Finite Group Databasebeta

← #49130620

Subgroups — #49130620

Order 36224.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583656C211yesmaximal, characteristic, complemented
2#31827265C211yesmaximal, characteristic, complemented
3#31827263C211yesmaximal, characteristic, complemented
4#31827034C211yesmaximal, characteristic, complemented
5#31826644C214yesmaximal, complemented
6#31826634C214yesmaximal, complemented
7#31826483C211yesmaximal, characteristic, complemented
8C2×#22583526C211yesmaximal, characteristic, complemented
9#421×C283C211yesmaximal, characteristic, complemented
10#22583656—22no—
11#22583656—22no—
12#22583656—22no—
13#22583656C2214yes—
14C2×#17985393—21no—
15C2×#17985392—21no—
16C2×#17985392C2211yescharacteristic
17C2×#17985392C2211yescharacteristic, complemented
18C22×D1132—21no—
19C22×D1132C2211yescharacteristic
20C2×D4×D283—22no—
21C2×D4×D283—21no—
22#22583643—21no—
23#22583643C2212yescomplemented
24#22583641—21no—
25#22583640C2212yescomplemented
26#22583638—21no—
27#22583638C2212yescomplemented
28#22583636—21no—
29#22583635C2212yescomplemented
30#22583632—22no—
31#22583630—22no—
32C2×#17985376C2211yescharacteristic, complemented
33#22583607C2212yescomplemented
34#22583598C2212yescomplemented
35#22583585—22no—
36#22583580—22no—
37#22583538C2214yescomplemented
38#22583526C2214yescomplemented
39C2×#41×C283C2211yescharacteristic
40C2×M16×C283C2211yescharacteristic, complemented
41#117×C283C2211yescharacteristic, complemented
42#104×C283C2214yescomplemented
43C23×D283—22no—
44C23×D283D412yes—
45#17985393—24no—
46#17985393—24no—
47#17985392—24no—
48#17985392—24no—
49#17985392—21no—
50#17985392—21no—
51#17985392—21no—
52#17985392—21no—
53#17985392—21no—
54#17985392—21no—
55#17985392D412yescomplemented
56#17985392C2312yes—
57C2×Dic566D411yescharacteristic, complemented
58C2×Dic566C2311yescharacteristic, complemented
59C2×D1132—22no—
60C2×D1132—24no—
61C2×D1132—24no—
62C2×D1132—24no—
63C2×D1132—21no—
64C2×D1132—21no—
65C2×D1132—21no—
66C2×D1132—21no—
67C2×D1132D412yes—
68C2×D1132D411yescharacteristic, complemented
69C2×D1132C2312yes—
70C2×D1132C2311yescharacteristic, complemented
71C2×C22⋊D283—22no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74D4×D283—28no—
75D4×D283—24no—
76D4×D283—28no—
77D4×D283—24no—
78C2×C4×D283—22no—
79C2×C4×D283—21no—
80C2×C4×D283—21no—
81Dic1132—22no—
82#17985382—22no—
83#17985381—22no—
84#17985381—22no—
85#17985380—22no—
86#17985379—22no—
87D2264—22no—
88#17985377—22no—
89#17985376—21no—
90#17985376C2312yescomplemented
91C2×C283⋊C8—21no—
92(C2×D283)⋊C4—24no—
93#41×C283—21no—
94#41×C283—21no—
95#41×C283D412yescomplemented
96#41×C283C2312yes—
97C2×Q8×C283D411yescharacteristic, complemented
98C2×D4×C283—21no—
99C2×D4×C283D411yescharacteristic, complemented
100C22×C4×C283—21no—
101C22×C4×C283C2311yescharacteristic
102M16×C283—21no—
103M16×C283C2312yescomplemented
104C2×C8×C283—21no—
105C4⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C42×C283C2312yescomplemented
108C22×D283—22no—
109C22×D283—21no—
110C22×D283—28no—
111C22×D283—22no—
112C22×D283—21no—
113C22×D283—82no—
114C22×D283—84no—
115C22×D283C2×D414yes—
116Dic566—21no—
117Dic566—21no—
118Dic566—21no—
119Dic566C2×D412yes—
120D1132—28no—
121D1132—22no—
122D1132—22no—
123D1132—21no—
124D1132—22no—
125D1132—22no—
126D1132—21no—
127D1132—21no—
128D1132—84no—
129D1132C2×D412yes—
130C22⋊D283—28no—
131C22⋊D283—22no—
132C22⋊D283—22no—
133C22⋊D283—41no—
134C22⋊D283—41no—
135C2×Dic283—21no—
136C2×Dic283—21no—
137C4×D283—28no—
138C4×D283—22no—
139C4×D283—22no—
140C4×D283—41no—
141C4×D283—41no—
142C283⋊C8—22no—
143C23×C283—21no—
144C23×C283C2×D411yescharacteristic
145Q8×C283—21no—
146Q8×C283C2×D412yes—
147D4×C283—22no—
148D4×C283—22no—
149D4×C283—21no—
150D4×C283C2×D412yes—
151C2×C4×C283—21no—
152C2×C4×C283—22no—
153C2×C4×C283—22no—
154C2×C4×C283—22no—
155C2×C4×C283C2×D411yescharacteristic
156C2×C4×C283C2×D411yescharacteristic
157C2×C4×C283C2×D411yescharacteristic
158C2×C4×C283C2×D411yescharacteristic
159C2×C4×C283C2×D411yescharacteristic
160C2×C4×C283C2411yescharacteristic
161C8×C283—22no—
162C22×C283—22no—
163C22×C283—41no—
164C22×C283—42no—
165C22×C283#12011yescharacteristic
166C22×C283#12011yescharacteristic
167C22×C283C22×D411yescharacteristic
168C2×D283—24no—
169C2×D283—22no—
170C2×D283—44no—
171C2×D283—42no—
172C2×D283—44no—
173C2×D283—42no—
174C2×D283—41no—
175C2×D283—82no—
176C2×D283—88no—
177C4×C283—22no—
178C4×C283—42no—
179C4×C283#12011yescharacteristic
180C4×C283#12011yescharacteristic
181C4×C283C22×D411yescharacteristic
182C4×C283C22×D411yescharacteristic
183Dic283—22no—
184Dic283—41no—
185C2×C283—21no—
186C2×C283—21no—
187C2×C283—21no—
188C2×C283—42no—
189C2×C283C2×#12011yescharacteristic
190D283—44no—
191D283—42no—
192D283—81no—
193C283#301911yescharacteristic, complemented, Sylow
194#3019—2831nomaximal, Sylow
195C2×H(5,𝔽2)—2831no—
196#575—2831no—
197#575—2831no—
198#535—2831no—
199#455—2834no—
200#453—2834no—
201#421—2831no—
202#421D28311yescharacteristic, complemented
203C2×#100—2831no—
204H(5,𝔽2)—2834no—
205H(5,𝔽2)—5662no—
206H(5,𝔽2)—5662no—
207H(5,𝔽2)—5662no—
208C2×#41—2831no—
209C2×#41—2831no—
210C2×#41—5661no—
211C2×#41—5661no—
212C2×#41C2×D28311yescharacteristic
213C22×D4—2831no—
214C22×D4—5661no—
215C22×D4—5662no—
216C22×D4—5661no—
217#137—2832no—
218#137—2832no—
219#137—5661no—
220#137—5661no—
221AGL(1,ℤ/8)—2832no—
222AGL(1,ℤ/8)—2832no—
223AGL(1,ℤ/8)—5661no—
224AGL(1,ℤ/8)—5661no—
225#135—5662no—
226#135—5662no—
227C2×M16—2831no—
228C2×M16C2×D28311yescharacteristic, complemented
229#127—2832no—
230#124—2832no—
231#121—5662no—
232#120—5662no—
233#117C2×D28311yescharacteristic, complemented
234#104—2834no—
235#104C2×D28314yescomplemented
236#100—2834no—
237C24—2832no—
238C24—5662no—
239#41—21no—
240#41—21no—
241#41—2832no—
242#41—2832no—
243#41—5664no—
244#41—5664no—
245#41—5664no—
246#41—5664no—
247#41—5661no—
248#41—5661no—
249#41—5661no—
250#41—5661no—
251#41—5661no—
252#41—5661no—
253#41D113212yescomplemented
254#41C22×D28312yes—
255C2×Q8—2831no—
256C2×Q8—2831no—
257C2×Q8D113211yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—2831no—
260C2×D4—2832no—
261C2×D4—2832no—
262C2×D4—2831no—
263C2×D4—5664no—
264C2×D4—5662no—
265C2×D4—5664no—
266C2×D4—5664no—
267C2×D4—5668no—
268C2×D4—5668no—
269C2×D4—5664no—
270C2×D4—5662no—
271C2×D4—5661no—
272C2×D4—5664no—
273C2×D4—5661no—
274C2×D4—5661no—
275C2×D4—5661no—
276C2×D4—5661no—
277C2×D4—5661no—
278C2×D4D113211yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—5662no—
281C22×C4—5661no—
282C22×C4—5661no—
283C22×C4C22×D28311yescharacteristic
284Q16—5662no—
285Q16—5662no—
286SD16—5662no—
287SD16—5662no—
288SD16—5662no—
289SD16—5662no—
290D8—5662no—
291D8—5662no—
292M16—21no—
293M16—2832no—
294M16—5661no—
295M16C22×D28312yescomplemented
296C2×C8—21no—
297C2×C8—5661no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—5664no—
301C42C22×D28312yescomplemented
302C23—21no—
303C23—2834no—
304C23—5668no—
305C23—5661no—
306C23—5662no—
307C23—5662no—
308C23—5661no—
309C23—22642no—
310C23—22644no—
311C23D4×D28311yescharacteristic
312Q8—21no—
313Q8—2832no—
314Q8—5661no—
315Q8—5661no—
316Q8—5661no—
317Q8C2×D113212yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—2832no—
322D4—5668no—
323D4—5668no—
324D4—5662no—
325D4—5662no—
326D4—5661no—
327D4—5662no—
328D4—5662no—
329D4—5662no—
330D4—5661no—
331D4—5662no—
332D4—5661no—
333D4—11321no—
334D4—11321no—
335D4—22644no—
336D4C2×D113212yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—5668no—
342C2×C4—5661no—
343C2×C4—5662no—
344C2×C4—5662no—
345C2×C4—5661no—
346C2×C4—11321no—
347C2×C4—11321no—
348C2×C4D4×D28311yescharacteristic
349C2×C4D4×D28311yescharacteristic
350C2×C4D4×D28311yescharacteristic
351C2×C4C2×D113211yescharacteristic
352C2×C4C2×D113211yescharacteristic
353C2×C4C23×D28311yescharacteristic
354C8—22no—
355C8—5662no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—5662no—
360C22—5664no—
361C22—11324no—
362C22—11324no—
363C22—11322no—
364C22—11322no—
365C22—11321no—
366C22—22642no—
367C22—22648no—
368C22#2258358011yescharacteristic
369C22#2258358011yescharacteristic
370C22C2×D4×D28311yescharacteristic
371C4—22no—
372C4—42no—
373C4—5662no—
374C4—11321no—
375C4#2258358011yescharacteristic
376C4#2258358011yescharacteristic
377C4C2×D4×D28311yescharacteristic
378C4C22×D113211yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—11324no—
384C2—11322no—
385C2—22641no—
386C2C2×#2258358011yescharacteristic
subgroupG/H
centerC2C2×#22583580
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#421×C283C2
socleC2×C283C2×#120
hypercenter#421D283
hypocenterC283#3019
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#22583656C2semidirect
2. #31827265C2semidirect
3. #31827263C2semidirect
4. #31827034C2semidirect
5. #31826644C2semidirect
6. #31826634C2semidirect
7. #31826483C2semidirect
8. C2×#22583526C2semidirect
9. #421×C283C2semidirect
17. C2×#17985392C22semidirect
23. #22583643C22semidirect
25. #22583640C22semidirect
27. #22583638C22semidirect
29. #22583635C22semidirect
32. C2×#17985376C22semidirect
33. #22583607C22semidirect
34. #22583598C22semidirect
37. #22583538C22semidirect
38. #22583526C22semidirect
40. C2×M16×C283C22semidirect
41. #117×C283C22semidirect
42. #104×C283C22semidirect
55. #17985392D4semidirect
57. C2×Dic566D4semidirect
58. C2×Dic566C23semidirect
68. C2×D1132D4semidirect
70. C2×D1132C23semidirect
90. #17985376C23semidirect
95. #41×C283D4semidirect
97. C2×Q8×C283D4semidirect
99. C2×D4×C283D4semidirect
103. M16×C283C23semidirect
107. C42×C283C23semidirect
193. C283#3019semidirect
202. #421D283semidirect
228. C2×M16C2×D283semidirect
233. #117C2×D283semidirect
235. #104C2×D283semidirect
253. #41D1132semidirect
257. C2×Q8D1132semidirect
278. C2×D4D1132semidirect
295. M16C22×D283semidirect
301. C42C22×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
#318272652
#318272632
#318270342
#318266442
#318266342
#318264832
C2×#225835262
#421×C2832
#3019283