Finite Group Databasebeta

← #49131732

Subgroups — #49131732

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×#22583642C211yesmaximal, characteristic, complemented
2C2×#22583634C211yesmaximal, characteristic, complemented
3C2×#22583631C211yesmaximal, characteristic, complemented
4C2×SD16×D283C211yesmaximal, characteristic, complemented
5#31827035C211yesmaximal, characteristic, complemented
6#31827033C211yesmaximal, characteristic, complemented
7#31826860C211yesmaximal, characteristic, complemented
8#31826858C211yesmaximal, characteristic, complemented
9#31826855C211yesmaximal, characteristic, complemented
10#31826853C211yesmaximal, characteristic, complemented
11#31826846C211yesmaximal, characteristic, complemented
12#31826838C211yesmaximal, characteristic, complemented
13#31826836C211yesmaximal, characteristic, complemented
14#31826419C211yesmaximal, characteristic, complemented
15#495×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic
20#22583642—24no—
21#22583634—24no—
22#22583631—24no—
23C2×#17985382C2211yescharacteristic, complemented
24C2×#17985382C2211yescharacteristic, complemented
25C2×#17985381C2211yescharacteristic
26C2×#17985381C2211yescharacteristic
27C2×#17985381C2211yescharacteristic, complemented
28C2×#17985381C2211yescharacteristic, complemented
29C2×#17985380C2211yescharacteristic
30C2×#17985380C2211yescharacteristic
31C2×#17985379C2211yescharacteristic
32C2×#17985379C2211yescharacteristic, complemented
33SD16×D283—24no—
34C2×#17985377C2211yescharacteristic
35C2×#17985377C2211yescharacteristic, complemented
36C2×#17985375C2211yescharacteristic
37C2×C8×D283C2211yescharacteristic
38#22583610C2211yescharacteristic, complemented
39#22583608C2211yescharacteristic, complemented
40#22583607C2211yescharacteristic
41#22583606C2211yescharacteristic
42#22583599C2211yescharacteristic
43#22583597C2211yescharacteristic, complemented
44#22583592—22no—
45#22583587—22no—
46#22583583—22no—
47#22583580—22no—
48#22583567C2211yescharacteristic
49#22583521C2211yescharacteristic, complemented
50#22583520C2211yescharacteristic, complemented
51C8×Dic283C2211yescharacteristic, 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×D1132—22no—
71C2×D1132C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—22no—
76D4×D283—22no—
77D4×D283—22no—
78C22×Dic283—21no—
79C2×C4×D283—21no—
80C2×C4×D283—21no—
81C2×C4×D283C2311yescharacteristic
82#17985382—22no—
83#17985382—22no—
84#17985381—22no—
85#17985381—22no—
86#17985381—22no—
87#17985381—22no—
88#17985380—22no—
89#17985380—22no—
90#17985379—22no—
91#17985379—22no—
92#17985377—22no—
93#17985377—22no—
94#17985375D414yescomplemented
95C2×C283⋊C8C2311yescharacteristic
96C2×C283⋊C8C2311yescharacteristic
97C8×D283D414yescomplemented
98C4⋊Dic283—22no—
99Dic283⋊C4—22no—
100Dic283⋊C4D412yes—
101C22⋊Dic283—22no—
102(C2×D283)⋊C4—22no—
103(C2×D283)⋊C4D412yes—
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—42no—
121C22×D283—82no—
122C22×D283C2×D411yescharacteristic
123Dic566—21no—
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—42no—
128D1132—21no—
129D1132—21no—
130D1132—21no—
131D1132—21no—
132D1132—42no—
133D1132—82no—
134C22⋊D283—42no—
135C22⋊D283—42no—
136C22⋊D283—82no—
137C2×Dic283—21no—
138C2×Dic283—21no—
139C2×Dic283—42no—
140C2×Dic283C2×D412yes—
141C2×Dic283C2×D411yescharacteristic
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#57712yes—
189D283—22no—
190D283—81no—
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#496—2831no—
204#495—2831no—
205#495D28311yescharacteristic, complemented
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×Q16C2×D28311yescharacteristic, complemented
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×D8—2831no—
232C2×D8C2×D28311yescharacteristic
233C2×M16—2831no—
234C22×C8—2831no—
235C4⋊Q8—2831no—
236C4⋊Q8—2831no—
237#127—2831no—
238#127—2831no—
239#124—2831no—
240#124—2831no—
241#124C2×D28311yescharacteristic
242#124C2×D28311yescharacteristic, complemented
243#123—5662no—
244C22⋊Q8—5662no—
245#121—5662no—
246#120—5662no—
247#117—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—5661no—
270C2×D4—5662no—
271C2×D4—5661no—
272C2×D4—5662no—
273C2×D4C22×D28311yescharacteristic
274C2×D4C22×D28311yescharacteristic
275C22×C4—2831no—
276C22×C4—5661no—
277C22×C4—5661no—
278C22×C4—5661no—
279Q16—22no—
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—5662no—
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—22no—
306C22⋊C4—2832no—
307C22⋊C4—5662no—
308C22⋊C4—5662no—
309C42—2831no—
310C42C22×D28311yescharacteristic
311C23—21no—
312C23—21no—
313C23—2831no—
314C23—5661no—
315C23—5661no—
316C23—11322no—
317C23—22642no—
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—11322no—
338D4—22642no—
339D4—22642no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2831no—
343C2×C4—2832no—
344C2×C4—2832no—
345C2×C4—2832no—
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—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#3182727212yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#495×C283C2
socleC22×C283C22×D4
hypercenter#495D283
hypocenterC283#3154
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×#22583634C2semidirect
3. C2×#22583631C2semidirect
4. C2×SD16×D283C2semidirect
5. #31827035C2semidirect
6. #31827033C2semidirect
7. #31826860C2semidirect
8. #31826858C2semidirect
9. #31826855C2semidirect
10. #31826853C2semidirect
11. #31826846C2semidirect
12. #31826838C2semidirect
13. #31826836C2semidirect
14. #31826419C2semidirect
15. #495×C283C2semidirect
23. C2×#17985382C22semidirect
24. C2×#17985382C22semidirect
27. C2×#17985381C22semidirect
28. C2×#17985381C22semidirect
32. C2×#17985379C22semidirect
35. C2×#17985377C22semidirect
38. #22583610C22semidirect
39. #22583608C22semidirect
43. #22583597C22semidirect
49. #22583521C22semidirect
50. #22583520C22semidirect
51. C8×Dic283C22semidirect
52. C2×Q16×C283C22semidirect
54. C2×SD16×C283C22semidirect
57. #124×C283C22semidirect
58. C4×C8×C283C22semidirect
94. #17985375D4semidirect
97. C8×D283D4semidirect
192. C283#3154semidirect
205. #495D283semidirect
218. C2×Q16C2×D283semidirect
229. C2×SD16C2×D283semidirect
242. #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×#225836342
C2×#225836312
C2×SD16×D2832
#318270352
#318270332
#318268602
#318268582
#318268552
#318268532
#318268462
#318268382
#318268362
#318264192
#495×C2832
#3154283