Finite Group Databasebeta

← #49131736

Subgroups — #49131736

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×#22583637C211yesmaximal, characteristic, complemented
3C2×#22583633C211yesmaximal, characteristic, complemented
4C2×SD16×D283C211yesmaximal, characteristic, complemented
5#31827040C211yesmaximal, characteristic, complemented
6#31827033C211yesmaximal, characteristic, complemented
7#31826858C211yesmaximal, characteristic, complemented
8#31826854C211yesmaximal, characteristic, complemented
9#31826851C211yesmaximal, characteristic, complemented
10#31826850C211yesmaximal, characteristic, complemented
11#31826847C211yesmaximal, characteristic, complemented
12#31826839C211yesmaximal, characteristic, complemented
13#31826836C211yesmaximal, characteristic, complemented
14#31826422C211yesmaximal, characteristic, complemented
15#497×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985393C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic
20#22583642—24no—
21#22583637—24no—
22#22583633—24no—
23C2×#17985382C2211yescharacteristic, complemented
24C2×#17985381C2211yescharacteristic
25C2×#17985381C2211yescharacteristic, complemented
26C2×#17985380C2211yescharacteristic
27C2×#17985379C2211yescharacteristic
28C2×#17985379C2211yescharacteristic
29C2×#17985379C2211yescharacteristic, complemented
30C2×#17985379C2211yescharacteristic, complemented
31C2×#17985379C2211yescharacteristic, complemented
32SD16×D283—24no—
33C2×D2264C2211yescharacteristic
34C2×D2264C2211yescharacteristic, complemented
35C2×#17985377C2211yescharacteristic
36C2×#17985375C2211yescharacteristic
37C2×C8×D283C2211yescharacteristic
38#22583608C2211yescharacteristic, complemented
39#22583607C2211yescharacteristic
40#22583606C2211yescharacteristic
41#22583599C2211yescharacteristic
42#22583599C2211yescharacteristic, complemented
43#22583598C2211yescharacteristic, complemented
44#22583595—22no—
45#22583588—22no—
46#22583583—22no—
47#22583580—22no—
48#22583567C2211yescharacteristic
49#22583520C2211yescharacteristic, complemented
50C4×C283⋊C8C2211yescharacteristic, complemented
51C8×Dic283C2211yescharacteristic, complemented
52C2×Q16×C283C2211yescharacteristic, complemented
53C2×Q16×C283C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic
56C4⋊Q8×C283C2211yescharacteristic, complemented
57#124×C283C2211yescharacteristic
58#97×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60#17985393—22no—
61#17985393—22no—
62#17985393—22no—
63#17985393—22no—
64C2×Dic566—21no—
65C2×Dic566C2311yescharacteristic
66Q8×D283—22no—
67Q8×D283—22no—
68C2×D1132—21no—
69C2×D1132—21no—
70C2×D1132—22no—
71C2×D1132C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×D1132C2311yescharacteristic
74C2×C22⋊D283—21no—
75C2×C22⋊D283—22no—
76D4×D283—22no—
77D4×D283—22no—
78C2×C4×D283—21no—
79C2×C4×D283—21no—
80C2×C4×D283—21no—
81C2×C4×D283C2311yescharacteristic
82#17985382—22no—
83#17985381—22no—
84#17985381—22no—
85#17985380—22no—
86#17985379—22no—
87#17985379—22no—
88#17985379—22no—
89#17985379—22no—
90#17985379—22no—
91D2264—22no—
92D2264—22no—
93#17985377—22no—
94#17985375D414yescomplemented
95C2×C283⋊C8C2311yescharacteristic
96C2×C283⋊C8C2311yescharacteristic
97C8×D283D414yescomplemented
98Dic283⋊C4—22no—
99Dic283⋊C4D412yes—
100(C2×D283)⋊C4—22no—
101(C2×D283)⋊C4—22no—
102(C2×D283)⋊C4—22no—
103(C2×D283)⋊C4D412yes—
104C4×Dic283C2311yescharacteristic
105C2×Q8×C283C2311yescharacteristic
106C2×Q8×C283C2311yescharacteristic
107C2×Q8×C283C2311yescharacteristic
108C2×D4×C283C2311yescharacteristic
109Q16×C283—22no—
110Q16×C283—22no—
111SD16×C283—22no—
112SD16×C283—22no—
113C2×C8×C283C2311yescharacteristic
114C2×C8×C283C2311yescharacteristic
115C4⋊C4×C283—22no—
116C22⋊C4×C283—22no—
117C42×C283C2311yescharacteristic
118C22×D283—21no—
119C22×D283—21no—
120C22×D283—21no—
121C22×D283—42no—
122C22×D283—82no—
123C22×D283C2×D411yescharacteristic
124Dic566—21no—
125Dic566—21no—
126Dic566—42no—
127D1132—21no—
128D1132—21no—
129D1132—21no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—42no—
134D1132—42no—
135D1132—82no—
136C22⋊D283—42no—
137C22⋊D283—82no—
138C2×Dic283—21no—
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—
149Q8×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152Q8×C283—21no—
153Q8×C283—21no—
154Q8×C283—21no—
155D4×C283—21no—
156D4×C283—21no—
157C2×C4×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—42no—
166C22×C283C22×D411yescharacteristic
167C2×D283—22no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—42no—
172C2×D283—42no—
173C2×D283—42no—
174C2×D283—84no—
175C2×D283#12712yes—
176C4×C283—22no—
177C4×C283—41no—
178C4×C283—41no—
179C4×C283—41no—
180C4×C283C22×D411yescharacteristic
181C4×C283C22×D411yescharacteristic
182Dic283—22no—
183Dic283—41no—
184Dic283#12712yes—
185C2×C283—81no—
186C2×C283C2×#12711yescharacteristic
187C2×C283#57712yes—
188D283—22no—
189D283—81no—
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#497D28311yescharacteristic, complemented
202#496—2831no—
203#496—2831no—
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×Q16C2×D28311yescharacteristic, complemented
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
230C2×D8—2831no—
231C2×D8—2831no—
232C2×D8—2831no—
233C2×M16—2831no—
234C22×C8—2831no—
235C4⋊Q8—2831no—
236C4⋊Q8C2×D28311yescharacteristic, complemented
237#127—2831no—
238#127—2831no—
239#124—2831no—
240#124—2831no—
241#124—2831no—
242#124C2×D28311yescharacteristic
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—5661no—
259C2×Q8—5662no—
260C2×Q8—5662no—
261C2×Q8C22×D28311yescharacteristic
262C2×Q8C22×D28311yescharacteristic
263C2×Q8C22×D28311yescharacteristic
264C2×D4—2831no—
265C2×D4—2831no—
266C2×D4—2831no—
267C2×D4—5661no—
268C2×D4—5661no—
269C2×D4—5662no—
270C2×D4—5662no—
271C2×D4—5661no—
272C2×D4—5662no—
273C2×D4—5662no—
274C2×D4C22×D28311yescharacteristic
275C22×C4—2831no—
276C22×C4—5661no—
277C22×C4—5661no—
278C22×C4—5661no—
279Q16—22no—
280Q16—22no—
281Q16—5662no—
282SD16—22no—
283SD16—22no—
284SD16—5662no—
285SD16—5662no—
286SD16—5662no—
287SD16—5662no—
288SD16—5662no—
289SD16—5662no—
290SD16—5662no—
291SD16—5662no—
292D8—5662no—
293D8—5662no—
294D8—5662no—
295M16—2834no—
296C2×C8—2831no—
297C2×C8—2831no—
298C2×C8—2834no—
299C2×C8C22×D28311yescharacteristic
300C2×C8C22×D28311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—2832no—
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—2831no—
313C23—5661no—
314C23—5661no—
315C23—5661no—
316C23—11322no—
317C23—22642no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—5661no—
325Q8—5661no—
326Q8—11322no—
327D4—21no—
328D4—21no—
329D4—5661no—
330D4—5661no—
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—21no—
343C2×C4—2831no—
344C2×C4—2832no—
345C2×C4—2832no—
346C2×C4—2832no—
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—42no—
359C22—2832no—
360C22—5662no—
361C22—11321no—
362C22—11321no—
363C22—11321no—
364C22—11322no—
365C22—11322no—
366C22—11322no—
367C22—22644no—
368C22C2×D4×D28311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2832no—
374C4—5662no—
375C4—11321no—
376C4C2×D4×D28311yescharacteristic
377C4C2×D4×D28311yescharacteristic
378C2—81no—
379C2—5662no—
380C2—22641no—
381C2—22641no—
382C2—22641no—
383C2#127×D28311yescharacteristic
384C2#3182727412yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#497×C283C2
socleC22×C283C22×D4
hypercenter#497D283
hypocenterC283#3154
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22583642C2semidirect
2. C2×#22583637C2semidirect
3. C2×#22583633C2semidirect
4. C2×SD16×D283C2semidirect
5. #31827040C2semidirect
6. #31827033C2semidirect
7. #31826858C2semidirect
8. #31826854C2semidirect
9. #31826851C2semidirect
10. #31826850C2semidirect
11. #31826847C2semidirect
12. #31826839C2semidirect
13. #31826836C2semidirect
14. #31826422C2semidirect
15. #497×C283C2semidirect
23. C2×#17985382C22semidirect
25. C2×#17985381C22semidirect
29. C2×#17985379C22semidirect
30. C2×#17985379C22semidirect
31. C2×#17985379C22semidirect
34. C2×D2264C22semidirect
38. #22583608C22semidirect
42. #22583599C22semidirect
43. #22583598C22semidirect
49. #22583520C22semidirect
50. C4×C283⋊C8C22semidirect
51. C8×Dic283C22semidirect
52. C2×Q16×C283C22semidirect
53. C2×Q16×C283C22semidirect
56. C4⋊Q8×C283C22semidirect
58. #97×C283C22semidirect
94. #17985375D4semidirect
97. C8×D283D4semidirect
192. C283#3154semidirect
201. #497D283semidirect
217. C2×Q16C2×D283semidirect
218. C2×Q16C2×D283semidirect
236. C4⋊Q8C2×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×#225836422
C2×#225836372
C2×#225836332
C2×SD16×D2832
#318270402
#318270332
#318268582
#318268542
#318268512
#318268502
#318268472
#318268392
#318268362
#318264222
#497×C2832
#3154283