Finite Group Databasebeta

← #49131734

Subgroups — #49131734

Order 36224.

564 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×#22583639C211yesmaximal, characteristic, complemented
2C2×#22583634C211yesmaximal, characteristic, complemented
3C2×#22583629C211yesmaximal, characteristic, complemented
4C2×SD16×D283C211yesmaximal, characteristic, complemented
5#31827035C211yesmaximal, characteristic, complemented
6#31827028C211yesmaximal, characteristic, complemented
7#31826859C211yesmaximal, characteristic, complemented
8#31826856C211yesmaximal, characteristic, complemented
9#31826853C211yesmaximal, characteristic, complemented
10#31826848C211yesmaximal, characteristic, complemented
11#31826847C211yesmaximal, characteristic, complemented
12#31826837C211yesmaximal, characteristic, complemented
13#31826836C211yesmaximal, characteristic, complemented
14#31826422C211yesmaximal, characteristic, complemented
15#496×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic
20#22583639—24no—
21#22583634—24no—
22#22583629—24no—
23C2×Dic1132C2211yescharacteristic, complemented
24C2×Dic1132C2211yescharacteristic, complemented
25C2×#17985382C2211yescharacteristic, complemented
26C2×#17985381C2211yescharacteristic
27C2×#17985381C2211yescharacteristic
28C2×#17985380C2211yescharacteristic
29C2×#17985380C2211yescharacteristic
30C2×#17985380C2211yescharacteristic, complemented
31C2×#17985380C2211yescharacteristic, complemented
32C2×#17985380C2211yescharacteristic, complemented
33C2×#17985379C2211yescharacteristic
34SD16×D283—24no—
35C2×#17985377C2211yescharacteristic
36C2×#17985375C2211yescharacteristic
37C2×C8×D283C2211yescharacteristic
38#22583610C2211yescharacteristic, complemented
39#22583608C2211yescharacteristic, complemented
40#22583606C2211yescharacteristic
41#22583598C2211yescharacteristic
42#22583597C2211yescharacteristic
43#22583597C2211yescharacteristic, complemented
44#22583592—22no—
45#22583587—22no—
46#22583582—22no—
47#22583578—22no—
48#22583567C2211yescharacteristic
49#22583520C2211yescharacteristic, complemented
50C4×C283⋊C8C2211yescharacteristic, complemented
51C8×Dic283C2211yescharacteristic, complemented
52C2×SD16×C283C2211yescharacteristic
53C2×SD16×C283C2211yescharacteristic, complemented
54C2×D8×C283C2211yescharacteristic
55C2×D8×C283C2211yescharacteristic, complemented
56#127×C283C2211yescharacteristic
57#124×C283C2211yescharacteristic, complemented
58#97×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60#17985391—22no—
61#17985391—22no—
62#17985391—22no—
63#17985391—22no—
64C2×Dic566—21no—
65C2×Dic566C2311yescharacteristic
66C2×Dic566C2311yescharacteristic
67C2×Dic566C2311yescharacteristic
68Q8×D283—22no—
69Q8×D283—22no—
70C2×D1132C2311yescharacteristic
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—22no—
75D4×D283—22no—
76D4×D283—22no—
77C22×Dic283—21no—
78C22×Dic283—21no—
79C2×C4×D283—21no—
80C2×C4×D283C2311yescharacteristic
81Dic1132—22no—
82Dic1132—22no—
83#17985382—22no—
84#17985381—22no—
85#17985381—22no—
86#17985380—22no—
87#17985380—22no—
88#17985380—22no—
89#17985380—22no—
90#17985380—22no—
91#17985379—22no—
92#17985377—22no—
93#17985375D414yescomplemented
94C2×C283⋊C8C2311yescharacteristic
95C2×C283⋊C8C2311yescharacteristic
96C8×D283D414yescomplemented
97C4⋊Dic283—22no—
98Dic283⋊C4—22no—
99Dic283⋊C4D412yes—
100C22⋊Dic283—22no—
101C22⋊Dic283—22no—
102(C2×D283)⋊C4—22no—
103(C2×D283)⋊C4D412yes—
104C4×Dic283C2311yescharacteristic
105C2×Q8×C283C2311yescharacteristic
106C2×D4×C283—22no—
107C2×D4×C283C2311yescharacteristic
108C2×D4×C283C2311yescharacteristic
109C2×D4×C283C2311yescharacteristic
110SD16×C283—22no—
111SD16×C283—22no—
112D8×C283—22no—
113D8×C283—22no—
114C2×C8×C283C2311yescharacteristic
115C2×C8×C283C2311yescharacteristic
116C22⋊C4×C283—22no—
117C42×C283C2311yescharacteristic
118C22×D283—21no—
119C22×D283—42no—
120C22×D283—82no—
121C22×D283C2×D411yescharacteristic
122Dic566—21no—
123Dic566—21no—
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—21no—
128Dic566—42no—
129D1132—21no—
130D1132—21no—
131C22⋊D283—42no—
132C22⋊D283—42no—
133C22⋊D283—42no—
134C22⋊D283—82no—
135C2×Dic283—21no—
136C2×Dic283—21no—
137C2×Dic283—21no—
138C2×Dic283—42no—
139C2×Dic283—42no—
140C2×Dic283C2×D412yes—
141C2×Dic283C2×D411yescharacteristic
142C4×D283—42no—
143C4×D283C2×D412yes—
144C4×D283C2×D412yes—
145C283⋊C8C2×D412yes—
146C283⋊C8C2×D412yes—
147C23×C283—21no—
148C23×C283—21no—
149C23×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152D4×C283—21no—
153D4×C283—21no—
154D4×C283—21no—
155D4×C283—21no—
156D4×C283—21no—
157D4×C283—21no—
158D4×C283—82no—
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—41no—
167C22×C283—42no—
168C22×C283—42no—
169C22×C283—42no—
170C22×C283C22×D411yescharacteristic
171C2×D283—22no—
172C2×D283—41no—
173C2×D283—42no—
174C2×D283—84no—
175C2×D283#12712yes—
176C4×C283—22no—
177C4×C283—41no—
178C4×C283C22×D411yescharacteristic
179C4×C283C22×D411yescharacteristic
180Dic283—22no—
181Dic283—41no—
182Dic283—41no—
183Dic283—41no—
184Dic283#12712yes—
185C2×C283—81no—
186C2×C283—81no—
187C2×C283—81no—
188C2×C283C2×#12711yescharacteristic
189C2×C283#57712yes—
190D283—22no—
191D283—81no—
192C283#315411yescharacteristic, complemented, Sylow
193#3154—2831nomaximal, Sylow
194C2×#137—2831no—
195C2×AGL(1,ℤ/8)—2831no—
196C2×#135—2831no—
197C22×SD16—2831no—
198#536—2831no—
199#535—2831no—
200#497—2831no—
201#497—2831no—
202#496—2831no—
203#496D28311yescharacteristic, complemented
204#495—2831no—
205#495—2831no—
206#492—2831no—
207#492—2831no—
208#405—2831no—
209C2×#41—2831no—
210C2×#41—2831no—
211C22×Q8—2831no—
212C22×D4—2831no—
213#137—5664no—
214AGL(1,ℤ/8)—5664no—
215#135—5664no—
216C2×Q16—2831no—
217C2×Q16—2831no—
218C2×Q16—2831no—
219C2×SD16—2831no—
220C2×SD16—2831no—
221C2×SD16—2831no—
222C2×SD16—2831no—
223C2×SD16—2831no—
224C2×SD16—2831no—
225C2×SD16—2831no—
226C2×SD16—2831no—
227C2×SD16—5664no—
228C2×SD16C2×D28311yescharacteristic
229C2×SD16C2×D28311yescharacteristic, complemented
230C2×D8—2831no—
231C2×D8C2×D28311yescharacteristic
232C2×D8C2×D28311yescharacteristic, complemented
233C2×M16—2831no—
234C22×C8—2831no—
235C4⋊Q8—2831no—
236C4⋊Q8—2831no—
237#127—2831no—
238#127C2×D28311yescharacteristic
239#124—2831no—
240#124—2831no—
241#124—2831no—
242#124C2×D28311yescharacteristic, complemented
243#123—5662no—
244C22⋊Q8—5662no—
245#121—5662no—
246#120—5662no—
247#117—2831no—
248#97—2831no—
249#97C2×D28311yescharacteristic, complemented
250C4×C8—2831no—
251C4×C8—2831no—
252C24—5661no—
253#41—5662no—
254#41—5662no—
255#41—5662no—
256#41—5662no—
257C2×Q8—2831no—
258C2×Q8—2831no—
259C2×Q8—2831no—
260C2×Q8—5661no—
261C2×Q8—5662no—
262C2×Q8—5662no—
263C2×Q8C22×D28311yescharacteristic
264C2×D4—22no—
265C2×D4—2831no—
266C2×D4—5662no—
267C2×D4—5662no—
268C2×D4—5661no—
269C2×D4—5662no—
270C2×D4—5661no—
271C2×D4—5661no—
272C2×D4C22×D28311yescharacteristic
273C2×D4C22×D28311yescharacteristic
274C2×D4C22×D28311yescharacteristic
275C22×C4—2831no—
276C22×C4—5661no—
277C22×C4—5661no—
278C22×C4—5661no—
279Q16—5662no—
280Q16—5662no—
281Q16—5662no—
282SD16—22no—
283SD16—22no—
284SD16—5662no—
285SD16—5662no—
286SD16—5662no—
287SD16—5662no—
288SD16—5662no—
289SD16—5662no—
290SD16—5662no—
291SD16—5662no—
292D8—22no—
293D8—22no—
294D8—5662no—
295M16—2834no—
296C2×C8—2831no—
297C2×C8—2831no—
298C2×C8—2834no—
299C2×C8C22×D28311yescharacteristic
300C2×C8C22×D28311yescharacteristic
301C4⋊C4—2832no—
302C4⋊C4—5662no—
303C4⋊C4—5662no—
304C22⋊C4—22no—
305C22⋊C4—2832no—
306C22⋊C4—5662no—
307C22⋊C4—5662no—
308C22⋊C4—5662no—
309C42—2831no—
310C42C22×D28311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—2831no—
315C23—5661no—
316C23—11322no—
317C23—22642no—
318Q8—21no—
319Q8—21no—
320Q8—5661no—
321Q8—5661no—
322Q8—5661no—
323Q8—5661no—
324Q8—5661no—
325Q8—5661no—
326Q8—11322no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—82no—
334D4—5661no—
335D4—5661no—
336D4—11322no—
337D4—11322no—
338D4—11322no—
339D4—22642no—
340C2×C4—21no—
341C2×C4—2831no—
342C2×C4—2832no—
343C2×C4—2832no—
344C2×C4—2832no—
345C2×C4—5661no—
346C2×C4—5661no—
347C2×C4—5661no—
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—41no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—2832no—
364C22—5662no—
365C22—11321no—
366C22—11322no—
367C22—22644no—
368C22C2×D4×D28311yescharacteristic
369C4—22no—
370C4—41no—
371C4—2832no—
372C4—5662no—
373C4—11321no—
374C4—11321no—
375C4—11321no—
376C4C2×D4×D28311yescharacteristic
377C4C2×D4×D28311yescharacteristic
378C2—81no—
379C2—81no—
380C2—81no—
381C2—5662no—
382C2—22641no—
383C2#127×D28311yescharacteristic
384C2#3182727512yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3154
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22583639C2semidirect
2. C2×#22583634C2semidirect
3. C2×#22583629C2semidirect
4. C2×SD16×D283C2semidirect
5. #31827035C2semidirect
6. #31827028C2semidirect
7. #31826859C2semidirect
8. #31826856C2semidirect
9. #31826853C2semidirect
10. #31826848C2semidirect
11. #31826847C2semidirect
12. #31826837C2semidirect
13. #31826836C2semidirect
14. #31826422C2semidirect
15. #496×C283C2semidirect
23. C2×Dic1132C22semidirect
24. C2×Dic1132C22semidirect
25. C2×#17985382C22semidirect
30. C2×#17985380C22semidirect
31. C2×#17985380C22semidirect
32. C2×#17985380C22semidirect
38. #22583610C22semidirect
39. #22583608C22semidirect
43. #22583597C22semidirect
49. #22583520C22semidirect
50. C4×C283⋊C8C22semidirect
51. C8×Dic283C22semidirect
53. C2×SD16×C283C22semidirect
55. C2×D8×C283C22semidirect
57. #124×C283C22semidirect
58. #97×C283C22semidirect
93. #17985375D4semidirect
96. C8×D283D4semidirect
192. C283#3154semidirect
203. #496D283semidirect
229. C2×SD16C2×D283semidirect
232. C2×D8C2×D283semidirect
242. #124C2×D283semidirect
249. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836392
C2×#225836342
C2×#225836292
C2×SD16×D2832
#318270352
#318270282
#318268592
#318268562
#318268532
#318268482
#318268472
#318268372
#318268362
#318264222
#496×C2832
#3154283