Finite Group Databasebeta

← #49131722

Subgroups — #49131722

Order 36224.

572 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583642C211yesmaximal, characteristic, complemented
2C2×#22583639C211yesmaximal, characteristic, complemented
3C2×#22583637C211yesmaximal, characteristic, complemented
4C2×#22583634C211yesmaximal, characteristic, complemented
5#31827014C211yesmaximal, characteristic, complemented
6#124×D283C211yesmaximal, characteristic, complemented
7#31826858C211yesmaximal, characteristic, complemented
8#31826856C211yesmaximal, characteristic, complemented
9#31826853C211yesmaximal, characteristic, complemented
10#31826851C211yesmaximal, characteristic, complemented
11#31826849C211yesmaximal, characteristic, complemented
12#31826847C211yesmaximal, characteristic, complemented
13#31826847C211yesmaximal, characteristic, complemented
14C4×#17985375C211yesmaximal, characteristic, complemented
15#495×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic
20#22583642—24no—
21#22583639—24no—
22#22583637—24no—
23#22583634—24no—
24C2×Dic1132C2211yescharacteristic, complemented
25C2×#17985382C2211yescharacteristic, complemented
26C2×#17985382C2211yescharacteristic, complemented
27C2×#17985381C2211yescharacteristic
28C2×#17985381C2211yescharacteristic, complemented
29C2×#17985380C2211yescharacteristic
30C2×#17985380C2211yescharacteristic, complemented
31C2×#17985379C2211yescharacteristic
32C2×#17985379C2211yescharacteristic, complemented
33C2×D2264C2211yescharacteristic
34C2×#17985377C2211yescharacteristic
35C2×#17985377C2211yescharacteristic
36C2×#17985375C2211yescharacteristic
37C2×#17985375C2211yescharacteristic
38#22583608C2211yescharacteristic, complemented
39#22583606C2211yescharacteristic, complemented
40#22583599C2211yescharacteristic
41#22583598C2211yescharacteristic
42#22583598C2211yescharacteristic, complemented
43#22583597C2211yescharacteristic
44#22583596—22no—
45#22583583—22no—
46(C22⋊D283)⋊C4—22no—
47C22⋊C4×D283—22no—
48C42×D283C2211yescharacteristic
49#22583520C2211yescharacteristic, complemented
50#22583520C2211yescharacteristic, complemented
51C4×C283⋊C8C2211yescharacteristic, complemented
52C2×Q16×C283C2211yescharacteristic, complemented
53C2×SD16×C283C2211yescharacteristic
54C2×SD16×C283C2211yescharacteristic, complemented
55C2×D8×C283C2211yescharacteristic
56#124×C283C2211yescharacteristic
57#124×C283C2211yescharacteristic, complemented
58C4×C8×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60#17985393—22no—
61#17985393—22no—
62#17985391—22no—
63#17985391—22no—
64C2×Dic566—21no—
65C2×Dic566C2311yescharacteristic
66C2×Dic566C2311yescharacteristic
67Q8×D283—22no—
68Q8×D283—22no—
69C2×D1132—21no—
70C2×D1132C2311yescharacteristic
71C2×D1132C2311yescharacteristic
72C2×C22⋊D283—21no—
73C2×C22⋊D283—22no—
74C2×C22⋊D283—21no—
75D4×D283—22no—
76D4×D283—22no—
77C22×Dic283—21no—
78C2×C4×D283—21no—
79C2×C4×D283—21no—
80C2×C4×D283D412yes—
81C2×C4×D283C2311yescharacteristic
82Dic1132—22no—
83#17985382—22no—
84#17985382—22no—
85#17985381—22no—
86#17985381—22no—
87#17985380—22no—
88#17985380—22no—
89#17985379—22no—
90#17985379—22no—
91D2264—22no—
92#17985377—22no—
93#17985377—22no—
94#17985375D414yescomplemented
95#17985375D414yescomplemented
96C2×C283⋊C8C2311yescharacteristic
97C2×C283⋊C8C2311yescharacteristic
98Dic283⋊C4—22no—
99C22⋊Dic283—22no—
100(C2×D283)⋊C4—22no—
101(C2×D283)⋊C4—22no—
102(C2×D283)⋊C4—22no—
103C4×Dic283D412yes—
104C4×Dic283C2311yescharacteristic
105C2×Q8×C283C2311yescharacteristic
106C2×Q8×C283C2311yescharacteristic
107C2×D4×C283C2311yescharacteristic
108C2×D4×C283C2311yescharacteristic
109Q16×C283—22no—
110SD16×C283—22no—
111SD16×C283—22no—
112D8×C283—22no—
113C2×C8×C283C2311yescharacteristic
114C2×C8×C283C2311yescharacteristic
115C22⋊C4×C283—22no—
116C22⋊C4×C283—22no—
117C42×C283C2311yescharacteristic
118C22×D283—21no—
119C22×D283—21no—
120C22×D283—44no—
121C22×D283—42no—
122C22×D283C2×D411yescharacteristic
123Dic566—21no—
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—42no—
128D1132—21no—
129D1132—21no—
130D1132—21no—
131D1132—21no—
132D1132—42no—
133C22⋊D283—44no—
134C22⋊D283—42no—
135C22⋊D283—42no—
136C2×Dic283—21no—
137C2×Dic283—21no—
138C2×Dic283—42no—
139C2×Dic283C2×D412yes—
140C2×Dic283C2×D411yescharacteristic
141C4×D283—42no—
142C4×D283—42no—
143C4×D283—42no—
144C4×D283C2×D412yes—
145C4×D283C2×D412yes—
146C283⋊C8C2×D412yes—
147C283⋊C8C2×D412yes—
148C23×C283—21no—
149C23×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152Q8×C283—21no—
153Q8×C283—21no—
154D4×C283—21no—
155D4×C283—21no—
156D4×C283—21no—
157D4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283C2×D412yes—
161C2×C4×C283C2411yescharacteristic
162C8×C283C2×D412yes—
163C8×C283C2×D412yes—
164C22×C283—41no—
165C22×C283—41no—
166C22×C283—42no—
167C22×C283—42no—
168C22×C283C22×D411yescharacteristic
169C2×D283—22no—
170C2×D283—41no—
171C2×D283—41no—
172C2×D283—42no—
173C2×D283—42no—
174C2×D283—84no—
175C2×D283#12712yes—
176C4×C283—22no—
177C4×C283—41no—
178C4×C283—41no—
179C4×C283C22×D411yescharacteristic
180C4×C283C22×D411yescharacteristic
181Dic283—22no—
182Dic283—41no—
183Dic283—41no—
184Dic283#12712yes—
185C2×C283—81no—
186C2×C283—81no—
187C2×C283C2×#12711yescharacteristic
188C2×C283#57512yes—
189D283—22no—
190D283—81no—
191D283—81no—
192C283#315311yescharacteristic, complemented, Sylow
193#3153—2831nomaximal, Sylow
194C2×#137—2831no—
195C2×#137—2831no—
196C2×AGL(1,ℤ/8)—2831no—
197C2×AGL(1,ℤ/8)—2831no—
198#532—2831no—
199C2×#124—2831no—
200#497—2831no—
201#497—2831no—
202#496—2831no—
203#496—2831no—
204#495—2831no—
205#495—2831no—
206#495—2831no—
207#495D28311yescharacteristic, complemented
208C4×M16—2831no—
209C2×#41—2831no—
210C2×#41—2831no—
211C22×Q8—2831no—
212C22×D4—2831no—
213#137—5664no—
214#137—5664no—
215AGL(1,ℤ/8)—5664no—
216AGL(1,ℤ/8)—5664no—
217C2×Q16—2831no—
218C2×Q16—2831no—
219C2×Q16—2831no—
220C2×Q16C2×D28311yescharacteristic, complemented
221C2×SD16—2831no—
222C2×SD16—2831no—
223C2×SD16—2831no—
224C2×SD16—2831no—
225C2×SD16—2831no—
226C2×SD16—2831no—
227C2×SD16C2×D28311yescharacteristic
228C2×SD16C2×D28311yescharacteristic, complemented
229C2×D8—2831no—
230C2×D8—2831no—
231C2×D8—2831no—
232C2×D8C2×D28311yescharacteristic
233C2×M16—2831no—
234C2×M16—2831no—
235C4⋊Q8—2831no—
236#127—2831no—
237#124—2831no—
238#124—2831no—
239#124—2831no—
240#124—2831no—
241#124—5662no—
242#124C2×D28311yescharacteristic
243#124C2×D28311yescharacteristic, complemented
244#121—5662no—
245C4×D4—5662no—
246C2×C22⋊C4—5662no—
247C2×C42—2831no—
248#97—2831no—
249#97—2831no—
250C4×C8—2831no—
251C4×C8C2×D28311yescharacteristic, complemented
252C24—5661no—
253#41—5662no—
254#41—5662no—
255#41—5662no—
256#41—5662no—
257C2×Q8—2831no—
258C2×Q8—2831no—
259C2×Q8—5661no—
260C2×Q8—5662no—
261C2×Q8—5662no—
262C2×Q8C22×D28311yescharacteristic
263C2×Q8C22×D28311yescharacteristic
264C2×D4—2831no—
265C2×D4—2831no—
266C2×D4—5661no—
267C2×D4—5662no—
268C2×D4—5662no—
269C2×D4—5662no—
270C2×D4—5661no—
271C2×D4—5661no—
272C2×D4C22×D28311yescharacteristic
273C2×D4C22×D28311yescharacteristic
274C22×C4—2831no—
275C22×C4—2832no—
276C22×C4—5661no—
277C22×C4—5661no—
278C22×C4—5661no—
279Q16—22no—
280Q16—5662no—
281Q16—5662no—
282Q16—5662no—
283SD16—22no—
284SD16—22no—
285SD16—5662no—
286SD16—5662no—
287SD16—5662no—
288SD16—5662no—
289SD16—5662no—
290SD16—5662no—
291D8—22no—
292D8—5662no—
293D8—5662no—
294D8—5662no—
295M16—2834no—
296M16—2834no—
297C2×C8—2831no—
298C2×C8—2831no—
299C2×C8C22×D28311yescharacteristic
300C2×C8C22×D28311yescharacteristic
301C4⋊C4—5662no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—5662no—
305C22⋊C4—5662no—
306C22⋊C4—5662no—
307C22⋊C4—5662no—
308C42—2832no—
309C42—2831no—
310C42C22×D28311yescharacteristic
311C23—21no—
312C23—21no—
313C23—2831no—
314C23—5661no—
315C23—5661no—
316C23—11322no—
317C23—11324no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—5661no—
323Q8—5661no—
324Q8—5661no—
325Q8—5661no—
326Q8—11322no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—5661no—
332D4—5661no—
333D4—5661no—
334D4—5661no—
335D4—11322no—
336D4—11322no—
337D4—11324no—
338D4—11322no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—2831no—
342C2×C4—2832no—
343C2×C4—2832no—
344C2×C4—2832no—
345C2×C4—5661no—
346C2×C4—5661no—
347C2×C4—11322no—
348C2×C4—11322no—
349C2×C4—11322no—
350C2×C4—11322no—
351C2×C4D4×D28312yes—
352C2×C4C23×D28311yescharacteristic
353C8—2832no—
354C8—2832no—
355C8D4×D28312yes—
356C8D4×D28312yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—2832no—
362C22—5662no—
363C22—11321no—
364C22—11321no—
365C22—11322no—
366C22—11322no—
367C22—22644no—
368C22C2×D4×D28311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—2832no—
373C4—5662no—
374C4—11321no—
375C4—11321no—
376C4C2×D4×D28311yescharacteristic
377C4C2×D4×D28311yescharacteristic
378C2—81no—
379C2—81no—
380C2—5662no—
381C2—22641no—
382C2—22641no—
383C2#127×D28311yescharacteristic
384C2#3182726212yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#495×C283C2
socleC22×C283C22×D4
hypercenter#495D283
hypocenterC283#3153
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22583642C2semidirect
2. C2×#22583639C2semidirect
3. C2×#22583637C2semidirect
4. C2×#22583634C2semidirect
5. #31827014C2semidirect
6. #124×D283C2semidirect
7. #31826858C2semidirect
8. #31826856C2semidirect
9. #31826853C2semidirect
10. #31826851C2semidirect
11. #31826849C2semidirect
12. #31826847C2semidirect
13. #31826847C2semidirect
14. C4×#17985375C2semidirect
15. #495×C283C2semidirect
24. C2×Dic1132C22semidirect
25. C2×#17985382C22semidirect
26. C2×#17985382C22semidirect
28. C2×#17985381C22semidirect
30. C2×#17985380C22semidirect
32. C2×#17985379C22semidirect
38. #22583608C22semidirect
39. #22583606C22semidirect
42. #22583598C22semidirect
49. #22583520C22semidirect
50. #22583520C22semidirect
51. C4×C283⋊C8C22semidirect
52. C2×Q16×C283C22semidirect
54. C2×SD16×C283C22semidirect
57. #124×C283C22semidirect
58. C4×C8×C283C22semidirect
94. #17985375D4semidirect
95. #17985375D4semidirect
192. C283#3153semidirect
207. #495D283semidirect
220. C2×Q16C2×D283semidirect
228. C2×SD16C2×D283semidirect
243. #124C2×D283semidirect
251. C4×C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836422
C2×#225836392
C2×#225836372
C2×#225836342
#318270142
#124×D2832
#318268582
#318268562
#318268532
#318268512
#318268492
#318268472
C4×#179853752
#495×C2832
#3153283