Finite Group Databasebeta

← #49131745

Subgroups — #49131745

Order 36224.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583637C211yesmaximal, characteristic, complemented
2C2×#22583634C211yesmaximal, characteristic, complemented
3C2×#22583631C211yesmaximal, characteristic, complemented
4C2×D8×D283C211yesmaximal, characteristic, complemented
5#31827033C211yesmaximal, characteristic, complemented
6#31827028C211yesmaximal, characteristic, complemented
7#31826854C211yesmaximal, characteristic, complemented
8#31826853C211yesmaximal, characteristic, complemented
9#31826851C211yesmaximal, characteristic, complemented
10#31826847C211yesmaximal, characteristic, complemented
11#31826846C211yesmaximal, characteristic, complemented
12#31826841C211yesmaximal, characteristic, complemented
13#31826840C211yesmaximal, characteristic, complemented
14#31826422C211yesmaximal, characteristic, complemented
15#496×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×D4×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic
20#22583637—24no—
21#22583634—24no—
22#22583631—24no—
23C2×#17985382C2211yescharacteristic, complemented
24C2×#17985381C2211yescharacteristic, complemented
25C2×#17985381C2211yescharacteristic, complemented
26C2×#17985380C2211yescharacteristic, complemented
27C2×#17985379C2211yescharacteristic, complemented
28C2×D2264C2211yescharacteristic
29C2×D2264C2211yescharacteristic, complemented
30C2×#17985377C2211yescharacteristic
31C2×#17985377C2211yescharacteristic
32C2×#17985377C2211yescharacteristic, complemented
33C2×#17985377C2211yescharacteristic, complemented
34C2×#17985377C2211yescharacteristic, complemented
35D8×D283—24no—
36C2×#17985375C2211yescharacteristic
37C2×C8×D283C2211yescharacteristic
38#22583607C2211yescharacteristic, complemented
39#22583606C2211yescharacteristic, complemented
40#22583606C2211yescharacteristic, complemented
41#22583599C2211yescharacteristic, complemented
42#22583598C2211yescharacteristic, complemented
43#22583597C2211yescharacteristic, complemented
44#22583583—22no—
45#22583582—22no—
46#22583580—22no—
47#22583578—22no—
48#22583567C2211yescharacteristic
49#22583520C2211yescharacteristic, complemented
50C4×C283⋊C8C2211yescharacteristic, complemented
51C8×Dic283C2211yescharacteristic, complemented
52C2×SD16×C283C2211yescharacteristic, complemented
53C2×SD16×C283C2211yescharacteristic, complemented
54C2×D8×C283C2211yescharacteristic
55C2×D8×C283C2211yescharacteristic, complemented
56#127×C283C2211yescharacteristic, complemented
57#124×C283C2211yescharacteristic, complemented
58#97×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60C23×D283—21no—
61#17985393—22no—
62#17985393—22no—
63#17985391—22no—
64#17985391—22no—
65C2×Dic566C2311yescharacteristic
66C2×D1132—21no—
67C2×D1132—22no—
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×D1132C2311yescharacteristic
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—22no—
75C2×C22⋊D283—22no—
76D4×D283—22no—
77D4×D283—22no—
78D4×D283—22no—
79D4×D283—22no—
80C22×Dic283—21no—
81C2×C4×D283—21no—
82C2×C4×D283C2311yescharacteristic
83#17985382—22no—
84#17985381—22no—
85#17985381—22no—
86#17985380—22no—
87#17985379—22no—
88D2264—22no—
89D2264—22no—
90#17985377—22no—
91#17985377—22no—
92#17985377—22no—
93#17985377—22no—
94#17985377—22no—
95#17985375D414yescomplemented
96C2×C283⋊C8C2311yescharacteristic
97C2×C283⋊C8C2311yescharacteristic
98C8×D283D414yescomplemented
99Dic283⋊C4D412yes—
100C22⋊Dic283—22no—
101(C2×D283)⋊C4—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—21no—
120C22×D283—21no—
121C22×D283—42no—
122C22×D283—42no—
123C22×D283—82no—
124C22×D283—82no—
125C22×D283C2×D411yescharacteristic
126Dic566—21no—
127Dic566—21no—
128D1132—21no—
129D1132—21no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—21no—
134D1132—42no—
135D1132—82no—
136C22⋊D283—42no—
137C22⋊D283—42no—
138C22⋊D283—42no—
139C22⋊D283—82no—
140C22⋊D283—82no—
141C2×Dic283—21no—
142C2×Dic283—42no—
143C2×Dic283C2×D412yes—
144C2×Dic283C2×D411yescharacteristic
145C4×D283—42no—
146C4×D283C2×D412yes—
147C4×D283C2×D412yes—
148C283⋊C8C2×D412yes—
149C283⋊C8C2×D412yes—
150C23×C283—21no—
151C23×C283—21no—
152C23×C283—21no—
153Q8×C283—21no—
154Q8×C283—21no—
155D4×C283—21no—
156D4×C283—21no—
157D4×C283—21no—
158D4×C283—21no—
159D4×C283—21no—
160D4×C283—21no—
161D4×C283—82no—
162C2×C4×C283—21no—
163C2×C4×C283C2×D412yes—
164C2×C4×C283C2411yescharacteristic
165C8×C283C2×D412yes—
166C8×C283C2×D412yes—
167C22×C283—41no—
168C22×C283—41no—
169C22×C283—41no—
170C22×C283—42no—
171C22×C283—42no—
172C22×C283—42no—
173C22×C283C22×D411yescharacteristic
174C2×D283—22no—
175C2×D283—41no—
176C2×D283—41no—
177C2×D283—41no—
178C2×D283—42no—
179C2×D283—42no—
180C2×D283—42no—
181C2×D283—84no—
182C2×D283—84no—
183C2×D283#12712yes—
184C4×C283—22no—
185C4×C283—41no—
186C4×C283C22×D411yescharacteristic
187C4×C283C22×D411yescharacteristic
188Dic283—22no—
189Dic283—41no—
190Dic283#12712yes—
191C2×C283—81no—
192C2×C283—81no—
193C2×C283—81no—
194C2×C283C2×#12711yescharacteristic
195C2×C283#57612yes—
196D283—22no—
197D283—81no—
198D283—81no—
199D283—81no—
200C283#315511yescharacteristic, complemented, Sylow
201#3155—2831nomaximal, Sylow
202C2×AGL(1,ℤ/8)—2831no—
203C2×AGL(1,ℤ/8)—2831no—
204C2×#135—2831no—
205C22×D8—2831no—
206#535—2831no—
207#535—2831no—
208#496—2831no—
209#496—2831no—
210#496—2831no—
211#496D28311yescharacteristic, complemented
212#495—2831no—
213#495—2831no—
214#493—2831no—
215#493—2831no—
216#405—2831no—
217C2×#41—2831no—
218C2×#41—2831no—
219C22×D4—2831no—
220C22×D4—2831no—
221AGL(1,ℤ/8)—5664no—
222AGL(1,ℤ/8)—5664no—
223#135—5664no—
224C2×Q16—2831no—
225C2×SD16—2831no—
226C2×SD16—2831no—
227C2×SD16—2831no—
228C2×SD16—2831no—
229C2×SD16C2×D28311yescharacteristic, complemented
230C2×SD16C2×D28311yescharacteristic, complemented
231C2×D8—2831no—
232C2×D8—2831no—
233C2×D8—2831no—
234C2×D8—2831no—
235C2×D8—2831no—
236C2×D8—2831no—
237C2×D8—2831no—
238C2×D8—5664no—
239C2×D8C2×D28311yescharacteristic
240C2×D8C2×D28311yescharacteristic, complemented
241C2×M16—2831no—
242C22×C8—2831no—
243#127—2831no—
244#127—2831no—
245#127—2831no—
246#127C2×D28311yescharacteristic, complemented
247#124—2831no—
248#124—2831no—
249#124—2831no—
250#124C2×D28311yescharacteristic, complemented
251#121—5662no—
252#121—5662no—
253#120—5662no—
254#120—5662no—
255#117—2831no—
256#97—2831no—
257#97C2×D28311yescharacteristic, complemented
258C4×C8—2831no—
259C4×C8—2831no—
260C24—5661no—
261C24—5661no—
262#41—5662no—
263#41—5662no—
264#41—5662no—
265#41—5662no—
266C2×Q8—2831no—
267C2×Q8C22×D28311yescharacteristic
268C2×D4—22no—
269C2×D4—2831no—
270C2×D4—2831no—
271C2×D4—2831no—
272C2×D4—5661no—
273C2×D4—5662no—
274C2×D4—5662no—
275C2×D4—5662no—
276C2×D4—5662no—
277C2×D4—5661no—
278C2×D4—5661no—
279C2×D4—5662no—
280C2×D4—5662no—
281C2×D4—5661no—
282C2×D4—5662no—
283C2×D4C22×D28311yescharacteristic
284C2×D4C22×D28311yescharacteristic
285C2×D4C22×D28311yescharacteristic
286C22×C4—2831no—
287C22×C4—5661no—
288C22×C4—5661no—
289Q16—5662no—
290SD16—22no—
291SD16—22no—
292SD16—5662no—
293SD16—5662no—
294SD16—5662no—
295SD16—5662no—
296D8—22no—
297D8—22no—
298D8—5662no—
299D8—5662no—
300D8—5662no—
301D8—5662no—
302D8—5662no—
303D8—5662no—
304D8—5662no—
305M16—2834no—
306C2×C8—2831no—
307C2×C8—2831no—
308C2×C8—2834no—
309C2×C8C22×D28311yescharacteristic
310C2×C8C22×D28311yescharacteristic
311C4⋊C4—2832no—
312C22⋊C4—22no—
313C22⋊C4—2832no—
314C22⋊C4—5662no—
315C22⋊C4—5662no—
316C22⋊C4—5662no—
317C42—2831no—
318C42C22×D28311yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—2831no—
323C23—5661no—
324C23—5661no—
325C23—5661no—
326C23—11322no—
327C23—11322no—
328C23—22642no—
329C23—22642no—
330Q8—21no—
331Q8—21no—
332Q8—5661no—
333Q8—5661no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—5661no—
342D4—5661no—
343D4—5661no—
344D4—5661no—
345D4—5661no—
346D4—5661no—
347D4—11322no—
348D4—11322no—
349D4—11322no—
350D4—11322no—
351D4—22642no—
352D4—22642no—
353D4—22642no—
354C2×C4—21no—
355C2×C4—2831no—
356C2×C4—2832no—
357C2×C4—2832no—
358C2×C4—2832no—
359C2×C4—5661no—
360C2×C4—11322no—
361C2×C4—11322no—
362C2×C4D4×D28312yes—
363C2×C4C23×D28311yescharacteristic
364C8—2832no—
365C8—2832no—
366C8D4×D28312yes—
367C8D4×D28312yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—2832no—
375C22—5662no—
376C22—11321no—
377C22—11321no—
378C22—11321no—
379C22—11322no—
380C22—11322no—
381C22—11322no—
382C22—22644no—
383C22—22644no—
384C22C2×D4×D28311yescharacteristic
385C4—22no—
386C4—41no—
387C4—2832no—
388C4—5662no—
389C4—11321no—
390C4C2×D4×D28311yescharacteristic
391C4C2×D4×D28311yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—5662no—
396C2—22641no—
397C2—22641no—
398C2—22641no—
399C2#127×D28311yescharacteristic
400C2#3182726912yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3155
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×#22583637C2semidirect
2. C2×#22583634C2semidirect
3. C2×#22583631C2semidirect
4. C2×D8×D283C2semidirect
5. #31827033C2semidirect
6. #31827028C2semidirect
7. #31826854C2semidirect
8. #31826853C2semidirect
9. #31826851C2semidirect
10. #31826847C2semidirect
11. #31826846C2semidirect
12. #31826841C2semidirect
13. #31826840C2semidirect
14. #31826422C2semidirect
15. #496×C283C2semidirect
23. C2×#17985382C22semidirect
24. C2×#17985381C22semidirect
25. C2×#17985381C22semidirect
26. C2×#17985380C22semidirect
27. C2×#17985379C22semidirect
29. C2×D2264C22semidirect
32. C2×#17985377C22semidirect
33. C2×#17985377C22semidirect
34. C2×#17985377C22semidirect
38. #22583607C22semidirect
39. #22583606C22semidirect
40. #22583606C22semidirect
41. #22583599C22semidirect
42. #22583598C22semidirect
43. #22583597C22semidirect
49. #22583520C22semidirect
50. C4×C283⋊C8C22semidirect
51. C8×Dic283C22semidirect
52. C2×SD16×C283C22semidirect
53. C2×SD16×C283C22semidirect
55. C2×D8×C283C22semidirect
56. #127×C283C22semidirect
57. #124×C283C22semidirect
58. #97×C283C22semidirect
95. #17985375D4semidirect
98. C8×D283D4semidirect
200. C283#3155semidirect
211. #496D283semidirect
229. C2×SD16C2×D283semidirect
230. C2×SD16C2×D283semidirect
240. C2×D8C2×D283semidirect
246. #127C2×D283semidirect
250. #124C2×D283semidirect
257. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×#225836342
C2×#225836312
C2×D8×D2832
#318270332
#318270282
#318268542
#318268532
#318268512
#318268472
#318268462
#318268412
#318268402
#318264222
#496×C2832
#3155283