—81no—
218C173#327211yescharacteristic, complemented, Sylow
219#3272—1731nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1731no—
221C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
222#547—1731no—
223D42—1731no—
224#492—1731no—
225#468—1731no—
226#468—1731no—
227#461—1731no—
228#459—1731no—
229#450—1731no—
230#450—1731no—
231#449—1731no—
232#449—1731no—
233C4×D8—1731no—
234#436—1731no—
235C2×#41—1731no—
236C2×#41C2×D17311yescharacteristic, complemented
237C22×D4—3461no—
238C22×D4—1731no—
239C22×D4C2×D17311yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—3464no—
242C2×SD16—1731no—
243C2×SD16—1731no—
244C2×SD16—1731no—
245C2×SD16—1731no—
246C2×SD16C2×D17311yescharacteristic
247C2×SD16C2×D17311yescharacteristic, complemented
248C2×D8—1731no—
249C2×D8C2×D17311yescharacteristic
250C2×D8C2×D17311yescharacteristic, complemented
251C2×M16—1731no—
252C2×M16C2×D17311yescharacteristic, complemented
253C4⋊Q8—1731no—
254#127—1731no—
255#123—3461no—
256#123—3461no—
257C22⋊Q8—1731no—
258C22⋊Q8—1731no—
259#121—3461no—
260#121—3461no—
261#121—1731no—
262#121—1731no—
263#120—3461no—
264#120—3461no—
265C4×D4—1731no—
266C4×D4—1731no—
267C4×D4—1731no—
268C2×C4⋊C4—3461no—
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—
279C4×C8—1731no—
280C24—21no—
281C24—3461no—
282#41—22no—
283#41—41no—
284#41—3462no—
285#41—6921no—
286C2×Q8—1731no—
287C2×Q8C22×D17311yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
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—6924no—
303C2×D4C22×D17311yescharacteristic
304C2×D4C22×D17311yescharacteristic
305C2×D4C22×D17311yescharacteristic
306C22×C4—21no—
307C22×C4—3461no—
308C22×C4—3461no—
309C22×C4—3461no—
310C22×C4—1731no—
311C22×C4C22×D17311yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—3462no—
315SD16—3462no—
316SD16—6921no—
317SD16—6921no—
318D8—22no—
319D8—3462no—
320D8C22⋊D17314yescomplemented
321M16—22no—
322M16—3462no—
323C2×C8—1731no—
324C2×C8—1731no—
325C2×C8C22×D17311yescharacteristic
326C2×C8C22×D17311yescharacteristic
327C4⋊C4—3461no—
328C4⋊C4—3461no—
329C4⋊C4—3461no—
330C4⋊C4—3461no—
331C4⋊C4—1731no—
332C4⋊C4—1731no—
333C4⋊C4—1731no—
334C22⋊C4—3461no—
335C22⋊C4—3461no—
336C22⋊C4—3461no—
337C22⋊C4—3461no—
338C22⋊C4—3461no—
339C22⋊C4—3461no—
340C22⋊C4—1731no—
341C22⋊C4—1731no—
342C42—1731no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—3461no—
350C23—13841no—
351C23—1731no—
352C23—6922no—
353C23—6922no—
354C23D4×D17311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3461no—
358Q8—3461no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—3462no—
366D4—3461no—
367D4—3461no—
368D4—13841no—
369D4—6922no—
370D4—6922no—
371D4—6922no—
372D4—6922no—
373D4—6922no—
374D4C2×C22⋊D17312yes—
375D4C2×C22⋊D17312yescomplemented
376C2×C4—21no—
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—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—3461no—
395C8—3461no—
396C8C2×C22⋊D17312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—3461no—
411C22—3461no—
412C22—3461no—
413C22—13844no—
414C22—6921no—
415C22—6921no—
416C22—6921no—
417C22C2×D4×D17311yescharacteristic
418C4—21no—
419C4—41no—
420C4—3461no—
421C4—3461no—
422C4—1732no—
423C4—6921no—
424C4—6921no—
425C4—6921no—
426C4C22×C22⋊D17311yescharacteristic
427C4#1914972411yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—13841no—
433C2—6921no—
434C2D4×C22⋊D17311yescharacteristic
435C2AGL(1,ℤ/8)×D17311yescharacteristic
436C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. #26012140C2semidirect
3. D4×C22⋊D173C2semidirect
4. #26011862C2semidirect
5. #26011737C2semidirect
6. #26011734C2semidirect
7. #26011694C2semidirect
8. #26011684C2semidirect
9. #26011642C2semidirect
10. #26011640C2semidirect
11. #26011636C2semidirect
12. #26011633C2semidirect
13. D8×Dic173C2semidirect
14. #26011567C2semidirect
15. C2×AGL(1,ℤ/8)×C173C2semidirect
16. C2×#14583825C22semidirect
20. C2×#14583815C22semidirect
22. C2×#14583814C22semidirect
23. C2×#14583813C22semidirect
24. C2×#14583811C22semidirect
25. C2×#14583809C22semidirect
26. #19149677C22semidirect
30. #19149658C22semidirect
31. #19149655C22semidirect
39. D4×Dic173C22semidirect
42. #19149613C22semidirect
43. #19149608C22semidirect
44. #19149604C22semidirect
45. #19149603C22semidirect
46. #19149601C22semidirect
48. #19149600C22semidirect
49. #19149600C22semidirect
50. C22⋊#13063436C22semidirect
51. #19149591C22semidirect
52. C8×Dic173C22semidirect
53. C2×#41×C173C22semidirect
57. C2×SD16×C173C22semidirect
59. C2×D8×C173C22semidirect
60. C2×M16×C173C22semidirect
120. D8×C173D4semidirect
169. D4×C173C2×D4semidirect
177. C8×C173C2×D4semidirect
208. Dic173AGL(1,ℤ/8)semidirect
218. C173#3272semidirect
221. C2×AGL(1,ℤ/8)D173semidirect
236. C2×#41C2×D173semidirect
247. C2×SD16C2×D173semidirect
250. C2×D8C2×D173semidirect
252. C2×M16C2×D173semidirect
320. D8C22⋊D173semidirect
375. D4C2×C22⋊D173semidirect
396. C8C2×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
#260121402
D4×C22⋊D1732
#260118622
#260117372
#260117342
#260116942
#260116842
#260116422
#260116402
#260116362
#260116332
D8×Dic1732
#260115672
C2×AGL(1,ℤ/8)×C1732
#3272173