Finite Group Databasebeta

← #49130884

Subgroups — #49130884

Order 36224.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583639C211yesmaximal, characteristic, complemented
2C22×#17985375C211yesmaximal, characteristic, complemented
3#31826975C211yesmaximal, characteristic
4C22⋊Q8×D283C211yesmaximal, characteristic, complemented
5#31826724C211yesmaximal, characteristic, complemented
6#31826722C211yesmaximal, characteristic, complemented
7#31826710C211yesmaximal, characteristic, complemented
8#31826708C211yesmaximal, characteristic, complemented
9#31826706C211yesmaximal, characteristic, complemented
10#31826704C211yesmaximal, characteristic, complemented
11#31826692C211yesmaximal, characteristic, complemented
12#31826504C211yesmaximal, characteristic, complemented
13#31826472C211yesmaximal, characteristic, complemented
14#31826463C211yesmaximal, characteristic, complemented
15#465×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18C22×C4×D283C2211yescharacteristic
19#22583639—24no—
20C2×Dic1132C2211yescharacteristic, complemented
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985380C2211yescharacteristic, complemented
23C2×#17985375—24no—
24C2×#17985375C2211yescharacteristic
25C2×#17985375C2211yescharacteristic
26C2×#17985375C2211yescharacteristic
27C2×#17985375C2211yescharacteristic
28C22×C283⋊C8C2211yescharacteristic
29#22583592—21no—
30#22583590C2211yescharacteristic
31#22583590C2211yescharacteristic
32#22583589C2211yescharacteristic
33#22583589C2211yescharacteristic
34#22583588—21no—
35#22583587—21no—
36#22583586C2211yescharacteristic
37#22583581C2211yescharacteristic
38#22583578—21no—
39D4×Dic283—21no—
40(C22⋊D283)⋊C4—21no—
41#22583568C2211yescharacteristic
42C4⋊C4×D283—21no—
43C4⋊C4×D283C2211yescharacteristic
44C22⋊C4×D283—21no—
45#22583544C2211yescharacteristic, complemented
46#22583543C2211yescharacteristic, complemented
47#22583541C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583535C2211yescharacteristic, complemented
50#22583534C2211yescharacteristic, complemented
51#22583533C2211yescharacteristic, complemented
52#22583533C2211yescharacteristic, complemented
53#22583531C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic, complemented
55C22×C8×C283C2211yescharacteristic
56C22⋊Q8×C283C2211yescharacteristic
57#121×C283C2211yescharacteristic
58#106×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#102×C283C2211yescharacteristic, complemented
61C23×D283D411yescharacteristic
62#17985391—22no—
63#17985391—22no—
64C2×Dic566—21no—
65C2×Dic566C2311yescharacteristic
66C2×Dic566C2311yescharacteristic
67Q8×D283—22no—
68Q8×D283—22no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C22×Dic283—21no—
72C22×Dic283D411yescharacteristic
73C2×C4×D283—22no—
74C2×C4×D283—22no—
75C2×C4×D283—21no—
76C2×C4×D283—21no—
77C2×C4×D283D411yescharacteristic
78C2×C4×D283D411yescharacteristic
79C2×C4×D283C2311yescharacteristic
80C2×C4×D283C2311yescharacteristic
81Dic1132—22no—
82#17985382—22no—
83#17985380—22no—
84#17985375—22no—
85#17985375—22no—
86#17985375—22no—
87#17985375D414yescomplemented
88C2×C283⋊C8—22no—
89C2×C283⋊C8C2311yescharacteristic
90C2×C283⋊C8C2311yescharacteristic
91C4⋊Dic283—21no—
92C4⋊Dic283—21no—
93C4⋊Dic283C2311yescharacteristic
94C4⋊Dic283C2311yescharacteristic
95Dic283⋊C4—21no—
96Dic283⋊C4—21no—
97Dic283⋊C4—21no—
98Dic283⋊C4—21no—
99C22⋊Dic283—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283—21no—
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106C4×Dic283—21no—
107C2×Q8×C283C2311yescharacteristic
108C2×D4×C283—21no—
109C2×D4×C283C2311yescharacteristic
110C22×C4×C283C2311yescharacteristic
111SD16×C283—22no—
112C2×C8×C283—22no—
113C2×C8×C283C2311yescharacteristic
114C2×C8×C283C2311yescharacteristic
115C4⋊C4×C283—21no—
116C4⋊C4×C283C2311yescharacteristic
117C4⋊C4×C283C2311yescharacteristic
118C22⋊C4×C283—21no—
119C22⋊C4×C283—21no—
120C22×D283—22no—
121C22×D283—22no—
122C22×D283—41no—
123C22×D283C2×D411yescharacteristic
124C22×D283C2×D411yescharacteristic
125Dic566—21no—
126Dic566—21no—
127Dic566—21no—
128Dic566—21no—
129Dic566—42no—
130C22⋊D283—42no—
131C22⋊D283—42no—
132C2×Dic283—22no—
133C2×Dic283—21no—
134C2×Dic283—21no—
135C2×Dic283—21no—
136C2×Dic283—21no—
137C2×Dic283—42no—
138C2×Dic283C2×D411yescharacteristic
139C2×Dic283C2×D411yescharacteristic
140C4×D283—22no—
141C4×D283—21no—
142C4×D283—21no—
143C4×D283—22no—
144C4×D283—42no—
145C4×D283—42no—
146C4×D283C2×D412yes—
147C4×D283#4112yes—
148C283⋊C8—21no—
149C283⋊C8C2×D412yes—
150C23×C283—21no—
151C23×C283C2×D411yescharacteristic
152Q8×C283—21no—
153Q8×C283—21no—
154D4×C283—21no—
155D4×C283—21no—
156D4×C283—42no—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283—21no—
161C2×C4×C283—21no—
162C2×C4×C283C2×D411yescharacteristic
163C2×C4×C283C2411yescharacteristic
164C8×C283—21no—
165C8×C283C2×D412yes—
166C22×C283—21no—
167C22×C283—21no—
168C22×C283—41no—
169C22×C283—41no—
170C22×C283—41no—
171C22×C283#13712yescomplemented
172C22×C283C22×D411yescharacteristic
173C2×D283—21no—
174C2×D283—21no—
175C2×D283—21no—
176C2×D283—21no—
177C2×D283—21no—
178C2×D283—44no—
179C2×D283#12112yes—
180C4×C283—21no—
181C4×C283—41no—
182C4×C283—41no—
183C4×C283—41no—
184C4×C283C22×D411yescharacteristic
185C4×C283C2×#4111yescharacteristic
186Dic283—21no—
187Dic283—41no—
188Dic283—41no—
189Dic283—41no—
190Dic283—41no—
191Dic283#12112yes—
192C2×C283—22no—
193C2×C283—81no—
194C2×C283C2×#12111yescharacteristic
195C2×C283C2×#13711yescharacteristic
196C2×C283#57511yescharacteristic
197D283—22no—
198D283—41no—
199C283#305611yescharacteristic, complemented, Sylow
200#3056—2831nomaximal, Sylow
201C2×#137—2831no—
202C22×M16—2831no—
203#525—2831no—
204C2×C22⋊Q8—2831no—
205#470—2831no—
206#470—2831no—
207#468—2831no—
208#468—2831no—
209#467—2831no—
210#467—2831no—
211#465—2831no—
212#465D28311yescharacteristic, complemented
213#428—2831no—
214#419—2831no—
215#417—2831no—
216C2×#41—2831no—
217C22×Q8—2831no—
218C23×C4—2831no—
219#137—5664no—
220C2×Q16—2831no—
221C2×Q16—2831no—
222C2×SD16—2831no—
223C2×SD16C2×D28311yescharacteristic, complemented
224C2×M16—2831no—
225C2×M16—2831no—
226C2×M16—2831no—
227C2×M16—2831no—
228C2×M16—5664no—
229C22×C8—2831no—
230C22×C8C2×D28311yescharacteristic
231#123—5661no—
232C22⋊Q8—2831no—
233C22⋊Q8—2831no—
234C22⋊Q8—2831no—
235C22⋊Q8—2831no—
236C22⋊Q8—2831no—
237C22⋊Q8—5661no—
238C22⋊Q8—5661no—
239C22⋊Q8C2×D28311yescharacteristic
240#121—2831no—
241#121C2×D28311yescharacteristic
242#120—5661no—
243C4×D4—5661no—
244C4×D4—5661no—
245#117—2831no—
246C2×C4⋊C4—2831no—
247C2×C4⋊C4—5661no—
248C2×C22⋊C4—5661no—
249#107—2831no—
250#107—2831no—
251#106—2831no—
252#106C2×D28311yescharacteristic, complemented
253#103—2831no—
254#103—2831no—
255#103—2831no—
256#103—2831no—
257#103—2831no—
258#103C2×D28311yescharacteristic, complemented
259#102—2831no—
260#102C2×D28311yescharacteristic, complemented
261C24—2831no—
262#41—5662no—
263#41—5662no—
264C2×Q8—2831no—
265C2×Q8—2831no—
266C2×Q8—5661no—
267C2×Q8—5662no—
268C2×Q8—5662no—
269C2×Q8C22×D28311yescharacteristic
270C2×D4—21no—
271C2×D4—5661no—
272C2×D4—5661no—
273C2×D4C22×D28311yescharacteristic
274C22×C4—2831no—
275C22×C4—2831no—
276C22×C4—2831no—
277C22×C4—2831no—
278C22×C4—2831no—
279C22×C4—5662no—
280C22×C4—5662no—
281C22×C4—5661no—
282C22×C4—5661no—
283C22×C4—5661no—
284C22×C4C22×D28311yescharacteristic
285Q16—5662no—
286Q16—5662no—
287SD16—22no—
288SD16—5662no—
289M16—2834no—
290M16—5662no—
291M16—5662no—
292M16—5662no—
293C2×C8—22no—
294C2×C8—2831no—
295C2×C8—2831no—
296C2×C8—5662no—
297C2×C8C22×D28311yescharacteristic
298C2×C8C22×D28311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—2831no—
301C4⋊C4—2831no—
302C4⋊C4—5661no—
303C4⋊C4—5661no—
304C4⋊C4—5661no—
305C4⋊C4—5661no—
306C4⋊C4—5661no—
307C4⋊C4—5661no—
308C4⋊C4C22×D28311yescharacteristic
309C4⋊C4C22×D28311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—5661no—
313C22⋊C4—5661no—
314C22⋊C4—5661no—
315C22⋊C4—5661no—
316C22⋊C4—5661no—
317C22⋊C4—5661no—
318C22⋊C4—5661no—
319C42—5661no—
320C23—21no—
321C23—2831no—
322C23—2831no—
323C23—5662no—
324C23—5662no—
325C23—11321no—
326C23D4×D28311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—5661no—
330Q8—5661no—
331Q8—5661no—
332Q8—5661no—
333Q8—11322no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—11322no—
338D4—11322no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2831no—
345C2×C4—2832no—
346C2×C4—2832no—
347C2×C4—2831no—
348C2×C4—5662no—
349C2×C4—5662no—
350C2×C4—5662no—
351C2×C4—5661no—
352C2×C4—5661no—
353C2×C4—5661no—
354C2×C4—5661no—
355C2×C4—5661no—
356C2×C4—5661no—
357C2×C4—11322no—
358C2×C4—11322no—
359C2×C4—11322no—
360C2×C4D4×D28311yescharacteristic
361C2×C4C23×D28311yescharacteristic
362C8—21no—
363C8—2832no—
364C8—5661no—
365C8D4×D28312yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2832no—
372C22—5661no—
373C22—5661no—
374C22—5661no—
375C22—5661no—
376C22—5661no—
377C22—11324no—
378C22#2258363912yescomplemented
379C22C2×D4×D28311yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—2832no—
385C4—5661no—
386C4—11321no—
387C4—11321no—
388C4—11321no—
389C4—11321no—
390C4C2×D4×D28311yescharacteristic
391C4#41×D28311yescharacteristic
392C2—22no—
393C2—81no—
394C2—5662no—
395C2—11321no—
396C2#121×D28311yescharacteristic
397C2C2×#2258363911yescharacteristic
398C2#3182726211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#465×C283C2
socleC22×C283C22×D4
prime radical#31826975C2
hypercenter#465D283
hypocenterC283#3056
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#22583639C2semidirect
2. C22×#17985375C2semidirect
4. C22⋊Q8×D283C2semidirect
5. #31826724C2semidirect
6. #31826722C2semidirect
7. #31826710C2semidirect
8. #31826708C2semidirect
9. #31826706C2semidirect
10. #31826704C2semidirect
11. #31826692C2semidirect
12. #31826504C2semidirect
13. #31826472C2semidirect
14. #31826463C2semidirect
15. #465×C283C2semidirect
20. C2×Dic1132C22semidirect
21. C2×#17985382C22semidirect
22. C2×#17985380C22semidirect
45. #22583544C22semidirect
46. #22583543C22semidirect
47. #22583541C22semidirect
48. #22583535C22semidirect
49. #22583535C22semidirect
50. #22583534C22semidirect
51. #22583533C22semidirect
52. #22583533C22semidirect
53. #22583531C22semidirect
54. C2×SD16×C283C22semidirect
58. #106×C283C22semidirect
59. #103×C283C22semidirect
60. #102×C283C22semidirect
87. #17985375D4semidirect
171. C22×C283#137semidirect
199. C283#3056semidirect
212. #465D283semidirect
223. C2×SD16C2×D283semidirect
252. #106C2×D283semidirect
258. #103C2×D283semidirect
260. #102C2×D283semidirect
378. C22#22583639semidirect

Maximal subgroups

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

subgroupindex
C2×#225836392
C22×#179853752
#318269752
C22⋊Q8×D2832
#318267242
#318267222
#318267102
#318267082
#318267062
#318267042
#318266922
#318265042
#318264722
#318264632
#465×C2832
#3056283